research article modeling and optimization of the multiobjective...

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 916057 11 pageshttpdxdoiorg1011552013916057

Research ArticleModeling and Optimization of the MultiobjectiveStochastic Joint Replenishment and Delivery Problem underSupply Chain Environment

Lin Wang1 Hui Qu1 Shan Liu2 and Cai-xia Dun1

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

Correspondence should be addressed to Shan Liu 279266384qqcom

Received 21 July 2013 Accepted 18 September 2013

Academic Editors S Choi and S Comai

Copyright copy 2013 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

As a practical inventory and transportation problem it is important to synthesize several objectives for the joint replenishmentand delivery (JRD) decision In this paper a new multiobjective stochastic JRD (MSJRD) of the one-warehouse and 119899-retailersystems considering the balance of service level and total cost simultaneously is proposed The goal of this problem is to decidethe reasonable replenishment interval safety stock factor and traveling routing Secondly two approaches are designed to handlethis complex multi-objective optimization problem Linear programming (LP) approach converts the multi-objective to singleobjective while a multi-objective evolution algorithm (MOEA) solves a multi-objective problem directly Thirdly three intelligentoptimization algorithms differential evolution algorithm (DE) hybrid DE (HDE) and genetic algorithm (GA) are utilized inLP-based and MOEA-based approaches Results of the MSJRD with LP-based and MOEA-based approaches are compared by acontrastive numerical example To analyses the nondominated solution of MOEA a metric is also used to measure the distributionof the last generation solution Results show that HDE outperforms DE and GA whenever LP or MOEA is adopted

1 Introduction

The joint replenishment problem (JRP) is a practical inven-tory problem of a group of products that can be jointlyordered from a single supplier (Goyal [1] Wang et al [2])which can help save the ordering costs and inventory holdingcosts According to the characteristic of demand the existingstudy of JRPs can be divided into two categories (1) constantdemand (2) stochastic or dynamic demand An extensiveliterature review is available in Khouja and Goyal [3] andNarayanan et al [4] Many scholars also discussed morerealistic JRPs (J-M Chen and T-H Chen [5] Axsater et al[6] Hsu [7] Abdul-Jalbar et al [8])

Many companies have realized that a joint replenishmentand delivery scheduling (JRD) policy can result in consid-erable cost savings But the literature on the JRDs undersupply chain environment is limited A stochastic JRD ofthe one-warehouse 119899-retailer system has been formulated(Qu et al [9]) Wang et al [10] studied the same JRD but

reduced the decision variables using specific mathematicalmethod and provided a new differential evolution algorithmSindhuchao et al [11] studied the coordinated inventory andtransportation decisions with the vehicle capacity limitationin an inbound commodity collection system Chan et al [12]addressed issues in scheduling of the multi-item multibuyerand single supplier system Cha et al [13] handled the JRDof the one-warehouse and 119899-retailer system in which thewarehouse supplies items from the supplier and deliversthem to retailers Moon et al [14] modified the modelof [13] by utilizing a consolidated freight delivery policiesWang et al [2] extended the JRD model of Cha et al [13]under fuzzy environment and used the widely used signeddistance method to ranking fuzzy numbers Wang et al[15] studied the JRD with deterministic demand and fuzzycost using the graded mean integration representation andcentroid approaches to defuzzify the total costs A commonlimitation in all the literature studies mentioned above is thatthey only consider a single objective

2 The Scientific World Journal

However managers are usually faced with complex mul-tiobjective optimization problems (MOPs) in reality For theJRD policy it is necessary to decrease the total cost whileimproving the service level Although there are several papersthat studied multiobjective inventory models (Roy and Maiti[16] Rong et al [17] Islam [18] Wee et al [19]) no study onthe multiobjective JRD can be found

