Research ArticleThe Synchronized and Integrated Two-LevelLot Sizing and Scheduling ProblemEvaluating the Generalized Mathematical Model
Claudio F M Toledo1 Alf Kimms2 Paulo M Franccedila3 and Reinaldo Morabito4
1Department of Computer Systems Institute for Science Mathematics and Computing Universidade de Sao Paulo13566-590 Sao Carlos SP Brazil2Department of Technology and Operations Management Mercator School of Management University of Duisburg-Essen47057 Duisburg Germany3Department of Mathematics and Computer Science Universidade Estadual Paulista 19060-900 Presidente Prudente SP Brazil4Department of Production Engineering Universidade Federal de Sao Carlos 13565-905 Sao Carlos SP Brazil
Correspondence should be addressed to Reinaldo Morabito morabitoufscarbr
Received 4 December 2014 Revised 16 April 2015 Accepted 23 April 2015
Academic Editor Joaquim Joao Judice
Copyright copy 2015 Claudio F M Toledo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper presents the synchronized and integrated two-level lot sizing and scheduling problem (SITLSP) This problem isfound in beverage production foundry glass industry and electrofused grains where the production processes have usually twointerdependent levels with sequence-dependent setups in each level For instance in the first level of soft drink production rawmaterials are stored in tanks flowing to production lines in the second level The amount and the time the raw materials andproducts have to be stored and produced should be determined A synchronization problem occurs because the production in linesand the storage in tanks have to be compatible with each other throughout the time horizon The SITLSP and its mathematicalmodel are described in detail by this paper The lack of similar models in the literature has led us to also propose a set of instancesfor the SITLSP based on data provided by a soft drink company Thus a set of benchmark results for these problem instancesare established using an exact method available in an optimization package Moreover results for two relaxations proved that themodeling methodology could be useful in real-world applications
1 Introduction
The problem studied in this paper is motivated by a real situ-ation found in soft drink companies In the bottling industryproduction involves two interdependent levels with decisionsabout raw material storage in tanks and soft drink bottling inlines In the first level decisions regarding the amount andtime the raw materials have to be stored in each one of theavailable tanks should be made Similarly in the second levelthe lot size of each demanded item and its correspondingscheduling in each line should be determined A capacitatedlot sizing and scheduling problem has to be solved on eachone of these two levels (Figure 1)This problem covers variousissues of lot sizing and scheduling that have beendealtwith on
isolation only in the literature before The challenging aspectis the combination of all these issues in one interdependenttwo-level problem Capacitated lot sizing (and scheduling) isa topic of broad interest and has attracted many researchersWhile capacitated lot sizing (and scheduling) is an NP-hardoptimization problem [1 2] the problem of finding a feasiblesolution isNP-complete already [3] if setup times are present
One of the few optimal procedures for solving lot sizingand scheduling problems with sequence-dependent setuptimes can be found in Haase and Kimms [4] Kovacs et al(2009) presented a more compact model based on the pre-vious one proposed by Haase and Kimms [4] which allowssolving larger instances within a reasonable computationtime Luche et al [5] and Clark et al [6] presented mixed
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 182781 18 pageshttpdxdoiorg1011552015182781
2 Mathematical Problems in Engineering
Tanks
Productionlines
Products
Figure 1 The two-level production process
integer programming (MIP) models for lot sizing and sched-uling in the presence of setup times for problems found inelectrofused grain and animal nutrition industries Concern-ing multilevel lot sizing problems several publications exist(eg [7ndash10]) The present paper deals with a multilevel lotsizing and scheduling problem which is difficult to tackleeven if the capacity is unlimited [11 12] and there are parallelmachines represented by production lines (Figure 1) Thework in de Matta and Guignard [13] deals with rolling prod-uction schedules for lot sizingwith parallelmachines A studyon sequence-dependent setup costs with parallel machinesis found in Kang et al [14] A lot sizing problem in a semi-conductor assembly with parallel machines was described byKuhn and Quadt [15] and Quadt and Kuhn [16] A GRASPheuristic with path-relinking intensification was applied byNascimento et al [17] to solve instances of the capacitated lotsizing problem with parallel machines
A R Clark and S J Clark [18] proposed a MIP formu-lation for a lot sizing and scheduling problem on parallelmachines with sequence-dependent setup times Their com-plex MIP formulation was solved using approximate modelsin a rolling horizon basis with a reduced number of binaryvariables per period Model reformulations were also pre-sented by Wolsey [19] to make standard software applicableClark [20] presented a mathematical model for the planningof a canning line in a drinks manufacturer He proposedapproaches using local search integrated with the solution ofapproximate MIP models Gupta and Magnusson [21] pre-sented a model for the capacitated lot sizing and schedulingproblem with sequence-dependent setup costs and nonzerosetup timesThe solver CPLEXwas used to find optimal solu-tions for small sized instances The authors also determinedlower bounds for large sized instances using row aggregationand relaxing the integrality of somemodel variables Almada-Lobo et al [22] proposed models for the same problem tack-led by Gupta and Magnusson [21] which were solved usingspecific-purpose heuristics and the CPLEX solver
A synchronization problem arises in a two-level problemreported by Almada-Lobo et al [23] and Toledo et al [24]Their focus is a glass container industry where the first leveldeals with the glass color produced in the furnaces and
the second level deals with the color of the containers that areproduced afterwards MIP formulations are tailored to tacklethis two-level and synchronized production planning prob-lem where metaheuristic approaches are proposed Baldo etal [25] studied another two-level problem in the breweryindustry preparing the liquids including fermentation andmaturation inside the fermentation tanks and bottling theliquids on the filling linesThe problemdiffers from soft drinkproblemsmainly due to the relatively long lead times requiredfor the fermentation and maturation processes and becausethe ready liquid can remain in the tanks for some time beforebeing bottled The surveys in Drexl and Kimms [26] Karimiet al [27] Jans and Degraeve [28 29] and Ramezanian et al[30] can be referred to for complete overviews in lot sizingproblems and industrial modeling for lot sizing problems
Awork that comes close to our case is the one described inMeyr [31] which is an extension of Meyr [32] Meyr [31] con-sidered a lot sizing and scheduling problemwith nonidenticalparallel machines inventory costs production costs andsequence-dependent setup times and costs However the softdrink industry problem in the present paper is more complexin the sense that it has two lot sizing and scheduling problemseach of them similar to Meyrrsquos problem one for lines andanother for tanks The problem solution should find simulta-neously the lot sizing and scheduling of rawmaterials in tanksand soft drinks in the bottling lines because there is interde-pendence between these two levels which is not present inMeyrrsquos approach Thus new variables and constraints able tosynchronize and integrate these two levels need to be addedThere are studies dealing with supply chain synchronizationor integration applied to decentralized manufacturers [33]However the synchronization aspect dealt with by the presentpaper is closer to the work of Tempelmeier and Buschkuhl[34]These authors introduced a synchronization problem ina multi-item multimachine lot sizing and scheduling prob-lemThis problem is found in the production system of auto-mobile suppliers There is a predetermined assignment ofproducts to machines where a common resource is necessaryfor setup processes on all machines
Bearing in mind the mentioned aspects the problemstudied by the present paper is called synchronized and inte-grated two-level lot sizing and scheduling problem (SITLSP)Someworks relatedwith the specific softdrink industrial situ-ation considered by the SITLSP have been published recentlyFor instance in Ferreira et al [35] a simplified formulationwas presented for the SITLSP where each bottling line hasa dedicated tank and each tank can be filled with all liquidflavors needed by this line Single-stage formulations werepresented in Ferreira et al [36] based on the general lot sizingand scheduling problem (GLSP) and on the asymmetric trav-elling salesman problem (ATSP) The solutions found usingthese formulations outperformed those found by solving theformulation described in Ferreira et al [35] Toledo et al [37]introduced a genetic algorithmmathematical programmingapproach using Ferreira et alrsquos [35] model which reachedcompetitive or better solutions for the real-world instancessolved in Ferreira et al [36] All these recent results assumethe hypotheses of dedicated tanks In our formulation thereare no dedicated tanks as it will be explained later
Mathematical Problems in Engineering 3
There are no existing modeling approaches to be directlyapplicable that include all the issues simultaneously consid-ered lot sizing scheduling setup (time and cost) parallelmachines among others Thus formulating a specific modelseems to be in order In Toledo et al [38] we have already pre-sented a related formulation of this paper (in Portuguese) in avery compact form but nowwe spend a lot of effort to explainall themodel constraints providing details about their mean-ings and interpretations An effective method to deal with allthe aspects considered by the SITLSP was first proposed byToledo et al [39] This method based on a multipopulationgenetic algorithm with individuals hierarchically structuredin trees was able to solve instances with several levels ofdifficulty based on data provided by a soft drink industry andthe solutions were compared with AMPLCPLEX solutionsreported on that paper
To sum up the main contributions of this paper are topresent in detail a mathematical formulation for the SITLP toevaluate its performance solving real-world problem instan-ces and to define sets of benchmark instances for the SITLSPAmathematicalmodel servesmany purposes as to provide anunequivocal definition of the problem better than a textualdescription can do The present paper combines ideas fromthe general lot sizing and scheduling problem (GLSP) withthe continuous setup lot sizing problem (CSLP) [26] todescribe the SITLSPThemodel is codednext and sets of small(artificially created) instances are solved using commercialsoftware packages to find optimum solutions which can beused for example as benchmark to evaluate heuristics pro-posed for the same problem Finallymedium-to-large instan-ces are created and solved applying some relaxation over theformulation when the optimal or even a feasible solution canbe found by the exact method
In the next section we describe the industrial problemin detail with some examples that will help us to explainmany specific problem issues In the section onMathematicalModels a MIP model is presented for the SITLSP In sectionon Computational Results we show some computationalresults obtained by solving the model with the modeling lan-guage AMPL in combination with the CPLEX solver Finallyin Conclusions we present concluding remarks and discussperspectives for future research
2 The Industrial Problem
Tomake the explanation easier the problem is posed as a softdrink industrial problem Taking this into account what it isreferred to as product in the problem is the combination ofthe soft drink flavor and the type of container Each of thesoft drinks is available in one or more bottle types such asglass bottles plastic bottles cans or bag-in-boxes of differentsizes In general only a subset of the bottling lines is capable ofproducing a certain product and products can be producedusing production lines in parallel Various products sharea common production line which cannot produce morethan one product at a time However whenever the productswitches a setup time (of up to several hours) is required toprepare the production line for the next product Production
lines can maintain the setup state When the production ofa product is preempted for a while and continued after someidle time then no setup is required Setup times are sequence-dependent meaning that the order in which products areproduced affects the required time to perform a setup
Let us take an example considering a time horizon of2 periods for a production process with 5 raw materials 6products 3 tanks and 3 lines Each time period has 5 timeunits of capacity Suppose the following
(i) Rm119860 (raw material 119860) produces 1198751 (product 1)(ii) Rm119861 produces 1198752 and 1198753(iii) Rm119862 produces 1198754(iv) Rm119863 produces 1198755(v) Rm119864 produces 1198756(vi) 1198711 (line 1) can bottle 1198751 1198752 and 1198753(vii) 1198712 can bottle 1198751 1198754 and 1198755(viii) 1198713 can bottle 1198751 and 1198756(ix) Tk1 (tank 1) can store Rm119860 Rm119861 and Rm119862(x) Tk2 can store Rm119860 Rm119861 Rm119862 and Rm119863(xi) Tk3 can store Rm119864
In the first time period the demands of products are 100units of 1198751 1198754 and 1198756 and 50 units of 1198755 In the second timeperiod the demands are 100 units of 1198752 and 1198753 and 50 unitsof 1198754 and 1198755 We are mentioning only relevant data for thisexample Figure 2 shows a possible schedule for a productionline level taking into account the line constraints as explainedbefore
The processing time for each product to be produced isshown in Figure 2 The assignments satisfied the set of prod-ucts allowed to be produced in each line and the demands ineach period There are also sequence-dependent setup timesfrom product 1198752 to 1198753 in 1198711 and from 1198754 to 1198755 in 1198712 Thisschedule only pays attention to the line constraints but it doesnot take into account the tank constraints discussed next
The raw material (ie the soft drink) that is bottled ina production line comes from a storage tank with limitedstorage capacity Different soft drinks cannot be put in a tanksimultaneouslyHence from time to time a tankmust be filledwith a particular soft drink which might be another drink orthe same as before Nevertheless whenever a tank is filled asignificant (sequence-dependent) setup time occurs to cleanand fill the tank even if the same soft drink is filled into thetank as before For technical reasons a tank can only be filledup when it is empty During the time of filling up a tanknothing can be pumped to a production line from that tankThere is no fixed assignment between tanks and productionlines Every tank can be connected to every production lineand a tank can feed more than one production line
In the previous example let us assume that all tanks have100 L (liters) of maximum storage capacity Suppose that 2 Lof Rm119860 is necessary to produce 1 u (unit) of 1198751 05 L of Rm119861
for 1 u of 1198752 and 1 u of 1198753 1 L of Rm119862 for 1198754 2 L of Rm119863 for1198755 and 1 L of Rm119864 for 1198756 Figure 3 shows a possible scheduleconsidering the tank constraints
4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Macro-period 1 Macro-period 2
Time10
L1L2L3
P1 P2 P3P4P5P4
P6P5
Figure 2 Schedule for the production line level
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmA
RmD RmD
RmE
RmC
Tk1Tk2Tk3
Macro-period 1 Macro-period 2
Figure 3 Schedule for tank level
A total of 200 L of Rm119860 to produce 100 u of 1198751 isnecessaryTherefore tankTk1must be refilledwith Rm119860 andthe setup times are shown in Figure 3 The interdependencebetween the two levels of the problems becomes clear nowThe schedule in Figure 3 cannot be created without knowingthe line schedule in Figure 2 For instance it is necessaryto know when 100 u of 1198751 are scheduled on 1198711 to fill Tk1
with Rm119860 This also happens in Tk2 where it is necessaryto know when 1198754 is scheduled on 1198712 to fill Tk2 with Rm119863The schedules proposed in Figures 2 and 3 cannot be executedwithout repairs The same problem would happen if we hadscheduled the tank level first Figure 4 illustrates a possiblerepair that swapped product 1198754 for 1198755 from the schedules inFigures 2 and 3
