research article multi-item distribution policies with...

Research ArticleMulti-Item Distribution Policies with Supply Hub andLateral Transshipment

Zhong Jin-Hong Jiang Rui-Xuan and Zheng Gui

School of Management Hefei University of Technology Hefei 230009 China

Correspondence should be addressed to Zhong Jin-Hong jinhongzhonghfuteducn

Received 17 January 2015 Accepted 23 April 2015

Academic Editor Thomas Hanne

Copyright copy 2015 Zhong Jin-Hong et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Supply Hub is defined as the horizontal coordination among the suppliers while lateral transshipment is a horizontal coordinationpolicy among the retailers By considering the Supply Hub and lateral transshipment simultaneously ones can reduce the totalcost of the supply chain system and improve the response to customer requirement and the customersrsquo satisfaction We investigatethe distribution policies for the supply chain which consists of multisuppliers single Supply Hub and multidistributors In thesystem both Supply Hub and distributors adopt the (t S) policy Supply Hub will not be out of stock and backlogging is forbiddenCustomer requirements at distributors are assumed to be independent random variables complying with uniform distributiontransshipment is assumed to be bidirectional instantaneous and emergent We establish the distribution models respectively forthe cases of transshipment or no transshipment For the casewith transshipment we design aGA-based solutionmethod involving atwo-stage selection technology that is firstly selecting individuals from parent population to generate offspring chromosomes andsecondly selecting individuals from the interim population comprising all of the parent and offspring genomes to form the next-generation population We show that lateral transshipment can increase the overall profit of the supply chain by the comparisonexaminations between the models with and without transshipment

1 Introduction

The main objective of supply chain management (SCM) isto optimize the entire system [1] Inventory managementmodels of supply chain are the focuses of attention embody-ing the characteristic of integrated and joint managementVendor managed inventory (VMI) is a representative modelwhere both supply suppliers and their customers managetheir inventory in a win to win manner [2] VMI is beneficialfor suppliers to arrange inventory and production reasonablyaccording to the status of distributors or manufacturersMoreover it is helpful to reduce the inventory of the supplyand distribution parties to improve customer satisfaction soit has been widely applied by firms Supply Hub mode is theupgrading version of VMI which allows multiple suppliersto share the storage space of Supply Hub and pool the inven-tory cost of Supply Hub and is the horizontal cooperationamong suppliers Lateral transshipment is the horizontal

collaboration among distributors (retailers) which is prof-itable to reduce the overall inventory of distributors (retailers)and improve customer satisfaction Considering lateral trans-shipment policy under Supply Hub model could improve theflexibility of firmsrsquo response to the various customer demandsfurther

Initially Supply Hub is aimed at supply chain upstreamand is defined as a place which is physically close to man-ufacturerrsquos facility and is used to store the raw materials ofall or some of the suppliers and adopts the agreement thatthe materials will be paid for only when consumed [3 4]Supply Hub first appears in the practice of enterprise man-agement Kopczak [5] studies the strategies adopted by Applecomputers in the setting up of Supplier Hub at three of theirproduction sites Zuckerman [3] introduces the analogouspractice undertaken by Compaq Computer at its Houstonproduction facility Barnes et al [4] point out that SupplyHub evolves gradually fromManufacturer Owned Inventory

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 702482 12 pageshttpdxdoiorg1011552015702482

2 Mathematical Problems in Engineering

(MOI) VMI and Supplier Owned Inventory (SOI) particu-larizes the arguments for and against using Supply Hub theprerequisite of establishing Supply Hub and the typical casesof Supply Hub describes current operation models of SupplyHub discusses the information technology supporting theoperation of Supply Hub and identified general industryproblems and possible research issues Shah and Goh [6]address the coordination problem in a two-stage supplychain consisting of single supplier single Supply Hub andsingle customer under the deterministic demand of singleproduct where backlogging is allowed and minimal andmaximal inventory limitations at Supply Hub are consideredthey develop a structured hierarchy method Gaonkar andVisvanadham [7] investigate an electronic product supplychain composed of single Supply Hub 119873 first-tier suppliersand 119873 second-tier suppliers and establish the linear pro-grammingmodel of coordination schedulingwith the limitedproduction and inventory capacity Li et al [8] establishdecentralized inventorytransportation supply chain modeland centered inventorytransportation supply chain modelconsidering Supply Hub respectively through comparativeanalysis they find that the latter could significantly reducethe system total cost and the observation that the closer theSupply Hub is located to the upstream of supply chain themore cost will be saved Furthermore they also find thatthe latter could lessen the bullwhip effect better comparedwith the former Li et al [9] establish a mixed integerprogramming model for the supply chain design with twoSupply Hubs and put forward the genetic algorithm to solveit Huang andMa [10] address the coordinated replenishmentpolicy between suppliers considering milk run and SupplyHub and compared it with traditional replenishment policyLiu and Chen [11] study the single product supply chaincomprising multiple manufacturers and multiple retailersunder the asymmetric information environment for thecase of the nonlinear relationship between demand andprice they establish the models with and without SupplyHub respectively Lin [12] studies the supply chain involvingmultiple suppliers single VMI Hub multiple factories andmultiple customers and compares the two modes of supplyinventory managed by firm itself and by VMI Hub Qiuand Huang [13] put forward the concept of Supply Hub inIndustrial Park (SHIP) aimed at warehousing and logisticsservices and establish the productiondistribution modelswith and without SHIP respectively give a genetic algorithmto solve themodels and show that the freight consolidation isprofitable to the entire industrial park Qiu et al [14] presentthe thought enhancing the efficiency and effectiveness of thephysical assets and services sharing through the SHIP drivenby the Internet of things (IOT)

The earliest study of lateral transshipment is about thesingle period maximal inventory order policy [15] Up todate there have been many scholars to conduct extensiveresearch in this area [16ndash18] the existing researchmainly aimsat the single product and the research on multiproduct isscarce Archibald et al [19] study multi-item two-locationinventory system with lateral transshipment and emergencyorder where the periodic review order-up-to S policy isadopted Kranenburg and vanHoutum [20] studymulti-item

multilocation single-echelon spare parts system with basestock control which comprises two types of local warehouse(ie main and regular warehouses) onlymainwarehouse canbe the supplier of lateral transshipment Wong et al [21 22]studymulti-item repairable spare parts inventory systemwithlateral and emergency shipments where continuous reviewbase stock policy and the average waiting time limitation ofrequired parts are taken into account They develop a greedyheuristic method which can obtain the approximate optimalsolution for multilocation inventory system and a solutionprocedure based on Lagrangian relaxation for two-locationinventory system Inventory problem of substitutable prod-ucts can be viewed as lateral transshipment between prod-ucts Shumsky andZhang [23] study themultiperiod dynamicallocation problem of inventory capacity in the firm sellingmultiple product types of same class where demand of aproduct could bemet by a product from the next-higher classAvsar and Baykal-Gursoy [24] study an infinite period two-retailer inventory control system with two kinds of substi-tutable products using stochastic game theory Netessine andRudi [25] study the centered inventorymanagementmodel ofmultiple substitutable products and decentralized inventorymanagement model in which each firm manages a productrespectively Rottkemper et al [26] study the inventoryrelocation and distribution for relief items transshipmentnetwork comprising a global a central and a number ofregional depots build a multiperiodmixed-integer program-ming model with minimization of unsatisfied demand andminimization of operational costs and suggest a rollinghorizon solutionmethod Alvarez et al [27] consider amulti-item spare parts network of a central depot andmultiple localwarehouses where each warehouse has its own premiumand nonpremium customers with class-specific waiting timerestrictions lateral transshipment and emergency shipmentsare used as differentiation tool They develop a heuristicapproach similar to Dantzig-Wolfe decomposition to setstock levels and (trans)shipment strategies of multi-itemsystem Moghaddam and Nof [28] investigate demand andcapacity sharing decisions problem in collaborative networkof supply enterprises under uncertain multi-item demandwhere the bestmatching protocol is proposed tominimize thetotal cost of collaboration and the gap between the capacityof the supply enterprises and their allocated demands Theybuild a fuzzy mixed-integer planning model for the problem