ForMOPs direct comparison among the solutions is verydifficult because of the different measurements between eachcontradicted target In this study total cost and service levelare obviously two contradictory targets reducing total costmay result in the decline of service level and vice verse Sowe should coordinate two targets Different from a single-objective optimization problem which has unique optimalsolution an MOP has a set of optimal solutions called Paretooptimal solutions Due to the characteristics of MOPs theyare much more complex and the key is to find an effectivemethod to obtain Pareto optimal solutions Unfortunatelythe classical JRPs and JRDs are already NP hard problems(Arkin et al [20]) and the multiobjective makes the JRDsbecome much more difficult to handle

Many linear or nonlinear weighted methods (Rong et al[17] Islam [18] Wee et al [19] Roy and Maiti [16]) were usedto convert the multiobjective to a single one in the existingstudies These methods undoubtedly provide one easy wayto deal with the multiobjective JRD model However theseapproaches do not solve the MOPs intrinsically since thesolutions for MOPs are multiple rather than one On theother hand multiobjective optimization methods based onPareto-basedMOEAs are widely used such as multiobjectivegenetic algorithm (MOGA) (Aiello et al [21]) nondominatedsorting genetic algorithm (NSGA) (Lin [22]) and strengthPareto evolutionary algorithm (SPEA) (Zitzler and Thiele[23] Sheng et al [24])

In recent years several MOEAs based on Pareto differ-ential evolution (DE) were utilized to solve MOPs Santana-Quintero and Coello [25] presented a DE-based multiob-jective algorithm using a secondary population and theconcept of 120598-dominance The performance of the proposedalgorithm was also compared with NSGA-II and 120598-MOEAQian et al [26] proposed a memetic algorithm based onDE (MODEMA) for multiobjective job shop schedulingproblems Qian and li [27] proposed an adaptive DE (ADEA)and the results of five test functions showed that the ADEAwas very efficient to find out the true Pareto front Howeverthe majority of the above studies focused on the effectivenessverification of the algorithms and always analyzed standardtesting functions and ignored the practical applications ofthem in MOPs Due to the existing unpredictable and uncer-tain factors of inventorymanagement it is difficult to convertshortage ratequantity to shortage cost Therefore discussingshortage ratequantity independently is meaningful

The aim of this study is to propose a new multiobjectivestochastic JRD (MSJRD) model including two minimumobjectives that is the total cost and shortage quantity Themain difference between this study and [10] is that twoobjectives are handled simultaneously Moreover effectiveapproaches are provided to handle this MSJRD Having thesuccessful applications in engineering management as well

as the effectiveness of DE in solving MOPs and JRPsJRDs(Wang et al [28ndash30]) linear programming (LP) and MOEA-based approaches using DE are provided Results of anexample show that the proposed hybrid DE is more effectivethan the original DE and GA whatever LP or MOEAmethodis used

The rest of this paper is organized as follows Section 2describes the proposed multiobjective stochastic JRD modelSection 3 introduces the hybrid DE Section 4 presents twoapproaches to solve theMSJRD Section 5 contains numericalexamples and results Section 6 discusses conclusions andprovides future research directions

2 Formulation of the Proposed MSJRD Model

Consider an enterprise has a center warehouse in a properplace and several suppliers in decentralized locations Thecenter warehouse jointly replenishes items from its suppliersaccording to the market demand or historical data Thenthe center warehouse will collect replenished items fromsuppliers In this situation the center warehouse can jointlydetermine the replenishment and distribution policy toobtain the optimal decision

Refering to the model of Qu et al [9] the differencesbetween the JRD policy and typical JRP can be concludedas follows (a) the JRP supposes deterministic demandwhile our model considers stochastic demand and allowsfor shortage (b) the JRP just discusses the replenishmentpolicy while our model analyses not only replenishment butalso transportation decision (c) the JRP is a single objectivemodel while our model is a multiobjective model

For the MSJRD reducing the related total cost as well asdecreasing shortage quantity should be considered simulta-neously Therefore the first target is to minimize total costwhich consists of replenishment cost inventory holding costand distribution costMinimizing the shortage quantity is thesecond target