Bottling is usually done to meet a given demand (perweek) with a production planning horizon of four weeks inparticular soft drink companies The decision maker has tofind out howmany units of what product should be producedwhen and on what production line To be able to do so thetanks must be filled appropriately The objective can includethe minimization of the total sum of setups inventory hold-ing and production costs Thus there is a multilevel (twolevel) lot sizing and scheduling problem with parallel proces-sors (tanks on the first level and lines on the second) capacityconstraints and sequence-dependent setup times and costsThe mathematical model proposed by this paper in the nextsection has to integrate these two lot sizing and schedulingproblems The production in lines and the storage in tanksmust be compatible with each other
3 A Model Formulation
This section introduces the mathematical model formulationfor the SITLSP First the basic ideas used by our formulationare described Next the objective function as well as each oneof the model constraints is explained Finally an example isused to show the type of information returned by the modelsolution
RmA RmB
RmC
RmE
RmA
RmD RmC
Macro-period 1 Macro-period 2
RmD
Tk1
Tk2
Tk3
L1
L2
L3
P1
P4
P2 P3P1
P5 P5 P4
P6
Figure 4 Occupation for tank and line slots
31 Basic Ideas The underlying idea for modeling our appli-cation combines issues from the general lot sizing and sched-uling problem (GLSP) and the so-called continuous setup lotsizing problem (CSLP) (see [26] for a comparison of thesemodels) A set of additional variables and constraints are alsoincluded in the model which can describe the specific issuesof the SITLSP
From theGLSPwe adopt the idea of defining a fixed num-ber (119878 for the production lines and 119878 for the tanks) of slots per(macro-) period so that at most one product or raw materialcan be scheduled per slotWhat needs to be determined is forwhat product or raw material respectively a particular slotshould be reserved in lines and tanks and what lot size (a lotof size zero is possible) should be effectively scheduled givena valid slot reservation Let us take the schedule shown inFigures 2 and 3 Suppose now 2macro-periods with 119878 = 119878 = 2
slots per period Figure 4 shows only the simultaneous slotreservation for lines and tanks in each period
The number of products in lines and raw materials intanks cannot be greater than the total number of slots (119878 = 119878 =
2) in each macro-period There is not a full slot occupationin Figure 4 For instance only one slot is effectively occupiedby raw materials and products respectively in tank Tk3 andline 1198713 during the first macro-period In the second macro-period there is no slot occupied in Tk3 and 1198713 Followingthe GLSP idea these two slots are reserved (1198756 in line 1198713 andRm119864 in tank Tk3) but nothing is produced
The use of slots in the model enables us to describe linesand tank occupation using variables and constraints indexedby slotsTherefore we can define decision variables like 119909119895119897119904 =
1 if slot 119904 in line 119897 can be used to produce product 119895 0 other-wise and119906119897119904 = 1 if a positive production amount is effectivelyproduced in slot 119904 in production line 119897 (0 otherwise) In thesame way we have decision variables like 119909119895119896119904 = 1 if slot 119904 canbe used to fill rawmaterial 119895 in tank 119896 0 otherwise and 119906119896119904 =1 if slot 119904 is used to fill raw material in tank 119896 0 otherwiseThese variables (and others) will define constraints for thecorrect line (or tank) occupation the reservation of only oneproduct (raw material) per slot and the effective production(storage) of products (raw materials) (see model constraints(2)ndash(10) and (30)ndash(39) in the next subsections)
In the SITLSP the slots scheduled on one level with theslots scheduled on the other level need to be coordinatedFor instance a production line level which must wait for asetup time on the tank level requires the tank setup time to
Mathematical Problems in Engineering 5
Table 1 Guideline for line and tank constraints
Line constraints Tank constraints Meaning(2) (30) Assignments not allowed for lines and tanks(3) (31) Assignments of products or raw materials to slots
(4) (16) and (17) (32)ndash(34) Link between the products or raw material assignments and the respective lot sizeproduced or stored
(7) (35) and (36) Products or raw materials changeover(8) and (9) (37) and (38) Changeover setup time(10) (39) Macro-period capacity(5) and (6) (64) and (65) Inventory balance(11)ndash(13) (41)ndash(43) Available capacity between endings of two consecutive slots(14) and (15) (44)ndash(47) Define only one beginning and only one ending for each slot(18)ndash(21) (23)ndash(24) (51)ndash(57) (59)ndash(61) Define the beginning and the ending for a slot effectively used or not(22) (58) Define an order for slot occupation(25) (28) and (29) (40) (48) (64) and (65) Synchronize when a slot of raw material is taken from tanks and produced by lines(26) and (27) mdash Use of the capacity of the first micro-period of each slotmdash (62) (63) and (65) Refilling of tanks
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmC
RmE
RmA
RmD RmD RmC
Tk1Tk2Tk3L1
L2
L3
P 1 P 2 P 3P 1
P 5 P 4
P 6
Macro-period 1 Macro-period 2
Micro-periods
P 5P 4
Figure 5 Synchronization between slots in lines and tanks
be ready at least one small time unit before the productionto start in the line level For that reason we introduce smallmicro-periods in a CSLP-likemannerThemacro-periods aredivided into a fixed number of small micro-periods with thesame capacity This capacity can be partially used but thesame micro-period cannot be occupied by two slots Let usillustrate this idea in Figure 5 which synchronizes lines andtanks from Figure 4
Each macro-period was divided into 5 micro-periodswith the same length In the first period the productionof 1198751 1198754 and 1198756 must start after the initial setup time ofRm119860 Rm119862 and Rm119864 respectivelyThe rawmaterialmust bepreviously ready in the tank so that the production begins inlines The setup time needed to refill Tk1 with Rm119860 and Tk2with Rm119863 leads to an interruption of 1198751 and 1198755 productionin lines 1198711 and 1198712 respectively In the second period there isan idle time in1198712 after1198755 Line1198712 is ready to produce1198754 afterits setup time but the production has to wait for the Rm119862
setup time in Tk2Variables and constraints indexed by slots and micro-
periods will enable us to describe these situations Binary
variables 119909119861119897119904120591 and 119909
119864119897119904120591 indicate the micro-period 120591 when
the production of slot 119904 begins or when it ends in line 119897respectivelyThe analogous variables 119909
119861119896119904120591 and 119909
119864119896119904120591 are used to
indicate micro-period 120591 on slot 119904 of tank 119896 It is also necessaryto define linking variables 119902119896119895119897119904120591 which determine the amountof product 119895 produced in line 119897 in micro-period 120591 on slot 119904using raw material from tank 119896 These variables (and others)establish constraints which link production lines and tanksand synchronize raw material and product slots (see modelconstraints (25)ndash(27) (40) and (64) in the next subsections)
The model constraints are formally defined in the nextsubsections Table 1 has already separated them according tothe meaning making it easier to understand
32 Objective Function Suppose that we have 119869 raw mate-rials and 119869 products 119871 production lines and 119871 tanks Theplanning horizon is subdivided into119879macro-periods For themodeling reasons explained before we assume that at most 119878
slots per macro-period can be scheduled on each productionline and at most 119878 slots per macro-period and per tankcan be scheduled These slot values can be set to sufficientlylarge numbers so that this assumption is not restrictivefor practical purposes Furthermore for the same modelingreasons explained before we assume that each macro-periodis subdivided into119879
119898micro-periodsThenotation being usedin the model description is summarized in the appendix
As usual in lot sizing models the objective is to minimizethe total sum of setup costs inventory holding costs and pro-duction costsThe first three terms in expression (1) representsetup holding and production costs for production lines andthe next three the same costs for tanks Notice that a feasiblesolution which fulfills all the demand right in time withoutviolating all constraints to be describedmay not exist Hencewe allow that 119902
0119895 units of the demand for product 119895may not be
produced to ensure that the model can always be solved (lastterm in (1)) Of course a very high penalty 119872 is attached to
6 Mathematical Problems in Engineering
such shortages so that whenever there is a feasible solutionthat fulfills all the demands we would prefer this one
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119879
sum
119905=1ℎ119895119868119895119905 +
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1V119895119897119902119895119897119904
+
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119871
sum
119896=1
119879
sum
119905=1ℎ119895119868119895119896119905sdot119879119898
+
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1V119895119897119902119895119896119904 + 119872
119869
sum
119895=11199020119895
(1)
33 Production Line Usual Constraints First constraints forthe production line level are presented which are usuallyfound in lot sizing and scheduling problems
119909119895119897119904 = 0
119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(2)
119869
sum
119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)
119901119895119897119902119895119897119904 le 119862119909119895119897119904
119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(4)
1198681198951 = 1198681198950 + 1199020119895 +
119871
sum
119897=1
119878
sum
119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)
119868119895119905 = 119868119895119905minus1 +
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905
119895 = 1 119869 119905 = 2 119879
(6)
119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(7)
120590119897119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119897119911119894119895119897119904
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(8)
120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)
119905sdot119878
sum
119904=(119905minus1)119878+1(120590119897119904 +
119869
sum
119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905
119897 = 1 119871 119905 = 1 119879
(10)
A particular product can be produced on some lines butnot on all in general (2) and only one product 119895 is produced ineach slot 119904 of line 119897 (3)The available time permacro-period isan upper bound on the time for production (4)The inventorybalancing constraints (5) and (6) ensure that all demands aremet taking into account the dummy variable 119902
0119895 Constraint
(7) defines the setup times (119911119894119895119897119904 indicates this) based on slotreservations (119909119895119897119904) Thus it is possible to determine the setuptime 120590119897119904 to be included just in front of a certain slot (8) As aresult overlapping setup can happen and for the first slot ofa macro-period 119905 some setup time 120596119897119905 may be at the end ofthe previous macro-period (9) The capacity 119862 in constraint(10) has to be incremented by 120596119897119905 Thus the total sum of pro-duction and setup times in a certain macro-period must notexceed the macro-period capacity 119862 by constraint (10)
34 Production Line Time Slot Constraints The constraintsshowed next describe how to integrate the micro-period ideafrom CLSP and the slot idea from GLSPThemain point hereis to define specific constraints for the SITLSPwhich describewhen some slots in lines will be used throughout the timehorizon
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119905sdot119878120591 ge 120596119897119905+1
119897 = 1 119871 119905 = 1 119879 minus 1
(11)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904minus1120591 ge 120590119897119904
+
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(12)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897(119905minus1)119878+1120591 ge (119905 minus 1) 119862 + 120590119897(119905minus1)119878+1 minus 120596119897119905
+
119869
sum
119895=1119901119895119897119902119895119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
(13)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(14)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(15)
119869
sum
119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)
120576119906119897119904 le
119869
sum
119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(18)
Mathematical Problems in Engineering 7
Macro-period 1 Macro-period 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time
Time
L1
L1s1 s2 s3 s4
P1 P1 P2 P3
P1 P1 P2 P3
xEL1s38
xEL1s38
xEL1s41012059012
120590L1s3120596L12
Constraint (12)
Constraint (13)
10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1
8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +
s4
pP2L1qP2L1s3
Figure 6 Keeping enough time between slots
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(19)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878
(20)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879
119898+ 1) le 119879
119898119906119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879
(21)
119906119897119904 ge 119906119897119904+1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1
(22)
120576119906119897119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591
minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 le
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(23)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 ge
119869
sum
119895=1119901119895119897119902119895119897119904
minus 119862119898
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(24)
The end of the last lot in some previous macro-period 119905
must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5
There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909
119861119897119904120591 = 119909
119864119897119904120591must occur for the samemicro-period
120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints
The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862
119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)
35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines
119902119895119897119904 =
119871
sum
119896=1
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
120575119897119904 = (
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862
119898
minus
119869
sum
119895=1119901119895119897119902119895119897119904
(26)
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
Tanks
Productionlines
Products
Figure 1 The two-level production process
integer programming (MIP) models for lot sizing and sched-uling in the presence of setup times for problems found inelectrofused grain and animal nutrition industries Concern-ing multilevel lot sizing problems several publications exist(eg [7ndash10]) The present paper deals with a multilevel lotsizing and scheduling problem which is difficult to tackleeven if the capacity is unlimited [11 12] and there are parallelmachines represented by production lines (Figure 1) Thework in de Matta and Guignard [13] deals with rolling prod-uction schedules for lot sizingwith parallelmachines A studyon sequence-dependent setup costs with parallel machinesis found in Kang et al [14] A lot sizing problem in a semi-conductor assembly with parallel machines was described byKuhn and Quadt [15] and Quadt and Kuhn [16] A GRASPheuristic with path-relinking intensification was applied byNascimento et al [17] to solve instances of the capacitated lotsizing problem with parallel machines
A R Clark and S J Clark [18] proposed a MIP formu-lation for a lot sizing and scheduling problem on parallelmachines with sequence-dependent setup times Their com-plex MIP formulation was solved using approximate modelsin a rolling horizon basis with a reduced number of binaryvariables per period Model reformulations were also pre-sented by Wolsey [19] to make standard software applicableClark [20] presented a mathematical model for the planningof a canning line in a drinks manufacturer He proposedapproaches using local search integrated with the solution ofapproximate MIP models Gupta and Magnusson [21] pre-sented a model for the capacitated lot sizing and schedulingproblem with sequence-dependent setup costs and nonzerosetup timesThe solver CPLEXwas used to find optimal solu-tions for small sized instances The authors also determinedlower bounds for large sized instances using row aggregationand relaxing the integrality of somemodel variables Almada-Lobo et al [22] proposed models for the same problem tack-led by Gupta and Magnusson [21] which were solved usingspecific-purpose heuristics and the CPLEX solver
