Research ArticleOptimizing the Joint Replenishment andChannel Coordination Problem under Supply ChainEnvironment Using a Simple and EffectiveDifferential Evolution Algorithm

Lin Wang1 Hui Qu1 Shan Liu2 and Can Chen1

1 School of Management Huazhong University of Science amp Technology Wuhan 430074 China2 Economics and Management School Wuhan University Wuhan 430072 China

Correspondence should be addressed to Shan Liu 279266384qqcom

Received 12 April 2014 Revised 11 May 2014 Accepted 17 May 2014 Published 17 June 2014

Academic Editor Stefan Balint

Copyright copy 2014 Lin Wang et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a useful and practical procurement approach using the joint replenishment and channel coordination (JR-CC)policy in a two-echelon supply chain considering the coordination cost The objective is to determine a basic replenishment cycletime and the replenishment interval to minimize the total cost of the supply chain To solve this NP-hard problem a simple andimproved differential evolution algorithm (IDE) is developed The performance of the IDE is verified by benchmark functionsMoreover results of comparative numerical example show the effectiveness of the proposed IDE IDE can be used as a goodcandidate for the JR-CC model Results of numerical examples also indicate that the JR-CC policy can result in considerable costsaving and enhance the efficiency of a supply chain But not all members in the supply chain can benefit a lot using this policyMoreover results of sensitivity analysis show that retailers have more willingness to adopt the JR-CC policy than the manufacturersbecause of the different cost savings

1 Introduction

Over the past few decades the joint replenishment problem(JRP) has been heavily studied since not only is the JRP amulti-item inventory problem but also it is widely applied tolot sizing problems in manufacturing applications (Axsaterand Zhang [1] Wang et al [2]) The JRP means to groupitems into the same order from a supplier to achieve thepurpose of sharing the main preparation costs and savingthe procurement costs Moreover a quantity discount will beoffered by the supplier when the order amount is greater thana predefined quantity Olsen [3] developed an evolutionaryalgorithm to solve the JRP Moon and Cha [4] studied theJRPwith resource constraints Khouja andGoyal [5] reviewedthe literatures on JRPs from 1989 to 2005 and summarizedheuristics for JRPs special approaches to JRPs and somespecial applications of JRPs Since 2013 there are several

relevant papers related to JRPs and extensions such as Quet al [6] Wang et al [7 8] Buyukkaramikli et al [9] andCui et al [10] However these literatures mainly focus on thejoint replenishment of multi-items from a single supplier inan individual enterprise

In recent years many companies realized that consider-able cost savings can be achieved by the joint replenishmentpolicy Axsater and Zhang [1] considered a two-level supplychain with a central warehouse and a number of identicalretailers using the JR policy Cha and Moon [11] and Moonet al [12] developed efficient algorithms for solving the JRPsconsidering the quantity discounts Cha and Park [13] dealtwith the joint replenishment and delivery scheduling Hsu[14] investigated the joint replenishment decisions involv-ing combining different materialscomponents from severalsatellite factories to form a large shipment delivered to thecentral factory on a Just in Time (JIT) basis T-H Chen

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 709856 12 pageshttpdxdoiorg1011552014709856

2 Discrete Dynamics in Nature and Society

and J-M Chen [15] proposed four decision-making policiescharacterized by the joint replenishment and channel coor-dination (JR-CC) practice to determine optimal inventoryreplenishment and production policies in a supply chainHowever these literatures have not considered the potentialcoordination cost when the JR-CC policy is adopted in thesupply chain All the center of the supply chain in these lit-eratures is the suppliers because the suppliers can distributethe items to the retailers jointly to reduce the delivery costMoreover the suppliers can provide quantity discounts toinspire the retailers to adopt the joint replenishment policy

The aim of this paper is to study a useful and practicalprocurement approach using the JR-CC policy consideringthe coordination cost and provide a simple and effectivesolution In this study the advantage of joint replenishmentand channel coordination (JR-CC) policy is investigatedbased on the work of Hsu [14] Hsu [14] studied the jointreplenishment decisions for a central factory and satellitefactories under a Just in Time environment and regardedthe members of the supply chain as the same benefit groupAdversely we consider that they have independent financialaccounting systems and make their own decisions from theview of retailers for the efficient implementation of JR-CCpolicy Moreover we further compare costs to obtain usefulmanagerial insights Because the success of the JR-CC policyneeds a close cooperation with other enterprises it can beregarded as a practical and effective management methodfor most enterprises who only concentrate on its own benefitwithout consideration for the others

JRPs had been proven to be the NP-hard problems andthey were rather hard to find effective algorithms (Cha andMoon [11] Wang et al[16 17]) Hsu [14] designed a complexheuristic to handle this problem Several heuristics may giveacceptable solutions when the scale of optimization is smallHowever it is not easy to find a proper heuristic with a robustperformance However this heuristic is relatively complex fordecision-makers and cannot handle JR-CC model with thecost constrain conveniently With the increasing complexityof supply chain optimization problem (Geunes and Pardalos[18] Geunes et al [19] Pardalos et al [20]) and the devel-opment of the intelligent algorithms (Wang et al [21] Li etal [22] Wang et al [23] Cui et al [24]) they were widelyused for handling the practical supply chain optimizationproblems (Pei et al [25] Al-Anzi and Allahverdi [26] Wanget al [27]) Among these algorithms the differential evolutionalgorithm (DE) had an extensive application in JRPs (Wanget al [2 6 8 9]) So a simple and effective DE is utilized tohandle this problemTheDEwas proposed by Storn andPrice[28] for complex continuous nonlinear nondifferentiableand multimodal optimization problem This technique com-bines simple arithmetic operators with the classical events ofcrossover mutation and selection to evolve from a randomlygenerated starting population to a final solution Due to itssimple structure easy implementation quick convergenceand robustness the DE has been applied in a variety of fields(Al-Anzi and Allahverdi [26] Pan et al [29] Qu et al [6])To the best of our knowledge no work on the JR-CC modelusing DE can be found The DE has a good performance inconvergent speed but the faster convergence may cause the

diversity of population to descend quickly during the solutionprocess Moreover a closely clustered population yields apremature convergence or a local optimum and cannotreproduce a better individual So the classic DE should beimproved to find a good trade-off between convergence anddiversity

The rest of this paper is organized as follows In Section 2the independent replenishment (IR)model and JR-CCmodelare discussed Section 3 proposes a DE-based solution for theproposed model Section 4 contains numerical example andsensitivity analysis for the JR-CC models Conclusions andfuture research are presented in Section 5

2 Mathematical Formulation

21 Assumptions and Notations We consider two-echelonsupply chain consisting of a retailer replenishes 119899 materialsfrom multiple manufacturers which distributes in a cen-tralized location The retailer and manufacturers share theinformation of demand and inventory and both of them canhold inventory We assume that the demand of retailer isconstant and no shortage is allowed The manufacturers areassumed to be a ldquomake-to-orderrdquo producer using a lot-for-lotpolicy to fulfill customer demand Each time the retailer hasa need for replenishment a major ordering cost regardless ofthe number of the materials included and a minor orderingcost related tomaterial are incurred In each production runthemanufacturers have a constant production rate and spendcorresponding setup cost The quantity in one product cycleof material 119894 is split into multiple equal-size shipment lotsdelivered to the retailer

The following notations are used

119899 number of materials involved in the model119894 index of item 119894 = 1 2 119899119863119894 demand rate of material 119894

119875119894 production rate of material 119894 in a manufacturer

119875119894ge 119863119894

119896119894 the number of shipments in which the material

119894 is delivered from the manufacturers to the retailerwithin one production cycle a positive integer119878119894 setup cost of material 119894 for the manufacturers

119865 major ordering cost for the retailer119891119894 minor ordering cost of material 119894 for the retailer

119867119888119894 carrying cost rate of material 119894 for the retailer

119867119904119894 carrying cost rate of material 119894 for the manufac-

turers119879 the common shipment cycle time for the 119899materi-als119879119894 the shipment cycle time of the material 119894 with

independent replenishmentTC1199041198940 total relevant cost of material 119894 per year for the

manufacturers with independent replenishmentTC1198880 total relevant cost of 119899materials per year for the

retailer with independent replenishment

Discrete Dynamics in Nature and Society 3

TC119904119894 total relevant cost of material 119894 per year for the

manufacturers with joint replenishment and channelcoordinationTC119888 total relevant cost of 119899 materials per year for

the retailer with joint replenishment and channelcoordinationJTC0 total relevant cost for the supply chain per year

with independent replenishment of materialsJTC total relevant cost for the supply chain per yearwith joint replenishment of materials and channelcoordination

Two models are discussed one is independent replen-ishment (IR) model in which the materials have their ownshipment cycle time 119879

119894 the other is JR-CC model in which

the materials have a common shipment cycle time 119879

22 Formulation of the IR Model With the independentreplenishment policy each entity in the supply chain con-centrated on minimizing its own costs without consideringthe others The retailer makes the replenishment decision foreach itembased on an EOQpolicy (J-MChen andT-HChen[30]) Then the total cost of 119899 items per year for the retailer isexpressed as

TC1198880=

119899

sum

119894

(

1

119879119894

(119865 + 119891119894) +

119879119894

2

119863119894119867119888119894) (1)

Since the manufacturers adopt a ldquomake-to-orderrdquo policythe total cost for each manufacturer per year is

TC1199041198940=

119878119894

119879119894119896119894

+

119879119894119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894]

119894 = 1 2 119899

(2)

Then the total cost per year incurred to the retailer andthe manufacturers for independent replenishing 119899 materialscan be derived as

JTC0(119896119894 119879119894)

= TC1198880+

119899

sum

119894=1

TC1199041198940

=

119899

sum

119894=1

(

119865 + 119891119894

119879119894

+

119863119894119867119888119894119879119894

2

)

+

119899

sum

119894=1

119878119894

119879119894119896119894

+

119879119894119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894]

=

119899

sum

119894=1

1

119879119894

(119865 + 119891119894+

119878119894

119896119894

)

+

119899

sum

119894=1

119863119894119879119894

2

[119867119888119894+ 119867119904119894(

2119863119894

119875119894

minus 1) + 119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(3)

To simplify the expression we denote 119860119894= 119865 + 119891

119894 119861119894=

119863119894[119867119888119894+ 119867119904119894(2119863119894119875119894minus 1)] Then we can obtain the simplified

formulation of JTC0as follows

JTC0(119896119894 119879119894) =

119899

sum

119894=1

1

119879119894

(119860119894+

119878119894

119896119894

)

+

119899

sum

119894=1

119879119894

2

[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(4)

For a given set of 119870 = (1198961 1198962 119896

119899) taking the first

order of JTC0 with respect to 119879119894we have

120597JTC0(119896119894 119879119894)

120597119879119894

= minus

119860119894+ 119878119894119896119894

1198792

119894

+

1

2

[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(5)

Let 120597JTC0(119896119894 119879119894)120597119879119894= 0 the optimal 119879lowast

119894must satisfy

119879lowast

119894(119896119894) = radic

2 (119860119894+ (119878119894119896119894))

[119861119894+ 119863119894119867119904119894(1 minus 119863

119894119875119894) 119896119894]

(6)

Since 1205972JTC0(119896119894 119879119894)1205971198792

119894= 2(119860

119894+ 119878119894119896119894)1198793

119894gt 0 119879lowast

119894can

be determined uniquely by (6) after 119896119894are given

Substitute (6) into (4) the optimal JTC0can be obtained

by

JTClowast0(119896119894)

=

119899

sum

119894=1

radic2(119860119894+

119878119894

119896119894

)[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(7)

23 Formulation of the JR-CCModel The objective of the JR-CC model is to minimize the total cost in this two-echelonsupply chain It involves determining a basic replenishmentcycle time 119879 and the replenishment interval 119896

119894119879 for item i

where 119896119894is an integral number Then the total cost per year

for the retailer is

TC119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894 (8)

The total cost of material 119894 per year for the manufacturersis

TC119904119894=

119878119894

119896119894119879

+

119879119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894] (9)

4 Discrete Dynamics in Nature and Society

Then the total cost per year incurred to the retailer andthe manufacturers for joint replenishing 119899 materials can bederived as

JTC (119896119894 119879) = TC

119888+

119899

sum

119894=1

TC119904119894

=

1

119879

(119865 +

119899

sum

119894=1

119891119894+

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[

119899

sum

119894=1

119863119894(119867119888119894+ 119867119904119894(

2119863119894

119875119894

minus 1))

+

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

=

1

119879

(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(10)

where 119860119860 = 119865 + sum119899

119894=1119891119894 119861119861 = sum

119899

119894=1119863119894(119867119888119894+ 119867119904119894(2119863119894119875119894minus

1)) Since the formulation of JTC has the similar structure ofJTC0 for any fixed 119896

119894 the corresponding optimal common

shipment cycle time 119879lowast(119870) is given by

119879lowast(119896119894) = radic

2 (119860119860 + sum119899

119894=1119878119894119896119894)

119861119861 + sum119899

119894=1119863119894119867119904119894(1 minus 119863

119894119875119894) 119896119894

(11)

Then the minimal total cost of the supply chain underchannel cooperation is

JTClowast (119896119894)

= radic2(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(12)

Obviously both the JR-CC model and IR model havesimilar function structures Hsu [14] proposed a heuristicfor solving this model and obtaining the optimal solutionHowever the heuristic is not universal and relatively com-plex for decision-makers In addition the JR-CC model isextended considering the restriction of coordination cost inthe sensitivity analysis So the heuristic cannot solve thismodel with a cost constraint Therefore in order to analyzemodel more effectively a DE-based solution is provided

3 The Improved DE (IDE) forthe JR-CC Model

31 The Classical DE and Improvement

311TheClassical DE Unlike other evolutionary algorithmsDE does not make use of some probability distribution

function in order to introduce variations into the population(Wang et al [7] Salman et al [31]) DE initializes populationrandomly by uniform distribution over search space Eachindividual is called target vector A new vector is createdby adding the weighted difference of two random vectors totarget vector Then a trail vector is generated by mixing themutated vectors with the target vectors according to selectedrules Finally a one-to-one competition selects the best onebetween the trail vector and the corresponding target vectoraccording to its fitness (Wang et al [32] Brest et al [33])Three operations of the classical DE are introduced as follows

Mutation For each target vector 119909(119866)119905

of generation 119866 amutated vector V(119866+1)

119905is created according to the following

V(119866+1)119905

= 119909(119866)

1199031+ 119865119865lowast(119909(119866)

1199032minus 119909(119866)

1199033) (13)

where 119866 presents the present generation 119865119865 isin [0 2] isa mutation factor used to control the amplification of thedifferential variation119903

1 1199032 and 119903

3are three distinct random

numbers and none of them coincides with the current targetindividual 119905 (119903

1= 1199032= 1199033= 119905)