Although many authors have widely applied GA to solveSCM problems we cannot find the research which is directlyrelated to this work So we shall provide an overview forthe literature on the inventory management for one-to-manysupply chain under VMI mode and multilocation inventorysystem For the VMI system Michaelraj and Shahabudeen[29] research optimal replenishment policies of the distrib-utors working in the scenario that retailers are making thepayment to the distributors in number of unequal install-ments and use GA and the linear programming to solve the adistributor multiretailer single itemmultiperiodVMI systemto minimize balance payment and maximize sales Lan et al[30] establish two inventory control models of deterioratingitem for the single item multiperiod supply chain includinga manufacturer a vendor and many retailers located in

Mathematical Problems in Engineering 3

different regions respectively according to the time-basedand the quantity-based integrated delivery strategies forsuppliers under VMI model and design a GA to resolve themodels Nachiappan and Jawahar [31] present a GA-basedheuristic to deal with the operational issues of single vendor-multiple buyers single item supply chain model underVMI mode of operation Sue-Ann et al [32] research theoperational issues of a single vendormultiple buyers singleitem supply chain under VMI mode and propose a hybridof GA and Artificial Immune System to solve the problemDiabat [33] addresses the issue of VMI by considering a singlevendormultiple buyers single item supply chain network anddevelops a hybrid geneticsimulated annealing algorithm tocope with the nonlinear problem Sadeghi et al [34] establisha biobjective single item VMI model in a supply chain withone vendor and several retailers where the constraints arethe total budget required storage space vendorrsquos total replen-ishment frequencies and average inventory they employa nondominated sorting genetic algorithm-II (NSGA-II) tominimize the total system inventory cost and the total systemreliability Later Sadeghi and Niaki [35] consider the othertwo objectives of the minimization of the total inventorycost and the warehouse space for the same SC and developa NSGA-II as well to solve it For the multilocation systemChan et al [36] address single item production and distri-bution problems in multifactory multicustomer supply chainand develop a hybrid GA to determine demands allocationto suitable manufacturers Maiti et al [37] study a multi-itemtwo-storage multiple price breaks deterministic inventorymodel with a discount policy where the stocks of rentedwarehouse are transported to the owned warehouse in bulk-release rule and develop a real-coded GA with advanced GAoperators Pasandideh et al [38] deal with a two-echeloninventory system for nonrepairable items where the systemconsists of one warehouse andmultiple identical retailers anduses continuous review (119877 119876) ordering policy and developa parameter-tuned genetic algorithm to minimize the totalannual inventory investment Yang et al [39] suggest GA-based approach to resolve the optimal problem for a mul-tiproduct multiperiod supply chain system of multisuppliersingle warehouse and multiretailer with backlogging andtransportation capacity Paul and Rajendran [40] addressthe problem of determining the inventory control-policyor order-policy parameters and rationing mechanisms fora static divergent one distributor-many retailer multiperiodsingle item supply chain in a lost sales environment andpresent a GA-based heuristic methodology

This study concentrates onmultiple products distributionpolicy considering Supply Hub and lateral transshipmentsimultaneously To the best of our knowledge it still holdsopen The sole relative research is the work of Chen et al[41] They addressed vendorrsquos distribution policy with trans-shipment under the VMI environment for a centralizedtwo-echelon supply chain with one vendor and two retailersselling products with a short life cycle and facing stochasticdemand

The rest of the paper is organized as follows Sec-tion 2 gives the problem description and hypothesis Sec-tion 3 presents multi-item distribution model without

Supplier 1

Supplier 2

Supplier i minus 1

Supplier i

Supply Hub

Distributor 1

Distributor 2

Product flowInformation flow

Figure 1 Product distribution supply chain considering SupplyHuband lateral transshipment

transshipment Section 4 provides multi-item distributionmodel with transshipment Section 5 gives a GA-based algo-rithm to solve the transshipment model Section 6 is compu-tation example Section 7 concludes this study

2 Problem Description and Hypothesis

We consider a three-echelon supply chain consisting ofmultiple suppliers a Supply Hub and two distributors (seeFigure 1) Each supplier supplies only one product to theSupply Hub which holds a certain stock to supply thedistributors The demand of supplier is random but complieswith uniform distribution in each period During the replen-ishment period lateral transshipment between distributorscould be operated to balance their own inventory in order tolessen the overhigh stock or stockout

For the supply chain as shown in Figure 1 we make thefollowing assumptions

Assumption 1 The supply chain is composed of multiple sup-pliers Supply Hub and two distributors where each suppliersupplies only one product to the Supply Hub

Assumption 2 The transportation cost comprises fixedcharge and unit cost

Assumption 3 Both Supply Hub and distributor adopt the (119905119878) policy

Assumption 4 The inventory review cycle of SupplyHub is119872times of that of the distributor (119872 is an integer greater than1)

Assumption 5 The replenishment lead time of Supply Hub isnot more than that of distributors that is to say Supply Hubwill not be out of stock

Assumption 6 The demand at distributors is independentrandom variable of uniform distribution at each period

Assumption 7 Backlogging is not allowed


Assumption 8 The demand quantity within lead time canbe determined when distributors place replenishment ordersaccordingly two distributors could reduce overhigh stock orstockout

Assumption 9 Lateral transshipment can be completedinstantaneously This assumption is reasonable in real worldLateral transshipment commonly takes place between theretailers who are physically very close it takes very short time(eg within one hour) to ship goods from a retailer to anotherone while regular replenishment time may be several daysor longer Therefore transshipment time usually is ignoredin the literature on the transshipment especially for thegeneral or seasonal product only in some studies of inven-tory system on service parts spare parts repairable itemsand expensive low-demand items transshipment time wasconsidered

Define the following notations to describe the supplychain system

119894 supply (or product) number 119894 isin 1 2 119868119895 the replenishment period number of Supply Hub119895 isin 1 2 119869120591 the replenishment period number of distributors120591 isin 1 2 119872119869119899 distributors number 119899 isin 1 2 119873119875119894 unit procurement cost of product 119894 at Supply Hub

119865119894 the fixed cost transporting the product 119894 to Supply

Hub119860119894 the unit cost transporting the product 119894 to Supply

Hub119867119894 the unit holding cost of product 119894 at Supply Hub

119879 the length of replenishment cycle at Supply Hub119871 the length of replenishment lead time at SupplyHub119876119894119895 replenishment quantity of product 119894 in replenish-

ment period 119895 at Supply Hub119878119894 themaximal stock level of product 119894 at Supply Hub

119881119894119899 sale price of product 119894 at distributor 119899

119891119894119899 fixed costs transporting product 119894 to distributor 119899

119886119894119899 unit cost transporting product 119894 to distributor 119899

ℎ119894 unit holding cost of product 119894 which is the same

for the same product in different distributorLD the length of replenishment lead time at distrib-utor119868119894120591119899 initial inventory level of product 119894 at distributor 119899

in period 120591119889119894120591119899 the demand rate of product 119894 at distributor 119899 in

period 120591119876119894120591119899 replenishment quantity of product 119894 at distribu-

tor 119899 in period 120591

T T

TD TD

Si

T minus L

Figure 2 The change of inventory level of product 119894 at Supply Hub

TD TD TD

Sin

TD minus LD

Figure 3 The change of inventory level for product 119894 at distributor119899

119876119894120591 total replenishment quantity of product 119894 in

period 120591119887119894 unit shortage cost of product 119894

120587119894 unit lateral transshipment cost of product 119894

119878119894120591119899 the inventory level of product 119894 at distributor 119899 at

the outset of period 120591

119878119894120591119899 inventory level of product 119894 when distributor 119899

place an order in period 120591119878119894119899 the maximal inventory level of product 119894 at

distributor 119899

3 The Multi-Item Distribution Model withoutLateral Transshipment

The supplier 119894 supplies the goods of 119876119894119895to Supply Hub every

119879 time Supply Hub supplies the commodity of 119876119894120591119899

to otherdistributors every TD time The changes of inventory level atSupply Hub and distributors are showed in Figures 2 and 3respectively

From Figures 2 and 3 we can know that the inventorylevel of Supply Hub and distributors are replenished to theirmaximal inventory levels at the beginning of the cycle Theinventory level will decrease with the occurrence of demandWhen the time of (119879 minus 119871) or (TD minus LD) arrives order will beplaced according to the current inventory level and maximalinventory level


In the case of no lateral transshipment the maximal totalprofit model of the supply chain system is as follows

Max TSminus (TP+TT+TI119904+TI119889+TO) (1)

st TS =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1min (TD sdot 119889

119894120591119899 119878119894120591119899

) sdot 119881119894119899

(2)