A synchronization problem arises in a two-level problemreported by Almada-Lobo et al [23] and Toledo et al [24]Their focus is a glass container industry where the first leveldeals with the glass color produced in the furnaces and
the second level deals with the color of the containers that areproduced afterwards MIP formulations are tailored to tacklethis two-level and synchronized production planning prob-lem where metaheuristic approaches are proposed Baldo etal [25] studied another two-level problem in the breweryindustry preparing the liquids including fermentation andmaturation inside the fermentation tanks and bottling theliquids on the filling linesThe problemdiffers from soft drinkproblemsmainly due to the relatively long lead times requiredfor the fermentation and maturation processes and becausethe ready liquid can remain in the tanks for some time beforebeing bottled The surveys in Drexl and Kimms [26] Karimiet al [27] Jans and Degraeve [28 29] and Ramezanian et al[30] can be referred to for complete overviews in lot sizingproblems and industrial modeling for lot sizing problems
Awork that comes close to our case is the one described inMeyr [31] which is an extension of Meyr [32] Meyr [31] con-sidered a lot sizing and scheduling problemwith nonidenticalparallel machines inventory costs production costs andsequence-dependent setup times and costs However the softdrink industry problem in the present paper is more complexin the sense that it has two lot sizing and scheduling problemseach of them similar to Meyrrsquos problem one for lines andanother for tanks The problem solution should find simulta-neously the lot sizing and scheduling of rawmaterials in tanksand soft drinks in the bottling lines because there is interde-pendence between these two levels which is not present inMeyrrsquos approach Thus new variables and constraints able tosynchronize and integrate these two levels need to be addedThere are studies dealing with supply chain synchronizationor integration applied to decentralized manufacturers [33]However the synchronization aspect dealt with by the presentpaper is closer to the work of Tempelmeier and Buschkuhl[34]These authors introduced a synchronization problem ina multi-item multimachine lot sizing and scheduling prob-lemThis problem is found in the production system of auto-mobile suppliers There is a predetermined assignment ofproducts to machines where a common resource is necessaryfor setup processes on all machines
Bearing in mind the mentioned aspects the problemstudied by the present paper is called synchronized and inte-grated two-level lot sizing and scheduling problem (SITLSP)Someworks relatedwith the specific softdrink industrial situ-ation considered by the SITLSP have been published recentlyFor instance in Ferreira et al [35] a simplified formulationwas presented for the SITLSP where each bottling line hasa dedicated tank and each tank can be filled with all liquidflavors needed by this line Single-stage formulations werepresented in Ferreira et al [36] based on the general lot sizingand scheduling problem (GLSP) and on the asymmetric trav-elling salesman problem (ATSP) The solutions found usingthese formulations outperformed those found by solving theformulation described in Ferreira et al [35] Toledo et al [37]introduced a genetic algorithmmathematical programmingapproach using Ferreira et alrsquos [35] model which reachedcompetitive or better solutions for the real-world instancessolved in Ferreira et al [36] All these recent results assumethe hypotheses of dedicated tanks In our formulation thereare no dedicated tanks as it will be explained later
Mathematical Problems in Engineering 3
There are no existing modeling approaches to be directlyapplicable that include all the issues simultaneously consid-ered lot sizing scheduling setup (time and cost) parallelmachines among others Thus formulating a specific modelseems to be in order In Toledo et al [38] we have already pre-sented a related formulation of this paper (in Portuguese) in avery compact form but nowwe spend a lot of effort to explainall themodel constraints providing details about their mean-ings and interpretations An effective method to deal with allthe aspects considered by the SITLSP was first proposed byToledo et al [39] This method based on a multipopulationgenetic algorithm with individuals hierarchically structuredin trees was able to solve instances with several levels ofdifficulty based on data provided by a soft drink industry andthe solutions were compared with AMPLCPLEX solutionsreported on that paper
To sum up the main contributions of this paper are topresent in detail a mathematical formulation for the SITLP toevaluate its performance solving real-world problem instan-ces and to define sets of benchmark instances for the SITLSPAmathematicalmodel servesmany purposes as to provide anunequivocal definition of the problem better than a textualdescription can do The present paper combines ideas fromthe general lot sizing and scheduling problem (GLSP) withthe continuous setup lot sizing problem (CSLP) [26] todescribe the SITLSPThemodel is codednext and sets of small(artificially created) instances are solved using commercialsoftware packages to find optimum solutions which can beused for example as benchmark to evaluate heuristics pro-posed for the same problem Finallymedium-to-large instan-ces are created and solved applying some relaxation over theformulation when the optimal or even a feasible solution canbe found by the exact method
In the next section we describe the industrial problemin detail with some examples that will help us to explainmany specific problem issues In the section onMathematicalModels a MIP model is presented for the SITLSP In sectionon Computational Results we show some computationalresults obtained by solving the model with the modeling lan-guage AMPL in combination with the CPLEX solver Finallyin Conclusions we present concluding remarks and discussperspectives for future research
2 The Industrial Problem
Tomake the explanation easier the problem is posed as a softdrink industrial problem Taking this into account what it isreferred to as product in the problem is the combination ofthe soft drink flavor and the type of container Each of thesoft drinks is available in one or more bottle types such asglass bottles plastic bottles cans or bag-in-boxes of differentsizes In general only a subset of the bottling lines is capable ofproducing a certain product and products can be producedusing production lines in parallel Various products sharea common production line which cannot produce morethan one product at a time However whenever the productswitches a setup time (of up to several hours) is required toprepare the production line for the next product Production
lines can maintain the setup state When the production ofa product is preempted for a while and continued after someidle time then no setup is required Setup times are sequence-dependent meaning that the order in which products areproduced affects the required time to perform a setup
Let us take an example considering a time horizon of2 periods for a production process with 5 raw materials 6products 3 tanks and 3 lines Each time period has 5 timeunits of capacity Suppose the following
(i) Rm119860 (raw material 119860) produces 1198751 (product 1)(ii) Rm119861 produces 1198752 and 1198753(iii) Rm119862 produces 1198754(iv) Rm119863 produces 1198755(v) Rm119864 produces 1198756(vi) 1198711 (line 1) can bottle 1198751 1198752 and 1198753(vii) 1198712 can bottle 1198751 1198754 and 1198755(viii) 1198713 can bottle 1198751 and 1198756(ix) Tk1 (tank 1) can store Rm119860 Rm119861 and Rm119862(x) Tk2 can store Rm119860 Rm119861 Rm119862 and Rm119863(xi) Tk3 can store Rm119864
In the first time period the demands of products are 100units of 1198751 1198754 and 1198756 and 50 units of 1198755 In the second timeperiod the demands are 100 units of 1198752 and 1198753 and 50 unitsof 1198754 and 1198755 We are mentioning only relevant data for thisexample Figure 2 shows a possible schedule for a productionline level taking into account the line constraints as explainedbefore
The processing time for each product to be produced isshown in Figure 2 The assignments satisfied the set of prod-ucts allowed to be produced in each line and the demands ineach period There are also sequence-dependent setup timesfrom product 1198752 to 1198753 in 1198711 and from 1198754 to 1198755 in 1198712 Thisschedule only pays attention to the line constraints but it doesnot take into account the tank constraints discussed next
The raw material (ie the soft drink) that is bottled ina production line comes from a storage tank with limitedstorage capacity Different soft drinks cannot be put in a tanksimultaneouslyHence from time to time a tankmust be filledwith a particular soft drink which might be another drink orthe same as before Nevertheless whenever a tank is filled asignificant (sequence-dependent) setup time occurs to cleanand fill the tank even if the same soft drink is filled into thetank as before For technical reasons a tank can only be filledup when it is empty During the time of filling up a tanknothing can be pumped to a production line from that tankThere is no fixed assignment between tanks and productionlines Every tank can be connected to every production lineand a tank can feed more than one production line
In the previous example let us assume that all tanks have100 L (liters) of maximum storage capacity Suppose that 2 Lof Rm119860 is necessary to produce 1 u (unit) of 1198751 05 L of Rm119861
for 1 u of 1198752 and 1 u of 1198753 1 L of Rm119862 for 1198754 2 L of Rm119863 for1198755 and 1 L of Rm119864 for 1198756 Figure 3 shows a possible scheduleconsidering the tank constraints
4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Macro-period 1 Macro-period 2
Time10
L1L2L3
P1 P2 P3P4P5P4
P6P5
Figure 2 Schedule for the production line level
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmA
RmD RmD
RmE
RmC
Tk1Tk2Tk3
Macro-period 1 Macro-period 2
Figure 3 Schedule for tank level
A total of 200 L of Rm119860 to produce 100 u of 1198751 isnecessaryTherefore tankTk1must be refilledwith Rm119860 andthe setup times are shown in Figure 3 The interdependencebetween the two levels of the problems becomes clear nowThe schedule in Figure 3 cannot be created without knowingthe line schedule in Figure 2 For instance it is necessaryto know when 100 u of 1198751 are scheduled on 1198711 to fill Tk1
with Rm119860 This also happens in Tk2 where it is necessaryto know when 1198754 is scheduled on 1198712 to fill Tk2 with Rm119863The schedules proposed in Figures 2 and 3 cannot be executedwithout repairs The same problem would happen if we hadscheduled the tank level first Figure 4 illustrates a possiblerepair that swapped product 1198754 for 1198755 from the schedules inFigures 2 and 3
Bottling is usually done to meet a given demand (perweek) with a production planning horizon of four weeks inparticular soft drink companies The decision maker has tofind out howmany units of what product should be producedwhen and on what production line To be able to do so thetanks must be filled appropriately The objective can includethe minimization of the total sum of setups inventory hold-ing and production costs Thus there is a multilevel (twolevel) lot sizing and scheduling problem with parallel proces-sors (tanks on the first level and lines on the second) capacityconstraints and sequence-dependent setup times and costsThe mathematical model proposed by this paper in the nextsection has to integrate these two lot sizing and schedulingproblems The production in lines and the storage in tanksmust be compatible with each other
3 A Model Formulation
This section introduces the mathematical model formulationfor the SITLSP First the basic ideas used by our formulationare described Next the objective function as well as each oneof the model constraints is explained Finally an example isused to show the type of information returned by the modelsolution
RmA RmB
RmC
RmE
RmA
RmD RmC
Macro-period 1 Macro-period 2
RmD
Tk1
Tk2
Tk3
L1
L2
L3
P1
P4
P2 P3P1
P5 P5 P4
P6
Figure 4 Occupation for tank and line slots
31 Basic Ideas The underlying idea for modeling our appli-cation combines issues from the general lot sizing and sched-uling problem (GLSP) and the so-called continuous setup lotsizing problem (CSLP) (see [26] for a comparison of thesemodels) A set of additional variables and constraints are alsoincluded in the model which can describe the specific issuesof the SITLSP
From theGLSPwe adopt the idea of defining a fixed num-ber (119878 for the production lines and 119878 for the tanks) of slots per(macro-) period so that at most one product or raw materialcan be scheduled per slotWhat needs to be determined is forwhat product or raw material respectively a particular slotshould be reserved in lines and tanks and what lot size (a lotof size zero is possible) should be effectively scheduled givena valid slot reservation Let us take the schedule shown inFigures 2 and 3 Suppose now 2macro-periods with 119878 = 119878 = 2
slots per period Figure 4 shows only the simultaneous slotreservation for lines and tanks in each period
The number of products in lines and raw materials intanks cannot be greater than the total number of slots (119878 = 119878 =
2) in each macro-period There is not a full slot occupationin Figure 4 For instance only one slot is effectively occupiedby raw materials and products respectively in tank Tk3 andline 1198713 during the first macro-period In the second macro-period there is no slot occupied in Tk3 and 1198713 Followingthe GLSP idea these two slots are reserved (1198756 in line 1198713 andRm119864 in tank Tk3) but nothing is produced
The use of slots in the model enables us to describe linesand tank occupation using variables and constraints indexedby slotsTherefore we can define decision variables like 119909119895119897119904 =
1 if slot 119904 in line 119897 can be used to produce product 119895 0 other-wise and119906119897119904 = 1 if a positive production amount is effectivelyproduced in slot 119904 in production line 119897 (0 otherwise) In thesame way we have decision variables like 119909119895119896119904 = 1 if slot 119904 canbe used to fill rawmaterial 119895 in tank 119896 0 otherwise and 119906119896119904 =1 if slot 119904 is used to fill raw material in tank 119896 0 otherwiseThese variables (and others) will define constraints for thecorrect line (or tank) occupation the reservation of only oneproduct (raw material) per slot and the effective production(storage) of products (raw materials) (see model constraints(2)ndash(10) and (30)ndash(39) in the next subsections)
In the SITLSP the slots scheduled on one level with theslots scheduled on the other level need to be coordinatedFor instance a production line level which must wait for asetup time on the tank level requires the tank setup time to
Mathematical Problems in Engineering 5
Table 1 Guideline for line and tank constraints
Line constraints Tank constraints Meaning(2) (30) Assignments not allowed for lines and tanks(3) (31) Assignments of products or raw materials to slots
(4) (16) and (17) (32)ndash(34) Link between the products or raw material assignments and the respective lot sizeproduced or stored
(7) (35) and (36) Products or raw materials changeover(8) and (9) (37) and (38) Changeover setup time(10) (39) Macro-period capacity(5) and (6) (64) and (65) Inventory balance(11)ndash(13) (41)ndash(43) Available capacity between endings of two consecutive slots(14) and (15) (44)ndash(47) Define only one beginning and only one ending for each slot(18)ndash(21) (23)ndash(24) (51)ndash(57) (59)ndash(61) Define the beginning and the ending for a slot effectively used or not(22) (58) Define an order for slot occupation(25) (28) and (29) (40) (48) (64) and (65) Synchronize when a slot of raw material is taken from tanks and produced by lines(26) and (27) mdash Use of the capacity of the first micro-period of each slotmdash (62) (63) and (65) Refilling of tanks
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmC
RmE
RmA
RmD RmD RmC
Tk1Tk2Tk3L1
L2
L3
P 1 P 2 P 3P 1
P 5 P 4
P 6
Macro-period 1 Macro-period 2
Micro-periods
P 5P 4
Figure 5 Synchronization between slots in lines and tanks