CrossoverThe trail vector can be obtained using the followingrules

119906(119866+1)

119905119895=

V(119866+1)119905119895

if (rand le 119862119877) or 119895 = rnbr (119905)

119909(119866)

119905119895if (rand gt 119862119877) and 119895 = rnbr (119905)

(14)

where rand(119895) is a uniformly distributed random numberin range (0 1) rnbr(119905) is a randomly chosen integer in theset1 2 119873

119901 which ensures the trail vector gets at least

one parameter from the mutated vector 119862119877 isin [0 1] is acrossover constant

Selection The selection mechanism adopts one-to-one com-petition greedy strategy to decide which vector (trail vector119906(119866+1)

119905and target vector 119909(119866)

119905) is chosen into the next genera-

tion That is to say 119906(119866+1)119905

replaces 119909(119866)119905

only if its fitness (iethe value of objective function) is better than the fitness of119909(119866)

119905

312 The Improved DE (IDE) In order to improve DE anadaptive parameter 119865 is adopted and a new selection methodbased on DE and genetic algorithm (GA) is utilized

Adaptive Mutation Factor In mutation parameter 119865119865 playsan important role 119865119865affects the speed of convergence and-decides the search range Usually optimization algorithmsfavor global search at the early stage for exploring feasibledomain and local search at the latter stage for acceleratingconvergence Based on above features a parameter 119865119865 isdefined as follows

119865119865 = 119865119865min + (119865119865max minus 119865119865min) lowast 1198901minus(119866119890119899119872(119866119890119899119872minus119866+1))

(15)

where 119865119865min is the lower bound of 119865119865 119865119865max is the upperbound of 119865119865 119866119890119899119872 is maximum evolution generation 119866

Discrete Dynamics in Nature and Society 5

presents the evolution generation So 119865119865 can adjust its sizewith the change of the iterations The larger mutation factorensures greater diversity of population at the early stage andthe smaller mutation factor reserves excellent individual atthe latter stage

Combined SelectionOperation ofGAThe truncation selectionmethod of GA is adopted The trial solution is not comparedwith its parent but is kept in the trial setWhen all individualsin the population generate their trial vector a new populationsize of 2119873

119875is integrated by combining trial set and parent set

Then their fitness values are sorted in ascending order and theprevious 50 vectors are truncated for the next generationpopulation of the algorithm to ensure the excellent solution

313 An Example for IDE Here an example is given toillustrate the procedure of the proposed DE algorithm Forcurrent number of iteration119866 and target vector 119909(119866)

1 suppose

random generated numbers 1199031 1199032 and 119903

3are 23 40 and119873

119901

respectively

Target vectors 119909(119866) Vector x1 9 6 4 2 0085

Vector119909

23 4 1 3 7 0630

Vector119909

40 5 1 2 6 0845

Vector119909

119873119875 3 2 4 7 0437

Mutation If the calculated 119865119865 = 06 then the mutated vectorV(119866+1)1

can be obtained by (13)

Mutated vectors V(119866+1) Vector v1 52 04 18 64 0875

Crossover If 119862119877 = 03 rnbr(119905) = 3 (here 119905 = 1) and vectorrand = (01 04 05 02 06) the trial vector can be obtainedfrom (14)

Trial vectors 119906(119866+1) Vector u1 52 6 18 18 0085

Selection In this operation the superior 119873119901individuals

are chosen to next generation from the set of 119909(119866) 119906(119866+1)according to the improved selection scheme

0 50 100 150 200 250 300

0

10

20

30

40

50

60

70

80

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f1

Figure 1 Convergence results of 1198911

32 Comparative Study of the Classical DE andIDE by Benchmark Functions

321 Test Functions In order to verify the performance ofimproved DE four benchmark functions used by Brest et al[33] are utilized to test the performance of IDE To assure arelatively fair comparison the functions are selected accord-ing to their different property Function 119891

1is a unimodal

function 1198912is a step function and 119891

3-1198914are multimodal

functions which appear to be the most difficult class ofproblems for many intelligent algorithms Four benchmarkfunctions are given in Table 1

322 Comparative Results The comparative study involvesthe speed of convergence and the ability to obtain optimalsolution Two algorithms including the classical DE and IDEare compared In all cases the number of populations (119873

119901)

is 200 maximum number of iterations (GenM) is 300 and500 for 119871 = 10 and 119871 = 30 respectively For the classicalDE and IDE crossover rate CR is 03 The mutation rate119865119865 for the classical DE is 06 119865119865min is 02 and 119865119865max is 12for IDE The decision for using these values is based on theexperiences from literatures (Wang et al [17] Storn and Price[28] Brest et al [33]) Based onMatlab 70 all experiments areprogrammed on a computer (CPU Intel Core 2DuoT5870200GHz 200GHz RAM 992MB OS Microsoft WindowsXP) The convergence results are presented in Figures 1 2 3and 4

Figures 1 2 3 and 4 demonstrate that IDE convergesfaster than classical DE in all cases particularly when func-tions are difficult to converge such as 119891

3 Table 2 summarizes

values averaged over 50 independent runs for the average119891minand standard deviation 119891min of each algorithm for differenttest functionsThe average CPU time and standard derivationof CPU time are reported in Table 3

6 Discrete Dynamics in Nature and Society

Table 1 Four benchmark functions

Test functions DL Arr 119891min

1198911(119909) =

119871

sum

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816+

119871

prod

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816

10 [minus10 10]119871 0

1198912(119909) =

119871

sum

119897=1

(lfloor119909119897+ 05rfloor)

2 30 [minus100 100]119871 0

1198913(119909) =

119871

sum

119897=1

minus119909119897sin(radic100381610038161003816

1003816119909119897

1003816100381610038161003816) 30 [minus500 500]

119871minus125695

1198914(119909) = minus20 exp(minus02radic

(sum119871

119897=11199092

119897)

119871

) minus exp(1119871

119871

sum

119897=1

cos 2120587119909119897) + 20 + 119890 10 [minus32 32]

119871 0

DL denotes the dimensionality of the test problem Arr denotes the ranges of the variables and 119891min is a function value of the global optimum

Table 2 Results of two different algorithms under four test functions

Test function IDE Classical DEMean 119891min Std dev 119891min Mean 119891min Std dev 119891min

1198911

26794119864 minus 012 91466119864 minus 013 60456119864 minus 006 14096119864 minus 006

1198912

0 0 15 270801198913

minus125695 0 minus84739 16741741198914

34866119864 minus 010 11557119864 minus 010 44122119864 minus 005 77586119864 minus 006

0 100 200 300 400 500

0

1

2

3

4

5

6

times104

Number of iterations

Func

tion

valu

e

Convergence results for test function f2

Classical DEIDE

Figure 2 Convergence results of 1198912

Table 2 illustrates that IDE is more effective to obtainoptimal solutions especially for 119891

2and 119891

3 Moreover the

fluctuation from average 119891min of IDE is much smaller thanthat of classical DE which also indicates the stability of IDETables 2 and 3 show IDE always can find better solutions thanDE faster

33 IDE-Based Procedure for JR-CCModel Considering thatIR model and JR-CCmodel have similar property for a givenset of 119870 we can obtain the optimal 119879

119894according to (6) 119879

0 100 200 300 400 500

Number of iterations

Func

tion

valu

e

Convergence results for test function f3

minus13

minus12

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

times103

Classical DEIDE

Figure 3 Convergence results of 1198913

according to (11) and the optimal function value that is (7)for the IR model and (12) for the JR-CC model Thus theprocedure for two models is similar and in the followingthe JR-CC model is used as an example to introduce theprocedure

Step 1 (initialization) Set lower and upper bound ofmutationfactor119865119865 crossover factorCR and population scale119873

119875 Give

themaximumnumber of iterationsGenM and the dimensionof each individual DL Set the lower bound and the upperbound of 119896

119894 respectively (donated as 119896119871119861

119894 119896119880119861119894) Note that 119896

119894

Discrete Dynamics in Nature and Society 7

0 50 100 150 200 250 300

0

2

4

6

8

10

12

14

16

18

20

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f4

Figure 4 Convergence results of 1198914

Table 3 CPU time (s) of two different algorithms under four testfunctions

Testfunction

IDE Classical DEMean

CPU timeStd devCPU time

MeanCPU time

Std devCPU time

1198911

20781 00191 26048 001691198912

101757 02433 175211 039481198913

120578 03763 213064 058561198914

31686 00539 50259 01134

are integers so 119896119871119861119894

is obviously 1 According to the similarexperience of Wang et al [7] and Wang et al [17] 119896119880119861

119894is set

sufficiently large to guarantee that optimal solution does notescape such as 100 Generate the initial population 119909

(0)=

(119909(0)

1 119909(0)

2 119909

(0)

119873119875) randomly between 119896119871119861

119894and 119896119880119861119894

Step 2 While stopping criterion is not met (G lt GenM) goto Step 3 else go to Step 4

Step 3 For each individual 119909(119866)119905

= (119909(119866)

1199051 119909(119866)

1199052 119909

(119866)

119905119863) 119905 =

1 2 119873119875

Step 31 (mutation) Choose three vectors 119909(119866)

1199031 119909(119866)

1199032 119909(119866)

1199033

randomly from the current population Generate themutatedvectorV(119866)

119905by (13)

Step 32 (crossover) Generate the trail vector 119906(119866+1)119905

by (14)

Step 33 (selection) Sort the new population size of 2119873119901with

trial set and parent set by optimal objective function valueand then truncate the previous 50 into next generationpopulation 119909(119866+1)

Step 4 Output the best results

0 20 40 60 80 100

6

7

8

9

10

11

12

times103

Number of iterations

The b

est v

alue

of o

bjec

tive f

unct

ion

Convergence results

JR DEIR DE

JR IDEIR IDE

Figure 5 The convergence curves of DE and IDE

4 Numerical Examples andSensitivity Analysis

In this section three numerical examples are presented Toverify the accuracy of IDE and the classical DE for theproposed models a comparative example of Hsu [14] is con-ducted firstly In order to further test the performance of theproposed algorithm for the practical problem an extendedexample of the larger scale is presented in Section 42 For theanalysis of the impact of different parameters of the modelson the policy a sensitivity analysis is designed in Section 43

41 Comparative Example of Hsu [14] and Analysis For thesake of analysis and verifying the accuracy of the IDE and theclassical DE the same data as Hsu [14] is adoptedThe relatedparameters are reported in Table 4

Considering the advice of Neri and Tirronen [34] andWang et al [35] the following parameters are set 119865119865min =

02 119865119865max = 12 119865119865 = 06 119862119877 = 03 and 119873119901= 5119863119871

119866119890119899119872 is set to 100 and the dimension of each individual DLis as the same with the number of materials 119899 For a bettercomparison the cost of each entity and the total cost in thesupply chain are calculated Moreover the differences of eachcost between two models are shown in Table 5 Figure 5 givesthe convergence curves of the DE and IDE

Table 5 shows the optimal values of decision variables 119870of IR model and JR-CC model using IDE and the classicDE are the same with the heuristic of Hsu [14] and the littledifference of cost of the supply chain between Hsu [14] andours due to the number of decimal digits of decision variables119879119894 Compared with the IR policy the total cost of the system

is reduced by 2383 using the JR-CC policy In particularthe cost of the retailer is cut down by 3128 This indicatesthat the JR-CCmodel can reduce supply chain costs obviously

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

and J-M Chen [15] proposed four decision-making policiescharacterized by the joint replenishment and channel coor-dination (JR-CC) practice to determine optimal inventoryreplenishment and production policies in a supply chainHowever these literatures have not considered the potentialcoordination cost when the JR-CC policy is adopted in thesupply chain All the center of the supply chain in these lit-eratures is the suppliers because the suppliers can distributethe items to the retailers jointly to reduce the delivery costMoreover the suppliers can provide quantity discounts toinspire the retailers to adopt the joint replenishment policy

The aim of this paper is to study a useful and practicalprocurement approach using the JR-CC policy consideringthe coordination cost and provide a simple and effectivesolution In this study the advantage of joint replenishmentand channel coordination (JR-CC) policy is investigatedbased on the work of Hsu [14] Hsu [14] studied the jointreplenishment decisions for a central factory and satellitefactories under a Just in Time environment and regardedthe members of the supply chain as the same benefit groupAdversely we consider that they have independent financialaccounting systems and make their own decisions from theview of retailers for the efficient implementation of JR-CCpolicy Moreover we further compare costs to obtain usefulmanagerial insights Because the success of the JR-CC policyneeds a close cooperation with other enterprises it can beregarded as a practical and effective management methodfor most enterprises who only concentrate on its own benefitwithout consideration for the others

JRPs had been proven to be the NP-hard problems andthey were rather hard to find effective algorithms (Cha andMoon [11] Wang et al[16 17]) Hsu [14] designed a complexheuristic to handle this problem Several heuristics may giveacceptable solutions when the scale of optimization is smallHowever it is not easy to find a proper heuristic with a robustperformance However this heuristic is relatively complex fordecision-makers and cannot handle JR-CC model with thecost constrain conveniently With the increasing complexityof supply chain optimization problem (Geunes and Pardalos[18] Geunes et al [19] Pardalos et al [20]) and the devel-opment of the intelligent algorithms (Wang et al [21] Li etal [22] Wang et al [23] Cui et al [24]) they were widelyused for handling the practical supply chain optimizationproblems (Pei et al [25] Al-Anzi and Allahverdi [26] Wanget al [27]) Among these algorithms the differential evolutionalgorithm (DE) had an extensive application in JRPs (Wanget al [2 6 8 9]) So a simple and effective DE is utilized tohandle this problemTheDEwas proposed by Storn andPrice[28] for complex continuous nonlinear nondifferentiableand multimodal optimization problem This technique com-bines simple arithmetic operators with the classical events ofcrossover mutation and selection to evolve from a randomlygenerated starting population to a final solution Due to itssimple structure easy implementation quick convergenceand robustness the DE has been applied in a variety of fields(Al-Anzi and Allahverdi [26] Pan et al [29] Qu et al [6])To the best of our knowledge no work on the JR-CC modelusing DE can be found The DE has a good performance inconvergent speed but the faster convergence may cause the

diversity of population to descend quickly during the solutionprocess Moreover a closely clustered population yields apremature convergence or a local optimum and cannotreproduce a better individual So the classic DE should beimproved to find a good trade-off between convergence anddiversity

The rest of this paper is organized as follows In Section 2the independent replenishment (IR)model and JR-CCmodelare discussed Section 3 proposes a DE-based solution for theproposed model Section 4 contains numerical example andsensitivity analysis for the JR-CC models Conclusions andfuture research are presented in Section 5

