Omega 41 (2013) 510–516

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Omega

0305-04

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/omega

Multi-product lot scheduling with backordering and shelf-life constraints

Changyuan Yan a, Yi Liao b,n, Avijit Banerjee a

a Drexel University, Philadelphia, PA, USAb Southwestern University of Finance and Economics, Department of Information Systems and Supply Chain Management, School of Business Administration,

555 Liutai Avenue, Chengdu, Sichuan Province 611130, China

a r t i c l e i n f o

Article history:

Received 23 September 2010

Accepted 21 June 2012

Processed by B. Levfeasible production schedule, including the manufacturing batch size of each item. We assume that

total backordering is permissible and that each of the products has a limited post-production shelf life.

Available online 1 July 2012

Keywords:

Economic lot scheduling

Shelf-life constraint

Backordering

83/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.omega.2012.06.004

esponding author. Tel.: þ011861518439976

ail address: yiliao@swufe.edu.cn (Y. Liao).

a b s t r a c t

In this paper, we revisit the economic lot scheduling problem (ELSP), where a family of products is

produced on a single machine, or facility, on a continual basis. Our focus is on the determination of a

Several studies examining this problem have suggested a rotational common cycle approach, where

each item is produced exactly once every cycle. To ensure schedule feasibility, we resort to the

technique of reducing individual production rates and allow the flexibility of producing any item more

than once in every cycle, in conjunction with appropriate timing adjustments. In order to solve this

more generalized model, which is NP hard, we suggest a two-stage heuristic algorithm. A numerical

example demonstrates our solution approach.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

This paper examines the problem of determining the optimalproduction rate, manufacturing batch size and the productionfrequency for each item within a family of products that areprocessed on a single machine, or within a single capacitatedfacility. Total backordering is allowed for any of the products,each of which has a specified, finite post-production shelf-life.Our objective is to determine a feasible production schedule forthese items, while attempting to minimize the total relevant costpertaining to the entire family of products.

The economic lot scheduling problem (ELSP) with shelf-life con-straints has been examined earlier by Silver [9,10], Sarker and Babu[8], Goyal [1], Viswanathan [13], Viswanathan and Goyal [14,15], andSharma [3,4,5,6,7]. Silver [9] incorporates the characteristics of shelf-life constraints in his rotational cycle model and discusses two waysto satisfy these constraints. One approach involves slowing down theproduction rates and the other reduces the production cycle time.Silver [9] proves that slowing down the production rates is a moreeffective technique. Sarker and Babu [8], on the other hand, analyzethe same model outlined by Silver [9] and show that reducing thecycle time sometimes can be more effective, if a machine or facilityoperating cost is considered. Subsequently, Silver [10] deals with theshelf-life constraints in cyclic scheduling, by adjusting both the cycletime and the items’ production rates. He specifically examines the

ll rights reserved.

6.

situation where the cost-minimizing cycle time leads to the violationof one of the shelf-life constraints. Along similar lines, Viswanathanand Goyal [14] develop a model and provide an algorithm fordetermining the optimal production rate for each item and theoptimal cycle time for the product family. Furthermore, Viswanathanand Goyal [15] embellish their earlier model by allowing backorders.In recent years, this problem has received significant researchattention. For example, papers by Sharma [3,4] incorporate shortagesand fractional backordering, respectively; also, Sharma [5] includes ageneralized production cost; and Sharma [6,7] focus on the estima-tion of the inventory carrying cost and its modification. Recentstudies have focused on the development of optimal and heuristicsolution techniques for the ELSP. Notably, Grznar and Riggle [2]provide a global optimum solution for the basic period approach tothe ELSP, whereas Tempelmeier [11] develops a column generationheuristic to deal with the dynamic ELSP under stochastic demandswith a service level constraint.

All the studies mentioned above, however, assume that each ofthe items in question is produced exactly once in every rotationalmanufacturing cycle. Goyal [1] points out that producing some ofthe items more than once in a cycle may be more cost-effective.He assumes specific values of the production (or setup) frequencyof each item and determines the minimum total relevant cost,given the number of batches per cycle for each item. Nevertheless,Viswanathan [13] points out that this approach may sometimeslead to infeasible schedules. Furthermore, it is assumed that thesetup frequencies for the items are known a priori. These twoearlier studies do not explore the derivation of the appropriatenumber of setups for each item in a cycle.