be ready at least one small time unit before the productionto start in the line level For that reason we introduce smallmicro-periods in a CSLP-likemannerThemacro-periods aredivided into a fixed number of small micro-periods with thesame capacity This capacity can be partially used but thesame micro-period cannot be occupied by two slots Let usillustrate this idea in Figure 5 which synchronizes lines andtanks from Figure 4
Each macro-period was divided into 5 micro-periodswith the same length In the first period the productionof 1198751 1198754 and 1198756 must start after the initial setup time ofRm119860 Rm119862 and Rm119864 respectivelyThe rawmaterialmust bepreviously ready in the tank so that the production begins inlines The setup time needed to refill Tk1 with Rm119860 and Tk2with Rm119863 leads to an interruption of 1198751 and 1198755 productionin lines 1198711 and 1198712 respectively In the second period there isan idle time in1198712 after1198755 Line1198712 is ready to produce1198754 afterits setup time but the production has to wait for the Rm119862
setup time in Tk2Variables and constraints indexed by slots and micro-
periods will enable us to describe these situations Binary
variables 119909119861119897119904120591 and 119909
119864119897119904120591 indicate the micro-period 120591 when
the production of slot 119904 begins or when it ends in line 119897respectivelyThe analogous variables 119909
119861119896119904120591 and 119909
119864119896119904120591 are used to
indicate micro-period 120591 on slot 119904 of tank 119896 It is also necessaryto define linking variables 119902119896119895119897119904120591 which determine the amountof product 119895 produced in line 119897 in micro-period 120591 on slot 119904using raw material from tank 119896 These variables (and others)establish constraints which link production lines and tanksand synchronize raw material and product slots (see modelconstraints (25)ndash(27) (40) and (64) in the next subsections)
The model constraints are formally defined in the nextsubsections Table 1 has already separated them according tothe meaning making it easier to understand
32 Objective Function Suppose that we have 119869 raw mate-rials and 119869 products 119871 production lines and 119871 tanks Theplanning horizon is subdivided into119879macro-periods For themodeling reasons explained before we assume that at most 119878
slots per macro-period can be scheduled on each productionline and at most 119878 slots per macro-period and per tankcan be scheduled These slot values can be set to sufficientlylarge numbers so that this assumption is not restrictivefor practical purposes Furthermore for the same modelingreasons explained before we assume that each macro-periodis subdivided into119879
119898micro-periodsThenotation being usedin the model description is summarized in the appendix
As usual in lot sizing models the objective is to minimizethe total sum of setup costs inventory holding costs and pro-duction costsThe first three terms in expression (1) representsetup holding and production costs for production lines andthe next three the same costs for tanks Notice that a feasiblesolution which fulfills all the demand right in time withoutviolating all constraints to be describedmay not exist Hencewe allow that 119902
0119895 units of the demand for product 119895may not be
produced to ensure that the model can always be solved (lastterm in (1)) Of course a very high penalty 119872 is attached to
6 Mathematical Problems in Engineering
such shortages so that whenever there is a feasible solutionthat fulfills all the demands we would prefer this one
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119879
sum
119905=1ℎ119895119868119895119905 +
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1V119895119897119902119895119897119904
+
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119871
sum
119896=1
119879
sum
119905=1ℎ119895119868119895119896119905sdot119879119898
+
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1V119895119897119902119895119896119904 + 119872
119869
sum
119895=11199020119895
(1)
33 Production Line Usual Constraints First constraints forthe production line level are presented which are usuallyfound in lot sizing and scheduling problems
119909119895119897119904 = 0
119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(2)
119869
sum
119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)
119901119895119897119902119895119897119904 le 119862119909119895119897119904
119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(4)
1198681198951 = 1198681198950 + 1199020119895 +
119871
sum
119897=1
119878
sum
119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)
119868119895119905 = 119868119895119905minus1 +
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905
119895 = 1 119869 119905 = 2 119879
(6)
119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(7)
120590119897119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119897119911119894119895119897119904
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(8)
120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)
119905sdot119878
sum
119904=(119905minus1)119878+1(120590119897119904 +
119869
sum
119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905
119897 = 1 119871 119905 = 1 119879
(10)
A particular product can be produced on some lines butnot on all in general (2) and only one product 119895 is produced ineach slot 119904 of line 119897 (3)The available time permacro-period isan upper bound on the time for production (4)The inventorybalancing constraints (5) and (6) ensure that all demands aremet taking into account the dummy variable 119902
0119895 Constraint
(7) defines the setup times (119911119894119895119897119904 indicates this) based on slotreservations (119909119895119897119904) Thus it is possible to determine the setuptime 120590119897119904 to be included just in front of a certain slot (8) As aresult overlapping setup can happen and for the first slot ofa macro-period 119905 some setup time 120596119897119905 may be at the end ofthe previous macro-period (9) The capacity 119862 in constraint(10) has to be incremented by 120596119897119905 Thus the total sum of pro-duction and setup times in a certain macro-period must notexceed the macro-period capacity 119862 by constraint (10)
34 Production Line Time Slot Constraints The constraintsshowed next describe how to integrate the micro-period ideafrom CLSP and the slot idea from GLSPThemain point hereis to define specific constraints for the SITLSPwhich describewhen some slots in lines will be used throughout the timehorizon
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119905sdot119878120591 ge 120596119897119905+1
119897 = 1 119871 119905 = 1 119879 minus 1
(11)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904minus1120591 ge 120590119897119904
+
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(12)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897(119905minus1)119878+1120591 ge (119905 minus 1) 119862 + 120590119897(119905minus1)119878+1 minus 120596119897119905
+
119869
sum
119895=1119901119895119897119902119895119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
(13)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(14)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(15)
119869
sum
119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)
120576119906119897119904 le
119869
sum
119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(18)
Mathematical Problems in Engineering 7
Macro-period 1 Macro-period 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time
Time
L1
L1s1 s2 s3 s4
P1 P1 P2 P3
P1 P1 P2 P3
xEL1s38
xEL1s38
xEL1s41012059012
120590L1s3120596L12
Constraint (12)
Constraint (13)
10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1
8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +
s4
pP2L1qP2L1s3
Figure 6 Keeping enough time between slots
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(19)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878
(20)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879
119898+ 1) le 119879
119898119906119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879
(21)
119906119897119904 ge 119906119897119904+1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1
(22)
120576119906119897119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591
minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 le
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(23)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 ge
119869
sum
119895=1119901119895119897119902119895119897119904
minus 119862119898
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(24)
The end of the last lot in some previous macro-period 119905
must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5
There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909
119861119897119904120591 = 119909
119864119897119904120591must occur for the samemicro-period
120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints
The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862
119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)
35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines
119902119895119897119904 =
119871
sum
119896=1
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
120575119897119904 = (
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862
119898
minus
119869
sum
119895=1119901119895119897119902119895119897119904
(26)
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
There are no existing modeling approaches to be directlyapplicable that include all the issues simultaneously consid-ered lot sizing scheduling setup (time and cost) parallelmachines among others Thus formulating a specific modelseems to be in order In Toledo et al [38] we have already pre-sented a related formulation of this paper (in Portuguese) in avery compact form but nowwe spend a lot of effort to explainall themodel constraints providing details about their mean-ings and interpretations An effective method to deal with allthe aspects considered by the SITLSP was first proposed byToledo et al [39] This method based on a multipopulationgenetic algorithm with individuals hierarchically structuredin trees was able to solve instances with several levels ofdifficulty based on data provided by a soft drink industry andthe solutions were compared with AMPLCPLEX solutionsreported on that paper
To sum up the main contributions of this paper are topresent in detail a mathematical formulation for the SITLP toevaluate its performance solving real-world problem instan-ces and to define sets of benchmark instances for the SITLSPAmathematicalmodel servesmany purposes as to provide anunequivocal definition of the problem better than a textualdescription can do The present paper combines ideas fromthe general lot sizing and scheduling problem (GLSP) withthe continuous setup lot sizing problem (CSLP) [26] todescribe the SITLSPThemodel is codednext and sets of small(artificially created) instances are solved using commercialsoftware packages to find optimum solutions which can beused for example as benchmark to evaluate heuristics pro-posed for the same problem Finallymedium-to-large instan-ces are created and solved applying some relaxation over theformulation when the optimal or even a feasible solution canbe found by the exact method
In the next section we describe the industrial problemin detail with some examples that will help us to explainmany specific problem issues In the section onMathematicalModels a MIP model is presented for the SITLSP In sectionon Computational Results we show some computationalresults obtained by solving the model with the modeling lan-guage AMPL in combination with the CPLEX solver Finallyin Conclusions we present concluding remarks and discussperspectives for future research
2 The Industrial Problem
Tomake the explanation easier the problem is posed as a softdrink industrial problem Taking this into account what it isreferred to as product in the problem is the combination ofthe soft drink flavor and the type of container Each of thesoft drinks is available in one or more bottle types such asglass bottles plastic bottles cans or bag-in-boxes of differentsizes In general only a subset of the bottling lines is capable ofproducing a certain product and products can be producedusing production lines in parallel Various products sharea common production line which cannot produce morethan one product at a time However whenever the productswitches a setup time (of up to several hours) is required toprepare the production line for the next product Production
lines can maintain the setup state When the production ofa product is preempted for a while and continued after someidle time then no setup is required Setup times are sequence-dependent meaning that the order in which products areproduced affects the required time to perform a setup
Let us take an example considering a time horizon of2 periods for a production process with 5 raw materials 6products 3 tanks and 3 lines Each time period has 5 timeunits of capacity Suppose the following
(i) Rm119860 (raw material 119860) produces 1198751 (product 1)(ii) Rm119861 produces 1198752 and 1198753(iii) Rm119862 produces 1198754(iv) Rm119863 produces 1198755(v) Rm119864 produces 1198756(vi) 1198711 (line 1) can bottle 1198751 1198752 and 1198753(vii) 1198712 can bottle 1198751 1198754 and 1198755(viii) 1198713 can bottle 1198751 and 1198756(ix) Tk1 (tank 1) can store Rm119860 Rm119861 and Rm119862(x) Tk2 can store Rm119860 Rm119861 Rm119862 and Rm119863(xi) Tk3 can store Rm119864
In the first time period the demands of products are 100units of 1198751 1198754 and 1198756 and 50 units of 1198755 In the second timeperiod the demands are 100 units of 1198752 and 1198753 and 50 unitsof 1198754 and 1198755 We are mentioning only relevant data for thisexample Figure 2 shows a possible schedule for a productionline level taking into account the line constraints as explainedbefore
The processing time for each product to be produced isshown in Figure 2 The assignments satisfied the set of prod-ucts allowed to be produced in each line and the demands ineach period There are also sequence-dependent setup timesfrom product 1198752 to 1198753 in 1198711 and from 1198754 to 1198755 in 1198712 Thisschedule only pays attention to the line constraints but it doesnot take into account the tank constraints discussed next
The raw material (ie the soft drink) that is bottled ina production line comes from a storage tank with limitedstorage capacity Different soft drinks cannot be put in a tanksimultaneouslyHence from time to time a tankmust be filledwith a particular soft drink which might be another drink orthe same as before Nevertheless whenever a tank is filled asignificant (sequence-dependent) setup time occurs to cleanand fill the tank even if the same soft drink is filled into thetank as before For technical reasons a tank can only be filledup when it is empty During the time of filling up a tanknothing can be pumped to a production line from that tankThere is no fixed assignment between tanks and productionlines Every tank can be connected to every production lineand a tank can feed more than one production line
In the previous example let us assume that all tanks have100 L (liters) of maximum storage capacity Suppose that 2 Lof Rm119860 is necessary to produce 1 u (unit) of 1198751 05 L of Rm119861
for 1 u of 1198752 and 1 u of 1198753 1 L of Rm119862 for 1198754 2 L of Rm119863 for1198755 and 1 L of Rm119864 for 1198756 Figure 3 shows a possible scheduleconsidering the tank constraints
4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Macro-period 1 Macro-period 2
Time10
L1L2L3
P1 P2 P3P4P5P4
P6P5
Figure 2 Schedule for the production line level
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmA
RmD RmD
RmE
RmC
Tk1Tk2Tk3
Macro-period 1 Macro-period 2
Figure 3 Schedule for tank level
A total of 200 L of Rm119860 to produce 100 u of 1198751 isnecessaryTherefore tankTk1must be refilledwith Rm119860 andthe setup times are shown in Figure 3 The interdependencebetween the two levels of the problems becomes clear nowThe schedule in Figure 3 cannot be created without knowingthe line schedule in Figure 2 For instance it is necessaryto know when 100 u of 1198751 are scheduled on 1198711 to fill Tk1
with Rm119860 This also happens in Tk2 where it is necessaryto know when 1198754 is scheduled on 1198712 to fill Tk2 with Rm119863The schedules proposed in Figures 2 and 3 cannot be executedwithout repairs The same problem would happen if we hadscheduled the tank level first Figure 4 illustrates a possiblerepair that swapped product 1198754 for 1198755 from the schedules inFigures 2 and 3