TP =

119868

sum

119894=1

119869

sum

119895=1119875119894119876119894119895 (3)

TT =

119868

sum

119894=1

119869

sum

119895=1(119865119894+119860119894sdot 119876119894119895) +

119872119869

sum

120591=1

119873

sum

119899=1(119891119894119899

+ 119886119894119899

sdot 119876119894120591119899

)

(4)

TI119904=

119868

sum

119894=1

119869

sum

119895=1

(119878119894 minus1119872

119872119895

sum

120591=119872(119895minus1)+1(119872119895 minus 120591 + 1) sdot 119876

119894120591) sdot119879 sdot119867

119894

(5)

TI119889=

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

1198782119894120591119899

sdot ℎ119894

2119889119894120591119899

) if 119889119894120591119899

lt119878119894120591119899

TD119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

119878119894120591119899

sdot TD sdot ℎ119894minus

TD2sdot 119889119894120591119899

sdot ℎ119894

2

) otherwise(6)

TO =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1([TD sdot 119889

119894120591119899minus 119878119894120591119899

]+sdot 119887119894) (7)

119876119894119895

=

119872119895

sum

120591=119872(119895minus1)+1119876119894120591 (8)

119876119894120591

=

119873

sum

119899=1119876119894120591119899

(9)

where [119909]+ represents that [119909]

+= 119909 if 119909 gt 0 [119909]+ = 0

otherwise All of the parameters involved in the model arepositive integers The optimization objection (1) is to maxi-mize the total profit of supply chain system Equations (2)ndash(7) denotes the total sales (TS) the total ordering cost (TP)the total transportation cost (TT) the total holding cost atSupplyHub (TT

119904) the total holding cost at distributors (TT

119889)

and the total shortage cost (TO) respectively Equation (4)involves the transportation cost from suppliers to SupplyHuband one fromSupplyHub to distributorsThe two cases in (6)respectively consider two cases where distributor is in stockor out of stock in period 120591 Due to the assumption with noshortage at Supply Hub only stockout cost at distributor isconsidered in (7)

Note that the calculation of 119876119894120591119899

is the key to solve themodel In view of no backlogging and stock control policywe can obtain

119876119894120591119899

= 119878119894119899

minus 119878119894(120591minus1)119899 (10)

119878119894120591119899

= [119878119894120591119899

minus (TDminus LD) sdot 119889119894120591119899]+ (11)

119878119894120591119899

= 119878119894119899

minusmin (119878119894(120591minus1)119899 LD sdot 119889

119894(120591minus1)119899) (12)

4 The Multi-Item Distribution Model withLateral Transshipment

For the case with lateral transshipment at time (TD minus

LD) the distributors need to make a decision whetherlateral transshipment occurs or not and how much thetransshipment level of each product should be in order tocut down the unbalanced distribution of the distributorsrsquoinventory with respect to their demand Note that lateraltransshipment occurs solely when the inventory level of adistributor is greater than the demand within lead time andstatus of another distributor is just opposite Let 119883

1198941205911198991198991015840 be

the transshipment amount of product 119894 from distributor 119899 to


TD

The transferor

TD

The transferee

TransshipmentNo transshipment


Si120591n

Xi120591nn998400

Xi120591nn998400

Si120591n998400

TD minus LDTD minus LD

Figure 4 The inventory level change of distributor before and after transshipment

distributor 1198991015840 in period 1205911198831198941205911198991015840119899is the transshipment amount

of product 119894 from distributor 1198991015840 to distributor 119899 in period 120591

1198831198941205911198991198991015840 = min [119878

119894119899120591minus1 minus LD sdot 119889119894119899120591minus1]+

[LD sdot 1198891198941198991015840120591minus1 minus 119878

1198941198991015840120591minus1]+

1198831198941205911198991015840119899= min [LD sdot 119889

119894119899120591minus1 minus 119878119894119899120591minus1]+

[1198781198941198991015840120591minus1 minus LD

sdot 1198891198941198991015840120591minus1]+

(13)

Therefore the total transfer cost (TL) is

TL =

119872119869

sum

120591=1

119868

sum

119894=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840 =119899

120587119894sdot (1198831198941205911198991198991015840 + 1198831198941205911198991015840119899) (14)

Note that the transshipment amount from another dis-tributor at most is equal to the stockout level of a distributorwithin its lead time that is it is used to satisfy the demandwithin lead time So the transshipment amount should notbe counted into transfereersquos current inventory the transfereersquosinventory level at the outset of the next cycle is still calcu-lated by formula (12) But lateral transshipment will affect

the beginning inventory of the transferor in the next cycleIf distributor 119899 is the transferor there is

119878119894120591119899

= 119878119894119899

minusmin(119878119894(120591minus1)119899 minus

119873

sum

1198991015840=11198991015840=119899

119883119894(120591minus1)1198991198991015840 LD sdot 119889

119894(120591minus1)119899)

(15)

After calculating 119878119894120591119899 the total sales (TS1015840) the total

ordering cost (TP) and the total stockout cost (TO1015840) couldbe calculated by (2) (3) and (7) respectivelyThe total trans-portation cost (TT) has nothing with lateral transshipmentso it could be calculated still by (4) Here the most difficultthing is to compute the total inventory cost

The total holding cost (TI1015840119904) at Supply Hub will not be

affected by lateral transshipment and could still be calcu-lated by (5) The total inventory cost (TI

119889) at distributor

will change because the inventory level of the transferorwill drop down 119883

1198941205911198991198991015840 and the transfereersquos inventory level

will immediately increase by the amount of 1198831198941205911198991198991015840 at the

transshipment moment which is consumed gradually by thecustomerrsquos requirement within the lead time Assume thatdistributor 119899 is the transferor and distributor 1198991015840 is transfereethe inventory changes at the two sides for the transshipmentare shown in Figure 4

From Figure 4 the variation amount of inventory costΔTI at distributor is

ΔTI =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840=119899

(1198831198941205911198991198991015840 sdot LD minus

([1198781198941205911198991015840]+

+ 1198831198941205911198991198991015840)

2minus ([1198781198941205911198991015840]+

)

2

2119889119894120591119899

) sdot ℎ119894 (16)


Start

Initialize popSize pc pm maxGen

Generate the initial population Pk whosesize is popSize

Generate offspringpopulation Qk by reproduction

crossover and mutation operations

Unite Pk and Qk into an interim

population Mk

Make secondary selection operation onMk by ranking selection

k = k + 1

Generate the next-generationpopulation Pk

No

Yes

k ge maxGen

Output the best solution

End

and k = 0

Figure 5 Flow chart of the devised genetic algorithm

The total inventory cost TI1015840119889at distributor is

TI1015840119889= TI119889minusΔTI (17)

Consequently the total profit of supply chain with lateraltransshipment is

TV1015840 = TS1015840 minus (TP1015840 +TT1015840 +TI1015840119904+TI1015840119889+TL+TO1015840) (18)

5 The Algorithm to Solve the Model withLateral Transshipment

The supply chain system shown in Figure 1 considers Sup-ply Hub and lateral transshipment simultaneously It isintractable task to solve this model due to the complexitycomputing transshipment level Genetic algorithm (GA) is arandom search technique and global optimization algorithmbased on natural selection and genetic mechanism [42]Because GA has such advantages like independence of theapplication domain suitability for very general optimiza-tion problems (no requirements of linearity convexity anddifferentiability) robustness with respect to starting pointsand abilities of leaving local optima and finding the global

one it is a good method to solve production and operationsmanagement problems [43] So we select a GA with realnumber coding scheme to solve the problem in whichtwo stage selection technology is applied to improve theperformance of genetic algorithm Let popSize represent thepopulation size maxGen denotes the maximum iterationnumber 119896 denotes the number of current generations 119901

119888is

the crossover probability and 119901119898is the mutation probability

then flow chart of the algorithm can be expressed by Figure 5We use the real number coding scheme in GA A chro-

mosome consists of the inventory levels 119878119894119899at distributors for

instance

119866 = 11987811 11987812 1198781119873 1198781198941 1198781198942 119878119894119873 1198781198681 1198781198682

119878119868119873

(19)