C. Yan et al. / Omega 41 (2013) 510–516 511

In this paper, we develop an extended model for the ELSP,allowing for backorders and limited post-production shelf-livesfor the items in the family, while removing a restrictive assump-tion made in earlier studies and allow each of the items to beproduced more than once in every cycle. Based on our model, weattempt to determine the corresponding optimal productionpolicy, including each item’s production frequency, its lot size,as well as its manufacturing rate and a feasible overall productionschedule. The distinguishing feature of our work, compared toexisting studies, is that the number of setups for each item in arotational production cycle is treated as a decision variable and isdetermined through the solution of our model. We illustrate themodel developed and its solution, in order to indicate the efficacyof our approach, via a numerical example.

2. Assumptions and notation

2.1. Assumptions

We make the following major assumptions in developing ourmodel:

(a)

The demand rate for each item is known and constant. (b) The setup time for each item is known and constant. (c) Inventory transactions are based on the FIFO rule. (d) Total backordering is allowed for each item. (e) Each item has a limited post-production shelf life. (f) There is a machine operating cost per time unit. (g) The production rate of each item is treated as a decision variable.

The third assumption above, i.e. the FIFO rule, is needed forsimplicity of analysis in dealing with items having finite shelflives. Also, the assumption concerning the existence of a machineoperating cost allows the possibility of cycle time reduction to beeffective from a cost reduction standpoint.

2.2. Notation

Our model development process and subsequent analyses arebased on the following notational scheme.

(a) For the entire family:

N total number of items;T the production cycle time;O the machine operating cost per unit time;C the average total relevant cost per time unit.

(b) Parameters for item i (i¼1, 2,y, N)

di the item’s demand rate;pmax

i the item’s maximum possible production rate;

0

Fig. 1. Time conflict in

rmini ¼ di=pmax

i ;hi the item’s unit holding cost per time unit;bi the item’s unit backorder cost per time unit;li post-production shelf-life;ri setup time per batch;si setup cost per batch (excluding machine operating cost

during setup).

(c) Variables for item i (i¼1, 2,y, N)

pi the adopted production rate for the item;ri di=pi;f i production frequency per cycle;ti T=f i, the item’s cycle time;ki the item’s production start time;aj

i the start time advancement for item i in its jth produc-tion batch (1r jr f i);

ai

Pf i

j ¼ 1 aji, the total start time advancement for item i over

the entire production cycle;bj

i the start time delay for item i in its jth production batch(1r jr f i);

bi

Pf i

j ¼ 1 bji , the total start time delay for item i over the

entire production cycle;ai an adjustment cost during the total production

cycle time;ci the total relevant cost over the item’s production

cycle time.

3. Model development

As mentioned earlier, Viswanathan [13] points out that apolicy allowing for more than one setup per cycle for the itemsmay lead to an infeasible schedule. However, when this happens,the schedule can be adjusted by advancing or delaying the starttime(s) of one or more of the items, in order to achieve feasibility.We demonstrate this by a simple example, where we have 2 itemsin a family, with setup frequencies per cycle of 1 and 2,respectively. As shown in Fig. 1, during time T to T0, both item1 and 2 are scheduled for production simultaneously, whichmakes this schedule infeasible. In order to force schedule feasi-bility, we can advance the start time of the second batch of item2 in the production cycle by T0–T, as shown in Fig. 2. Analternative approach would be to delay the start time of item1 in the second production cycle by an amount T0–T, as shown inFig. 3.

3.1. Cost function

If a schedule is adjusted for attaining feasibility, the appro-priate adjustment costs should be considered. Based on ouranalysis shown in the Appendix, the total adjustment cost can

T T’

Item 1

Item 2

original schedule.

Fig. 2. Possible alternative schedule: advance start time of item 2 by T0–T.

Fig. 3. Possible alternative schedule: delay item 1’s start time by T0–T.