21The First Target Total Cost The total cost includes inven-tory holding cost replenishment cost and distribution costDistribution cost involves the stopover cost at suppliersand cost related to distance The infinite capacity of vehicleassumption makes the distribution problem a traveling sales-man problem (TSP)

Inventory cost can be given as

119862119867

=

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)] (1)

In (1) the first term is deterministic inventory and thesecond one is safety stock The following replenishment costis the same as the classical JRP

119862119878

=

119878

119879

+

119899

sum

119894=1

119904119894

119896119894119879

=

1

119879

(119878 +

119899

sum

119894=1

119904119894

119896119894

) (2)

When 119896119894are given taking the least common multiple of

119896119894we can get integer 119872 It means that the replenishment and

The Scientific World Journal 3

distribution behavior will be repeated every 119872 period Forexample if 119896

119894= (4 2 1 1) then 119872 = 4 That is to say in

period 1 all items should be replenished in period 2 items3 and 4 need to be replenished in period 3 items 2 3 and4 should be replenished in period 4 items 3 and 4 shouldbe replenished It is obvious that in period 5 the situation isthe same as in period 1 The cycle period is 119872 Therefore alimited horizon with 119872 periods is used to calculate annualdistribution cost

Jointly replenishment of items can take the advantageof scale economies not only in the replenishment but alsoin distribution process Four items specifically should bereplenished in period 1 and it is advisable for the centerwarehouse to traverse suppliers that supply these items atone time The shortest path in each period is consideredso distribution cost can be obtained by solving a travellingsalesman problem (TSP) The distribution cost is

119862119879

=

sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901

+ 119888119889 (119895))

119872119879

(3)

where 119909119901119895

= 1 if supplier 119901 is visited0 otherwise

Therefore the first target can be summarized as follows

1198651

(119879 119896119894 119911119894) = 119862119867

+ 119862119878

+ 119862119879

=

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

(119878 +

119899

sum

119894=1

119904119894

119896119894

)

+

sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901

+ 119888119889 (119895))

119872119879

(4)

22 The Second Target Stock-Out Quantity In this study thedemand is assumed to follow normal distribution at intervalof unit time which is widely used in the literature (seeQu et al [9] Eynan and Kropp [31 32]) This assumptionmeans that given 119896

119894and 119879 for item 119894 the demand will follow

normal distribution over the interval of length 119871 + 119896119894119879 with

the expectation 119864 = 119863119894(119871 + 119896

119894119879) variance Var = 120590

2

119894(119871 + 119896

119894119879)

and probability density function 119891(119909119894 119871 + 119896

119894119879)

With a periodic replenishment policy stockout couldoccur any time during replenishment intervals as long as thereal demand exceeds maximum inventory level 119877

119894 The total

annual stock-out quantity is

1198652

(119879 119896119894 119911119894) =

sum119899

119894=1(int

infin

119877119894

(119909119894minus 119877119894) 119891 (119909

119894 119871 + 119896

119894119879) 119889119909

119894)

119896119894119879

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 int

infin

119911119894

(119910 minus 119911119894) 119891 (119910) 119889119910

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871

times (int

infin

119911119894

119910119891 (119910) 119889119910 minus 119911119894int

infin

119911119894

119891 (119910) 119889119910)

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)])

(5)

where 119877119894= 119863119894(119896119894119879 + 119871) + 119911

119894120590119894radic(119896119894119879 + 119871)

23 The MSJRD Model The whole multiobjective model canbe written as

Min 1198651

(119879 119896119894 119911119894) =

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

(119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

)

Min 1198652

(119879 119896119894 119911119894) =

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871

times (119891 (119911119894) minus 119911119894[1 minus 119865 (119911

119894)])

(6)

where 119892(119896119894) = sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901