2 Mathematical Formulation

21 Assumptions and Notations We consider two-echelonsupply chain consisting of a retailer replenishes 119899 materialsfrom multiple manufacturers which distributes in a cen-tralized location The retailer and manufacturers share theinformation of demand and inventory and both of them canhold inventory We assume that the demand of retailer isconstant and no shortage is allowed The manufacturers areassumed to be a ldquomake-to-orderrdquo producer using a lot-for-lotpolicy to fulfill customer demand Each time the retailer hasa need for replenishment a major ordering cost regardless ofthe number of the materials included and a minor orderingcost related tomaterial are incurred In each production runthemanufacturers have a constant production rate and spendcorresponding setup cost The quantity in one product cycleof material 119894 is split into multiple equal-size shipment lotsdelivered to the retailer

The following notations are used

119899 number of materials involved in the model119894 index of item 119894 = 1 2 119899119863119894 demand rate of material 119894

119875119894 production rate of material 119894 in a manufacturer

119875119894ge 119863119894

119896119894 the number of shipments in which the material

119894 is delivered from the manufacturers to the retailerwithin one production cycle a positive integer119878119894 setup cost of material 119894 for the manufacturers

119865 major ordering cost for the retailer119891119894 minor ordering cost of material 119894 for the retailer

119867119888119894 carrying cost rate of material 119894 for the retailer

119867119904119894 carrying cost rate of material 119894 for the manufac-

turers119879 the common shipment cycle time for the 119899materi-als119879119894 the shipment cycle time of the material 119894 with

independent replenishmentTC1199041198940 total relevant cost of material 119894 per year for the

manufacturers with independent replenishmentTC1198880 total relevant cost of 119899materials per year for the

retailer with independent replenishment

Discrete Dynamics in Nature and Society 3

TC119904119894 total relevant cost of material 119894 per year for the

manufacturers with joint replenishment and channelcoordinationTC119888 total relevant cost of 119899 materials per year for

the retailer with joint replenishment and channelcoordinationJTC0 total relevant cost for the supply chain per year

with independent replenishment of materialsJTC total relevant cost for the supply chain per yearwith joint replenishment of materials and channelcoordination

Two models are discussed one is independent replen-ishment (IR) model in which the materials have their ownshipment cycle time 119879

119894 the other is JR-CC model in which

the materials have a common shipment cycle time 119879

22 Formulation of the IR Model With the independentreplenishment policy each entity in the supply chain con-centrated on minimizing its own costs without consideringthe others The retailer makes the replenishment decision foreach itembased on an EOQpolicy (J-MChen andT-HChen[30]) Then the total cost of 119899 items per year for the retailer isexpressed as

TC1198880=

119899

sum

119894

(

1

119879119894

(119865 + 119891119894) +

119879119894

2

119863119894119867119888119894) (1)

Since the manufacturers adopt a ldquomake-to-orderrdquo policythe total cost for each manufacturer per year is

TC1199041198940=

119878119894

119879119894119896119894

+

119879119894119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894]

119894 = 1 2 119899

(2)

Then the total cost per year incurred to the retailer andthe manufacturers for independent replenishing 119899 materialscan be derived as

JTC0(119896119894 119879119894)

= TC1198880+

119899

sum

119894=1

TC1199041198940

=

119899

sum

119894=1

(

119865 + 119891119894

119879119894

+

119863119894119867119888119894119879119894

2

)

+

119899

sum

119894=1

119878119894

119879119894119896119894

+

119879119894119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894]

=

119899

sum

119894=1

1

119879119894

(119865 + 119891119894+

119878119894

119896119894

)

+

119899

sum

119894=1

119863119894119879119894

2

[119867119888119894+ 119867119904119894(

2119863119894

119875119894

minus 1) + 119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(3)

To simplify the expression we denote 119860119894= 119865 + 119891

119894 119861119894=

119863119894[119867119888119894+ 119867119904119894(2119863119894119875119894minus 1)] Then we can obtain the simplified

formulation of JTC0as follows

JTC0(119896119894 119879119894) =

119899

sum

119894=1

1

119879119894

(119860119894+

119878119894

119896119894

)

+

119899

sum

119894=1

119879119894

2

[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(4)

For a given set of 119870 = (1198961 1198962 119896

119899) taking the first

order of JTC0 with respect to 119879119894we have

120597JTC0(119896119894 119879119894)

120597119879119894

= minus

119860119894+ 119878119894119896119894

1198792

119894

+

1

2

[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(5)

Let 120597JTC0(119896119894 119879119894)120597119879119894= 0 the optimal 119879lowast

119894must satisfy

119879lowast

119894(119896119894) = radic

2 (119860119894+ (119878119894119896119894))

[119861119894+ 119863119894119867119904119894(1 minus 119863

119894119875119894) 119896119894]

(6)

Since 1205972JTC0(119896119894 119879119894)1205971198792

119894= 2(119860

119894+ 119878119894119896119894)1198793

119894gt 0 119879lowast

119894can

be determined uniquely by (6) after 119896119894are given

Substitute (6) into (4) the optimal JTC0can be obtained

by

JTClowast0(119896119894)

=

119899

sum

119894=1

radic2(119860119894+

119878119894

119896119894

)[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(7)

23 Formulation of the JR-CCModel The objective of the JR-CC model is to minimize the total cost in this two-echelonsupply chain It involves determining a basic replenishmentcycle time 119879 and the replenishment interval 119896

119894119879 for item i

where 119896119894is an integral number Then the total cost per year

for the retailer is

TC119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894 (8)

The total cost of material 119894 per year for the manufacturersis

TC119904119894=

119878119894

119896119894119879

+

119879119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894] (9)

4 Discrete Dynamics in Nature and Society

Then the total cost per year incurred to the retailer andthe manufacturers for joint replenishing 119899 materials can bederived as

JTC (119896119894 119879) = TC

119888+

119899

sum

119894=1

TC119904119894

=

1

119879

(119865 +

119899

sum

119894=1

119891119894+

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[

119899

sum

119894=1

119863119894(119867119888119894+ 119867119904119894(

2119863119894

119875119894

minus 1))

+

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

=

1

119879

(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(10)

where 119860119860 = 119865 + sum119899

119894=1119891119894 119861119861 = sum

119899

119894=1119863119894(119867119888119894+ 119867119904119894(2119863119894119875119894minus

1)) Since the formulation of JTC has the similar structure ofJTC0 for any fixed 119896

119894 the corresponding optimal common

shipment cycle time 119879lowast(119870) is given by

119879lowast(119896119894) = radic

2 (119860119860 + sum119899

119894=1119878119894119896119894)

119861119861 + sum119899

119894=1119863119894119867119904119894(1 minus 119863

119894119875119894) 119896119894

(11)

Then the minimal total cost of the supply chain underchannel cooperation is

JTClowast (119896119894)

= radic2(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(12)

Obviously both the JR-CC model and IR model havesimilar function structures Hsu [14] proposed a heuristicfor solving this model and obtaining the optimal solutionHowever the heuristic is not universal and relatively com-plex for decision-makers In addition the JR-CC model isextended considering the restriction of coordination cost inthe sensitivity analysis So the heuristic cannot solve thismodel with a cost constraint Therefore in order to analyzemodel more effectively a DE-based solution is provided

3 The Improved DE (IDE) forthe JR-CC Model

31 The Classical DE and Improvement

311TheClassical DE Unlike other evolutionary algorithmsDE does not make use of some probability distribution

function in order to introduce variations into the population(Wang et al [7] Salman et al [31]) DE initializes populationrandomly by uniform distribution over search space Eachindividual is called target vector A new vector is createdby adding the weighted difference of two random vectors totarget vector Then a trail vector is generated by mixing themutated vectors with the target vectors according to selectedrules Finally a one-to-one competition selects the best onebetween the trail vector and the corresponding target vectoraccording to its fitness (Wang et al [32] Brest et al [33])Three operations of the classical DE are introduced as follows

Mutation For each target vector 119909(119866)119905

of generation 119866 amutated vector V(119866+1)

119905is created according to the following

V(119866+1)119905

= 119909(119866)

1199031+ 119865119865lowast(119909(119866)

1199032minus 119909(119866)

1199033) (13)

where 119866 presents the present generation 119865119865 isin [0 2] isa mutation factor used to control the amplification of thedifferential variation119903

1 1199032 and 119903

3are three distinct random

numbers and none of them coincides with the current targetindividual 119905 (119903

1= 1199032= 1199033= 119905)

CrossoverThe trail vector can be obtained using the followingrules

119906(119866+1)

119905119895=

V(119866+1)119905119895

if (rand le 119862119877) or 119895 = rnbr (119905)

119909(119866)

119905119895if (rand gt 119862119877) and 119895 = rnbr (119905)

(14)

where rand(119895) is a uniformly distributed random numberin range (0 1) rnbr(119905) is a randomly chosen integer in theset1 2 119873

119901 which ensures the trail vector gets at least

one parameter from the mutated vector 119862119877 isin [0 1] is acrossover constant

Selection The selection mechanism adopts one-to-one com-petition greedy strategy to decide which vector (trail vector119906(119866+1)

119905and target vector 119909(119866)

119905) is chosen into the next genera-

tion That is to say 119906(119866+1)119905

replaces 119909(119866)119905

only if its fitness (iethe value of objective function) is better than the fitness of119909(119866)

119905

312 The Improved DE (IDE) In order to improve DE anadaptive parameter 119865 is adopted and a new selection methodbased on DE and genetic algorithm (GA) is utilized

Adaptive Mutation Factor In mutation parameter 119865119865 playsan important role 119865119865affects the speed of convergence and-decides the search range Usually optimization algorithmsfavor global search at the early stage for exploring feasibledomain and local search at the latter stage for acceleratingconvergence Based on above features a parameter 119865119865 isdefined as follows

119865119865 = 119865119865min + (119865119865max minus 119865119865min) lowast 1198901minus(119866119890119899119872(119866119890119899119872minus119866+1))

(15)

where 119865119865min is the lower bound of 119865119865 119865119865max is the upperbound of 119865119865 119866119890119899119872 is maximum evolution generation 119866

Discrete Dynamics in Nature and Society 5

presents the evolution generation So 119865119865 can adjust its sizewith the change of the iterations The larger mutation factorensures greater diversity of population at the early stage andthe smaller mutation factor reserves excellent individual atthe latter stage

Combined SelectionOperation ofGAThe truncation selectionmethod of GA is adopted The trial solution is not comparedwith its parent but is kept in the trial setWhen all individualsin the population generate their trial vector a new populationsize of 2119873

119875is integrated by combining trial set and parent set

Then their fitness values are sorted in ascending order and theprevious 50 vectors are truncated for the next generationpopulation of the algorithm to ensure the excellent solution

313 An Example for IDE Here an example is given toillustrate the procedure of the proposed DE algorithm Forcurrent number of iteration119866 and target vector 119909(119866)

1 suppose

random generated numbers 1199031 1199032 and 119903

3are 23 40 and119873

119901

respectively

Target vectors 119909(119866) Vector x1 9 6 4 2 0085

Vector119909

23 4 1 3 7 0630

Vector119909

40 5 1 2 6 0845

Vector119909

119873119875 3 2 4 7 0437

Mutation If the calculated 119865119865 = 06 then the mutated vectorV(119866+1)1

can be obtained by (13)

Mutated vectors V(119866+1) Vector v1 52 04 18 64 0875

Crossover If 119862119877 = 03 rnbr(119905) = 3 (here 119905 = 1) and vectorrand = (01 04 05 02 06) the trial vector can be obtainedfrom (14)

Trial vectors 119906(119866+1) Vector u1 52 6 18 18 0085

Selection In this operation the superior 119873119901individuals

are chosen to next generation from the set of 119909(119866) 119906(119866+1)according to the improved selection scheme

0 50 100 150 200 250 300

0

10

20

30

40

50

60

70

80

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f1

Figure 1 Convergence results of 1198911

32 Comparative Study of the Classical DE andIDE by Benchmark Functions

321 Test Functions In order to verify the performance ofimproved DE four benchmark functions used by Brest et al[33] are utilized to test the performance of IDE To assure arelatively fair comparison the functions are selected accord-ing to their different property Function 119891

1is a unimodal

function 1198912is a step function and 119891

3-1198914are multimodal

functions which appear to be the most difficult class ofproblems for many intelligent algorithms Four benchmarkfunctions are given in Table 1

322 Comparative Results The comparative study involvesthe speed of convergence and the ability to obtain optimalsolution Two algorithms including the classical DE and IDEare compared In all cases the number of populations (119873

119901)

is 200 maximum number of iterations (GenM) is 300 and500 for 119871 = 10 and 119871 = 30 respectively For the classicalDE and IDE crossover rate CR is 03 The mutation rate119865119865 for the classical DE is 06 119865119865min is 02 and 119865119865max is 12for IDE The decision for using these values is based on theexperiences from literatures (Wang et al [17] Storn and Price[28] Brest et al [33]) Based onMatlab 70 all experiments areprogrammed on a computer (CPU Intel Core 2DuoT5870200GHz 200GHz RAM 992MB OS Microsoft WindowsXP) The convergence results are presented in Figures 1 2 3and 4

Figures 1 2 3 and 4 demonstrate that IDE convergesfaster than classical DE in all cases particularly when func-tions are difficult to converge such as 119891

3 Table 2 summarizes

values averaged over 50 independent runs for the average119891minand standard deviation 119891min of each algorithm for differenttest functionsThe average CPU time and standard derivationof CPU time are reported in Table 3

6 Discrete Dynamics in Nature and Society

Table 1 Four benchmark functions

Test functions DL Arr 119891min

1198911(119909) =

119871

sum

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816+

119871

prod

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816

10 [minus10 10]119871 0

1198912(119909) =

119871

sum

119897=1

(lfloor119909119897+ 05rfloor)

2 30 [minus100 100]119871 0

1198913(119909) =

119871

sum

119897=1

minus119909119897sin(radic100381610038161003816

1003816119909119897

1003816100381610038161003816) 30 [minus500 500]

119871minus125695

1198914(119909) = minus20 exp(minus02radic

(sum119871

119897=11199092

119897)

119871

) minus exp(1119871

119871

sum

119897=1

cos 2120587119909119897) + 20 + 119890 10 [minus32 32]

119871 0

DL denotes the dimensionality of the test problem Arr denotes the ranges of the variables and 119891min is a function value of the global optimum

Table 2 Results of two different algorithms under four test functions

Test function IDE Classical DEMean 119891min Std dev 119891min Mean 119891min Std dev 119891min

1198911

26794119864 minus 012 91466119864 minus 013 60456119864 minus 006 14096119864 minus 006

1198912

0 0 15 270801198913

minus125695 0 minus84739 16741741198914

34866119864 minus 010 11557119864 minus 010 44122119864 minus 005 77586119864 minus 006

0 100 200 300 400 500

0

1

2

3

4

5

6

times104

Number of iterations

Func

tion

valu

e

Convergence results for test function f2

Classical DEIDE

Figure 2 Convergence results of 1198912

Table 2 illustrates that IDE is more effective to obtainoptimal solutions especially for 119891