C. Yan et al. / Omega 41 (2013) 510–516512

be expressed as

ai ¼hiþbi

2di

Xf i

j ¼ 1

ðajiÞ

2þXf i

j ¼ 1

ðbjiÞ

2

0@

1A 1�

di

pi

� �ð1Þ

Note that in expression (1) above, the adjustment cost foradvancing an item’s start time results from the additional inven-tory holding cost, while backorder costs increase as a conse-quence of a delay in its start time. Needless to say, that an item’sstart time can be either advanced or delayed, but not both. Thus, ifai40, bi¼0, and vice versa.

We consider the total relevant cost over an item’s cycle timefor each item individually, which is provided in Viswanathan andGoyal [15]. Based on their results, the total cost for each item(produced exactly once every cycle) during its own productioncycle is

ci ¼ siþO riþO ri tiþti2 diUhi

2ð1�riÞ

bi

hiþbið2Þ

This expression captures the setup, machine operating, inven-tory holding and backordering costs, respectively, for item i. Adetailed description can be found in Viswanathan and Goyal [15].Considering all the N items, with an overall production cycle timeof T, where for item i, fi batches are produced with a scheduleadjustment cost, if any, of ai, the derivation of our objectivefunction (the total relevant cost per time unit), using (1) and (2),is straightforward, i.e.

C ¼1

T

XN

i ¼ 1

½ðcif iÞþai�

¼1

T

XN

i ¼ 1

24 siþOriþOritiþti

2 diUhi

2 ð1�riÞbi

hiþbi

� �f i

þhiþbi

2di

Xf i

j ¼ 1

ðajiÞ

2þXf i

j ¼ 1

ðbjiÞ

2

0@

1A 1�

di

pi

� �35 ð3Þ

3.2. Constraints

The constraints that apply to this problem are: (a) restrictionson the product shelf-lives, (b) total time available for productionand setups for all items in a cycle, (c) constraints on themaximum production rates, and (d) schedule feasibility, i.e. theschedule is free of conflicts indicated by overlapping productiontimes. These are outlined below in detail.

(a) Shelf-Life Constraints

According to Viswanathan and Goyal [15], the shelf-life con-straint for item i, before schedule adjustment, can be written as

tið1�riÞbi

hiþbir li

If there is any time advancement in the schedule, denoting thetime advanced for the jth batch of item i during the cycle as aj

i , theshelf-life constraints can be rewritten as follows:

tið1�riÞbi

hiþbiþaj

ir li

Similarly, in the case of a delay, denoting the time delayed forthe jth batch of item i during the cycle as bj

i, the shelf-lifeconstraints can be rewritten as

tið1�riÞbi

hiþbi�bj

ir li

Therefore, the operative constraints on the shelf-life of item i

can be expressed as

tið1�riÞbi

hiþbiþaj

i�bjir li, for j¼ 1,2,:::,f i

Xf i

j ¼ 1

aji ¼ ai;

Xf i

j ¼ 1

bji ¼ bi; aj

ibji ¼ 0 ð4Þ

where aji

40, when advancement occurs

¼ 0, otherwisebj

i

40, when delay occurs

¼ 0, otherwise

((

(b) The total time available for production and setups in aproduction cycle:

Tð1�XN

i ¼ 1

riÞZXN

i ¼ 1

f iri ð5Þ

(c) Constraints on the maximum production rates are

riZrmini , for i¼ 1,2,. . .,N ð6Þ

(d) Schedule feasibility:We define a set of Bernoulli variables cw

i , where 1r irN,1rwrF ¼

PNi ¼ 1 f i as follows.

cwi ¼

1, if item i is produced in the wth batch

0, otherwise

(

Therefore, for the schedule to be feasible, the following musthold:

for 8i,X

w

cwi ¼ f i and for 8w,

Xi

cwi ¼ 1

C. Yan et al. / Omega 41 (2013) 510–516 513

For the wth production batch within a production cycle,denoting the item produced asIðwÞ, which is the JðwÞth batch ofthis item, we have