+ 119888119889(119895))119872The goal of this multiobjective model is to find out the

optimal 119896119894 119911119894 and119879 to simultaneouslyminimize the total cost

and stock-out quantity and thus to achieve Pareto solutionsin which two objectives can be balanced Two targets havedifferent units of measurements and it is usually difficult toconvert the shortage quantity to stock-out cost In additionthey are often in conflict with each other that is decreasingshortage quantity may result in cost increasing

MOPs are much more complex but closer to realitySeveral traditional mathematic methods are used for solvingmultiobjectivemodels such as linear programming goal pro-gramming and analytic hierarchy process However they aresuccessful only in small scale problemsMathematicmethodsare too complex and too time consuming to solve largescale problems In the following we provide two commonapproaches based on an HDE to deal with the proposedMSJRD Then a numerical example and comparative studybetween the proposed LP and MOEA are presented

3 The Hybrid Differential EvolutionAlgorithm (HDE)

31 The Classical DE DE has been described as an effectiveand robust method to optimize some well-known nonlinearnondifferentiable and nonconvex functions Due to its easyimplementation quick convergence and robustness DE hasturned to be one of the best evolutionary algorithms in avariety of fields (Wang et al [33] Cui et al [34]) DE containsthree operations mutation crossover and selection


311 Mutation Themutation operation creates a new vectorby adding the weighted difference of two random vectors toa third one For each target vector 119909

119866

119905(119905 = 1 2 NP) the

mutated vector is created as follows

V119866+1119905

= 119909119866

1199031+ 119865 times (119909

119866

1199032minus 119909119866

1199033) (7)

In (7) 1199031 1199032 and 119903

3 are three serial numbers of vectors

which are randomly generated with different values and noneof them equals 119905Three vectors119909

119866

11990311199091198661199032 and119909

119866

1199033will be selected

from the population for mutation operation when 1199031 1199032 and

1199033are confirmed 119865 is a scaling factor and 119866 is the current

number of iteration

312 Crossover A trail vector is created by mixing themutated vector with the target vector according to thefollowing formula

119906119866+1

119905119895=

V119866+1119905119895

if 119903119886119899119889119898 (119895) le CR or 119895 = 119903119886119899119889119899 (119905)

119909119866

119905119895 otherwise

(8)

where 119895 represents the 119895th dimension 119903119886119899119889119898(119895) is randomlygenerated from 0 to 1 119903119886119899119889119899(119905) isin [1 2 119863] is a randomlyselected integer to ensure the effect of mutated vector CR isthe crossover probability and it is very important for DE sincethe larger CR is the more V119866+1

119905contributes to 119906

119866+1

119905

313 Selection The selection operation is implemented bycomparing the trial vector (obtained through mutation andcrossover operations) with the corresponding target vectorFor example to minimize the function the next generationis formed by

119909119866+1

119905=

119906119866+1

119905 if 119891 (119906

119866+1

119905) lt 119891 (119909

119866

119905)

119909119866

119905 otherwise

(9)

where the 119891( sdot ) is the fitness function of DEHere an example is given to illustrate the three operations

mentioned previously For the current number of iteration119866 and target vector 119909

119866

1 suppose that random generated

numbers 1199031 1199032 and 119903

3are 23 40 and NP respectively and

we obtain the following

Target kectors 119909119866

Vector x1 6 5 2 4 0090

sdot sdot sdot

Vector 11990923

5 4 2 1 0578

sdot sdot sdot

Vector 11990940

5 1 3 6 0745

sdot sdot sdot

Vector 119909NP 4 2 5 7 0024

(10)

Mutation if 119865 = 06 the mutated vector V119866+11

can beobtained by (7) as follows

Mutated kectors V119866+1

Vector k1 56 34 08 04 1010

sdot sdot sdot

(11)

Crossover if CR = 03 119903119886119899119889119899(119905) = 3 (here 119905 = 1) andvector 119903119886119899119889119898 = (01 04 05 02 06) the trial vector can beobtained by (8) as follows