2and 119891

3 Moreover the

fluctuation from average 119891min of IDE is much smaller thanthat of classical DE which also indicates the stability of IDETables 2 and 3 show IDE always can find better solutions thanDE faster

33 IDE-Based Procedure for JR-CCModel Considering thatIR model and JR-CCmodel have similar property for a givenset of 119870 we can obtain the optimal 119879

119894according to (6) 119879

0 100 200 300 400 500

Number of iterations

Func

tion

valu

e

Convergence results for test function f3

minus13

minus12

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

times103

Classical DEIDE

Figure 3 Convergence results of 1198913

according to (11) and the optimal function value that is (7)for the IR model and (12) for the JR-CC model Thus theprocedure for two models is similar and in the followingthe JR-CC model is used as an example to introduce theprocedure

Step 1 (initialization) Set lower and upper bound ofmutationfactor119865119865 crossover factorCR and population scale119873

119875 Give

themaximumnumber of iterationsGenM and the dimensionof each individual DL Set the lower bound and the upperbound of 119896

119894 respectively (donated as 119896119871119861

119894 119896119880119861119894) Note that 119896

119894

Discrete Dynamics in Nature and Society 7

0 50 100 150 200 250 300

0

2

4

6

8

10

12

14

16

18

20

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f4

Figure 4 Convergence results of 1198914

Table 3 CPU time (s) of two different algorithms under four testfunctions

Testfunction

IDE Classical DEMean

CPU timeStd devCPU time

MeanCPU time

Std devCPU time

1198911

20781 00191 26048 001691198912

101757 02433 175211 039481198913

120578 03763 213064 058561198914

31686 00539 50259 01134

are integers so 119896119871119861119894

is obviously 1 According to the similarexperience of Wang et al [7] and Wang et al [17] 119896119880119861

119894is set

sufficiently large to guarantee that optimal solution does notescape such as 100 Generate the initial population 119909

(0)=

(119909(0)

1 119909(0)

2 119909

(0)

119873119875) randomly between 119896119871119861

119894and 119896119880119861119894

Step 2 While stopping criterion is not met (G lt GenM) goto Step 3 else go to Step 4

Step 3 For each individual 119909(119866)119905

= (119909(119866)

1199051 119909(119866)

1199052 119909

(119866)

119905119863) 119905 =

1 2 119873119875

Step 31 (mutation) Choose three vectors 119909(119866)

1199031 119909(119866)

1199032 119909(119866)

1199033

randomly from the current population Generate themutatedvectorV(119866)

119905by (13)

Step 32 (crossover) Generate the trail vector 119906(119866+1)119905

by (14)

Step 33 (selection) Sort the new population size of 2119873119901with

trial set and parent set by optimal objective function valueand then truncate the previous 50 into next generationpopulation 119909(119866+1)

Step 4 Output the best results

0 20 40 60 80 100

6

7

8

9

10

11

12

times103

Number of iterations

The b

est v

alue

of o

bjec

tive f

unct

ion

Convergence results

JR DEIR DE

JR IDEIR IDE

Figure 5 The convergence curves of DE and IDE

4 Numerical Examples andSensitivity Analysis

In this section three numerical examples are presented Toverify the accuracy of IDE and the classical DE for theproposed models a comparative example of Hsu [14] is con-ducted firstly In order to further test the performance of theproposed algorithm for the practical problem an extendedexample of the larger scale is presented in Section 42 For theanalysis of the impact of different parameters of the modelson the policy a sensitivity analysis is designed in Section 43

41 Comparative Example of Hsu [14] and Analysis For thesake of analysis and verifying the accuracy of the IDE and theclassical DE the same data as Hsu [14] is adoptedThe relatedparameters are reported in Table 4

Considering the advice of Neri and Tirronen [34] andWang et al [35] the following parameters are set 119865119865min =

02 119865119865max = 12 119865119865 = 06 119862119877 = 03 and 119873119901= 5119863119871

119866119890119899119872 is set to 100 and the dimension of each individual DLis as the same with the number of materials 119899 For a bettercomparison the cost of each entity and the total cost in thesupply chain are calculated Moreover the differences of eachcost between two models are shown in Table 5 Figure 5 givesthe convergence curves of the DE and IDE

Table 5 shows the optimal values of decision variables 119870of IR model and JR-CC model using IDE and the classicDE are the same with the heuristic of Hsu [14] and the littledifference of cost of the supply chain between Hsu [14] andours due to the number of decimal digits of decision variables119879119894 Compared with the IR policy the total cost of the system

is reduced by 2383 using the JR-CC policy In particularthe cost of the retailer is cut down by 3128 This indicatesthat the JR-CCmodel can reduce supply chain costs obviously

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

TC119904119894 total relevant cost of material 119894 per year for the

manufacturers with joint replenishment and channelcoordinationTC119888 total relevant cost of 119899 materials per year for

the retailer with joint replenishment and channelcoordinationJTC0 total relevant cost for the supply chain per year

with independent replenishment of materialsJTC total relevant cost for the supply chain per yearwith joint replenishment of materials and channelcoordination

Two models are discussed one is independent replen-ishment (IR) model in which the materials have their ownshipment cycle time 119879

119894 the other is JR-CC model in which

the materials have a common shipment cycle time 119879

22 Formulation of the IR Model With the independentreplenishment policy each entity in the supply chain con-centrated on minimizing its own costs without consideringthe others The retailer makes the replenishment decision foreach itembased on an EOQpolicy (J-MChen andT-HChen[30]) Then the total cost of 119899 items per year for the retailer isexpressed as

TC1198880=

119899

sum

119894

(

1

119879119894

(119865 + 119891119894) +

119879119894

2

119863119894119867119888119894) (1)

Since the manufacturers adopt a ldquomake-to-orderrdquo policythe total cost for each manufacturer per year is

TC1199041198940=

119878119894

119879119894119896119894

+

119879119894119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894]

119894 = 1 2 119899

(2)

Then the total cost per year incurred to the retailer andthe manufacturers for independent replenishing 119899 materialscan be derived as

JTC0(119896119894 119879119894)

= TC1198880+

119899

sum

119894=1

TC1199041198940

=

119899

sum

119894=1

(

119865 + 119891119894

119879119894

+

119863119894119867119888119894119879119894

2

)

+

119899

sum

119894=1

119878119894

119879119894119896119894

+

119879119894119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894]

=

119899

sum

119894=1

1

119879119894

(119865 + 119891119894+

119878119894

119896119894

)

+

119899

sum

119894=1

119863119894119879119894

2

[119867119888119894+ 119867119904119894(

2119863119894

119875119894

minus 1) + 119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(3)

To simplify the expression we denote 119860119894= 119865 + 119891

119894 119861119894=

119863119894[119867119888119894+ 119867119904119894(2119863119894119875119894minus 1)] Then we can obtain the simplified

formulation of JTC0as follows

JTC0(119896119894 119879119894) =

119899

sum

119894=1

1

119879119894

(119860119894+

119878119894

119896119894

)

+

119899

sum

119894=1

119879119894

2

[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(4)

For a given set of 119870 = (1198961 1198962 119896

119899) taking the first

order of JTC0 with respect to 119879119894we have

120597JTC0(119896119894 119879119894)

120597119879119894

= minus

119860119894+ 119878119894119896119894

1198792

119894

+

1

2

[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(5)

Let 120597JTC0(119896119894 119879119894)120597119879119894= 0 the optimal 119879lowast

119894must satisfy

119879lowast

119894(119896119894) = radic

2 (119860119894+ (119878119894119896119894))

[119861119894+ 119863119894119867119904119894(1 minus 119863

119894119875119894) 119896119894]

(6)

Since 1205972JTC0(119896119894 119879119894)1205971198792

119894= 2(119860

119894+ 119878119894119896119894)1198793

119894gt 0 119879lowast

119894can

be determined uniquely by (6) after 119896119894are given

Substitute (6) into (4) the optimal JTC0can be obtained

by

JTClowast0(119896119894)

=

119899

sum

119894=1

radic2(119860119894+

119878119894

119896119894

)[119861119894+ 119863119894119867119904119894(1 minus

119863119894

119875119894

)119896119894]

(7)

23 Formulation of the JR-CCModel The objective of the JR-CC model is to minimize the total cost in this two-echelonsupply chain It involves determining a basic replenishmentcycle time 119879 and the replenishment interval 119896

119894119879 for item i

where 119896119894is an integral number Then the total cost per year

for the retailer is

TC119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894 (8)

The total cost of material 119894 per year for the manufacturersis

TC119904119894=

119878119894

119896119894119879

+

119879119863119894119867119904119894

2

[

2119863119894

119875119894

minus 1 + (1 minus

119863119894

119875119894

)119896119894] (9)

4 Discrete Dynamics in Nature and Society

Then the total cost per year incurred to the retailer andthe manufacturers for joint replenishing 119899 materials can bederived as

JTC (119896119894 119879) = TC

119888+

119899

sum

119894=1

TC119904119894

=

1

119879

(119865 +

119899

sum

119894=1

119891119894+

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[

119899

sum

119894=1

119863119894(119867119888119894+ 119867119904119894(

2119863119894

119875119894

minus 1))

+

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

=

1

119879

(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(10)

where 119860119860 = 119865 + sum119899

119894=1119891119894 119861119861 = sum

119899

119894=1119863119894(119867119888119894+ 119867119904119894(2119863119894119875119894minus

1)) Since the formulation of JTC has the similar structure ofJTC0 for any fixed 119896

119894 the corresponding optimal common

shipment cycle time 119879lowast(119870) is given by

119879lowast(119896119894) = radic

2 (119860119860 + sum119899

119894=1119878119894119896119894)

119861119861 + sum119899

119894=1119863119894119867119904119894(1 minus 119863

119894119875119894) 119896119894

(11)

Then the minimal total cost of the supply chain underchannel cooperation is

JTClowast (119896119894)

= radic2(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(12)

Obviously both the JR-CC model and IR model havesimilar function structures Hsu [14] proposed a heuristicfor solving this model and obtaining the optimal solutionHowever the heuristic is not universal and relatively com-plex for decision-makers In addition the JR-CC model isextended considering the restriction of coordination cost inthe sensitivity analysis So the heuristic cannot solve thismodel with a cost constraint Therefore in order to analyzemodel more effectively a DE-based solution is provided

3 The Improved DE (IDE) forthe JR-CC Model

31 The Classical DE and Improvement

311TheClassical DE Unlike other evolutionary algorithmsDE does not make use of some probability distribution

function in order to introduce variations into the population(Wang et al [7] Salman et al [31]) DE initializes populationrandomly by uniform distribution over search space Eachindividual is called target vector A new vector is createdby adding the weighted difference of two random vectors totarget vector Then a trail vector is generated by mixing themutated vectors with the target vectors according to selectedrules Finally a one-to-one competition selects the best onebetween the trail vector and the corresponding target vectoraccording to its fitness (Wang et al [32] Brest et al [33])Three operations of the classical DE are introduced as follows

Mutation For each target vector 119909(119866)119905

of generation 119866 amutated vector V(119866+1)

119905is created according to the following

V(119866+1)119905

= 119909(119866)

1199031+ 119865119865lowast(119909(119866)

1199032minus 119909(119866)

1199033) (13)

where 119866 presents the present generation 119865119865 isin [0 2] isa mutation factor used to control the amplification of thedifferential variation119903

1 1199032 and 119903

3are three distinct random

numbers and none of them coincides with the current targetindividual 119905 (119903

1= 1199032= 1199033= 119905)

CrossoverThe trail vector can be obtained using the followingrules

119906(119866+1)

119905119895=

V(119866+1)119905119895

if (rand le 119862119877) or 119895 = rnbr (119905)

119909(119866)

119905119895if (rand gt 119862119877) and 119895 = rnbr (119905)

(14)

where rand(119895) is a uniformly distributed random numberin range (0 1) rnbr(119905) is a randomly chosen integer in theset1 2 119873

119901 which ensures the trail vector gets at least

one parameter from the mutated vector 119862119877 isin [0 1] is acrossover constant

Selection The selection mechanism adopts one-to-one com-petition greedy strategy to decide which vector (trail vector119906(119866+1)

119905and target vector 119909(119866)

119905) is chosen into the next genera-

tion That is to say 119906(119866+1)119905

replaces 119909(119866)119905

only if its fitness (iethe value of objective function) is better than the fitness of119909(119866)

119905

312 The Improved DE (IDE) In order to improve DE anadaptive parameter 119865 is adopted and a new selection methodbased on DE and genetic algorithm (GA) is utilized

Adaptive Mutation Factor In mutation parameter 119865119865 playsan important role 119865119865affects the speed of convergence and-decides the search range Usually optimization algorithmsfavor global search at the early stage for exploring feasibledomain and local search at the latter stage for acceleratingconvergence Based on above features a parameter 119865119865 isdefined as follows

119865119865 = 119865119865min + (119865119865max minus 119865119865min) lowast 1198901minus(119866119890119899119872(119866119890119899119872minus119866+1))

(15)

where 119865119865min is the lower bound of 119865119865 119865119865max is the upperbound of 119865119865 119866119890119899119872 is maximum evolution generation 119866

Discrete Dynamics in Nature and Society 5

presents the evolution generation So 119865119865 can adjust its sizewith the change of the iterations The larger mutation factorensures greater diversity of population at the early stage andthe smaller mutation factor reserves excellent individual atthe latter stage

Combined SelectionOperation ofGAThe truncation selectionmethod of GA is adopted The trial solution is not comparedwith its parent but is kept in the trial setWhen all individualsin the population generate their trial vector a new populationsize of 2119873

119875is integrated by combining trial set and parent set

Then their fitness values are sorted in ascending order and theprevious 50 vectors are truncated for the next generationpopulation of the algorithm to ensure the excellent solution

313 An Example for IDE Here an example is given toillustrate the procedure of the proposed DE algorithm Forcurrent number of iteration119866 and target vector 119909(119866)

1 suppose

random generated numbers 1199031 1199032 and 119903

3are 23 40 and119873

119901

respectively

Target vectors 119909(119866) Vector x1 9 6 4 2 0085

Vector119909

23 4 1 3 7 0630

Vector119909

40 5 1 2 6 0845

Vector119909

119873119875 3 2 4 7 0437

Mutation If the calculated 119865119865 = 06 then the mutated vectorV(119866+1)1

can be obtained by (13)

Mutated vectors V(119866+1) Vector v1 52 04 18 64 0875

Crossover If 119862119877 = 03 rnbr(119905) = 3 (here 119905 = 1) and vectorrand = (01 04 05 02 06) the trial vector can be obtainedfrom (14)

Trial vectors 119906(119866+1) Vector u1 52 6 18 18 0085

Selection In this operation the superior 119873119901individuals

are chosen to next generation from the set of 119909(119866) 119906(119866+1)according to the improved selection scheme