IðwÞ ¼XN

i ¼ 1

cwi i, JðwÞ ¼

Xw

q ¼ 1

cqIðwÞ

Since the start time of the first production batch of item i is ki,the time to start its wth batch is

tw ¼ kIðwÞ þðJðwÞ�1ÞtIðwÞ�aJðwÞIðwÞ þb

JðwÞIðwÞ,

and its completion time is

Bw ¼ kIðwÞ þðJðwÞ�1ÞtIðwÞ�aJðwÞIðwÞ þb

JðwÞIðwÞ þrIðwÞtIðwÞ

The conditions that the start of an item’s production can occuronly after its previous batch is completed and that the completiontime of its last batch cannot exceed the cycle time are

twþ1ZBw, for 8wA ½1,F�1�

BF rT

In summary, the complete set of constraints for ensuringschedule feasibility isP

wcw

i ¼ f i, 1r irN

Xi

cwi ¼ 1, 1rwrF ¼

XN

i ¼ 1f i

IðwÞ ¼XN

i ¼ 1

cwi i

JðwÞ ¼Xw

q ¼ 1

cqIðwÞ

kIðwþ1Þ þðJðwþ1Þ�1ÞtIðwþ1Þ�aJðwþ1ÞIðwþ1Þ þb

Jðwþ1ÞIðwþ1ÞZkIðwÞ þðJðwÞ�1ÞtIðwÞ

�aJðwÞIðwÞ þb

JðwÞIðwÞ þrIðwÞtIðwÞ, 1rwr

XN

i ¼ 1

f i�1

TZkIðFÞ þðJðFÞ�1ÞtIðFÞ�aJðFÞIðFÞ þb

JðFÞIðFÞ þrIðFÞtIðFÞ

cwi ¼ 1 or 0 ð7Þ

3.3. Optimization model

Using objective function (3) and incorporating the constraints(4)–(7), we can express our complete optimization model asfollows:

min CðT ,ri,f iÞ ¼1

T

XN

i ¼ 1

siþOriþOritiþti2 diUhi

2 ð1�riÞbi

hiþbi

� �

f iþhiþbi

2 di

Xf i

j ¼ 1

ðajiÞ

2þXf i

j ¼ 1

ðbjiÞ

2

0@

1A 1� di

pi

� �266664

377775

Subject to

for 1r irN, 1r jr f i

tið1�riÞbi

hiþbiþaj

i�bjir li,

for 1r irN, 1r jr f i ajib

ji ¼ 0

for 1r irN riZrmini

for 1r irNX

w

cwi ¼ f i

for 1rwrFX

i

cwi ¼ 1

for 2rwrF

kIðwÞ þðJðwÞ�1ÞtIðwÞ�aJðwÞIðwÞ þb

JðwÞIðwÞZkIðw�1Þ

þðJðw�1Þ�1ÞtIðw�1Þ�aJðw�1ÞIðw�1Þ þb

Jðw�1ÞIðw�1Þ þrIðw�1ÞtIðw�1Þ

TZ ðXN

i ¼ 1

f iriÞ=ð1�XN

i ¼ 1

riÞ

TZkIðFÞ þðJðFÞ�1ÞtIðFÞ�aJðFÞIðFÞ þb

JðFÞIðFÞ þrIðFÞtIðFÞ

T,ri,aji,b

ji Z0; f i40 integer; cw

i ¼ 1 or 0

where

ti ¼ T=f i

rmini ¼ di=pmax

i

ai ¼Xf i

j ¼ 1aj

i

bi ¼Xf i

j ¼ 1bj

i

F ¼XN

i ¼ 1f i

IðwÞ ¼XN

i ¼ 1cw

i i, for1rwrF

JðwÞ ¼Xw

q ¼ 1cq

IðwÞ, for1rwrF

This represents a relatively complex non-linear, mixed integermathematical programming problem.

4. Solution algorithm and numerical example

The model shown above is not easy to solve. Therefore, wesuggest the following two-stage heuristic algorithm. In the firststage, we determine the cycle time, as well as the frequency andthe production rate for each item via a relaxed version of themodel outlined above. In this simplified model, which is solvableby means of appropriate software, the schedule adjustment costsand constraints are omitted. In the second stage, if the initialsolution is infeasible, we adjust the schedule for attainingfeasibility, with minimal additional total adjustment cost. Whenthe second step is necessary, the problem becomes a single-machine total weighted earliness and tardiness scheduling pro-blem with different due dates, which is NP hard (see, for example,Verma and Dessouky [12]). Consequently, we suggested a rela-tively simple schedule adjustment process using a greedy heur-istic of successively choosing the activities, one at a time, foreither an appropriate start time advancement or delay, based onthe lowest per time unit cost of advancing (or delaying), until afeasible schedule is obtained. The detailed steps of our heuristicprocedure are outlined below.