Trial kectors 119906119866+1

Vector u1 56 5 08 04 0090

sdot sdot sdot

(12)

Selection then target 1199091should be compared with 119906

1

Since 119891(119906119866+1

1) lt 119891(119909

119866

1) vector 119906

1should be selected to the

next generation as follows

Next generation 119909119866+1

Vector 1 56 5 08 04 0090

sdot sdot sdot

(13)

32The ProposedHybrid DE (HDE) The typical DE is simpleand easy to be implemented However it is likely to bepremature too early One-to-one competing is one of themain reasons Therefore improvements including dynamicparameter adjusting different mutation and crossover strate-gies or hybrid algorithms are necessary to be adopted

Similar to DE a genetic algorithm (GA) contains cross-over mutation and selection operations The crossoveroperation of GA is quite complicated and its complexitymay grow rapidly when the problem scale becomes largerFortunately GA has several efficient selection operationssuch as roulette wheel selection tournament selection andtruncation selection In this study an HDE that combines theadvantages of DE and GA is proposed The proposed HDEcan simplify the evolutionary process and it can overcomethe limitation of one-to-one selection of DE and thus preventpremature convergence

Actually several scholars also proposed hybridDEs basedon DE and GA (Hrstka and Kucerova [35] He et al [36]Lin [37]) but their mixing modes are quite different fromours In the proposed HDE the mutation and crossoveroperations are the same as inDEwhile the selection operationis from truncation selection of GA That is to say it will bereserved instead of comparing with the target vector whena trial vector is generated When all trail and target vectorsare determined top NP vectors with better performance areselected to the next generationThe HDE-based procedure isshown in Figure 1


4 Two Methods for Solving MSJRD

41 Linear Programming (LP) Approach for the MSJRD Thismethod is to summarize the weighted targets and thusconverts the multiobjective model to a single one Take intoconsideration that two targets have different measurementsit is necessary to standardize two targets beforehand

411 Model Analysis Using Linear Programming ApproachSuppose that the weights of two objectives are119908

1and119908

2 and

then the multiobjective problem can be described as

Max 120582 = 11990811199061

+ 11990821199062

st

1199061

=

119865max1

(119879 119896119894 119911119894) minus 1198651

(119879 119896119894 119911119894)

119865max1

(119879 119896119894 119911119894) minus 119865

min1

(119879 119896119894 119911119894)

1199062

=

119865max2

(119879 119896119894 119911119894) minus 1198652

(119879 119896119894 119911119894)

119865max2

(119879 119896119894 119911119894) minus 119865

min2

(119879 119896119894 119911119894)

1199081

+ 1199082

= 1

(14)

119865max1

(119879 119896119894 119911119894) 119865

min1

(119879 119896119894 119911119894) 119865

max2

(119879 119896119894 119911119894) and 119865

min2

(119879

119896119894 119911119894) are the tolerant maximum total cost minimum total

cost maximum stock-out quantity and minimum stock-outquantity respectively which can be given in advance bydecision makers (Wee et al [18])

Set 1199081015840

1= 1199081(119865

max1

minus 119865min1

) 11990810158402

= 1199082(119865

max2

minus 119865min2

) Thenthe objective function is changed to

Max 120582 = 119865119888

minus (1199081015840

11198651

+ 1199081015840

21198652) (15)

where 119865119888

= 1199081119865max1

(119865max1

minus 119865min1

) + 1199082119865max2

(119865max2

minus 119865min2

)

and

119865119908

= 1199081015840

11198651

+ 1199081015840

21198652

= 1199081015840

1(

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)]))

(16)

Let 120597119865119908

120597119911119894

= 0 that is 1199081015840

1ℎ119894120590119894radic119896119894119879 + 119871 + 119908

1015840

2(120590119894

119896119894119879)radic119896

119894119879 + 119871((119889119891(119911

119894)119889119911119894) minus [1 minus 119865(119911

119894)] + 119911119894119891(119911119894)) = 0

Note that for standard normal distribution 119889119891(119911119894)119889119911119894