0 50 100 150 200 250 300

0

10

20

30

40

50

60

70

80

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f1

Figure 1 Convergence results of 1198911

32 Comparative Study of the Classical DE andIDE by Benchmark Functions

321 Test Functions In order to verify the performance ofimproved DE four benchmark functions used by Brest et al[33] are utilized to test the performance of IDE To assure arelatively fair comparison the functions are selected accord-ing to their different property Function 119891

1is a unimodal

function 1198912is a step function and 119891

3-1198914are multimodal

functions which appear to be the most difficult class ofproblems for many intelligent algorithms Four benchmarkfunctions are given in Table 1

322 Comparative Results The comparative study involvesthe speed of convergence and the ability to obtain optimalsolution Two algorithms including the classical DE and IDEare compared In all cases the number of populations (119873

119901)

is 200 maximum number of iterations (GenM) is 300 and500 for 119871 = 10 and 119871 = 30 respectively For the classicalDE and IDE crossover rate CR is 03 The mutation rate119865119865 for the classical DE is 06 119865119865min is 02 and 119865119865max is 12for IDE The decision for using these values is based on theexperiences from literatures (Wang et al [17] Storn and Price[28] Brest et al [33]) Based onMatlab 70 all experiments areprogrammed on a computer (CPU Intel Core 2DuoT5870200GHz 200GHz RAM 992MB OS Microsoft WindowsXP) The convergence results are presented in Figures 1 2 3and 4

Figures 1 2 3 and 4 demonstrate that IDE convergesfaster than classical DE in all cases particularly when func-tions are difficult to converge such as 119891

3 Table 2 summarizes

values averaged over 50 independent runs for the average119891minand standard deviation 119891min of each algorithm for differenttest functionsThe average CPU time and standard derivationof CPU time are reported in Table 3

6 Discrete Dynamics in Nature and Society

Table 1 Four benchmark functions

Test functions DL Arr 119891min

1198911(119909) =

119871

sum

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816+

119871

prod

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816

10 [minus10 10]119871 0

1198912(119909) =

119871

sum

119897=1

(lfloor119909119897+ 05rfloor)

2 30 [minus100 100]119871 0

1198913(119909) =

119871

sum

119897=1

minus119909119897sin(radic100381610038161003816

1003816119909119897

1003816100381610038161003816) 30 [minus500 500]

119871minus125695

1198914(119909) = minus20 exp(minus02radic

(sum119871

119897=11199092

119897)

119871

) minus exp(1119871

119871

sum

119897=1

cos 2120587119909119897) + 20 + 119890 10 [minus32 32]

119871 0

DL denotes the dimensionality of the test problem Arr denotes the ranges of the variables and 119891min is a function value of the global optimum

Table 2 Results of two different algorithms under four test functions

Test function IDE Classical DEMean 119891min Std dev 119891min Mean 119891min Std dev 119891min

1198911

26794119864 minus 012 91466119864 minus 013 60456119864 minus 006 14096119864 minus 006

1198912

0 0 15 270801198913

minus125695 0 minus84739 16741741198914

34866119864 minus 010 11557119864 minus 010 44122119864 minus 005 77586119864 minus 006

0 100 200 300 400 500

0

1

2

3

4

5

6

times104

Number of iterations

Func

tion

valu

e

Convergence results for test function f2

Classical DEIDE

Figure 2 Convergence results of 1198912

Table 2 illustrates that IDE is more effective to obtainoptimal solutions especially for 119891

2and 119891

3 Moreover the

fluctuation from average 119891min of IDE is much smaller thanthat of classical DE which also indicates the stability of IDETables 2 and 3 show IDE always can find better solutions thanDE faster

33 IDE-Based Procedure for JR-CCModel Considering thatIR model and JR-CCmodel have similar property for a givenset of 119870 we can obtain the optimal 119879

119894according to (6) 119879

0 100 200 300 400 500

Number of iterations

Func

tion

valu

e

Convergence results for test function f3

minus13

minus12

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

times103

Classical DEIDE

Figure 3 Convergence results of 1198913

according to (11) and the optimal function value that is (7)for the IR model and (12) for the JR-CC model Thus theprocedure for two models is similar and in the followingthe JR-CC model is used as an example to introduce theprocedure

Step 1 (initialization) Set lower and upper bound ofmutationfactor119865119865 crossover factorCR and population scale119873

119875 Give

themaximumnumber of iterationsGenM and the dimensionof each individual DL Set the lower bound and the upperbound of 119896

119894 respectively (donated as 119896119871119861

119894 119896119880119861119894) Note that 119896

119894

Discrete Dynamics in Nature and Society 7

0 50 100 150 200 250 300

0

2

4

6

8

10

12

14

16

18

20

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f4

Figure 4 Convergence results of 1198914

Table 3 CPU time (s) of two different algorithms under four testfunctions

Testfunction

IDE Classical DEMean

CPU timeStd devCPU time

MeanCPU time

Std devCPU time

1198911

20781 00191 26048 001691198912

101757 02433 175211 039481198913

120578 03763 213064 058561198914

31686 00539 50259 01134

are integers so 119896119871119861119894

is obviously 1 According to the similarexperience of Wang et al [7] and Wang et al [17] 119896119880119861

119894is set

sufficiently large to guarantee that optimal solution does notescape such as 100 Generate the initial population 119909

(0)=

(119909(0)

1 119909(0)

2 119909

(0)

119873119875) randomly between 119896119871119861

119894and 119896119880119861119894

Step 2 While stopping criterion is not met (G lt GenM) goto Step 3 else go to Step 4

Step 3 For each individual 119909(119866)119905

= (119909(119866)

1199051 119909(119866)

1199052 119909

(119866)

119905119863) 119905 =

1 2 119873119875

Step 31 (mutation) Choose three vectors 119909(119866)

1199031 119909(119866)

1199032 119909(119866)

1199033

randomly from the current population Generate themutatedvectorV(119866)

119905by (13)

Step 32 (crossover) Generate the trail vector 119906(119866+1)119905

by (14)

Step 33 (selection) Sort the new population size of 2119873119901with

trial set and parent set by optimal objective function valueand then truncate the previous 50 into next generationpopulation 119909(119866+1)

Step 4 Output the best results

0 20 40 60 80 100

6

7

8

9

10

11

12

times103

Number of iterations

The b

est v

alue

of o

bjec

tive f

unct

ion

Convergence results

JR DEIR DE

JR IDEIR IDE

Figure 5 The convergence curves of DE and IDE

4 Numerical Examples andSensitivity Analysis

In this section three numerical examples are presented Toverify the accuracy of IDE and the classical DE for theproposed models a comparative example of Hsu [14] is con-ducted firstly In order to further test the performance of theproposed algorithm for the practical problem an extendedexample of the larger scale is presented in Section 42 For theanalysis of the impact of different parameters of the modelson the policy a sensitivity analysis is designed in Section 43

41 Comparative Example of Hsu [14] and Analysis For thesake of analysis and verifying the accuracy of the IDE and theclassical DE the same data as Hsu [14] is adoptedThe relatedparameters are reported in Table 4

Considering the advice of Neri and Tirronen [34] andWang et al [35] the following parameters are set 119865119865min =

02 119865119865max = 12 119865119865 = 06 119862119877 = 03 and 119873119901= 5119863119871

119866119890119899119872 is set to 100 and the dimension of each individual DLis as the same with the number of materials 119899 For a bettercomparison the cost of each entity and the total cost in thesupply chain are calculated Moreover the differences of eachcost between two models are shown in Table 5 Figure 5 givesthe convergence curves of the DE and IDE

Table 5 shows the optimal values of decision variables 119870of IR model and JR-CC model using IDE and the classicDE are the same with the heuristic of Hsu [14] and the littledifference of cost of the supply chain between Hsu [14] andours due to the number of decimal digits of decision variables119879119894 Compared with the IR policy the total cost of the system

is reduced by 2383 using the JR-CC policy In particularthe cost of the retailer is cut down by 3128 This indicatesthat the JR-CCmodel can reduce supply chain costs obviously

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Then the total cost per year incurred to the retailer andthe manufacturers for joint replenishing 119899 materials can bederived as

JTC (119896119894 119879) = TC

119888+

119899

sum

119894=1

TC119904119894

=

1

119879

(119865 +

119899

sum

119894=1

119891119894+

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[

119899

sum

119894=1

119863119894(119867119888119894+ 119867119904119894(

2119863119894

119875119894

minus 1))

+

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

=

1

119879

(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)

+

119879

2

[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(10)

where 119860119860 = 119865 + sum119899

119894=1119891119894 119861119861 = sum

119899

119894=1119863119894(119867119888119894+ 119867119904119894(2119863119894119875119894minus

1)) Since the formulation of JTC has the similar structure ofJTC0 for any fixed 119896

119894 the corresponding optimal common

shipment cycle time 119879lowast(119870) is given by

119879lowast(119896119894) = radic

2 (119860119860 + sum119899

119894=1119878119894119896119894)

119861119861 + sum119899

119894=1119863119894119867119904119894(1 minus 119863

119894119875119894) 119896119894

(11)

Then the minimal total cost of the supply chain underchannel cooperation is

JTClowast (119896119894)

= radic2(119860119860 +

119899

sum

119894=1

119878119894

119896119894

)[119861119861 +

119899

sum

119894=1

119863119894119867119904119894((1 minus

119863119894

119875119894

)119896119894)]

(12)

Obviously both the JR-CC model and IR model havesimilar function structures Hsu [14] proposed a heuristicfor solving this model and obtaining the optimal solutionHowever the heuristic is not universal and relatively com-plex for decision-makers In addition the JR-CC model isextended considering the restriction of coordination cost inthe sensitivity analysis So the heuristic cannot solve thismodel with a cost constraint Therefore in order to analyzemodel more effectively a DE-based solution is provided

3 The Improved DE (IDE) forthe JR-CC Model

31 The Classical DE and Improvement

311TheClassical DE Unlike other evolutionary algorithmsDE does not make use of some probability distribution

function in order to introduce variations into the population(Wang et al [7] Salman et al [31]) DE initializes populationrandomly by uniform distribution over search space Eachindividual is called target vector A new vector is createdby adding the weighted difference of two random vectors totarget vector Then a trail vector is generated by mixing themutated vectors with the target vectors according to selectedrules Finally a one-to-one competition selects the best onebetween the trail vector and the corresponding target vectoraccording to its fitness (Wang et al [32] Brest et al [33])Three operations of the classical DE are introduced as follows

Mutation For each target vector 119909(119866)119905

of generation 119866 amutated vector V(119866+1)

119905is created according to the following

V(119866+1)119905

= 119909(119866)

1199031+ 119865119865lowast(119909(119866)

1199032minus 119909(119866)

1199033) (13)

where 119866 presents the present generation 119865119865 isin [0 2] isa mutation factor used to control the amplification of thedifferential variation119903

1 1199032 and 119903

3are three distinct random

numbers and none of them coincides with the current targetindividual 119905 (119903

1= 1199032= 1199033= 119905)

CrossoverThe trail vector can be obtained using the followingrules

119906(119866+1)

119905119895=

V(119866+1)119905119895

if (rand le 119862119877) or 119895 = rnbr (119905)

119909(119866)

119905119895if (rand gt 119862119877) and 119895 = rnbr (119905)

(14)

where rand(119895) is a uniformly distributed random numberin range (0 1) rnbr(119905) is a randomly chosen integer in theset1 2 119873

119901 which ensures the trail vector gets at least

one parameter from the mutated vector 119862119877 isin [0 1] is acrossover constant

Selection The selection mechanism adopts one-to-one com-petition greedy strategy to decide which vector (trail vector119906(119866+1)

119905and target vector 119909(119866)

119905) is chosen into the next genera-

tion That is to say 119906(119866+1)119905

replaces 119909(119866)119905

only if its fitness (iethe value of objective function) is better than the fitness of119909(119866)

119905

312 The Improved DE (IDE) In order to improve DE anadaptive parameter 119865 is adopted and a new selection methodbased on DE and genetic algorithm (GA) is utilized

Adaptive Mutation Factor In mutation parameter 119865119865 playsan important role 119865119865affects the speed of convergence and-decides the search range Usually optimization algorithmsfavor global search at the early stage for exploring feasibledomain and local search at the latter stage for acceleratingconvergence Based on above features a parameter 119865119865 isdefined as follows

119865119865 = 119865119865min + (119865119865max minus 119865119865min) lowast 1198901minus(119866119890119899119872(119866119890119899119872minus119866+1))

(15)

where 119865119865min is the lower bound of 119865119865 119865119865max is the upperbound of 119865119865 119866119890119899119872 is maximum evolution generation 119866

Discrete Dynamics in Nature and Society 5

presents the evolution generation So 119865119865 can adjust its sizewith the change of the iterations The larger mutation factorensures greater diversity of population at the early stage andthe smaller mutation factor reserves excellent individual atthe latter stage

Combined SelectionOperation ofGAThe truncation selectionmethod of GA is adopted The trial solution is not comparedwith its parent but is kept in the trial setWhen all individualsin the population generate their trial vector a new populationsize of 2119873

119875is integrated by combining trial set and parent set

Then their fitness values are sorted in ascending order and theprevious 50 vectors are truncated for the next generationpopulation of the algorithm to ensure the excellent solution

313 An Example for IDE Here an example is given toillustrate the procedure of the proposed DE algorithm Forcurrent number of iteration119866 and target vector 119909(119866)

1 suppose

random generated numbers 1199031 1199032 and 119903

3are 23 40 and119873

119901

respectively

Target vectors 119909(119866) Vector x1 9 6 4 2 0085

Vector119909

23 4 1 3 7 0630

Vector119909

40 5 1 2 6 0845

Vector119909

119873119875 3 2 4 7 0437

Mutation If the calculated 119865119865 = 06 then the mutated vectorV(119866+1)1

can be obtained by (13)

Mutated vectors V(119866+1) Vector v1 52 04 18 64 0875

Crossover If 119862119877 = 03 rnbr(119905) = 3 (here 119905 = 1) and vectorrand = (01 04 05 02 06) the trial vector can be obtainedfrom (14)

Trial vectors 119906(119866+1) Vector u1 52 6 18 18 0085

Selection In this operation the superior 119873119901individuals

are chosen to next generation from the set of 119909(119866) 119906(119866+1)according to the improved selection scheme

0 50 100 150 200 250 300

0

10

20

30

40

50

60

70

80

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f1

Figure 1 Convergence results of 1198911

32 Comparative Study of the Classical DE andIDE by Benchmark Functions

321 Test Functions In order to verify the performance ofimproved DE four benchmark functions used by Brest et al[33] are utilized to test the performance of IDE To assure arelatively fair comparison the functions are selected accord-ing to their different property Function 119891