Bottling is usually done to meet a given demand (perweek) with a production planning horizon of four weeks inparticular soft drink companies The decision maker has tofind out howmany units of what product should be producedwhen and on what production line To be able to do so thetanks must be filled appropriately The objective can includethe minimization of the total sum of setups inventory hold-ing and production costs Thus there is a multilevel (twolevel) lot sizing and scheduling problem with parallel proces-sors (tanks on the first level and lines on the second) capacityconstraints and sequence-dependent setup times and costsThe mathematical model proposed by this paper in the nextsection has to integrate these two lot sizing and schedulingproblems The production in lines and the storage in tanksmust be compatible with each other
3 A Model Formulation
This section introduces the mathematical model formulationfor the SITLSP First the basic ideas used by our formulationare described Next the objective function as well as each oneof the model constraints is explained Finally an example isused to show the type of information returned by the modelsolution
RmA RmB
RmC
RmE
RmA
RmD RmC
Macro-period 1 Macro-period 2
RmD
Tk1
Tk2
Tk3
L1
L2
L3
P1
P4
P2 P3P1
P5 P5 P4
P6
Figure 4 Occupation for tank and line slots
31 Basic Ideas The underlying idea for modeling our appli-cation combines issues from the general lot sizing and sched-uling problem (GLSP) and the so-called continuous setup lotsizing problem (CSLP) (see [26] for a comparison of thesemodels) A set of additional variables and constraints are alsoincluded in the model which can describe the specific issuesof the SITLSP
From theGLSPwe adopt the idea of defining a fixed num-ber (119878 for the production lines and 119878 for the tanks) of slots per(macro-) period so that at most one product or raw materialcan be scheduled per slotWhat needs to be determined is forwhat product or raw material respectively a particular slotshould be reserved in lines and tanks and what lot size (a lotof size zero is possible) should be effectively scheduled givena valid slot reservation Let us take the schedule shown inFigures 2 and 3 Suppose now 2macro-periods with 119878 = 119878 = 2
slots per period Figure 4 shows only the simultaneous slotreservation for lines and tanks in each period
The number of products in lines and raw materials intanks cannot be greater than the total number of slots (119878 = 119878 =
2) in each macro-period There is not a full slot occupationin Figure 4 For instance only one slot is effectively occupiedby raw materials and products respectively in tank Tk3 andline 1198713 during the first macro-period In the second macro-period there is no slot occupied in Tk3 and 1198713 Followingthe GLSP idea these two slots are reserved (1198756 in line 1198713 andRm119864 in tank Tk3) but nothing is produced
The use of slots in the model enables us to describe linesand tank occupation using variables and constraints indexedby slotsTherefore we can define decision variables like 119909119895119897119904 =
1 if slot 119904 in line 119897 can be used to produce product 119895 0 other-wise and119906119897119904 = 1 if a positive production amount is effectivelyproduced in slot 119904 in production line 119897 (0 otherwise) In thesame way we have decision variables like 119909119895119896119904 = 1 if slot 119904 canbe used to fill rawmaterial 119895 in tank 119896 0 otherwise and 119906119896119904 =1 if slot 119904 is used to fill raw material in tank 119896 0 otherwiseThese variables (and others) will define constraints for thecorrect line (or tank) occupation the reservation of only oneproduct (raw material) per slot and the effective production(storage) of products (raw materials) (see model constraints(2)ndash(10) and (30)ndash(39) in the next subsections)
In the SITLSP the slots scheduled on one level with theslots scheduled on the other level need to be coordinatedFor instance a production line level which must wait for asetup time on the tank level requires the tank setup time to
Mathematical Problems in Engineering 5
Table 1 Guideline for line and tank constraints
Line constraints Tank constraints Meaning(2) (30) Assignments not allowed for lines and tanks(3) (31) Assignments of products or raw materials to slots
(4) (16) and (17) (32)ndash(34) Link between the products or raw material assignments and the respective lot sizeproduced or stored
(7) (35) and (36) Products or raw materials changeover(8) and (9) (37) and (38) Changeover setup time(10) (39) Macro-period capacity(5) and (6) (64) and (65) Inventory balance(11)ndash(13) (41)ndash(43) Available capacity between endings of two consecutive slots(14) and (15) (44)ndash(47) Define only one beginning and only one ending for each slot(18)ndash(21) (23)ndash(24) (51)ndash(57) (59)ndash(61) Define the beginning and the ending for a slot effectively used or not(22) (58) Define an order for slot occupation(25) (28) and (29) (40) (48) (64) and (65) Synchronize when a slot of raw material is taken from tanks and produced by lines(26) and (27) mdash Use of the capacity of the first micro-period of each slotmdash (62) (63) and (65) Refilling of tanks
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmC
RmE
RmA
RmD RmD RmC
Tk1Tk2Tk3L1
L2
L3
P 1 P 2 P 3P 1
P 5 P 4
P 6
Macro-period 1 Macro-period 2
Micro-periods
P 5P 4
Figure 5 Synchronization between slots in lines and tanks
be ready at least one small time unit before the productionto start in the line level For that reason we introduce smallmicro-periods in a CSLP-likemannerThemacro-periods aredivided into a fixed number of small micro-periods with thesame capacity This capacity can be partially used but thesame micro-period cannot be occupied by two slots Let usillustrate this idea in Figure 5 which synchronizes lines andtanks from Figure 4
Each macro-period was divided into 5 micro-periodswith the same length In the first period the productionof 1198751 1198754 and 1198756 must start after the initial setup time ofRm119860 Rm119862 and Rm119864 respectivelyThe rawmaterialmust bepreviously ready in the tank so that the production begins inlines The setup time needed to refill Tk1 with Rm119860 and Tk2with Rm119863 leads to an interruption of 1198751 and 1198755 productionin lines 1198711 and 1198712 respectively In the second period there isan idle time in1198712 after1198755 Line1198712 is ready to produce1198754 afterits setup time but the production has to wait for the Rm119862
setup time in Tk2Variables and constraints indexed by slots and micro-
periods will enable us to describe these situations Binary
variables 119909119861119897119904120591 and 119909
119864119897119904120591 indicate the micro-period 120591 when
the production of slot 119904 begins or when it ends in line 119897respectivelyThe analogous variables 119909
119861119896119904120591 and 119909
119864119896119904120591 are used to
indicate micro-period 120591 on slot 119904 of tank 119896 It is also necessaryto define linking variables 119902119896119895119897119904120591 which determine the amountof product 119895 produced in line 119897 in micro-period 120591 on slot 119904using raw material from tank 119896 These variables (and others)establish constraints which link production lines and tanksand synchronize raw material and product slots (see modelconstraints (25)ndash(27) (40) and (64) in the next subsections)
The model constraints are formally defined in the nextsubsections Table 1 has already separated them according tothe meaning making it easier to understand
32 Objective Function Suppose that we have 119869 raw mate-rials and 119869 products 119871 production lines and 119871 tanks Theplanning horizon is subdivided into119879macro-periods For themodeling reasons explained before we assume that at most 119878
slots per macro-period can be scheduled on each productionline and at most 119878 slots per macro-period and per tankcan be scheduled These slot values can be set to sufficientlylarge numbers so that this assumption is not restrictivefor practical purposes Furthermore for the same modelingreasons explained before we assume that each macro-periodis subdivided into119879
119898micro-periodsThenotation being usedin the model description is summarized in the appendix
As usual in lot sizing models the objective is to minimizethe total sum of setup costs inventory holding costs and pro-duction costsThe first three terms in expression (1) representsetup holding and production costs for production lines andthe next three the same costs for tanks Notice that a feasiblesolution which fulfills all the demand right in time withoutviolating all constraints to be describedmay not exist Hencewe allow that 119902
0119895 units of the demand for product 119895may not be
produced to ensure that the model can always be solved (lastterm in (1)) Of course a very high penalty 119872 is attached to
6 Mathematical Problems in Engineering
such shortages so that whenever there is a feasible solutionthat fulfills all the demands we would prefer this one
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119879
sum
119905=1ℎ119895119868119895119905 +
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1V119895119897119902119895119897119904
+
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119871
sum
119896=1
119879
sum
119905=1ℎ119895119868119895119896119905sdot119879119898
+
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1V119895119897119902119895119896119904 + 119872
119869
sum
119895=11199020119895
(1)
33 Production Line Usual Constraints First constraints forthe production line level are presented which are usuallyfound in lot sizing and scheduling problems
119909119895119897119904 = 0
119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(2)
119869
sum
119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)
119901119895119897119902119895119897119904 le 119862119909119895119897119904
119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(4)
1198681198951 = 1198681198950 + 1199020119895 +
119871
sum
119897=1
119878
sum
119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)
119868119895119905 = 119868119895119905minus1 +
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905
119895 = 1 119869 119905 = 2 119879
(6)
119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(7)
120590119897119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119897119911119894119895119897119904
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(8)
120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)
119905sdot119878
sum
119904=(119905minus1)119878+1(120590119897119904 +
119869
sum
119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905
119897 = 1 119871 119905 = 1 119879
(10)
A particular product can be produced on some lines butnot on all in general (2) and only one product 119895 is produced ineach slot 119904 of line 119897 (3)The available time permacro-period isan upper bound on the time for production (4)The inventorybalancing constraints (5) and (6) ensure that all demands aremet taking into account the dummy variable 119902
0119895 Constraint
(7) defines the setup times (119911119894119895119897119904 indicates this) based on slotreservations (119909119895119897119904) Thus it is possible to determine the setuptime 120590119897119904 to be included just in front of a certain slot (8) As aresult overlapping setup can happen and for the first slot ofa macro-period 119905 some setup time 120596119897119905 may be at the end ofthe previous macro-period (9) The capacity 119862 in constraint(10) has to be incremented by 120596119897119905 Thus the total sum of pro-duction and setup times in a certain macro-period must notexceed the macro-period capacity 119862 by constraint (10)
34 Production Line Time Slot Constraints The constraintsshowed next describe how to integrate the micro-period ideafrom CLSP and the slot idea from GLSPThemain point hereis to define specific constraints for the SITLSPwhich describewhen some slots in lines will be used throughout the timehorizon
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119905sdot119878120591 ge 120596119897119905+1
119897 = 1 119871 119905 = 1 119879 minus 1
(11)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904minus1120591 ge 120590119897119904
+
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(12)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897(119905minus1)119878+1120591 ge (119905 minus 1) 119862 + 120590119897(119905minus1)119878+1 minus 120596119897119905
+
119869
sum
119895=1119901119895119897119902119895119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
(13)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(14)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(15)
119869
sum
119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)
120576119906119897119904 le
119869
sum
119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(18)
Mathematical Problems in Engineering 7
Macro-period 1 Macro-period 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time
Time
L1
L1s1 s2 s3 s4
P1 P1 P2 P3
P1 P1 P2 P3
xEL1s38
xEL1s38
xEL1s41012059012
120590L1s3120596L12
Constraint (12)
Constraint (13)
10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1
8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +
s4
pP2L1qP2L1s3
Figure 6 Keeping enough time between slots
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(19)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878
(20)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879
119898+ 1) le 119879
119898119906119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879
(21)
119906119897119904 ge 119906119897119904+1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1
(22)
120576119906119897119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591
minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 le
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(23)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 ge
119869
sum
119895=1119901119895119897119902119895119897119904
minus 119862119898
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(24)
The end of the last lot in some previous macro-period 119905
must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5
There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909
119861119897119904120591 = 119909
119864119897119904120591must occur for the samemicro-period
120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints
The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862
119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)
35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines
119902119895119897119904 =
119871
sum
119896=1
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
120575119897119904 = (
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862
119898
minus
119869
sum
119895=1119901119895119897119902119895119897119904
(26)
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Macro-period 1 Macro-period 2
Time10
L1L2L3
P1 P2 P3P4P5P4
P6P5
Figure 2 Schedule for the production line level
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmA
RmD RmD
RmE
RmC
Tk1Tk2Tk3
Macro-period 1 Macro-period 2
Figure 3 Schedule for tank level
A total of 200 L of Rm119860 to produce 100 u of 1198751 isnecessaryTherefore tankTk1must be refilledwith Rm119860 andthe setup times are shown in Figure 3 The interdependencebetween the two levels of the problems becomes clear nowThe schedule in Figure 3 cannot be created without knowingthe line schedule in Figure 2 For instance it is necessaryto know when 100 u of 1198751 are scheduled on 1198711 to fill Tk1
with Rm119860 This also happens in Tk2 where it is necessaryto know when 1198754 is scheduled on 1198712 to fill Tk2 with Rm119863The schedules proposed in Figures 2 and 3 cannot be executedwithout repairs The same problem would happen if we hadscheduled the tank level first Figure 4 illustrates a possiblerepair that swapped product 1198754 for 1198755 from the schedules inFigures 2 and 3
Bottling is usually done to meet a given demand (perweek) with a production planning horizon of four weeks inparticular soft drink companies The decision maker has tofind out howmany units of what product should be producedwhen and on what production line To be able to do so thetanks must be filled appropriately The objective can includethe minimization of the total sum of setups inventory hold-ing and production costs Thus there is a multilevel (twolevel) lot sizing and scheduling problem with parallel proces-sors (tanks on the first level and lines on the second) capacityconstraints and sequence-dependent setup times and costsThe mathematical model proposed by this paper in the nextsection has to integrate these two lot sizing and schedulingproblems The production in lines and the storage in tanksmust be compatible with each other
3 A Model Formulation