Step 1: Without considering the adjustment costs, solvefollowing integer nonlinear problem:

min CðT,ri,f iÞ ¼1

T

XN

i ¼ 1

ðsiþOriþOritiþti2 diUhi

2ð1-riÞ

bi

hiþbiÞf i

� �

Subject to,

for 1r irN tið1�riÞbi

hiþbir li,

for 1r irN riZrmini

And TZ ðXN

i ¼ 1

f iriÞ=ð1�XN

i ¼ 1

riÞ

where T,riZ0; f i are positive integers;

Step 2: For determining the feasibility of the solution obtainedfrom step 1, we create a schedule such that the first N batches areproduced in increasing order of the frequencies of all the N items,without idle time. Based on the cycle time for each item, we can,thus, obtain an initial schedule. If this schedule is feasible,indicated by no production time overlaps for all the items, thealgorithm terminates; otherwise, for attaining feasibility, some

Fig. 5. Feasible schedule obtained from step 3.

Table 3Summary of results from step 1 when O is low (Or500).

Item i f i ri pi ti Shelf time Maximum backorder level T

1 1 0.33 3000 0.390 0.16 97

2 3 0.16 2500 0.130 0.07 14 0.39

3 2 0.49 1430 0.195 0.06 26

C. Yan et al. / Omega 41 (2013) 510–516514

adjustments in the current solution are necessary, as indicated instep 3.

Step 3: If the initial schedule is infeasible, there must be sometime conflict(s) in it, i.e. the production times of two or moreactivities overlap. We resolve the first time conflict that occurs inthe current schedule by adjusting the start time of the item whichhas the smallest advancement, or delay cost per time unit. Thisstart time is either advanced or delayed, as dictated by the lowestcost, by the appropriate amount of time, so that the existingproduction time overlap is resolved. This adjustment process isrepeated until all time conflicts are eliminated from the schedule,in order to render it feasible.

It is to be noted that the relaxed optimization problemoutlined in step 1 can be solved with relative ease for problemsof reasonable size with any commonly available non-linear,mixed integer programming software package. In case that thesolution obtained at this step does not generate a feasibleschedule, adjustments would be necessary. Under the worst case

scenario, the possible number of time conflicts among thePN

i ¼ 1 f i

batches isPN

i ¼ 1 f i

� � PNi ¼ 1 f i�1

� �=2. Thus, theoretically, at mostPN

i ¼ 1 f i

� � PNi ¼ 1 f i�1

� �=2 adjustments have to be made to

achieve feasibility. In practice, however, the number of adjust-ments needed is likely to be less than the worst case, since inmany instances, a single time advancement or delay can oftenresolve multiple conflicts in an existing schedule.

We now demonstrate our solution procedure via a three-itemexample adapted from Silver [9]. The problem data are summar-ized in Table 1 below.

O¼ machine operating cost per year¼$1000.

Table 1Problem Data.

Item i di ri si hi pmaxi bi li

1 1000 0.0005 125 3 3000 5 0.20

2 400 0.0010 25 25 2500 50 0.10

3 700 0.0015 75 15 2500 25 0.20

Table 2Summary of results from step 1.

Item i f i ri pi ti Shelf time Maximum backorder level T

1 1 0.33 3000 0.36 0.15 90

2 3 0.16 2500 0.12 0.07 13 0.36

3 2 0.28 2500 0.18 0.08 34

Fig. 4. Schedule obtained from step 1.

To make this problem practical, as well for limiting thecomputational effort, we restrict the items’ production frequen-cies to be no more than three. The results of applying step 1 of ouralgorithm are summarized in Table 2 below. The annual totalrelevant cost yielded by this solution is $2749. It is interesting tonote that in the presence of a machine operating cost, none of theproduction rates are reduced, since the reduction of cycle time ismore cost effective here. This is in agreement with the earlierfindings of Sarker and Babu [8].