Obviously the length of a chromosome is 119868 times 119873We take formula (18) as fitness function that is individual

fitness represents the total profit (TV1015840) of supply chainsystem

For the sake of simplicity let 119892 denote a gene of thechromosome 119866

119896= 119892

1119896 119892

2119896 119892

119868times119873

119896 and 119866

119898= 119892

1119898 119892

2119898

119892119868times119873

119898 are the chromosomes selected from parent population


Ranking selection

Merging

Merging

Ranking selectionCrossover mutation

Parentpopulation

Offspringpopulation

Interimpopulation

Next-generationpopulation

Figure 6 Two-stage selection policy

119897 (119897 = 1 2 119868 times119873) is the crossover point chosen randomly120582 is crossover coefficient which is a random number atthe interval (01) and 119901

119888is crossover probability Then the

single point nonuniform arithmetic crossover operator canbe formulated as follows

119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873

1198952 are offspring chromosomesWe adopt uniform mutation policy namely a gene is

replaced with the value chosen randomly within the valuerange of the gene

To prevent the phenomenon in the GA iteration thatsubsequent crossover and mutation operations may incurthe missing of excellent solutions or solution componentsfrom parent population we introduce two-stage selectionpolicy At the first stage the algorithm selects individualsfrom parent population by ranking selection which is usedto generate offspring population by genetic operations andthen the parent and offspring populations form a unionnamed interimpopulation At the second stage the algorithmcarries out a ranking-based selection procedure for theinterim population once again to form the next-generationpopulation For more clarity we demonstrate the two-stageprocedure by Figure 6

The multi-item distribution model without lateral trans-shipment is viewed as a comparison benchmark We applyLingo software to solve it

6 Computation Examples

61The Description of Example We consider a three-echelonsupply chain composed of two suppliers Supply Hub and twodistributors namely 119868 = 2 119873 = 2 Replenishment cycle atSupplyHub119879 = 6 replenishment cycle at distributor TD = 2that is119872 = 3 Parameters of Supply Hub and distributors areshown in Tables 1 and 2 respectivelyThe demand rates of theproduct 1 and product 2 comply with uniform distribution of119880(20 30) and 119880(16 30) respectively The demand rates usedby the experiment are shown in Table 3

Table 1 The parameters at Supply Hub

Parameters Product119894 = 1 119894 = 2

119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04

Table 2 The parameters at distributors

ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06

Table 3 The demand rates at distributors

120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28

62 The Effect of Two-Stage Selection Technology In thisexperiment the crossover andmutation probabilities respec-tively take the value of 075 and 005 the population size isset to be 50 Through many experiments we find that themaximum convergence generation of the proposed GA is 50Figure 7 shows the convergence of the algorithm in a running

To test the effect of two-stage selection the GA-basedalgorithms with and without two-stage selection are carriedout 20 times where the terminate conditions are set to be 50generations The results are shown in Table 4

From Table 4 two-stage selection technology has littleeffect on the run time it takes about 50ms for the two algo-rithms to run 50 iterations As a whole the algorithm withtwo-stage selectionwould converge at around 20 generations


Table 4 The effect of two-stage selection technology

Number of experiments Without two-stage selection With two-stage selectionRun time (ms) Convergence Run time (ms) Convergence

1 55 lowast 52 142 51 lowast 50 233 53 lowast 45 94 50 9 51 195 51 lowast 50 156 54 lowast 51 127 56 27 50 148 50 lowast 50 169 49 lowast 50 2210 52 34 52 1411 51 lowast 51 1312 49 46 51 1713 51 lowast 50 2514 50 7 51 2015 54 19 53 1316 52 26 56 717 51 lowast 51 1718 51 lowast 52 719 53 lowast 50 2120 51 32 54 15lowastRepresenting that the run is not convergent within 50 generations

2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV

Figure 7 The convergence of the GA-based algorithm

On the contrary in the 20 experiments for the algorithmwithout two-stage selection only 8 experiments are conver-gentwithin 50 generationsObviously the two-stage selectiontechnology can boost or accelerate the convergence of thealgorithm

63 The Effect of Lateral Transshipment For the model con-sidering lateral transshipment the optimal solution obtainedby GA is 119878

11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59

the total profit of the supply chain TV = 22772 Under

Table 5 The detailed experimental results without and with lateraltransshipment

Without lateraltransshipment

With lateraltransshipment

S11 52 52S12 42 48S21 50 54S22 57 59TP 58668 61656TT 5975 6050TI 3786 4039TS 90108 95208TO 1174 674TV 20505 22772

the same parameters settings we use Lingo to solve themodelwithout transshipment the results are 119878

11= 52 119878

12= 42

11987821

= 50 and 11987821

= 57 and the total profit of the supplychain TV = 20505 Comparing with the two results wefind that the total profit of the supply chain grows by 111 inthe case of lateral transshipment Table 5 shows the concretecomparisons of the experimental results

According to Table 5 the maximal stock levels of the casewith lateral transshipment are higher than that of the casewithout lateral transshipment This leads to the increase ofthe total order quantity and the corresponding growth ofthe ordering transportation and inventory costs However


Table 6 The computation results related to product 1

Period 120591

Distributor 119899Demand withinperiod 120591 (TD)

Initial inventorylevel at period 120591

Shortage levelwithin the period 120591

Shortage levelwithin lead time Transshipment level

119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1

Without transshipment1 52 34 52 42 0 0 0 0 0 02 40 54 50 41 0 13 0 3 0 03 58 42 50 42 8 0 2 0 0 0

With transshipment1 52 34 52 48 0 0 0 0 0 02 40 54 50 47 0 4 0 0 3 03 58 42 50 46 6 0 0 0 0 2

lateral transshipment could decrease the total shortage leveland the total stockout cost of the supply chain system At thesame time it makes the total sale volume increase Since theprofit growth exceeds the cost increase we see that the totalprofit of the former is higher than that of the latter

In order to make a comprehensive display and thoroughanalysis of the calculation results we separate out the calcu-lation results related to product 1 which is shown in Table 6Due to the uncertainty of demand the phenomenon oftenappears in that the inventory of a distributor is surplus whileone of another distributor is insufficient For example inperiod 2 distributor 1 possesses beginning inventory of 50units but itmerely faces the demandof 40 units whichmeansthere are 10 units leftHowever at the sameperiod distributor2 is of beginning inventory of 41 units and it needs to meetthe demand of 54 units whichmeans there is shortage level of13 units Hence if lateral transshipment between distributorsis allowed the stock at distributors could be rebalanced andthen the shortage possibility and total inventory level for thesystem bothwill be reducedThe conclusion has been verifiedby the data in Table 6 namely the shortage level within thereplenishment lead time drops down to zero owing to lateraltransshipment But for lateral transshipment that is allowedonly at the moment (TD minus LD) this results in the fact thatthe shortage level before themoment (TDminusLD) is copedwithonly by increasing the order quantity and inventory level Forinstance the initial inventory level of product 1 at distributor2 under the transshipping case is clearly higher than the onefor the no transshipment case If the transshipment rule ischanged to judging whether lateral transshipment is adoptedonce a distributor is out of stock the shortage level and costcan be decreased further

In order to further verify the influence of lateral trans-shipment on the system total profit we randomly generate49 sets of requirements according to the demand distributionstated at the fore The experimental results are shown inFigure 8

FromFigure 8we can see that in 49 groups of comparisonexperiments the total profit of supply chainwith lateral trans-shipment is apparently higher than that of supply chain withno transshipmentThe profit difference varies up to 162 and

2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits

The number of experiments


Figure 8 The effect of lateral transshipment on the system totalprofit

down to 61 Thus the policy of lateral transshipment hasobvious optimization effect on the total profit of supply chainwith Supply Hub under uncertain demand

7 Conclusion

This paper concentrated on the supply chain comprisingmultisupplier Supply Hub and multidistributor establishedmulti-item distribution model with lateral transshipmentTo the best of our knowledge this is the first study onthe multilocation inventory system with Supply Hub andlateral transshipment For purposes of comparison we builtthe maximal total profit model of the inventory systemrespectively for the cases without transshipment and withtransshipment The former is viewed as a comparison objectand is easy to solve so we adopted Lingo software to obtainits exact solution The latter is difficult to solve in view of


the transshipment and we devised a GA-based solution algo-rithm with two-stage selection technology In order to verifythe effectiveness and efficiency of the proposed model andsolve algorithmwe presented the comparison experiment Bymeans of the experiment we firstly showed that the two-stageselection technology could boost or accelerate the conver-gence of the algorithm Secondly we verified that the policy oflateral transshipment could improve the performance of themultilocation inventory system with Supply Hub From theexperiment we also found that the systemperformance couldbe further improved if transshipment policy was immediatelyconsidered once a distributor is out of stock This will beone of the future works of this research In addition anothervaluable future direction of this paper is to consider multipleproduct distribution problems with Supply Hub and lateraltransshipment under other demand distributions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by National Natural ScienceFoundation of China (71171072) The authors are thankful totwo anonymous referees and the editor for their constructivecomments which resulted in the subsequent improvement ofthis paper