This section introduces the mathematical model formulationfor the SITLSP First the basic ideas used by our formulationare described Next the objective function as well as each oneof the model constraints is explained Finally an example isused to show the type of information returned by the modelsolution
RmA RmB
RmC
RmE
RmA
RmD RmC
Macro-period 1 Macro-period 2
RmD
Tk1
Tk2
Tk3
L1
L2
L3
P1
P4
P2 P3P1
P5 P5 P4
P6
Figure 4 Occupation for tank and line slots
31 Basic Ideas The underlying idea for modeling our appli-cation combines issues from the general lot sizing and sched-uling problem (GLSP) and the so-called continuous setup lotsizing problem (CSLP) (see [26] for a comparison of thesemodels) A set of additional variables and constraints are alsoincluded in the model which can describe the specific issuesof the SITLSP
From theGLSPwe adopt the idea of defining a fixed num-ber (119878 for the production lines and 119878 for the tanks) of slots per(macro-) period so that at most one product or raw materialcan be scheduled per slotWhat needs to be determined is forwhat product or raw material respectively a particular slotshould be reserved in lines and tanks and what lot size (a lotof size zero is possible) should be effectively scheduled givena valid slot reservation Let us take the schedule shown inFigures 2 and 3 Suppose now 2macro-periods with 119878 = 119878 = 2
slots per period Figure 4 shows only the simultaneous slotreservation for lines and tanks in each period
The number of products in lines and raw materials intanks cannot be greater than the total number of slots (119878 = 119878 =
2) in each macro-period There is not a full slot occupationin Figure 4 For instance only one slot is effectively occupiedby raw materials and products respectively in tank Tk3 andline 1198713 during the first macro-period In the second macro-period there is no slot occupied in Tk3 and 1198713 Followingthe GLSP idea these two slots are reserved (1198756 in line 1198713 andRm119864 in tank Tk3) but nothing is produced
The use of slots in the model enables us to describe linesand tank occupation using variables and constraints indexedby slotsTherefore we can define decision variables like 119909119895119897119904 =
1 if slot 119904 in line 119897 can be used to produce product 119895 0 other-wise and119906119897119904 = 1 if a positive production amount is effectivelyproduced in slot 119904 in production line 119897 (0 otherwise) In thesame way we have decision variables like 119909119895119896119904 = 1 if slot 119904 canbe used to fill rawmaterial 119895 in tank 119896 0 otherwise and 119906119896119904 =1 if slot 119904 is used to fill raw material in tank 119896 0 otherwiseThese variables (and others) will define constraints for thecorrect line (or tank) occupation the reservation of only oneproduct (raw material) per slot and the effective production(storage) of products (raw materials) (see model constraints(2)ndash(10) and (30)ndash(39) in the next subsections)
In the SITLSP the slots scheduled on one level with theslots scheduled on the other level need to be coordinatedFor instance a production line level which must wait for asetup time on the tank level requires the tank setup time to
Mathematical Problems in Engineering 5
Table 1 Guideline for line and tank constraints
Line constraints Tank constraints Meaning(2) (30) Assignments not allowed for lines and tanks(3) (31) Assignments of products or raw materials to slots
(4) (16) and (17) (32)ndash(34) Link between the products or raw material assignments and the respective lot sizeproduced or stored
(7) (35) and (36) Products or raw materials changeover(8) and (9) (37) and (38) Changeover setup time(10) (39) Macro-period capacity(5) and (6) (64) and (65) Inventory balance(11)ndash(13) (41)ndash(43) Available capacity between endings of two consecutive slots(14) and (15) (44)ndash(47) Define only one beginning and only one ending for each slot(18)ndash(21) (23)ndash(24) (51)ndash(57) (59)ndash(61) Define the beginning and the ending for a slot effectively used or not(22) (58) Define an order for slot occupation(25) (28) and (29) (40) (48) (64) and (65) Synchronize when a slot of raw material is taken from tanks and produced by lines(26) and (27) mdash Use of the capacity of the first micro-period of each slotmdash (62) (63) and (65) Refilling of tanks
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmC
RmE
RmA
RmD RmD RmC
Tk1Tk2Tk3L1
L2
L3
P 1 P 2 P 3P 1
P 5 P 4
P 6
Macro-period 1 Macro-period 2
Micro-periods
P 5P 4
Figure 5 Synchronization between slots in lines and tanks
be ready at least one small time unit before the productionto start in the line level For that reason we introduce smallmicro-periods in a CSLP-likemannerThemacro-periods aredivided into a fixed number of small micro-periods with thesame capacity This capacity can be partially used but thesame micro-period cannot be occupied by two slots Let usillustrate this idea in Figure 5 which synchronizes lines andtanks from Figure 4
Each macro-period was divided into 5 micro-periodswith the same length In the first period the productionof 1198751 1198754 and 1198756 must start after the initial setup time ofRm119860 Rm119862 and Rm119864 respectivelyThe rawmaterialmust bepreviously ready in the tank so that the production begins inlines The setup time needed to refill Tk1 with Rm119860 and Tk2with Rm119863 leads to an interruption of 1198751 and 1198755 productionin lines 1198711 and 1198712 respectively In the second period there isan idle time in1198712 after1198755 Line1198712 is ready to produce1198754 afterits setup time but the production has to wait for the Rm119862
setup time in Tk2Variables and constraints indexed by slots and micro-
periods will enable us to describe these situations Binary
variables 119909119861119897119904120591 and 119909
119864119897119904120591 indicate the micro-period 120591 when
the production of slot 119904 begins or when it ends in line 119897respectivelyThe analogous variables 119909
119861119896119904120591 and 119909
119864119896119904120591 are used to
indicate micro-period 120591 on slot 119904 of tank 119896 It is also necessaryto define linking variables 119902119896119895119897119904120591 which determine the amountof product 119895 produced in line 119897 in micro-period 120591 on slot 119904using raw material from tank 119896 These variables (and others)establish constraints which link production lines and tanksand synchronize raw material and product slots (see modelconstraints (25)ndash(27) (40) and (64) in the next subsections)
The model constraints are formally defined in the nextsubsections Table 1 has already separated them according tothe meaning making it easier to understand
32 Objective Function Suppose that we have 119869 raw mate-rials and 119869 products 119871 production lines and 119871 tanks Theplanning horizon is subdivided into119879macro-periods For themodeling reasons explained before we assume that at most 119878
slots per macro-period can be scheduled on each productionline and at most 119878 slots per macro-period and per tankcan be scheduled These slot values can be set to sufficientlylarge numbers so that this assumption is not restrictivefor practical purposes Furthermore for the same modelingreasons explained before we assume that each macro-periodis subdivided into119879
119898micro-periodsThenotation being usedin the model description is summarized in the appendix
As usual in lot sizing models the objective is to minimizethe total sum of setup costs inventory holding costs and pro-duction costsThe first three terms in expression (1) representsetup holding and production costs for production lines andthe next three the same costs for tanks Notice that a feasiblesolution which fulfills all the demand right in time withoutviolating all constraints to be describedmay not exist Hencewe allow that 119902
0119895 units of the demand for product 119895may not be
produced to ensure that the model can always be solved (lastterm in (1)) Of course a very high penalty 119872 is attached to
6 Mathematical Problems in Engineering
such shortages so that whenever there is a feasible solutionthat fulfills all the demands we would prefer this one
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119879
sum
119905=1ℎ119895119868119895119905 +
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1V119895119897119902119895119897119904
+
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119871
sum
119896=1
119879
sum
119905=1ℎ119895119868119895119896119905sdot119879119898
+
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1V119895119897119902119895119896119904 + 119872
119869
sum
119895=11199020119895
(1)
33 Production Line Usual Constraints First constraints forthe production line level are presented which are usuallyfound in lot sizing and scheduling problems
119909119895119897119904 = 0
119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(2)
119869
sum
119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)
119901119895119897119902119895119897119904 le 119862119909119895119897119904
119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(4)
1198681198951 = 1198681198950 + 1199020119895 +
119871
sum
119897=1
119878
sum
119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)
119868119895119905 = 119868119895119905minus1 +
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905
119895 = 1 119869 119905 = 2 119879
(6)
119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(7)
120590119897119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119897119911119894119895119897119904
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(8)
120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)
119905sdot119878
sum
119904=(119905minus1)119878+1(120590119897119904 +
119869
sum
119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905
119897 = 1 119871 119905 = 1 119879
(10)
A particular product can be produced on some lines butnot on all in general (2) and only one product 119895 is produced ineach slot 119904 of line 119897 (3)The available time permacro-period isan upper bound on the time for production (4)The inventorybalancing constraints (5) and (6) ensure that all demands aremet taking into account the dummy variable 119902
0119895 Constraint
(7) defines the setup times (119911119894119895119897119904 indicates this) based on slotreservations (119909119895119897119904) Thus it is possible to determine the setuptime 120590119897119904 to be included just in front of a certain slot (8) As aresult overlapping setup can happen and for the first slot ofa macro-period 119905 some setup time 120596119897119905 may be at the end ofthe previous macro-period (9) The capacity 119862 in constraint(10) has to be incremented by 120596119897119905 Thus the total sum of pro-duction and setup times in a certain macro-period must notexceed the macro-period capacity 119862 by constraint (10)
34 Production Line Time Slot Constraints The constraintsshowed next describe how to integrate the micro-period ideafrom CLSP and the slot idea from GLSPThemain point hereis to define specific constraints for the SITLSPwhich describewhen some slots in lines will be used throughout the timehorizon
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119905sdot119878120591 ge 120596119897119905+1
119897 = 1 119871 119905 = 1 119879 minus 1
(11)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904minus1120591 ge 120590119897119904
+
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(12)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897(119905minus1)119878+1120591 ge (119905 minus 1) 119862 + 120590119897(119905minus1)119878+1 minus 120596119897119905
+
119869
sum
119895=1119901119895119897119902119895119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
(13)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(14)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(15)
119869
sum
119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)
120576119906119897119904 le
119869
sum
119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(18)
Mathematical Problems in Engineering 7
Macro-period 1 Macro-period 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time
Time
L1
L1s1 s2 s3 s4
P1 P1 P2 P3
P1 P1 P2 P3
xEL1s38
xEL1s38
xEL1s41012059012
120590L1s3120596L12
Constraint (12)
Constraint (13)
10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1
8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +
s4
pP2L1qP2L1s3
Figure 6 Keeping enough time between slots
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(19)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878
(20)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879
119898+ 1) le 119879
119898119906119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879
(21)
119906119897119904 ge 119906119897119904+1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1
(22)
120576119906119897119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591
minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 le
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(23)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 ge
119869
sum
119895=1119901119895119897119902119895119897119904
minus 119862119898
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(24)
The end of the last lot in some previous macro-period 119905
must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5
There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909
119861119897119904120591 = 119909
119864119897119904120591must occur for the samemicro-period
120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints
The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862
119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)
35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines
119902119895119897119904 =
119871
sum
119896=1
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
120575119897119904 = (
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862
119898
minus
119869
sum
119895=1119901119895119897119902119895119897119904
(26)
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
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Mathematical Problems in Engineering 5
Table 1 Guideline for line and tank constraints
Line constraints Tank constraints Meaning(2) (30) Assignments not allowed for lines and tanks(3) (31) Assignments of products or raw materials to slots
(4) (16) and (17) (32)ndash(34) Link between the products or raw material assignments and the respective lot sizeproduced or stored
(7) (35) and (36) Products or raw materials changeover(8) and (9) (37) and (38) Changeover setup time(10) (39) Macro-period capacity(5) and (6) (64) and (65) Inventory balance(11)ndash(13) (41)ndash(43) Available capacity between endings of two consecutive slots(14) and (15) (44)ndash(47) Define only one beginning and only one ending for each slot(18)ndash(21) (23)ndash(24) (51)ndash(57) (59)ndash(61) Define the beginning and the ending for a slot effectively used or not(22) (58) Define an order for slot occupation(25) (28) and (29) (40) (48) (64) and (65) Synchronize when a slot of raw material is taken from tanks and produced by lines(26) and (27) mdash Use of the capacity of the first micro-period of each slotmdash (62) (63) and (65) Refilling of tanks
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmC
RmE
RmA
RmD RmD RmC
Tk1Tk2Tk3L1
L2
L3
P 1 P 2 P 3P 1
P 5 P 4
P 6
Macro-period 1 Macro-period 2
Micro-periods
P 5P 4
Figure 5 Synchronization between slots in lines and tanks
be ready at least one small time unit before the productionto start in the line level For that reason we introduce smallmicro-periods in a CSLP-likemannerThemacro-periods aredivided into a fixed number of small micro-periods with thesame capacity This capacity can be partially used but thesame micro-period cannot be occupied by two slots Let usillustrate this idea in Figure 5 which synchronizes lines andtanks from Figure 4
Each macro-period was divided into 5 micro-periodswith the same length In the first period the productionof 1198751 1198754 and 1198756 must start after the initial setup time ofRm119860 Rm119862 and Rm119864 respectivelyThe rawmaterialmust bepreviously ready in the tank so that the production begins inlines The setup time needed to refill Tk1 with Rm119860 and Tk2with Rm119863 leads to an interruption of 1198751 and 1198755 productionin lines 1198711 and 1198712 respectively In the second period there isan idle time in1198712 after1198755 Line1198712 is ready to produce1198754 afterits setup time but the production has to wait for the Rm119862
setup time in Tk2Variables and constraints indexed by slots and micro-
periods will enable us to describe these situations Binary
variables 119909119861119897119904120591 and 119909
119864119897119904120591 indicate the micro-period 120591 when
the production of slot 119904 begins or when it ends in line 119897respectivelyThe analogous variables 119909