For evaluating the feasibility of this solution, we apply step2 of our algorithm and the resulting Gantt chart is shown belowin Fig. 4. This figure details the production schedule over the firsttwo cycles. The dark areas in Fig. 4 represent the three items’respective production times on the single available machine. It isclear, that item 1, the third batch of item 2 and the second batchof item 3 have time conflicts (i.e. time overlaps), indicating theinfeasibility of the current schedule. For example, item 2 needs tobe produced from time 0.36 to 0.38, while item 1 needs to beproduced during the interval 0.36–0.48 and item 3 needs to beproduced during the interval 0.32–0.37.

Therefore, for deriving a feasible schedule, some adjustments,based on step 3, are necessary for eliminating the overlappingproduction times in the initial schedule. We eliminate the firsttime conflict between items 2 and 1, based on the minimum costfor changing the start time, which pertains to advancing the startfor item 2. Consequently, the start time for third batch of item 2 isadvanced from 0.36 to 0.34. After this adjustment, we focus onthe second time conflict between products 1 and 3. Again,following the procedure described in step 3, we resolve thisconflict by advancing the start time for the second batch of item3 from 0.32 to 0.29. These changes are shown in Fig. 5, represent-ing the revised schedule, which, in the absence of overlappingproduction times, is now feasible.

For this schedule, the batch sizes for items 1, 2 and 3 are,respectively, 360, 48 and 126 units; and their maximum back-order levels are 90, 13 and 34 units, respectively. Also, as shownin Fig. 5, in each cycle, we have advanced the start time for thirdbatch of item 2 by 0.02 and the start time for second batch of item3 by 0.03 time units, based on the lowest advancement or delaycost per time unit at each step of adjustment. Furthermore, since

Table 4Comparison of our model with existing models.

This paper Ref. [15] Ref. [9]

Initial

cost ($)

Adjustment

cost ($)

Total

cost ($)

O¼1000 2749 14 2763 3084 3678

O¼750 2552 14 2566 2881 3428

O¼500 2331 48 2379 2638 3178

O¼250 2081 48 2129 2388 2928

O¼0 1831 48 1879 2138 2678 Fig. 6. Inventory-time plot of an item after a start time advance.

C. Yan et al. / Omega 41 (2013) 510–516 515

a32 ¼ 0:02 and a2

3 ¼ 0:03, the total adjustment cost is $14 and theresulting annual total relevant cost is $2749þ$14¼$2763. Impos-ing the restriction that each item is produced exactly once everycycle, the total relevant cost would be $3084 per year. Thus, ouranalysis and results confirm the findings of an earlier paper byGoyal [1] and indicates that allowing for the production of eachitem more than one in every cycle yields a lower total cost.Furthermore, deriving the appropriate production frequencies forthe items concerned, albeit heuristically, instead of adoptingarbitrary, predetermined values of these frequencies (as in theprevious paper by Goyal [1]), is likely to be a more effectiveapproach for cost reduction.

In the above example, with a relatively high machine operat-ing cost (i.e. O¼$1000), reduction of cycle time appears to bemore cost effective than decreasing the production rates. Whenthe machine operating cost is low, however, production ratereduction could be more economical. For example, if the annualmachine operating cost is less than $500, we obtain the followinginitial results.

Table 3 shows that the production rate of item 3 has beenreduced from 2500 to 1430, indicating that production rate reduc-tion is more desirable than decreasing the cycle time. Table 4 belowshows a comparison of results obtained by our model with thoseyielded by earlier models developed by Viswanathan and Goyal[15] and Silver [9] for different values of annual machine operatingcost, O. From Table 4, it is clear that allowing for the production ofeach item more than once in every cycle yields a lower total cost forall instances of machine operating cost examined. Thus, it appearsthat our model is an improvement over existing methodologies.