References

[1] L E Cardenas-Barron ldquoOptimizing inventory decisions in amulti-stage multi-customer supply chain a noterdquo Transporta-tion Research Part E Logistics and Transportation Review vol43 no 5 pp 647ndash654 2007

[2] L E Cardenas-Barron G Trevino-Garza and H M WeeldquoA simple and better algorithm to solve the vendor managedinventory control system of multi-product multi-constrainteconomic order quantity modelrdquo Expert Systems with Applica-tions vol 39 no 3 pp 3888ndash3895 2012

[3] A Zuckerman ldquoCompaq switches to pre-position inventorymodelrdquoWorld Trade vol 13 no 4 pp 72ndash74 2000

[4] E Barnes J Dai S Deng D Down M Goh and L HChuin ldquoOn the strategy of supply hubs for cost reduction andresponsivenessrdquoThe ICFAI Journal of Operations Managementpp 1ndash27 2003

[5] L R Kopczak Apple Computerrsquos Supplier Hubs A Tale of ThreeCities StanfordGlobal Supply ChainManagement ForumCaseStanford Calif USA 1997

[6] J Shah andMGoh ldquoSetting operating policies for supply hubsrdquoInternational Journal of Production Economics vol 100 no 2pp 239ndash252 2006

[7] R Gaonkar and N Visvanadham ldquoCollaborative schedulingmodel for supply-hub managementrdquo in Proceedings of the 3rdAEGEAN International conference on Analysis and Modeling ofManufacturing Systems pp 1ndash6 2001

[8] J Li S Ma P Guo and Z Zuo ldquoAnalysis on design of supplychain embedding supply-hub with BOMrdquo in Proceedings of

the 4th International Conference on Wireless CommunicationsNetworking andMobile Computing (WiCOM rsquo08) pp 1ndash4 IEEEDalian China October 2008

[9] J Li N Xiong L Sun A Yuan J Chen and M Cao ldquoSupplychain designmodel based onmulti-supply hubsrdquo in Proceedingsof the 12th International Conference on Computational Scienceand Engineering (CSE rsquo09) vol 1 pp 449ndash454 August 2009

[10] K Huang and S-H Ma ldquoCoordinated replenishment withinformation sharing between two suppliers based on supply-hubrdquo in Proceedings of the 17th International Conference onManagement Science and Engineering (ICMSE rsquo10) pp 403ndash409November 2010

[11] Q G Liu and J X Chen ldquoA coordination mechanism of supplychain under asymmetric information based on supply hubrdquoin Proceedings of the International Conference on E-ProductE-Service and E-Entertainment (ICEEE rsquo10) pp 1ndash4 HenanChina November 2010

[12] W Lin ldquoAn effective lean supply inventory management modelusing VMI hubrdquo in Proceedings of the 18th IEEE InternationalConference on Industrial Engineering and Engineering Manage-ment (IEEM rsquo11) pp 950ndash954 December 2011

[13] XQiu andGQHuang ldquoSupplyHub in Industrial Park (SHIP)the value of freight consolidationrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 16ndash27 2013

[14] X Qiu H Luo G Xu R Zhong and G Q Huang ldquoPhysicalassets and service sharing for IoT-enabled Supply Hub inIndustrial Park (SHIP)rdquo International Journal of ProductionEconomics vol 159 pp 4ndash15 2015

[15] K S Krishnan and V R K Rao ldquoInventory control in Nwarehousesrdquo Journal of Industrial Engineering vol 16 no 3 pp212ndash215 1965

[16] C C Chiou ldquoTransshipment problems in supply chain sys-tems review and extensionsrdquo in Supply Chain Theory andApplications V Kordic Ed pp 558ndash579 I-Tech Education andPublishing Vienna Austria 2008

[17] C Paterson G Kiesmuller R Teunter and K GlazebrookldquoInventory models with lateral transshipments a reviewrdquo Euro-pean Journal of Operational Research vol 210 no 2 pp 125ndash1362011

[18] X Huang ldquoA review on policies and supply chain relationshipsunder inventory transshipmentrdquo Bulletin of Statistics and Oper-ations Research vol 29 no 1 pp 21ndash42 2013

[19] T W Archibald S A E Sassen and L CThomas ldquoAn optimalpolicy for a two depot inventory problem with stock transferrdquoManagement Science vol 43 no 2 pp 173ndash183 1997

[20] A A Kranenburg and G J van Houtum ldquoA new partialpooling structure for spare parts networksrdquo European Journalof Operational Research vol 199 no 3 pp 908ndash921 2009

[21] HWong G J van Houtum D Cattrysse and D van Oudheus-den ldquoSimple efficient heuristics for multi-item multi-locationspare parts systems with lateral transshipments and waitingtime constraintsrdquo Journal of the Operational Research Societyvol 56 no 12 pp 1419ndash1430 2005

[22] H Wong G J van Houtum D Cattrysse and D van Oud-heusden ldquoMulti-item spare parts systems with lateral trans-shipments and waiting time constraintsrdquo European Journal ofOperational Research vol 171 no 3 pp 1071ndash1093 2006

[23] R A Shumsky and F Zhang ldquoDynamic capacity managementwith substitutionrdquo Operations Research vol 57 no 3 pp 671ndash684 2009


[24] Z M Avsar and M Baykal-Gursoy ldquoInventory control undersubstitutable demand a stochastic game applicationrdquo NavalResearch Logistics vol 49 no 4 pp 359ndash375 2002

[25] S Netessine and N Rudi ldquoCentralized and competitive inven-tory models with demand substitutionrdquo Operations Researchvol 51 no 2 pp 329ndash335 2003

[26] B Rottkemper K Fischer and A Blecken ldquoA transshipmentmodel for distribution and inventory relocation under uncer-tainty in humanitarian operationsrdquo Socio-Economic PlanningSciences vol 46 no 1 pp 98ndash109 2012

[27] E M Alvarez M C van der Heijden I M H Vliegen andW H M Zijm ldquoService differentiation through selective lateraltransshipmentsrdquo European Journal of Operational Research vol237 no 3 pp 824ndash835 2014

[28] MMoghaddamand S YNof ldquoCombined demand and capacitysharing with best matching decisions in enterprise collabora-tionrdquo International Journal of Production Economics vol 148pp 93ndash109 2014

[29] L A Michaelraj and P Shahabudeen ldquoReplenishment policiesfor sustainable business development in a continuous creditbased vendormanaged inventory distribution systemrdquoComput-ers amp Industrial Engineering vol 56 no 1 pp 260ndash266 2009

[30] H Lan R Li Z Liu and R Wang ldquoStudy on the inventorycontrol of deteriorating items under VMI model based on bi-level programmingrdquo Expert Systems with Applications vol 38no 8 pp 9287ndash9295 2011

[31] S P Nachiappan and N Jawahar ldquoA genetic algorithm foroptimal operating parameters of VMI system in a two-echelonsupply chainrdquo European Journal of Operational Research vol182 no 3 pp 1433ndash1452 2007

[32] G Sue-Ann S G Ponnambalam and N Jawahar ldquoEvolution-ary algorithms for optimal operating parameters of vendormanaged inventory systems in a two-echelon supply chainrdquoAdvances in Engineering Software vol 52 pp 47ndash54 2012

[33] A Diabat ldquoHybrid algorithm for a vendor managed inventorysystem in a two-echelon supply chainrdquo European Journal ofOperational Research vol 238 no 1 pp 114ndash121 2014

[34] J Sadeghi S Sadeghi and S T A Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquoComputers ampOperations Research vol 41 no 1 pp53ndash64 2014

[35] J Sadeghi and S T Niaki ldquoTwo parameter tuned multi-objective evolutionary algorithms for a bi-objective vendormanaged inventory model with trapezoidal fuzzy demandrdquoApplied Soft Computing vol 30 pp 567ndash576 2015