119861119896119904120591 and 119909
119864119896119904120591 are used to
indicate micro-period 120591 on slot 119904 of tank 119896 It is also necessaryto define linking variables 119902119896119895119897119904120591 which determine the amountof product 119895 produced in line 119897 in micro-period 120591 on slot 119904using raw material from tank 119896 These variables (and others)establish constraints which link production lines and tanksand synchronize raw material and product slots (see modelconstraints (25)ndash(27) (40) and (64) in the next subsections)
The model constraints are formally defined in the nextsubsections Table 1 has already separated them according tothe meaning making it easier to understand
32 Objective Function Suppose that we have 119869 raw mate-rials and 119869 products 119871 production lines and 119871 tanks Theplanning horizon is subdivided into119879macro-periods For themodeling reasons explained before we assume that at most 119878
slots per macro-period can be scheduled on each productionline and at most 119878 slots per macro-period and per tankcan be scheduled These slot values can be set to sufficientlylarge numbers so that this assumption is not restrictivefor practical purposes Furthermore for the same modelingreasons explained before we assume that each macro-periodis subdivided into119879
119898micro-periodsThenotation being usedin the model description is summarized in the appendix
As usual in lot sizing models the objective is to minimizethe total sum of setup costs inventory holding costs and pro-duction costsThe first three terms in expression (1) representsetup holding and production costs for production lines andthe next three the same costs for tanks Notice that a feasiblesolution which fulfills all the demand right in time withoutviolating all constraints to be describedmay not exist Hencewe allow that 119902
0119895 units of the demand for product 119895may not be
produced to ensure that the model can always be solved (lastterm in (1)) Of course a very high penalty 119872 is attached to
6 Mathematical Problems in Engineering
such shortages so that whenever there is a feasible solutionthat fulfills all the demands we would prefer this one
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119879
sum
119905=1ℎ119895119868119895119905 +
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1V119895119897119902119895119897119904
+
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119871
sum
119896=1
119879
sum
119905=1ℎ119895119868119895119896119905sdot119879119898
+
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1V119895119897119902119895119896119904 + 119872
119869
sum
119895=11199020119895
(1)
33 Production Line Usual Constraints First constraints forthe production line level are presented which are usuallyfound in lot sizing and scheduling problems
119909119895119897119904 = 0
119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(2)
119869
sum
119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)
119901119895119897119902119895119897119904 le 119862119909119895119897119904
119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(4)
1198681198951 = 1198681198950 + 1199020119895 +
119871
sum
119897=1
119878
sum
119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)
119868119895119905 = 119868119895119905minus1 +
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905
119895 = 1 119869 119905 = 2 119879
(6)
119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(7)
120590119897119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119897119911119894119895119897119904
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(8)
120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)
119905sdot119878
sum
119904=(119905minus1)119878+1(120590119897119904 +
119869
sum
119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905
119897 = 1 119871 119905 = 1 119879
(10)
A particular product can be produced on some lines butnot on all in general (2) and only one product 119895 is produced ineach slot 119904 of line 119897 (3)The available time permacro-period isan upper bound on the time for production (4)The inventorybalancing constraints (5) and (6) ensure that all demands aremet taking into account the dummy variable 119902
0119895 Constraint
(7) defines the setup times (119911119894119895119897119904 indicates this) based on slotreservations (119909119895119897119904) Thus it is possible to determine the setuptime 120590119897119904 to be included just in front of a certain slot (8) As aresult overlapping setup can happen and for the first slot ofa macro-period 119905 some setup time 120596119897119905 may be at the end ofthe previous macro-period (9) The capacity 119862 in constraint(10) has to be incremented by 120596119897119905 Thus the total sum of pro-duction and setup times in a certain macro-period must notexceed the macro-period capacity 119862 by constraint (10)
34 Production Line Time Slot Constraints The constraintsshowed next describe how to integrate the micro-period ideafrom CLSP and the slot idea from GLSPThemain point hereis to define specific constraints for the SITLSPwhich describewhen some slots in lines will be used throughout the timehorizon
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119905sdot119878120591 ge 120596119897119905+1
119897 = 1 119871 119905 = 1 119879 minus 1
(11)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904minus1120591 ge 120590119897119904
+
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(12)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897(119905minus1)119878+1120591 ge (119905 minus 1) 119862 + 120590119897(119905minus1)119878+1 minus 120596119897119905
+
119869
sum
119895=1119901119895119897119902119895119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
(13)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(14)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(15)
119869
sum
119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)
120576119906119897119904 le
119869
sum
119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(18)
Mathematical Problems in Engineering 7
Macro-period 1 Macro-period 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time
Time
L1
L1s1 s2 s3 s4
P1 P1 P2 P3
P1 P1 P2 P3
xEL1s38
xEL1s38
xEL1s41012059012
120590L1s3120596L12
Constraint (12)
Constraint (13)
10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1
8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +
s4
pP2L1qP2L1s3
Figure 6 Keeping enough time between slots
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(19)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878
(20)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879
119898+ 1) le 119879
119898119906119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879
(21)
119906119897119904 ge 119906119897119904+1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1
(22)
120576119906119897119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591
minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 le
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(23)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 ge
119869
sum
119895=1119901119895119897119902119895119897119904
minus 119862119898
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(24)
The end of the last lot in some previous macro-period 119905
must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5
There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909
119861119897119904120591 = 119909
119864119897119904120591must occur for the samemicro-period
120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints
The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862
119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)
35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines
119902119895119897119904 =
119871
sum
119896=1
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
120575119897119904 = (
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862
119898
minus
119869
sum
119895=1119901119895119897119902119895119897119904
(26)
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
such shortages so that whenever there is a feasible solutionthat fulfills all the demands we would prefer this one
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119879
sum
119905=1ℎ119895119868119895119905 +
119869
sum
119895=1
119871
sum
119897=1
119879sdot119878
sum
119904=1V119895119897119902119895119897119904
+
119869
sum
119894=1
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1119904119894119895119897119911119894119895119897119904 +
119869
sum
119895=1
119871
sum
119896=1
119879
sum
119905=1ℎ119895119868119895119896119905sdot119879119898
+
119869
sum
119895=1
119871
sum
119896=1
119879sdot119878
sum
119904=1V119895119897119902119895119896119904 + 119872
119869
sum
119895=11199020119895
(1)
33 Production Line Usual Constraints First constraints forthe production line level are presented which are usuallyfound in lot sizing and scheduling problems
119909119895119897119904 = 0
119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(2)
119869
sum
119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)
119901119895119897119902119895119897119904 le 119862119909119895119897119904
119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(4)
1198681198951 = 1198681198950 + 1199020119895 +
119871
sum
119897=1
119878
sum
119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)
119868119895119905 = 119868119895119905minus1 +
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905
119895 = 1 119869 119905 = 2 119879
(6)
119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(7)
120590119897119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119897119911119894119895119897119904
119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878
(8)
120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)
119905sdot119878
sum
119904=(119905minus1)119878+1(120590119897119904 +
119869
sum
119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905
119897 = 1 119871 119905 = 1 119879
(10)
A particular product can be produced on some lines butnot on all in general (2) and only one product 119895 is produced ineach slot 119904 of line 119897 (3)The available time permacro-period isan upper bound on the time for production (4)The inventorybalancing constraints (5) and (6) ensure that all demands aremet taking into account the dummy variable 119902
0119895 Constraint
(7) defines the setup times (119911119894119895119897119904 indicates this) based on slotreservations (119909119895119897119904) Thus it is possible to determine the setuptime 120590119897119904 to be included just in front of a certain slot (8) As aresult overlapping setup can happen and for the first slot ofa macro-period 119905 some setup time 120596119897119905 may be at the end ofthe previous macro-period (9) The capacity 119862 in constraint(10) has to be incremented by 120596119897119905 Thus the total sum of pro-duction and setup times in a certain macro-period must notexceed the macro-period capacity 119862 by constraint (10)
34 Production Line Time Slot Constraints The constraintsshowed next describe how to integrate the micro-period ideafrom CLSP and the slot idea from GLSPThemain point hereis to define specific constraints for the SITLSPwhich describewhen some slots in lines will be used throughout the timehorizon
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119905sdot119878120591 ge 120596119897119905+1
119897 = 1 119871 119905 = 1 119879 minus 1
(11)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904minus1120591 ge 120590119897119904
+
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(12)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897(119905minus1)119878+1120591 ge (119905 minus 1) 119862 + 120590119897(119905minus1)119878+1 minus 120596119897119905
+
119869
sum
119895=1119901119895119897119902119895119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
(13)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(14)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(15)
119869
sum
119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)
120576119906119897119904 le
119869
sum
119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(18)
Mathematical Problems in Engineering 7
Macro-period 1 Macro-period 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time
Time
L1
L1s1 s2 s3 s4
P1 P1 P2 P3
P1 P1 P2 P3
xEL1s38
xEL1s38
xEL1s41012059012
120590L1s3120596L12
Constraint (12)
Constraint (13)
10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1
8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +
s4
pP2L1qP2L1s3
Figure 6 Keeping enough time between slots
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(19)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878
(20)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879
119898+ 1) le 119879
119898119906119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879
(21)
119906119897119904 ge 119906119897119904+1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1
(22)
120576119906119897119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591
minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 le
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(23)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 ge
119869
sum
119895=1119901119895119897119902119895119897119904
minus 119862119898
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(24)
The end of the last lot in some previous macro-period 119905
must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5
There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909
119861119897119904120591 = 119909
119864119897119904120591must occur for the samemicro-period
120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints
The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862
119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)
35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines
119902119895119897119904 =
119871
sum
119896=1
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
120575119897119904 = (
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862
119898
minus
119869
sum
119895=1119901119895119897119902119895119897119904
(26)
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Macro-period 1 Macro-period 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time
Time
L1
L1s1 s2 s3 s4
P1 P1 P2 P3
P1 P1 P2 P3
xEL1s38
xEL1s38
xEL1s41012059012
120590L1s3120596L12
Constraint (12)
Constraint (13)
10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1
8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +
s4
pP2L1qP2L1s3
Figure 6 Keeping enough time between slots
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(19)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879
119898119906119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878
(20)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879
119898+ 1) le 119879
119898119906119897(119905minus1)119878+1
119897 = 1 119871 119905 = 1 119879
(21)
119906119897119904 ge 119906119897119904+1
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1
(22)
120576119906119897119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591
minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 le
119869
sum
119895=1119901119895119897119902119895119897119904
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(23)
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119897119904120591 ge
119869
sum
119895=1119901119895119897119902119895119897119904
minus 119862119898
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878
(24)
The end of the last lot in some previous macro-period 119905
must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5
There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909
119861119897119904120591 = 119909
119864119897119904120591must occur for the samemicro-period
120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints
The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862
119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)
35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines
119902119895119897119904 =
119871
sum
119896=1
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
120575119897119904 = (
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862
119898
minus
119869
sum
119895=1119901119895119897119902119895119897119904
(26)
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
xBL1s37 xEL1s38
xBL1s49 xEL1s49
120590L1s2s3 s4
120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3
120576 lowast uL1s4 + 9xEL1s49 minus 9xBL1s49 le pP3L1qP3L1s4
8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3
9xEL1s49 minus 9
minus 1
minus 1xBL1s49 ge pP3L1qP3L1s4
Macro-period 2
Constraint (23)
Constraint (24)
L1 P2 P3
6 7 8 9 10
Time
Figure 7 Time window for production
Macro-period 2 Constraint (28)
Constraint (29)
Tk1
L1
L2
s3
s3
s3
RmA