1is a unimodal

function 1198912is a step function and 119891

3-1198914are multimodal

functions which appear to be the most difficult class ofproblems for many intelligent algorithms Four benchmarkfunctions are given in Table 1

322 Comparative Results The comparative study involvesthe speed of convergence and the ability to obtain optimalsolution Two algorithms including the classical DE and IDEare compared In all cases the number of populations (119873

119901)

is 200 maximum number of iterations (GenM) is 300 and500 for 119871 = 10 and 119871 = 30 respectively For the classicalDE and IDE crossover rate CR is 03 The mutation rate119865119865 for the classical DE is 06 119865119865min is 02 and 119865119865max is 12for IDE The decision for using these values is based on theexperiences from literatures (Wang et al [17] Storn and Price[28] Brest et al [33]) Based onMatlab 70 all experiments areprogrammed on a computer (CPU Intel Core 2DuoT5870200GHz 200GHz RAM 992MB OS Microsoft WindowsXP) The convergence results are presented in Figures 1 2 3and 4

Figures 1 2 3 and 4 demonstrate that IDE convergesfaster than classical DE in all cases particularly when func-tions are difficult to converge such as 119891

3 Table 2 summarizes

values averaged over 50 independent runs for the average119891minand standard deviation 119891min of each algorithm for differenttest functionsThe average CPU time and standard derivationof CPU time are reported in Table 3

6 Discrete Dynamics in Nature and Society

Table 1 Four benchmark functions

Test functions DL Arr 119891min

1198911(119909) =

119871

sum

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816+

119871

prod

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816

10 [minus10 10]119871 0

1198912(119909) =

119871

sum

119897=1

(lfloor119909119897+ 05rfloor)

2 30 [minus100 100]119871 0

1198913(119909) =

119871

sum

119897=1

minus119909119897sin(radic100381610038161003816

1003816119909119897

1003816100381610038161003816) 30 [minus500 500]

119871minus125695

1198914(119909) = minus20 exp(minus02radic

(sum119871

119897=11199092

119897)

119871

) minus exp(1119871

119871

sum

119897=1

cos 2120587119909119897) + 20 + 119890 10 [minus32 32]

119871 0

DL denotes the dimensionality of the test problem Arr denotes the ranges of the variables and 119891min is a function value of the global optimum

Table 2 Results of two different algorithms under four test functions

Test function IDE Classical DEMean 119891min Std dev 119891min Mean 119891min Std dev 119891min

1198911

26794119864 minus 012 91466119864 minus 013 60456119864 minus 006 14096119864 minus 006

1198912

0 0 15 270801198913

minus125695 0 minus84739 16741741198914

34866119864 minus 010 11557119864 minus 010 44122119864 minus 005 77586119864 minus 006

0 100 200 300 400 500

0

1

2

3

4

5

6

times104

Number of iterations

Func

tion

valu

e

Convergence results for test function f2

Classical DEIDE

Figure 2 Convergence results of 1198912

Table 2 illustrates that IDE is more effective to obtainoptimal solutions especially for 119891

2and 119891

3 Moreover the

fluctuation from average 119891min of IDE is much smaller thanthat of classical DE which also indicates the stability of IDETables 2 and 3 show IDE always can find better solutions thanDE faster

33 IDE-Based Procedure for JR-CCModel Considering thatIR model and JR-CCmodel have similar property for a givenset of 119870 we can obtain the optimal 119879

119894according to (6) 119879

0 100 200 300 400 500

Number of iterations

Func

tion

valu

e

Convergence results for test function f3

minus13

minus12

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

times103

Classical DEIDE

Figure 3 Convergence results of 1198913

according to (11) and the optimal function value that is (7)for the IR model and (12) for the JR-CC model Thus theprocedure for two models is similar and in the followingthe JR-CC model is used as an example to introduce theprocedure

Step 1 (initialization) Set lower and upper bound ofmutationfactor119865119865 crossover factorCR and population scale119873

119875 Give

themaximumnumber of iterationsGenM and the dimensionof each individual DL Set the lower bound and the upperbound of 119896

119894 respectively (donated as 119896119871119861

119894 119896119880119861119894) Note that 119896

119894

Discrete Dynamics in Nature and Society 7

0 50 100 150 200 250 300

0

2

4

6

8

10

12

14

16

18

20

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f4

Figure 4 Convergence results of 1198914

Table 3 CPU time (s) of two different algorithms under four testfunctions

Testfunction

IDE Classical DEMean

CPU timeStd devCPU time

MeanCPU time

Std devCPU time

1198911

20781 00191 26048 001691198912

101757 02433 175211 039481198913

120578 03763 213064 058561198914

31686 00539 50259 01134

are integers so 119896119871119861119894

is obviously 1 According to the similarexperience of Wang et al [7] and Wang et al [17] 119896119880119861

119894is set

sufficiently large to guarantee that optimal solution does notescape such as 100 Generate the initial population 119909

(0)=

(119909(0)

1 119909(0)

2 119909

(0)

119873119875) randomly between 119896119871119861

119894and 119896119880119861119894

Step 2 While stopping criterion is not met (G lt GenM) goto Step 3 else go to Step 4

Step 3 For each individual 119909(119866)119905

= (119909(119866)

1199051 119909(119866)

1199052 119909

(119866)

119905119863) 119905 =

1 2 119873119875

Step 31 (mutation) Choose three vectors 119909(119866)

1199031 119909(119866)

1199032 119909(119866)

1199033

randomly from the current population Generate themutatedvectorV(119866)

119905by (13)

Step 32 (crossover) Generate the trail vector 119906(119866+1)119905

by (14)

Step 33 (selection) Sort the new population size of 2119873119901with

trial set and parent set by optimal objective function valueand then truncate the previous 50 into next generationpopulation 119909(119866+1)

Step 4 Output the best results

0 20 40 60 80 100

6

7

8

9

10

11

12

times103

Number of iterations

The b

est v

alue

of o

bjec

tive f

unct

ion

Convergence results

JR DEIR DE

JR IDEIR IDE

Figure 5 The convergence curves of DE and IDE

4 Numerical Examples andSensitivity Analysis

In this section three numerical examples are presented Toverify the accuracy of IDE and the classical DE for theproposed models a comparative example of Hsu [14] is con-ducted firstly In order to further test the performance of theproposed algorithm for the practical problem an extendedexample of the larger scale is presented in Section 42 For theanalysis of the impact of different parameters of the modelson the policy a sensitivity analysis is designed in Section 43

41 Comparative Example of Hsu [14] and Analysis For thesake of analysis and verifying the accuracy of the IDE and theclassical DE the same data as Hsu [14] is adoptedThe relatedparameters are reported in Table 4

Considering the advice of Neri and Tirronen [34] andWang et al [35] the following parameters are set 119865119865min =

02 119865119865max = 12 119865119865 = 06 119862119877 = 03 and 119873119901= 5119863119871

119866119890119899119872 is set to 100 and the dimension of each individual DLis as the same with the number of materials 119899 For a bettercomparison the cost of each entity and the total cost in thesupply chain are calculated Moreover the differences of eachcost between two models are shown in Table 5 Figure 5 givesthe convergence curves of the DE and IDE

Table 5 shows the optimal values of decision variables 119870of IR model and JR-CC model using IDE and the classicDE are the same with the heuristic of Hsu [14] and the littledifference of cost of the supply chain between Hsu [14] andours due to the number of decimal digits of decision variables119879119894 Compared with the IR policy the total cost of the system

is reduced by 2383 using the JR-CC policy In particularthe cost of the retailer is cut down by 3128 This indicatesthat the JR-CCmodel can reduce supply chain costs obviously

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

presents the evolution generation So 119865119865 can adjust its sizewith the change of the iterations The larger mutation factorensures greater diversity of population at the early stage andthe smaller mutation factor reserves excellent individual atthe latter stage

Combined SelectionOperation ofGAThe truncation selectionmethod of GA is adopted The trial solution is not comparedwith its parent but is kept in the trial setWhen all individualsin the population generate their trial vector a new populationsize of 2119873

119875is integrated by combining trial set and parent set

Then their fitness values are sorted in ascending order and theprevious 50 vectors are truncated for the next generationpopulation of the algorithm to ensure the excellent solution

313 An Example for IDE Here an example is given toillustrate the procedure of the proposed DE algorithm Forcurrent number of iteration119866 and target vector 119909(119866)

1 suppose

random generated numbers 1199031 1199032 and 119903

3are 23 40 and119873

119901

respectively

Target vectors 119909(119866) Vector x1 9 6 4 2 0085

Vector119909

23 4 1 3 7 0630

Vector119909

40 5 1 2 6 0845

Vector119909

119873119875 3 2 4 7 0437

Mutation If the calculated 119865119865 = 06 then the mutated vectorV(119866+1)1

can be obtained by (13)

Mutated vectors V(119866+1) Vector v1 52 04 18 64 0875

Crossover If 119862119877 = 03 rnbr(119905) = 3 (here 119905 = 1) and vectorrand = (01 04 05 02 06) the trial vector can be obtainedfrom (14)

Trial vectors 119906(119866+1) Vector u1 52 6 18 18 0085

Selection In this operation the superior 119873119901individuals

are chosen to next generation from the set of 119909(119866) 119906(119866+1)according to the improved selection scheme

0 50 100 150 200 250 300

0

10

20

30

40

50

60

70

80

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f1

Figure 1 Convergence results of 1198911

32 Comparative Study of the Classical DE andIDE by Benchmark Functions

321 Test Functions In order to verify the performance ofimproved DE four benchmark functions used by Brest et al[33] are utilized to test the performance of IDE To assure arelatively fair comparison the functions are selected accord-ing to their different property Function 119891

1is a unimodal

function 1198912is a step function and 119891

3-1198914are multimodal

functions which appear to be the most difficult class ofproblems for many intelligent algorithms Four benchmarkfunctions are given in Table 1

322 Comparative Results The comparative study involvesthe speed of convergence and the ability to obtain optimalsolution Two algorithms including the classical DE and IDEare compared In all cases the number of populations (119873

119901)

is 200 maximum number of iterations (GenM) is 300 and500 for 119871 = 10 and 119871 = 30 respectively For the classicalDE and IDE crossover rate CR is 03 The mutation rate119865119865 for the classical DE is 06 119865119865min is 02 and 119865119865max is 12for IDE The decision for using these values is based on theexperiences from literatures (Wang et al [17] Storn and Price[28] Brest et al [33]) Based onMatlab 70 all experiments areprogrammed on a computer (CPU Intel Core 2DuoT5870200GHz 200GHz RAM 992MB OS Microsoft WindowsXP) The convergence results are presented in Figures 1 2 3and 4

Figures 1 2 3 and 4 demonstrate that IDE convergesfaster than classical DE in all cases particularly when func-tions are difficult to converge such as 119891

3 Table 2 summarizes

values averaged over 50 independent runs for the average119891minand standard deviation 119891min of each algorithm for differenttest functionsThe average CPU time and standard derivationof CPU time are reported in Table 3

6 Discrete Dynamics in Nature and Society

Table 1 Four benchmark functions

Test functions DL Arr 119891min

1198911(119909) =

119871

sum

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816+

119871

prod

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816

10 [minus10 10]119871 0

1198912(119909) =

119871

sum

119897=1

(lfloor119909119897+ 05rfloor)

2 30 [minus100 100]119871 0

1198913(119909) =

119871

sum

119897=1

minus119909119897sin(radic100381610038161003816

1003816119909119897

1003816100381610038161003816) 30 [minus500 500]

119871minus125695

1198914(119909) = minus20 exp(minus02radic

(sum119871

119897=11199092

119897)

119871

) minus exp(1119871

119871

sum

119897=1

cos 2120587119909119897) + 20 + 119890 10 [minus32 32]

119871 0

DL denotes the dimensionality of the test problem Arr denotes the ranges of the variables and 119891min is a function value of the global optimum

Table 2 Results of two different algorithms under four test functions

Test function IDE Classical DEMean 119891min Std dev 119891min Mean 119891min Std dev 119891min

1198911

26794119864 minus 012 91466119864 minus 013 60456119864 minus 006 14096119864 minus 006

1198912

0 0 15 270801198913

minus125695 0 minus84739 16741741198914

34866119864 minus 010 11557119864 minus 010 44122119864 minus 005 77586119864 minus 006

0 100 200 300 400 500

0

1

2

3

4

5

6

times104

Number of iterations

Func

tion

valu

e

Convergence results for test function f2

Classical DEIDE

Figure 2 Convergence results of 1198912

Table 2 illustrates that IDE is more effective to obtainoptimal solutions especially for 119891

2and 119891

3 Moreover the

fluctuation from average 119891min of IDE is much smaller thanthat of classical DE which also indicates the stability of IDETables 2 and 3 show IDE always can find better solutions thanDE faster

33 IDE-Based Procedure for JR-CCModel Considering thatIR model and JR-CCmodel have similar property for a givenset of 119870 we can obtain the optimal 119879

119894according to (6) 119879

0 100 200 300 400 500

Number of iterations

Func

tion

valu

e

Convergence results for test function f3

minus13

minus12

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

times103

Classical DEIDE

Figure 3 Convergence results of 1198913

according to (11) and the optimal function value that is (7)for the IR model and (12) for the JR-CC model Thus theprocedure for two models is similar and in the followingthe JR-CC model is used as an example to introduce theprocedure

Step 1 (initialization) Set lower and upper bound ofmutationfactor119865119865 crossover factorCR and population scale119873

119875 Give

themaximumnumber of iterationsGenM and the dimensionof each individual DL Set the lower bound and the upperbound of 119896

119894 respectively (donated as 119896119871119861

119894 119896119880119861119894) Note that 119896

119894

Discrete Dynamics in Nature and Society 7

0 50 100 150 200 250 300

0

2

4

6

8

10

12

14

16

18

20

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f4

Figure 4 Convergence results of 1198914

Table 3 CPU time (s) of two different algorithms under four testfunctions

Testfunction

IDE Classical DEMean

CPU timeStd devCPU time

MeanCPU time

Std devCPU time

1198911

20781 00191 26048 001691198912

101757 02433 175211 039481198913

120578 03763 213064 058561198914

31686 00539 50259 01134

are integers so 119896119871119861119894

is obviously 1 According to the similarexperience of Wang et al [7] and Wang et al [17] 119896119880119861

119894is set

sufficiently large to guarantee that optimal solution does notescape such as 100 Generate the initial population 119909

(0)=

(119909(0)

1 119909(0)

2 119909

(0)

119873119875) randomly between 119896119871119861

119894and 119896119880119861119894

Step 2 While stopping criterion is not met (G lt GenM) goto Step 3 else go to Step 4

Step 3 For each individual 119909(119866)119905

= (119909(119866)

1199051 119909(119866)

1199052 119909

(119866)