[36] F T S Chan S H Chung and S Wadhwa ldquoA hybrid geneticalgorithm for production and distributionrdquo Omega vol 33 no4 pp 345ndash355 2005

[37] A K Maiti A K Bhunia andMMaiti ldquoAn application of real-coded genetic algorithm (RCGA) for mixed integer non-linearprogramming in two-storage multi-item inventory model withdiscount policyrdquo Applied Mathematics and Computation vol183 no 2 pp 903ndash915 2006

[38] S H R Pasandideh S T A Niaki and N Tokhmehchi ldquoAparameter-tuned genetic algorithm to optimize two-echeloncontinuous review inventory systemsrdquo Expert Systems withApplications vol 38 no 9 pp 11708ndash11714 2011

[39] W Yang F T S Chan and V Kumar ldquoOptimizing replen-ishment polices using Genetic Algorithm for single-warehousemulti-retailer systemrdquo Expert Systems with Applications vol 39no 3 pp 3081ndash3086 2012

[40] B Paul and C Rajendran ldquoRationing mechanisms and inven-tory control-policy parameters for a divergent supply chainoperating with lost sales and costs of reviewrdquo Computers ampOperations Research vol 38 no 8 pp 1117ndash1130 2011

[41] X Chen G Hao X Li and K F C Yiu ldquoThe impact of demandvariability and transshipment on vendorrsquos distribution policiesunder vendor managed inventory strategyrdquo International Jour-nal of Production Economics vol 139 no 1 pp 42ndash48 2012

[42] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[43] H Aytug M Khouja and F E Vergara ldquoUse of genetic algo-rithms to solve production and operations management prob-lems a reviewrdquo International Journal of ProductionResearch vol41 no 17 pp 3955ndash4009 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(MOI) VMI and Supplier Owned Inventory (SOI) particu-larizes the arguments for and against using Supply Hub theprerequisite of establishing Supply Hub and the typical casesof Supply Hub describes current operation models of SupplyHub discusses the information technology supporting theoperation of Supply Hub and identified general industryproblems and possible research issues Shah and Goh [6]address the coordination problem in a two-stage supplychain consisting of single supplier single Supply Hub andsingle customer under the deterministic demand of singleproduct where backlogging is allowed and minimal andmaximal inventory limitations at Supply Hub are consideredthey develop a structured hierarchy method Gaonkar andVisvanadham [7] investigate an electronic product supplychain composed of single Supply Hub 119873 first-tier suppliersand 119873 second-tier suppliers and establish the linear pro-grammingmodel of coordination schedulingwith the limitedproduction and inventory capacity Li et al [8] establishdecentralized inventorytransportation supply chain modeland centered inventorytransportation supply chain modelconsidering Supply Hub respectively through comparativeanalysis they find that the latter could significantly reducethe system total cost and the observation that the closer theSupply Hub is located to the upstream of supply chain themore cost will be saved Furthermore they also find thatthe latter could lessen the bullwhip effect better comparedwith the former Li et al [9] establish a mixed integerprogramming model for the supply chain design with twoSupply Hubs and put forward the genetic algorithm to solveit Huang andMa [10] address the coordinated replenishmentpolicy between suppliers considering milk run and SupplyHub and compared it with traditional replenishment policyLiu and Chen [11] study the single product supply chaincomprising multiple manufacturers and multiple retailersunder the asymmetric information environment for thecase of the nonlinear relationship between demand andprice they establish the models with and without SupplyHub respectively Lin [12] studies the supply chain involvingmultiple suppliers single VMI Hub multiple factories andmultiple customers and compares the two modes of supplyinventory managed by firm itself and by VMI Hub Qiuand Huang [13] put forward the concept of Supply Hub inIndustrial Park (SHIP) aimed at warehousing and logisticsservices and establish the productiondistribution modelswith and without SHIP respectively give a genetic algorithmto solve themodels and show that the freight consolidation isprofitable to the entire industrial park Qiu et al [14] presentthe thought enhancing the efficiency and effectiveness of thephysical assets and services sharing through the SHIP drivenby the Internet of things (IOT)

The earliest study of lateral transshipment is about thesingle period maximal inventory order policy [15] Up todate there have been many scholars to conduct extensiveresearch in this area [16ndash18] the existing researchmainly aimsat the single product and the research on multiproduct isscarce Archibald et al [19] study multi-item two-locationinventory system with lateral transshipment and emergencyorder where the periodic review order-up-to S policy isadopted Kranenburg and vanHoutum [20] studymulti-item

multilocation single-echelon spare parts system with basestock control which comprises two types of local warehouse(ie main and regular warehouses) onlymainwarehouse canbe the supplier of lateral transshipment Wong et al [21 22]studymulti-item repairable spare parts inventory systemwithlateral and emergency shipments where continuous reviewbase stock policy and the average waiting time limitation ofrequired parts are taken into account They develop a greedyheuristic method which can obtain the approximate optimalsolution for multilocation inventory system and a solutionprocedure based on Lagrangian relaxation for two-locationinventory system Inventory problem of substitutable prod-ucts can be viewed as lateral transshipment between prod-ucts Shumsky andZhang [23] study themultiperiod dynamicallocation problem of inventory capacity in the firm sellingmultiple product types of same class where demand of aproduct could bemet by a product from the next-higher classAvsar and Baykal-Gursoy [24] study an infinite period two-retailer inventory control system with two kinds of substi-tutable products using stochastic game theory Netessine andRudi [25] study the centered inventorymanagementmodel ofmultiple substitutable products and decentralized inventorymanagement model in which each firm manages a productrespectively Rottkemper et al [26] study the inventoryrelocation and distribution for relief items transshipmentnetwork comprising a global a central and a number ofregional depots build a multiperiodmixed-integer program-ming model with minimization of unsatisfied demand andminimization of operational costs and suggest a rollinghorizon solutionmethod Alvarez et al [27] consider amulti-item spare parts network of a central depot andmultiple localwarehouses where each warehouse has its own premiumand nonpremium customers with class-specific waiting timerestrictions lateral transshipment and emergency shipmentsare used as differentiation tool They develop a heuristicapproach similar to Dantzig-Wolfe decomposition to setstock levels and (trans)shipment strategies of multi-itemsystem Moghaddam and Nof [28] investigate demand andcapacity sharing decisions problem in collaborative networkof supply enterprises under uncertain multi-item demandwhere the bestmatching protocol is proposed tominimize thetotal cost of collaboration and the gap between the capacityof the supply enterprises and their allocated demands Theybuild a fuzzy mixed-integer planning model for the problem

Although many authors have widely applied GA to solveSCM problems we cannot find the research which is directlyrelated to this work So we shall provide an overview forthe literature on the inventory management for one-to-manysupply chain under VMI mode and multilocation inventorysystem For the VMI system Michaelraj and Shahabudeen[29] research optimal replenishment policies of the distrib-utors working in the scenario that retailers are making thepayment to the distributors in number of unequal install-ments and use GA and the linear programming to solve the adistributor multiretailer single itemmultiperiodVMI systemto minimize balance payment and maximize sales Lan et al[30] establish two inventory control models of deterioratingitem for the single item multiperiod supply chain includinga manufacturer a vendor and many retailers located in





Supplier 1

Supplier 2

Supplier i minus 1

Supplier i

Supply Hub

Distributor 1

Distributor 2
































T T

TD TD

Si

T minus L


TD TD TD

Sin

TD minus LD









distributor 119899









st TS =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1min (TD sdot 119889

119894120591119899 119878119894120591119899

) sdot 119881119894119899

(2)

TP =

119868

sum

119894=1

119869

sum

119895=1119875119894119876119894119895 (3)

TT =

119868

sum

119894=1

119869

sum

119895=1(119865119894+119860119894sdot 119876119894119895) +

119872119869

sum

120591=1

119873

sum

119899=1(119891119894119899

+ 119886119894119899

sdot 119876119894120591119899

)

(4)

TI119904=

119868

sum

119894=1

119869

sum

119895=1

(119878119894 minus1119872

119872119895

sum

120591=119872(119895minus1)+1(119872119895 minus 120591 + 1) sdot 119876

119894120591) sdot119879 sdot119867

119894

(5)

TI119889=

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

1198782119894120591119899

sdot ℎ119894

2119889119894120591119899

) if 119889119894120591119899

lt119878119894120591119899

TD119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

119878119894120591119899


TD2sdot 119889119894120591119899

sdot ℎ119894

2

) otherwise(6)