P2
P3
xBL2s37
xEL1s37
xBL1s37
xEL2s38
pP2L1qTk1P2L1s37 le xEL1s37
pP3L2qTk1P3L2s37 le xEL2s38
pP3L2qTk1P3L1s38 le xEL2s38
pP2L1qTk1P2L1s37 le xBL1s37
pP3L2qTk1P3L2s37 le xBL2s37
pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10
Time
Figure 8 Synchronization of line productions with tank storage
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le 119862
119898minus 120575119897119904 + (1minus 119909
119861119897119904120591) 119862119898
(27)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198641198971199041205911015840 (28)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119871
sum
119896=1
119869
sum
119895=1119901119895119897119902119896119895119897119904120591 le
119905sdot119879119898
sum
1205911015840=(119905minus1)119879119898+1119862119898
1199091198611198971199041205911015840 (29)
119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862
119898= 1 Let us suppose that Rm119860 is used to
produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to
120591 = 8 This is considered by constraints (28) and (29)
36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level
119909119895119896119904 = 0
119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878
(30)
119869
sum
119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)
119902119895119896119904 le 119876119896119909119895119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(32)
119902119895119896119904 le 119876119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(33)
119869
sum
119895=1119902119895119896119904 ge 119876
119896119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(34)
119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(35)
119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904
119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(36)
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Tk1
Tk2
RmC
RmB RmB
RmDs3 s4
6 7 8 9 10
Time
From constraints (31)ndash(34)
Constraint (35)
Constraint (36)
Macro-period 2
xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1
xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1
zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4
xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1
xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1
zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1
zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1
Figure 9 Raw material changeover
120590119896119904 =
119869
sum
119894=1
119869
sum
119895=11205901015840119894119895119896119911119894119895119896119904
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(37)
120596119896119905119904 le 120590119896(119905minus1)119878+1
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(38)
119905sdot119878
sum
119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905
119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878
(39)
A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints
From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862
37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time
horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials
119902119895119896119904 =
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878
119905 sdot 119862 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896119905sdot119878120591ge 120603119896119905+1 (41)
119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904minus1120591 ge 120590119896119904 (42)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905
(43)
119896 = 1 119871 119905 = 1 119879119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119909119861
119896(119905minus1)119878+1120591 = 1 (46)
119896 = 1 119871 119905 = 2 119879
10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
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10 Mathematical Problems in Engineering
119879119898
sum
120591=11199091198611198961120591 = 1 (47)
119896 = 1 119871 119905 = 2 119879
119869
sum
119895=1119902119895119896119904120591 le 119876119896119909
119864119896119904120591 (48)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =
(119905 minus 1)119879119898
+ 1 119905 sdot 119879119898
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879
119898119906119896119904 (49)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591
le 119879119898
119906119896(119905minus1)119878+1
(50)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198641198961120591 minus119879119898
sum
120591=11205911199091198611198961120591 le 119879
1198981199061198961 (51)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864
119896(119905minus1)119878+1120591
le 119906119896(119905minus1)119878+1
(53)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus119879119898
sum
120591=11205911199091198641198961120591 le 1199061198961 (54)
119896 = 1 119871
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879
119898119906119896119904 (55)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1120591119909119861
119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898
+ 1)
le 119879119898
119906119896(119905minus1)119878+1
(56)
119896 = 1 119871 119905 = 2 119879
119879119898
sum
120591=11205911199091198611198961120591 minus 1 le 119879
1198981199061198961 (57)
119896 = 1 119871
119906119896119904 ge 119906119896119904+1 (58)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1
119862119898
119906119896119904 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864119896119904120591 minus
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119861119896119904120591
= 120590119896119904
(59)
119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878
119862119898
119906119896(119905minus1)119878+1 +
119905sdot119879119898
sum
120591=(119905minus1)119879119898+1119862119898
120591119909119864
119896(119905minus1)119878+1120591
minus
119905sdot119879119898
sum
120591=(119905minus2)119879119898+1119862119898
120591119909119861
119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1
(60)
119896 = 1 119871 119905 = 2 119879
119862119898
1199061198961 +
119879119898
sum
120591=1119862119898
1205911199091198641198961120591 minus119879119898
sum
120591=1119862119898
1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)
119896 = 1 119871
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (62)
119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879
119898+ 1 119905 sdot 119879
119898
119869
sum
119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909
119861119896119904120591) + (1minus 119906119896119904)) (63)
119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898
+
1 119879 sdot 119879119898
Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909
119864Tk211990437 = 119909
119864Tk211990449 = 1 in constraint (48) for
the example shown in Figure 10
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Tk 2
6 7 8 9 10
Time
Macro-period 2
RmC RmB
xETk2s37xETk2s49
Constraint (48) QTk2 = 100l
qRmCTk2s36 le 100 middot xTk2s37E
qRmCTk2s37 le 100 middot xTk2s37E
Figure 10 Tank refill before the end of the setup time
The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)
Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879
119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909
119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes
this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account
Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period
38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks
119868119895119896120591 = 119868119895119896120591minus1 +
119905sdot119878
sum
119904=(119905minus1)119878+1119902119895119896119904120591
minus
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591
(64)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
119868119895119896120591 ge
119869
sum
119894=1
119871
sum
119897=1
119905sdot119878
sum
119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)
119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898
+1 119905 sdot
119879119898
minus 1What is in the tank at the end of a micro-period must be
what was in the tank at the beginning of that micro-period
0 101 2 3 4 5 6 7 8 9Time
RmA RmBRmC
RmE
RmARmD RmD RmC
Macro-period 1 Macro-period 2
Tk1Tk2Tk3L1
L3
P1 P2 P3P1
P5P5 P4
P6
L2
Micro-periods
P4
Figure 11 Possible solution for SITLSP
0 101 2 3 4 5 6 7 8 9Time
RmA RmB
RmCRmE
RmA
RmD RmD RmCs4
s4
s4
s3Tk1
Tk2
Tk3
L1
L3
P1
P4
P2 P3P1
P5P5 P4
P6
L2
Macro-period 1 Macro-period 2
Figure 12 Possible solution provided by the model
plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)
39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions
According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861
12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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12 Mathematical Problems in Engineering
Table 2 Parameter values
Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879
119898 5 119876119896
1000 l 119876119896 5000 l119862 5 tu 119862
119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u
Table 3 Parameter combination for each set of instances
Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869
1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342
(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))
The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909
119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909
119864119896119904120591
must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods
For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =
8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks
4 Computational Results forSmall-to-Moderate Size Instances
The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted
A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances
found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function
The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used
Gap() = 100(119885 minus 119885
119885
) (66)
where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination
Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
Table 4 Parameter ranges
Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897
1205901015840
119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896
119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost
119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840
119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897
119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895
Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2
Comb 119879 = 1 119879 = 2
Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196
Table 6 Model figures for the set of instances with 119879 = 1 2
Comb119879 = 1 119879 = 2
Var Const Var ConstBin Cont Bin Cont
1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865
TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances
These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These
instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem
5 Computational Results forModerate-to-Large Size Instances
The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
Table 7 AMPLCPLEX results for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Opt Gap() CPU Opt Gap() CPU1198781 10 072 1064 9 013 535171198782 4 338 217773 3 500 2814481198783 5 184 224924 2 455 3159171198784 9 020 46135 7 015 1808171198785 3 472 307643 1 388 3276781198786 2 331 320899 0 652 3600451198787 7 033 115023 4 057 2922491198788 1 470 325773 0 497 3600331198789 2 555 296303 0 506 360032
Table 8 Model figures for the set of instances with 119879 = 3 4
Comb 119879 = 3 119879 = 4
Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735
Table 9 Parameter combination for each set of instances
Comb 119871119871119869119869119879 Comb 119871119871119869119869119879
1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812
solutions found in shorter times than optimal solutionswhichrequire much longer runtimes
For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values
While the model parameters were changed to 119862 = 10 tu119879119898
= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first
Table 10 Relaxation approaches results
Comb 1198771 1198772
Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6
relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem
These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
0
500
1000
1500
2000
2500
3000
Aver
age C
PU (s
)
Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9
Optimal solutionsAll solutions
Figure 13 Average CPU time to find solutions
nonsatisfied demands are penalized in the objective functionof the model (see (1))
Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772
Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772
approaches returned approximate solutions within a reason-able computational time for these sets of large size instances
6 Conclusions
A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before
An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very
well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research
Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed
Appendix
Symbols for Parameters andDecision Variables
General Parameters
119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period
119862119898 capacity (in time units) within a micro-period
(119862 = 119879119898
times 119862119898)
120576 a very small positive real-valued number119872 a very large number
Parameters for Linking Production Lines and Tanks
119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894
Decision Variables for Linking Production Lines and Tanks
119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897
in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
Parameters for the Production Lines
119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862
119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to
product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)
1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895
Decision Variables for the Production Lines
119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-
period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895
Parameters for the Tanks
119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored
119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial
119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862
119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896
119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896
Decision Variables for the Tanks
119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895
at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896
in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591
119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4
References
[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982
[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990
[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991
[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000
[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009
[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010
[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005
[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011
[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013
[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015
[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997
[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997
[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995
[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999
[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002
[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003
[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010
[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000
[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002
[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003
[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005
[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007
[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008
[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013
[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014
[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997
[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003
[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007
[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Mathematical Problems in Engineering
[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013
[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002
[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000
[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014
[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008
[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009
[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012
[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014
[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007
[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of