119905119863) 119905 =

1 2 119873119875

Step 31 (mutation) Choose three vectors 119909(119866)

1199031 119909(119866)

1199032 119909(119866)

1199033

randomly from the current population Generate themutatedvectorV(119866)

119905by (13)

Step 32 (crossover) Generate the trail vector 119906(119866+1)119905

by (14)

Step 33 (selection) Sort the new population size of 2119873119901with

trial set and parent set by optimal objective function valueand then truncate the previous 50 into next generationpopulation 119909(119866+1)

Step 4 Output the best results

0 20 40 60 80 100

6

7

8

9

10

11

12

times103

Number of iterations

The b

est v

alue

of o

bjec

tive f

unct

ion

Convergence results

JR DEIR DE

JR IDEIR IDE

Figure 5 The convergence curves of DE and IDE

4 Numerical Examples andSensitivity Analysis

In this section three numerical examples are presented Toverify the accuracy of IDE and the classical DE for theproposed models a comparative example of Hsu [14] is con-ducted firstly In order to further test the performance of theproposed algorithm for the practical problem an extendedexample of the larger scale is presented in Section 42 For theanalysis of the impact of different parameters of the modelson the policy a sensitivity analysis is designed in Section 43

41 Comparative Example of Hsu [14] and Analysis For thesake of analysis and verifying the accuracy of the IDE and theclassical DE the same data as Hsu [14] is adoptedThe relatedparameters are reported in Table 4

Considering the advice of Neri and Tirronen [34] andWang et al [35] the following parameters are set 119865119865min =

02 119865119865max = 12 119865119865 = 06 119862119877 = 03 and 119873119901= 5119863119871

119866119890119899119872 is set to 100 and the dimension of each individual DLis as the same with the number of materials 119899 For a bettercomparison the cost of each entity and the total cost in thesupply chain are calculated Moreover the differences of eachcost between two models are shown in Table 5 Figure 5 givesthe convergence curves of the DE and IDE

Table 5 shows the optimal values of decision variables 119870of IR model and JR-CC model using IDE and the classicDE are the same with the heuristic of Hsu [14] and the littledifference of cost of the supply chain between Hsu [14] andours due to the number of decimal digits of decision variables119879119894 Compared with the IR policy the total cost of the system

is reduced by 2383 using the JR-CC policy In particularthe cost of the retailer is cut down by 3128 This indicatesthat the JR-CCmodel can reduce supply chain costs obviously

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Table 1 Four benchmark functions

Test functions DL Arr 119891min

1198911(119909) =

119871

sum

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816+

119871

prod

119897=1

1003816100381610038161003816119909119897

1003816100381610038161003816

10 [minus10 10]119871 0

1198912(119909) =

119871

sum

119897=1

(lfloor119909119897+ 05rfloor)

2 30 [minus100 100]119871 0

1198913(119909) =

119871

sum

119897=1

minus119909119897sin(radic100381610038161003816

1003816119909119897

1003816100381610038161003816) 30 [minus500 500]

119871minus125695

1198914(119909) = minus20 exp(minus02radic

(sum119871

119897=11199092

119897)

119871

) minus exp(1119871

119871

sum

119897=1

cos 2120587119909119897) + 20 + 119890 10 [minus32 32]

119871 0

DL denotes the dimensionality of the test problem Arr denotes the ranges of the variables and 119891min is a function value of the global optimum

Table 2 Results of two different algorithms under four test functions

Test function IDE Classical DEMean 119891min Std dev 119891min Mean 119891min Std dev 119891min

1198911

26794119864 minus 012 91466119864 minus 013 60456119864 minus 006 14096119864 minus 006

1198912

0 0 15 270801198913

minus125695 0 minus84739 16741741198914

34866119864 minus 010 11557119864 minus 010 44122119864 minus 005 77586119864 minus 006

0 100 200 300 400 500

0

1

2

3

4

5

6

times104

Number of iterations

Func

tion

valu

e

Convergence results for test function f2

Classical DEIDE

Figure 2 Convergence results of 1198912

Table 2 illustrates that IDE is more effective to obtainoptimal solutions especially for 119891

2and 119891

3 Moreover the

fluctuation from average 119891min of IDE is much smaller thanthat of classical DE which also indicates the stability of IDETables 2 and 3 show IDE always can find better solutions thanDE faster

33 IDE-Based Procedure for JR-CCModel Considering thatIR model and JR-CCmodel have similar property for a givenset of 119870 we can obtain the optimal 119879

119894according to (6) 119879

0 100 200 300 400 500

Number of iterations

Func

tion

valu

e

Convergence results for test function f3

minus13

minus12

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

times103

Classical DEIDE

Figure 3 Convergence results of 1198913

according to (11) and the optimal function value that is (7)for the IR model and (12) for the JR-CC model Thus theprocedure for two models is similar and in the followingthe JR-CC model is used as an example to introduce theprocedure

Step 1 (initialization) Set lower and upper bound ofmutationfactor119865119865 crossover factorCR and population scale119873

119875 Give

themaximumnumber of iterationsGenM and the dimensionof each individual DL Set the lower bound and the upperbound of 119896

119894 respectively (donated as 119896119871119861

119894 119896119880119861119894) Note that 119896

119894

Discrete Dynamics in Nature and Society 7

0 50 100 150 200 250 300

0

2

4

6

8

10

12

14

16

18

20

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f4

Figure 4 Convergence results of 1198914

Table 3 CPU time (s) of two different algorithms under four testfunctions

Testfunction

IDE Classical DEMean

CPU timeStd devCPU time

MeanCPU time

Std devCPU time

1198911

20781 00191 26048 001691198912

101757 02433 175211 039481198913

120578 03763 213064 058561198914

31686 00539 50259 01134

are integers so 119896119871119861119894

is obviously 1 According to the similarexperience of Wang et al [7] and Wang et al [17] 119896119880119861

119894is set

sufficiently large to guarantee that optimal solution does notescape such as 100 Generate the initial population 119909

(0)=

(119909(0)

1 119909(0)

2 119909

(0)

119873119875) randomly between 119896119871119861

119894and 119896119880119861119894

Step 2 While stopping criterion is not met (G lt GenM) goto Step 3 else go to Step 4

Step 3 For each individual 119909(119866)119905

= (119909(119866)

1199051 119909(119866)

1199052 119909

(119866)

119905119863) 119905 =

1 2 119873119875

Step 31 (mutation) Choose three vectors 119909(119866)

1199031 119909(119866)

1199032 119909(119866)

1199033

randomly from the current population Generate themutatedvectorV(119866)

119905by (13)

Step 32 (crossover) Generate the trail vector 119906(119866+1)119905

by (14)

Step 33 (selection) Sort the new population size of 2119873119901with

trial set and parent set by optimal objective function valueand then truncate the previous 50 into next generationpopulation 119909(119866+1)

Step 4 Output the best results

0 20 40 60 80 100

6

7

8

9

10

11

12

times103

Number of iterations

The b

est v

alue

of o

bjec

tive f

unct

ion

Convergence results

JR DEIR DE

JR IDEIR IDE

Figure 5 The convergence curves of DE and IDE

4 Numerical Examples andSensitivity Analysis

In this section three numerical examples are presented Toverify the accuracy of IDE and the classical DE for theproposed models a comparative example of Hsu [14] is con-ducted firstly In order to further test the performance of theproposed algorithm for the practical problem an extendedexample of the larger scale is presented in Section 42 For theanalysis of the impact of different parameters of the modelson the policy a sensitivity analysis is designed in Section 43

41 Comparative Example of Hsu [14] and Analysis For thesake of analysis and verifying the accuracy of the IDE and theclassical DE the same data as Hsu [14] is adoptedThe relatedparameters are reported in Table 4

Considering the advice of Neri and Tirronen [34] andWang et al [35] the following parameters are set 119865119865min =

02 119865119865max = 12 119865119865 = 06 119862119877 = 03 and 119873119901= 5119863119871

119866119890119899119872 is set to 100 and the dimension of each individual DLis as the same with the number of materials 119899 For a bettercomparison the cost of each entity and the total cost in thesupply chain are calculated Moreover the differences of eachcost between two models are shown in Table 5 Figure 5 givesthe convergence curves of the DE and IDE

Table 5 shows the optimal values of decision variables 119870of IR model and JR-CC model using IDE and the classicDE are the same with the heuristic of Hsu [14] and the littledifference of cost of the supply chain between Hsu [14] andours due to the number of decimal digits of decision variables119879119894 Compared with the IR policy the total cost of the system

is reduced by 2383 using the JR-CC policy In particularthe cost of the retailer is cut down by 3128 This indicatesthat the JR-CCmodel can reduce supply chain costs obviously

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

0 50 100 150 200 250 300

0

2

4

6

8

10

12

14

16

18

20

Number of iterations

Func

tion

valu

e

Classical DEIDE

Convergence results for test function f4

Figure 4 Convergence results of 1198914

Table 3 CPU time (s) of two different algorithms under four testfunctions

Testfunction

IDE Classical DEMean

CPU timeStd devCPU time

MeanCPU time

Std devCPU time

1198911

20781 00191 26048 001691198912

101757 02433 175211 039481198913

120578 03763 213064 058561198914

31686 00539 50259 01134

are integers so 119896119871119861119894

is obviously 1 According to the similarexperience of Wang et al [7] and Wang et al [17] 119896119880119861

119894is set

sufficiently large to guarantee that optimal solution does notescape such as 100 Generate the initial population 119909

(0)=

(119909(0)

1 119909(0)

2 119909

(0)

119873119875) randomly between 119896119871119861

119894and 119896119880119861119894

Step 2 While stopping criterion is not met (G lt GenM) goto Step 3 else go to Step 4

Step 3 For each individual 119909(119866)119905

= (119909(119866)

1199051 119909(119866)

1199052 119909

(119866)

119905119863) 119905 =

1 2 119873119875

Step 31 (mutation) Choose three vectors 119909(119866)

1199031 119909(119866)

1199032 119909(119866)

1199033

randomly from the current population Generate themutatedvectorV(119866)

119905by (13)

Step 32 (crossover) Generate the trail vector 119906(119866+1)119905

by (14)

Step 33 (selection) Sort the new population size of 2119873119901with

trial set and parent set by optimal objective function valueand then truncate the previous 50 into next generationpopulation 119909(119866+1)

Step 4 Output the best results

0 20 40 60 80 100

6

7

8

9

10

11

12

times103

Number of iterations

The b

est v

alue

of o

bjec

tive f

unct

ion

Convergence results

JR DEIR DE

JR IDEIR IDE

Figure 5 The convergence curves of DE and IDE

4 Numerical Examples andSensitivity Analysis

In this section three numerical examples are presented Toverify the accuracy of IDE and the classical DE for theproposed models a comparative example of Hsu [14] is con-ducted firstly In order to further test the performance of theproposed algorithm for the practical problem an extendedexample of the larger scale is presented in Section 42 For theanalysis of the impact of different parameters of the modelson the policy a sensitivity analysis is designed in Section 43

41 Comparative Example of Hsu [14] and Analysis For thesake of analysis and verifying the accuracy of the IDE and theclassical DE the same data as Hsu [14] is adoptedThe relatedparameters are reported in Table 4

Considering the advice of Neri and Tirronen [34] andWang et al [35] the following parameters are set 119865119865min =

02 119865119865max = 12 119865119865 = 06 119862119877 = 03 and 119873119901= 5119863119871

119866119890119899119872 is set to 100 and the dimension of each individual DLis as the same with the number of materials 119899 For a bettercomparison the cost of each entity and the total cost in thesupply chain are calculated Moreover the differences of eachcost between two models are shown in Table 5 Figure 5 givesthe convergence curves of the DE and IDE

Table 5 shows the optimal values of decision variables 119870of IR model and JR-CC model using IDE and the classicDE are the same with the heuristic of Hsu [14] and the littledifference of cost of the supply chain between Hsu [14] andours due to the number of decimal digits of decision variables119879119894 Compared with the IR policy the total cost of the system

is reduced by 2383 using the JR-CC policy In particularthe cost of the retailer is cut down by 3128 This indicatesthat the JR-CCmodel can reduce supply chain costs obviously

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Table 4 Parameter settings

Production rate 119875119894

Setup cost 119878119894

Carrying cost rate119867119904119894

Manufacturer 1 18000 45 2Manufacturer 2 18000 45 1Manufacturer 3 18000 45 1

Demand rate119863119894

Minor ordering cost 119891119894

Carrying cost rate119867119888119894

Retailer

10000 8 812000 5 49000 10 8

Major ordering cost 119865 = 30

0 50 100 150

18

2

22

24

26

28

3

32

34

36

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 10)

JR DEIR DE

JR IDEIR IDE

Figure 6 The average convergence curves (119899 = 10)

and enhance supply chainrsquos efficiency Furthermore the con-vergence curves in Figure 5 demonstrate IDE can converge tothe optimal results faster than DE

42 An Extended Example with Larger Scale Since theproblem scale of Hsu [14] is too small an extended examplewith 10 30 and 50 materials is designed in this section toverify the performance of the proposed algorithm furtherThe input parameters are generated from Table 6

The parameters 119863119894 119891119894 119867119888119894 and 119867

119904119894are generated ran-

domly from the given ranges in Table 6 The algorithmrsquosparameters 119865119865min 119865119865max 119865119865 CR and 119873119901 are set the sameas in Section 41 the maximum iteration number GenM isset to 150 300 and 600 respectively for 119899 = 10 119899 = 30and 119899 = 50 Each scale is running 20 times The results arereported in Table 7 and the average convergence curves arelisted in Figures 6 7 and 8

The results shown in Table 7 and the curves in Figures 67 and 8 show the following interesting conclusions

(1) For different scales IDE can always converge to theoptimal solution with the convergence rate of 100

0 50 100 150 200 250 30055

6

65

7

75

8

85

9

95

10

105

times104

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 30)

JR DEIR DE

JR IDEIR IDE

Figure 7 The average convergence curves (119899 = 30)

0 100 200 300 400 500 600

08

09

1

11

12

13

14

15

16

17

times105

Number of iterations

The a

vera

ge b

est v

alue

of T

C

Convergence results (n = 50)

JR DEIR DE

JR IDEIR IDE

Figure 8 The average convergence curves (119899 = 50)

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

Table 5 Comparison between independent replenishment and JR-CC policy

IR JR-CC Cost reductionDE (IDE) Hsu [14] DE (IDE) Hsu [14]

Cost of retailer 67003 46044 3128Cost of manufacturer 1 9314354 9236882 083Cost of manufacturer 2 6738471 6473242 394Cost of manufacturer 3 6370856 6366200 007Cost of the supply chain 89427lowast 894264lowast 68120 681199 2383