TO =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1([TD sdot 119889

119894120591119899minus 119878119894120591119899

]+sdot 119887119894) (7)

119876119894119895

=

119872119895

sum

120591=119872(119895minus1)+1119876119894120591 (8)

119876119894120591

=

119873

sum

119899=1119876119894120591119899

(9)


+= 119909 if 119909 gt 0 [119909]+ = 0



119889)




119876119894120591119899

= 119878119894119899

minus 119878119894(120591minus1)119899 (10)

119878119894120591119899

= [119878119894120591119899


119878119894120591119899

= 119878119894119899


119894(120591minus1)119899) (12)




1198941205911198991198991015840 be



TD

The transferor

TD

The transferee



Si120591n

Xi120591nn998400

Xi120591nn998400

Si120591n998400





1198831198941205911198991198991015840 = min [119878



1198941198991015840120591minus1]+

1198831198941205911198991015840119899= min [LD sdot 119889


[1198781198941198991015840120591minus1 minus LD

sdot 1198891198941198991015840120591minus1]+

(13)


TL =

119872119869

sum

120591=1

119868

sum

119894=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840 =119899

120587119894sdot (1198831198941205911198991198991015840 + 1198831198941205911198991015840119899) (14)



119878119894120591119899

= 119878119894119899


119873

sum

1198991015840=11198991015840=119899

119883119894(120591minus1)1198991198991015840 LD sdot 119889

119894(120591minus1)119899)

(15)











ΔTI =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840=119899

(1198831198941205911198991198991015840 sdot LD minus

([1198781198941205911198991015840]+

+ 1198831198941205911198991198991015840)

2minus ([1198781198941205911198991015840]+

)

2

2119889119894120591119899

) sdot ℎ119894 (16)


Start






population Mk


k = k + 1


No

Yes

k ge maxGen


End

and k = 0



TI1015840119889= TI119889minusΔTI (17)






119888is




instance

119866 = 11987811 11987812 1198781119873 1198781198941 1198781198942 119878119894119873 1198781198681 1198781198682

119878119868119873

(19)




119896= 119892

1119896 119892

2119896 119892

119868times119873

119896 and 119866

119898= 119892

1119898 119892

2119898

119892119868times119873



Ranking selection

Merging

Merging


Parentpopulation

Offspringpopulation

Interimpopulation






119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873









119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04


ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06


120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28








2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Supplier 1

Supplier 2

Supplier i minus 1

Supplier i

Supply Hub

Distributor 1

Distributor 2
































T T

TD TD

Si

T minus L


TD TD TD

Sin

TD minus LD









distributor 119899









st TS =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1min (TD sdot 119889

119894120591119899 119878119894120591119899

) sdot 119881119894119899

(2)

TP =

119868

sum

119894=1

119869

sum

119895=1119875119894119876119894119895 (3)

TT =

119868

sum

119894=1

119869

sum

119895=1(119865119894+119860119894sdot 119876119894119895) +

119872119869

sum

120591=1

119873

sum

119899=1(119891119894119899

+ 119886119894119899

sdot 119876119894120591119899

)

(4)

TI119904=

119868

sum

119894=1

119869

sum

119895=1

(119878119894 minus1119872

119872119895

sum

120591=119872(119895minus1)+1(119872119895 minus 120591 + 1) sdot 119876

119894120591) sdot119879 sdot119867

119894

(5)

TI119889=

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

1198782119894120591119899

sdot ℎ119894

2119889119894120591119899

) if 119889119894120591119899

lt119878119894120591119899

TD119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

119878119894120591119899


TD2sdot 119889119894120591119899

sdot ℎ119894

2

) otherwise(6)

TO =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1([TD sdot 119889

119894120591119899minus 119878119894120591119899

]+sdot 119887119894) (7)

119876119894119895

=

119872119895

sum

120591=119872(119895minus1)+1119876119894120591 (8)

119876119894120591

=

119873

sum

119899=1119876119894120591119899

(9)


+= 119909 if 119909 gt 0 [119909]+ = 0



119889)




119876119894120591119899

= 119878119894119899

minus 119878119894(120591minus1)119899 (10)

119878119894120591119899

= [119878119894120591119899


119878119894120591119899

= 119878119894119899


119894(120591minus1)119899) (12)




1198941205911198991198991015840 be



TD

The transferor

TD

The transferee



Si120591n

Xi120591nn998400

Xi120591nn998400

Si120591n998400





1198831198941205911198991198991015840 = min [119878



1198941198991015840120591minus1]+

1198831198941205911198991015840119899= min [LD sdot 119889


[1198781198941198991015840120591minus1 minus LD

sdot 1198891198941198991015840120591minus1]+

(13)


TL =

119872119869

sum

120591=1

119868

sum

119894=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840 =119899

120587119894sdot (1198831198941205911198991198991015840 + 1198831198941205911198991015840119899) (14)



119878119894120591119899

= 119878119894119899


119873

sum

1198991015840=11198991015840=119899

119883119894(120591minus1)1198991198991015840 LD sdot 119889

119894(120591minus1)119899)

(15)











ΔTI =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840=119899

(1198831198941205911198991198991015840 sdot LD minus

([1198781198941205911198991015840]+

+ 1198831198941205911198991198991015840)

2minus ([1198781198941205911198991015840]+

)

2

2119889119894120591119899

) sdot ℎ119894 (16)


Start






population Mk


k = k + 1


No

Yes

k ge maxGen


End

and k = 0



TI1015840119889= TI119889minusΔTI (17)






119888is




instance

119866 = 11987811 11987812 1198781119873 1198781198941 1198781198942 119878119894119873 1198781198681 1198781198682

119878119868119873

(19)




119896= 119892

1119896 119892

2119896 119892

119868times119873

119896 and 119866

119898= 119892

1119898 119892

2119898

119892119868times119873



Ranking selection

Merging

Merging


Parentpopulation

Offspringpopulation

Interimpopulation






119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873









119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04


ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06


120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28








2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























T T

TD TD

Si

T minus L


TD TD TD

Sin

TD minus LD









distributor 119899









st TS =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1min (TD sdot 119889

119894120591119899 119878119894120591119899

) sdot 119881119894119899

(2)

TP =

119868

sum

119894=1

119869

sum

119895=1119875119894119876119894119895 (3)

TT =

119868

sum

119894=1

119869

sum

119895=1(119865119894+119860119894sdot 119876119894119895) +

119872119869

sum

120591=1

119873

sum

119899=1(119891119894119899

+ 119886119894119899

sdot 119876119894120591119899

)

(4)

TI119904=

119868

sum

119894=1

119869

sum

119895=1

(119878119894 minus1119872

119872119895

sum

120591=119872(119895minus1)+1(119872119895 minus 120591 + 1) sdot 119876

119894120591) sdot119879 sdot119867

119894

(5)

TI119889=

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

1198782119894120591119899

sdot ℎ119894

2119889119894120591119899

) if 119889119894120591119899

lt119878119894120591119899

TD119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

119878119894120591119899


TD2sdot 119889119894120591119899

sdot ℎ119894

2

) otherwise(6)

TO =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1([TD sdot 119889

119894120591119899minus 119878119894120591119899

]+sdot 119887119894) (7)

119876119894119895

=

119872119895

sum

120591=119872(119895minus1)+1119876119894120591 (8)

119876119894120591

=

119873

sum

119899=1119876119894120591119899

(9)


+= 119909 if 119909 gt 0 [119909]+ = 0



119889)




119876119894120591119899

= 119878119894119899

minus 119878119894(120591minus1)119899 (10)

119878119894120591119899

= [119878119894120591119899


119878119894120591119899

= 119878119894119899


119894(120591minus1)119899) (12)




1198941205911198991198991015840 be



TD

The transferor

TD

The transferee



Si120591n

Xi120591nn998400

Xi120591nn998400

Si120591n998400





1198831198941205911198991198991015840 = min [119878



1198941198991015840120591minus1]+

1198831198941205911198991015840119899= min [LD sdot 119889


[1198781198941198991015840120591minus1 minus LD

sdot 1198891198941198991015840120591minus1]+

(13)


TL =

119872119869

sum

120591=1

119868

sum

119894=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840 =119899

120587119894sdot (1198831198941205911198991198991015840 + 1198831198941205911198991015840119899) (14)