5. Summary and conclusions

This paper examines the problem of determining the optimalproduction rates and frequencies for multiple products in a familyproduction context, where each item has a limited post-produc-tion shelf life and total backordering is allowable. This is anembellished version of the classical ELSP, where schedule feasi-bility is a major complicating factor. Our study is different fromexisting research in this area in two important respects. First, werelax the restriction, adopted in much of previous work, that eachproduct is produced exactly once during each production cycle. Afew existing papers relax this restriction, albeit with the assump-tion that the production frequencies of the items are given. Thispaper relaxes this assumption and treats these production fre-quencies as decision variables. Our model is, thus, based on thepremise that multiple setups are possible for any product over afamily production cycle. In fact, we demonstrate that allowingsuch multiple setups leads to improved solutions. Furthermore,the appropriate number of setups for each item in a cycle isdetermined analytically via the model, rather than being known a

priori. This is a notable distinction of our approach in comparisonwith extant related studies.

We develop a non-linear, mixed integer optimization modelfor solving the above mentioned embellished ELSP, which hasbeen shown to be NP hard. Therefore, we suggest a two-stageheuristic approach towards obtaining an acceptable solution, withreasonable requirements on effort and time. Although the firststep of our solution algorithm requires the use of non-linear,mixed integer programming software, the current state-of-the-artconcerning the capabilities of such software indicates that ourfirst stage model can be solved with relative ease for mostreasonably sized problems. In addition, the schedule adjustmentprocess in the second stage of the solution procedure is relativelysimple and quick.

In view of this and the simplicity resulting from the premisethat each item is produced in equal batch sizes and that theproduction cycles are identical, our work is likely to have someappeal from a managerial perspective, due to its relatively ease ofimplementation in practice. This simplifying premise, however,also represents a limitation of this paper. Therefore, it is reason-able to state that future research should consider solutionsexploring the possibility of unequal batches for each product, aswell as dissimilar production cycles, for achieving further costsavings. We hope that this work represents a step forwardtowards enhancing the existing literature concerning an impor-tant problem area and, thus, has been able to shed some light inunderstanding the essential characteristics and complexities ofthis embellished version of the ELSP. In conclusion, our work islikely to be helpful for future researchers towards developingbetter models and improved solution methodologies.

Appendix

Derivation of schedule adjustment cost:Fig. 6 illustrates the inventory pattern change when a schedule

adjustment of advancing the start time occurs for the j-th batch ofan item i. With such an adjustment, the inventory replenishmentdepicted by the line segment AC will move to FE. Clearly, this starttime advancement increases the holding cost by hiUSCEHG, whilereducing the backorder cost by bi SAFHG, where SCEHG and SAFHG arethe inventory-time unit areas CEHG and AFHG, respectively. Thus,the cost increase resulting from the adjustment in the start timeof this batch is given by

aji ¼ hi SCEHG�bi SAFHG ðA1Þ

Denote the maximum inventory level and maximum back-order level of this item as M1 and M2, respectively. Then,according to Viswanathan and Goyal [15],

M1 ¼ tidið1�riÞbi

hiþbi

M2 ¼ tidið1�riÞhi

hiþbi

C. Yan et al. / Omega 41 (2013) 510–516516

Thus, hi M1 ¼ biM2 and hi SCDHG ¼ bi SABHG. Also, it can be easilyshown that SCDE ¼ SABF . Consequently, Eq. (A1) can be rewritten asfollows:

aji ¼ ðhiþbiÞ SABF ðA2Þ

Note that, the line segment AB represents aji and the height of

F from AB depicts diaji 1� di

pi

� �. Therefore, the cost of this schedule

adjustment is

aji ¼ ðhiþbiÞ

1

2diðaj

iÞ2 1�

di

pi

� �ðA3Þ

Following the same line of reasoning, we can show that whenthe start time of the jth batch of item i is delayed, the resultingschedule adjustment cost can be expressed as

aji ¼ ðhiþbiÞ

1

2diðb

jiÞ

2 1�di

pi

� �ðA4Þ

Considering both types of schedule adjustments, the totaladjustment cost for item i over an entire family production cycleis given by

ai ¼hiþbi

2di

Xf i

j ¼ 1

ðajiÞ

2þXf i

j ¼ 1

ðbjiÞ

2

0@

1A 1�

di

pi

� �

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