Shipment cycle time 1198791198791= 00312 119879

1= 0031

119879 = 0023 119879 = 00231198792= 00369 119879

2= 0037

1198793= 00337 119879

3= 0034

119870lowast

1198961= 3 119896

1= 3 119896

1= 4 119896

1= 4

1198962= 4 119896

2= 4 119896

2= 7 119896

2= 7

1198963= 4 119896

3= 4 119896

3= 6 119896

3= 6

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 6 Parameters for the extended problem

Parameters Value119899 Number of materials 10 30 50119863119894 119894 = 1 2 119899 Demand rate 119880[6000 15000]

119875119894 119894 = 1 2 119899 Production rate 18000

119878119894 119894 = 1 2 119899 Setup cost of material 119894 for the manufacturers 45

119865 Major ordering cost for the retailer 30119891119894 119894 = 1 2 119899 Minor ordering cost of material 119894 for the retailer 119880[4 12]

119867119888119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the retailer 119880[4 8]

119867119904119894 119894 = 1 2 119899 Carrying cost rate of material 119894 for the manufacturers 119880[1 2]

(2) The CPU time to obtain the optimal solution of IDEis always faster than DE

(3) The convergence speed of IDE is faster than DEtoo So whatever in robustness computing time orconvergence speed IDE is always better thanDE IDEis a better candidate algorithm for the JR-CC modeland IR model

43 Sensitivity Analysis Since the fundamental of jointreplenishment is sharing themajor ordering cost119865 to achievecost saving and the value of coordination cost119871 to implementJR policy has a direct impact on the selection of the policieswe mainly examine the impact of these two parameters onthe JR-CC model and IR model Based on the numericalexamples above IDE is adopted for the sensitivity analysis

431 The Impact of Misestimates of 119865on the Cost Reduction ofthe Supply Chain The results obtained by IDEwith119865 varyingfrom 0 to 2F based on the basic data in Table 4 are shown inTable 8

Table 8 shows the cost reduction of the whole supplychain and almost every entity is increasing when the valueof 119865 becomes larger The retailer especially has more cost

savings In order to observe this trend more clearly Figure 9is provided using the data of Table 8

It is obvious that the changing of major ordering cost 119865only causes a little increase in all manufacturers (less than7) but very tremendous in the retailer and the wholesupply chain (more than 17) This indicates that majorordering cost causes more influence on the retailer thanthe manufacturers In other words the retailer has morewillingness to adopt the joint replenishment and channelcoordination policy because of the difference of the costreduction value But the manufacturers may be reluctant toaccept the policy We also find that if the value of 119865 is too lowthe retailer also has no interest in JR-CC policy because thecost reduction is too little

432 The Impact of the Coordination Cost on the CostReduction of the Supply Chain When the retailer combinesthe manufacturers to adopt the JR-CC policy the relevantresponsible staff and certain resources including telephonecommunication mail communication and business tripsshould be needed Since the cost of these resources is hard tobe estimated or is small almost theoretical researches ignorethe coordination cost in the existing JR-CC model In orderto be more practical here we will consider a new parameter119871 representing the coordination cost From Table 5 we know

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

Table 7 The results of different scales under different algorithms

Problemscale Algorithm

IR JR-CCCost reduction

JTClowast0

The number toobtain JTClowast

0

Average CPUtime (s) JTClowast The number to

obtain JTClowastAverage CPU

time (s)

119899 = 10

DE 275111198644 20 07450 181611198644 20 07472 3399IDE 275111198644 20 04037 181611198644 20 04062

119899 = 30

DE 885281198644 9 75445 552141198644 20 73905 3763IDE 885281198644 20 41093 552141198644 20 39868

119899 = 50

DE 147701198645 0 145153 896471198644 0 131147 3930IDE 147691198645 20 78123 896441198644 20 76407

lowastRepresents the meaning of ldquothe optimalrdquo which had been hinted in Section 22 and Section 23

Table 8 Sensitivity analysis of major ordering cost 119865

Parameter variable Model Retailer Manufacturer 1 Manufacturer 2 Manufacturer 3 Supply chain

119865

(1 minus 100)

IR 30252 9102592 6275493 6368673 519988JR-CC 30343 9111031 6295798 636953 521194Cost reduction minus030 minus009 minus032 minus001 minus023

119865

(1 minus 50)

IR 52019 9220195 6576866 6372893 741890JR-CC 38988 9165907 6391153 6369343 609144Cost reduction 2505 059 282 006 1789

119865

(1 minus 25)

IR 60019 9254559 6647484 6370900 822919JR-CC 42673 9181299 6419134 6367032 646405Cost reduction 2890 079 344 006 2145

119865

IR 67003 9314354 6738471 6370856 894465JR-CC 46044 9236882 6473242 63662 681203Cost reduction 3128 083 394 007 2383

119865

(1 + 25)

IR 73369 9314831 6805486 6365465 958548JR-CC 49196 9219574 6488993 6367276 712718Cost reduction 3295 102 465 minus003 2565

119865

(1 + 50)

IR 79191 9349171 6933752 6385272 1018592JR-CC 52173 9226537 6508138 6381269 742889Cost reduction 3412 131 614 006 2707

119865

(1 + 100)

IR 89785 9481541 700000 6373805 1126403JR-CC 57637 9311428 6574859 6364094 798874Cost reduction 3581 179 607 015 2908

the cost of retailer is 67003 under IR policy and then the JR-CC model for the retailer will contain a constraint which canbe derived as

119879119862119888=

1

119879

(119865 +

119899

sum

119894=1

119891119894) +

119879

2

119899

sum

119894=1

119863119894119867119888119894lt 67003 minus 119871 (16)

when 119879 = radic2(119865 + sum119899

119894=1119891119894)sum119899

119894=1119863119894119867119888119894 the left of constraint

has minimal value radic2(119865 + sum119899119894=1119891119894)(sum119899

119894=1119863119894119867119888119894) = 46043

Then we can calculate 119871 less than 20945 if the retailerimplements the JR-CC policy Since 119871 is only concluded inthe private cost calculation of the retailer it has no evidentinfluence on the cost of manufacturers Next the impactof misestimates of 119871 on the cost reduction of the retailer

who advocates the JR-CC policy is analyzed using the DEapproach The results are shown in Table 9

Table 9 shows the cost reduction of the retailer is decreas-ing when 119871 is larger When 119871 is small (119871 = 10 50 100) theresults change a little and the retailer can neglect it for the JR-CC decision In realistic circumstances 119871 is not insignificantBut once 119871 becomes very large it will have a huge impact onthe results when 119871 = 1500 or 119871 = 2000

5 Conclusion and Future Research

This paper is an interdisciplinary research of an operationmanagement problem under supply chain environment andan intelligent optimization algorithm We discussed a prac-tical and useful JR-CC model and proposed an effective and

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

Table 9 The sensitivity analysis of coordination cost 119871 for the retailer

Model 119871 = 0 119871 = 10 119871 = 50 119871 = 100 119871 = 500 119871 = 1000 119871 = 1500 119871 = 2000

IR 67003JR-CC 46044 46144 46544 47044 51044 56044 61044 66044Cost reduction 3128 3113 3053 2979 2382 1636 890 143

minus100 minus50 0 50 100

minus5

0

5

10

15

20

25

30

35

40

Change rate of F ()

()

RetailerManufacture 1

Manufacture 2

Manufacture 3

Supply chain

Figure 9 Impact of variation with 119865 on the cost reduction of thesupply chain

simple algorithm to handle this NP-hard problem The maincontributions are as follows

(1) We consider the coordination cost when the retailermade JR-CC decisions while no existing works payattention on it Results of numerical examples showthat the JR-CC policy can reduce total costs obviouslyand enhance the efficiency of a supply chain But notall members in the supply chain can benefit a lotfrom this policy The discussed JR-CC models canbe applied in these industries such as manufacturingandwholesale supplies under JIT purchasing environ-ment

(2) An IDE with an adaptive parameter FF is utilizedbased on the classic DE and GA Results of bench-mark functions and comparative numerical exampleverify the effectiveness of the proposed IDE IDE is agood candidate for the JR-CC model

(3) The impacts of misestimates of major ordering costand the coordination cost on the cost reduction of thesupply chain are analyzed respectively Correspond-ing managerial insights are given Results of sensitiv-ity analysis show the retailers have more willingnessto adopt the JR-CC policy than the manufacturersbecause of the different cost savings

However the decision makers often have to face vagueoperational conditions In this case the fuzzy set theory isa useful approach to deal with this kind of problem [36] Inthe future integrated joint replenishment and channel coor-dination models under fuzzy environment can be developedThe DEs still can be used to provide a good solution to theseproblems

Conflict of Interests

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

Acknowledgments

The authors are very grateful for the constructive commentsof editors and referees This research is partially supportedby National Natural Science Foundation of China (7137108071131004 71101060) Humanities and Social Sciences Founda-tion of ChineseMinistry of Education (no 11YJC630275) andFundamental Research Funds for the Central Universities(HUST 2014QN201)

References

[1] S Axsater and W-F Zhang ldquoJoint replenishment policy formulti-echelon inventory controlrdquo International Journal of Pro-duction Economics vol 59 no 1 pp 243ndash250 1999

[2] L Wang J He D Wu and Y-R Zeng ldquoA novel differentialevolution algorithm for joint replenishment problem underinterdependence and its applicationrdquo International Journal ofProduction Economics vol 135 no 1 pp 190ndash198 2012

[3] A L Olsen ldquoAn evolutionary algorithm to solve the jointreplenishment problem using direct groupingrdquo Computers andIndustrial Engineering vol 48 no 2 pp 223ndash235 2005

[4] I K Moon and B C Cha ldquoThe joint replenishment problemwith resource restrictionrdquo European Journal of OperationalResearch vol 173 no 1 pp 190ndash198 2006

[5] M Khouja and S Goyal ldquoA review of the joint replenishmentproblem literature 1989ndash2005rdquoEuropean Journal of OperationalResearch vol 186 no 1 pp 1ndash16 2008

[6] H Qu L Wang and Y R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

[7] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[8] L Wang H Qu S Liu and C X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishment

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

and delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

[9] N C Buyukkaramikli U Gurler and O Alp ldquoCoordinatedlogistics joint replenishment with capacitated transportationfor a supply chainrdquo Production and Operations Managementvol 23 no 1 pp 110ndash126 2013

[10] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications vol 41 no4 pp 1792ndash1805 2014

[11] B C Cha and I K Moon ldquoThe joint replenishment problemwith quantity discounts under constant demandrdquoOR Spectrumvol 27 no 4 pp 569ndash581 2005

[12] I K Moon S K Goyal and B C Cha ldquoThe joint replenish-ment problem involving multiple suppliers offering quantitydiscountsrdquo International Journal of Systems Science vol 39 no6 pp 629ndash637 2008

[13] B C Cha and J H Park ldquoThe joint replenishment and deliveryscheduling involving multiplesuppliers offering different quan-tity discountsrdquo in Proceedings of the International Conferenceon Computers and Industrial Engineering (CIE 09) pp 52ndash56Troyes France July 2009

[14] S-L Hsu ldquoOptimal joint replenishment decisions for a centralfactory with multiple satellite factoriesrdquo Expert Systems withApplications vol 36 no 2 pp 2494ndash2502 2009

[15] T-H Chen and J-M Chen ldquoOptimizing supply chain collabo-ration based on joint replenishment and channel coordinationrdquoTransportation Research Part E Logistics and TransportationReview vol 41 no 4 pp 261ndash285 2005

[16] L Wang J He and Y-R Zeng ldquoA differential evolution algo-rithm for joint replenishment problem using direct groupingand its applicationrdquo Expert Systems vol 29 no 5 pp 429ndash4412012

[17] L Wang C-X Dun W-J Bi and Y-R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[18] J Geunes and P M Pardalos Supply Chain OptimizationKluwer Academic Publishers 2004

[19] J Geunes P M Pardalos and E Romeijn Supply ChainManagement Models Applications and Research DirectionsKluwer Academic Publishers 2002

[20] PM Pardalos AMigdalas andG Baourakis Supply Chain andFinance World Scientific 2004

[21] L Wang Q-L Fu and Y-R Zeng ldquoContinuous review inven-tory models with a mixture of backorders and lost sales underfuzzy demand and different decision situationsrdquo Expert Systemswith Applications vol 39 no 4 pp 4181ndash4189 2012

[22] YH Li H Guo LWang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] L Wang R Yang P M Pardalos L Qian and M Fei ldquoAnadaptive fuzzy controller based on harmony search and itsapplication to power plant controlrdquo International Journal ofElectrical Power and Energy Systems vol 53 no 1 pp 272ndash2782013

[24] L Cui L Wang J Deng and J Zhang ldquoA new improvedquantum evolution algorithm with local search procedure for

capacitated vehicle routing problemrdquoMathematical Problems inEngineering vol 2013 Article ID 159495 17 pages 2013

[25] J Pei X B Liu P M Pardalos W J Fan S Yang andL Wang ldquoApplication of an effective modified gravitationalsearch algorithm for the coordinated scheduling problem ina two-stage supply chainrdquo International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 335ndash348 2014

[26] F S Al-Anzi and A Allahverdi ldquoA self-adaptive differentialevolution heuristic for two-stage assembly scheduling problemto minimize maximum lateness with setup timesrdquo EuropeanJournal of Operational Research vol 182 no 1 pp 80ndash94 2007

[27] L Wang H Qu T Chen and F P Yan ldquoAn effective hybridself-adapting differential evolution algorithm for the jointreplenishment and location-inventory problem in a three-levelsupply chainrdquoThe Scientific World Journal vol 2013 Article ID270249 11 pages 2013

[28] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[29] Q-K Pan L Wang and B Qian ldquoA novel differential evo-lution algorithm for bi-criteria no-wait flow shop schedulingproblemsrdquo Computers amp Operations Research vol 36 no 8 pp2498ndash2511 2009

[30] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of cen-tralization versus decentralizationrdquo Computers and OperationsResearch vol 32 no 12 pp 3191ndash3207 2005

[31] A Salman A P Engelbrecht and M G H Omran ldquoEmpiricalanalysis of self-adaptive differential evolutionrdquo European Jour-nal of Operational Research vol 183 no 2 pp 785ndash804 2007

[32] Y Wang B Li and T Weise ldquoEstimation of distribution anddifferential evolution cooperation for large scale economic loaddispatch optimization of power systemsrdquo Information Sciencesvol 180 no 12 pp 2405ndash2420 2010

[33] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[34] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[35] L Wang C-X Dun C-G Lee Q-L Fu and Y-R ZengldquoModel and algorithm for fuzzy joint replenishment anddelivery scheduling without explicit membership functionrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 9-12 pp 1907ndash1920 2013

[36] Y-R Zeng LWang and J He ldquoA novel approach for evaluatingcontrol criticality of spare parts using fuzzy comprehensiveevaluation and GRArdquo International Journal of Fuzzy Systemsvol 14 no 3 pp 392ndash401 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of