119878119894120591119899

= 119878119894119899


119873

sum

1198991015840=11198991015840=119899

119883119894(120591minus1)1198991198991015840 LD sdot 119889

119894(120591minus1)119899)

(15)











ΔTI =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840=119899

(1198831198941205911198991198991015840 sdot LD minus

([1198781198941205911198991015840]+

+ 1198831198941205911198991198991015840)

2minus ([1198781198941205911198991015840]+

)

2

2119889119894120591119899

) sdot ℎ119894 (16)


Start






population Mk


k = k + 1


No

Yes

k ge maxGen


End

and k = 0



TI1015840119889= TI119889minusΔTI (17)






119888is




instance

119866 = 11987811 11987812 1198781119873 1198781198941 1198781198942 119878119894119873 1198781198681 1198781198682

119878119868119873

(19)




119896= 119892

1119896 119892

2119896 119892

119868times119873

119896 and 119866

119898= 119892

1119898 119892

2119898

119892119868times119873



Ranking selection

Merging

Merging


Parentpopulation

Offspringpopulation

Interimpopulation






119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873









119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04


ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06


120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28








2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












st TS =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1min (TD sdot 119889

119894120591119899 119878119894120591119899

) sdot 119881119894119899

(2)

TP =

119868

sum

119894=1

119869

sum

119895=1119875119894119876119894119895 (3)

TT =

119868

sum

119894=1

119869

sum

119895=1(119865119894+119860119894sdot 119876119894119895) +

119872119869

sum

120591=1

119873

sum

119899=1(119891119894119899

+ 119886119894119899

sdot 119876119894120591119899

)

(4)

TI119904=

119868

sum

119894=1

119869

sum

119895=1

(119878119894 minus1119872

119872119895

sum

120591=119872(119895minus1)+1(119872119895 minus 120591 + 1) sdot 119876

119894120591) sdot119879 sdot119867

119894

(5)

TI119889=

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

1198782119894120591119899

sdot ℎ119894

2119889119894120591119899

) if 119889119894120591119899

lt119878119894120591119899

TD119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1(

119878119894120591119899


TD2sdot 119889119894120591119899

sdot ℎ119894

2

) otherwise(6)

TO =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1([TD sdot 119889

119894120591119899minus 119878119894120591119899

]+sdot 119887119894) (7)

119876119894119895

=

119872119895

sum

120591=119872(119895minus1)+1119876119894120591 (8)

119876119894120591

=

119873

sum

119899=1119876119894120591119899

(9)


+= 119909 if 119909 gt 0 [119909]+ = 0



119889)




119876119894120591119899

= 119878119894119899

minus 119878119894(120591minus1)119899 (10)

119878119894120591119899

= [119878119894120591119899


119878119894120591119899

= 119878119894119899


119894(120591minus1)119899) (12)




1198941205911198991198991015840 be



TD

The transferor

TD

The transferee



Si120591n

Xi120591nn998400

Xi120591nn998400

Si120591n998400





1198831198941205911198991198991015840 = min [119878



1198941198991015840120591minus1]+

1198831198941205911198991015840119899= min [LD sdot 119889


[1198781198941198991015840120591minus1 minus LD

sdot 1198891198941198991015840120591minus1]+

(13)


TL =

119872119869

sum

120591=1

119868

sum

119894=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840 =119899

120587119894sdot (1198831198941205911198991198991015840 + 1198831198941205911198991015840119899) (14)



119878119894120591119899

= 119878119894119899


119873

sum

1198991015840=11198991015840=119899

119883119894(120591minus1)1198991198991015840 LD sdot 119889

119894(120591minus1)119899)

(15)











ΔTI =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840=119899

(1198831198941205911198991198991015840 sdot LD minus

([1198781198941205911198991015840]+

+ 1198831198941205911198991198991015840)

2minus ([1198781198941205911198991015840]+

)

2

2119889119894120591119899

) sdot ℎ119894 (16)


Start






population Mk


k = k + 1


No

Yes

k ge maxGen


End

and k = 0



TI1015840119889= TI119889minusΔTI (17)






119888is




instance

119866 = 11987811 11987812 1198781119873 1198781198941 1198781198942 119878119894119873 1198781198681 1198781198682

119878119868119873

(19)




119896= 119892

1119896 119892

2119896 119892

119868times119873

119896 and 119866

119898= 119892

1119898 119892

2119898

119892119868times119873



Ranking selection

Merging

Merging


Parentpopulation

Offspringpopulation

Interimpopulation






119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873









119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04


ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06


120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28








2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










TD

The transferor

TD

The transferee



Si120591n

Xi120591nn998400

Xi120591nn998400

Si120591n998400





1198831198941205911198991198991015840 = min [119878



1198941198991015840120591minus1]+

1198831198941205911198991015840119899= min [LD sdot 119889


[1198781198941198991015840120591minus1 minus LD

sdot 1198891198941198991015840120591minus1]+

(13)


TL =

119872119869

sum

120591=1

119868

sum

119894=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840 =119899

120587119894sdot (1198831198941205911198991198991015840 + 1198831198941205911198991015840119899) (14)



119878119894120591119899

= 119878119894119899


119873

sum

1198991015840=11198991015840=119899

119883119894(120591minus1)1198991198991015840 LD sdot 119889

119894(120591minus1)119899)

(15)











ΔTI =

119868

sum

119894=1

119872119869

sum

120591=1

119873

sum

119899=1

119873

sum

1198991015840=11198991015840=119899

(1198831198941205911198991198991015840 sdot LD minus

([1198781198941205911198991015840]+

+ 1198831198941205911198991198991015840)

2minus ([1198781198941205911198991015840]+

)

2

2119889119894120591119899

) sdot ℎ119894 (16)


Start






population Mk


k = k + 1


No

Yes

k ge maxGen


End

and k = 0



TI1015840119889= TI119889minusΔTI (17)






119888is




instance

119866 = 11987811 11987812 1198781119873 1198781198941 1198781198942 119878119894119873 1198781198681 1198781198682

119878119868119873

(19)




119896= 119892

1119896 119892

2119896 119892

119868times119873

119896 and 119866

119898= 119892

1119898 119892

2119898

119892119868times119873



Ranking selection

Merging

Merging


Parentpopulation

Offspringpopulation

Interimpopulation






119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873









119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04


ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06


120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28








2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start






population Mk


k = k + 1


No

Yes

k ge maxGen


End

and k = 0



TI1015840119889= TI119889minusΔTI (17)






119888is




instance

119866 = 11987811 11987812 1198781119873 1198781198941 1198781198942 119878119894119873 1198781198681 1198781198682

119878119868119873

(19)




119896= 119892

1119896 119892

2119896 119892

119868times119873

119896 and 119866

119898= 119892

1119898 119892

2119898

119892119868times119873



Ranking selection

Merging

Merging


Parentpopulation

Offspringpopulation

Interimpopulation






119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873









119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04


ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06


120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28








2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ranking selection

Merging

Merging


Parentpopulation

Offspringpopulation

Interimpopulation






119888119897

1198951= 120582119892119897

119895+ (1minus120582) 119892

119897

119898

119888119897

1198952= (1minus120582) 119892

119897

119895+120582119892119897

119898

(20)

where 1198621198951

= 11988811198951 119888

119894

1198951 119888

119868times119873

1198951 and 119862

1198952= 119888

11198952 119888

119894

1198952

119888119868times119873









119875119894

62 104119865119894

100 110119860119894

01 01119867119894

01 01120587119894

04 04


ParametersSuppliers

119899 = 1 119899 = 2

119894 = 1 119894 = 2 119894 = 1 119894 = 2

119881119894119899

134 186 136 188119891119894119899

40 45 50 55119886119894119899

01 015 01 015ℎ119894

04 06 04 06


120591 = 1 120591 = 2 120591 = 3

d11 26 20 29d12 24 26 25d21 17 27 21d22 29 18 28








2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













2280

2270

2260

2250

2240

2230

2220

2210

2200

2190

21800 5 10 15 20 25 30 35 40 45 50

Iterations

TV




11= 52 119878

12= 48 119878

21= 54 and 119878

21= 59







11= 52 119878

12= 42

11987821

= 50 and 11987821





Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Period 120591





119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 119899 = 1 119899 = 2 1 rarr 2 2 rarr 1







2800

0 5 10 15 20 25 30 35 40 45 50

2600

2400

2200

2000

1800

1600

Tota

l pro

fits





7 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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