=

minus119911119894119891(119911119894) so that

1 minus 119865 [119911119894(119896119894119879)] =

1199081015840

1ℎ119894

1199081015840

2

119896119894119879 (17)

That is to say when 119896119894and 119879 is known the optimal value

of 119911119894must satisfy (17)

Taking the second derivation of 119879119862(119879 119896119894 119911119894) with

respect to 119911119894 we obtain 120597

2119879119862(119879 119896

119894 119911119894)1205971199112

119894= (119908

1015840

2120590119894

119896119894119879)radic119896

119894119879 + 119871119891(119911

119894) gt 0 which means that the optimal 119911

119894is

derived from (17) and 119911lowast

119894(119896119894119879) = 119865

minus1(1 minus (119908

1015840

1ℎ1198941199081015840

2)119896119894119879)

Substituting 119911lowast

119894(119896119894119879) into (16) the optimal value of 119865

119908for

given 119896119894and 119879 is

119865119908

= 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894))

(18)

Finally the linear programming model can be written as

Max 120582 (119879 119896119894)

= 119865119888

minus 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

minus 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894

))

st

119865119888

=

1199081119865max1

119865max1

minus 119865min1

+

1199082119865max2

119865max2

minus 119865min2

1199081015840

1=

1199081

119865max1

minus 119865min1

1199081015840

2=

1199082

119865max2

minus 119865min2

1199081

+ 1199082

= 1

119911lowast

119894

(119896119894119879) = 119865

minus1(1 minus

1199081015840

1ℎ119894

1199081015840

2

119896119894119879)

(19)

The goal is to determine the best 119896119894and 119879 to maximize 120582

for the given 1199081and 119908

1

412 HDE-Based Procedures for MSJRD Using LP ApproachWhen 119896

119894are determined the optimal delivery cost can be

calculated by solving a TSP 119911lowast

119894(119896119894119879) can be calculated by

119911lowast

119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879) when 119896

119894and 119879 are known

Change 119896119894and 119879 with the following steps until maximum 120582 is

obtained

Step 1 Initialization set related parameters (CR F and NP)for HDE Set the lower bound (119896LB

119894) and the upper bound

(119896UB119894) of 119896

119894respectively Note that 119896

119894are integers so 119896

LB119894

isobviously 1 According to experience of [2 10 15] 119896

UB119894

is setsufficiently large to guarantee that the optimal solution doesnot escape In this study it can be set to 100 119879 is randomlygenerated in the range of 0 and 1 Combining 119896

119894and 119879 we

get the 119905th individual 119909119905

= (119896119894 119879) Create initial population

randomly

Step 2 For a given1199081 calculate the objective functionWhen

119909119905is determined 119911lowast

119894(119896119894119879) can be derived accordingly 119909

119905and

119911lowast

119894(119896119894119879) jointly determine 120582


Start

Given parametersNP FCR and GenM

Initialization create individualsrandomly and set G = 1

G gt GenM Yes

No

Mutation a mutant individual can begenerated by (7)

Crossover the trial individual can beproduced by (8)

Selection only the individualswith better performance will beselected by truncation selection

G = G + 1

Output optimal results

Stop

NP

NP

Figure 1 Flow chart of HDE

Step 3 Differential operations while stopping criterion is notmet implement mutation and crossover for each individualAfter that the number of population is two times the originalone

Step 4 Genetic operations select the individuals accordingto120582Thosewith larger120582will be chosen to the next generationThen the number of population is the same as the originalone

Step 5 When the number of iteration reaches a predefinedmaximum number output the optimal results otherwiserepeat Steps 2ndash4

42 Multiobjective Evolution Algorithm (MOEA) Approach forthe MSJRD In this section a brief introduction of MOP isgiven Then an HDE-based procedure to handle the MSJRDusing noninferior and crowding distance is designed

421 Some Definitions of MOP

Definition 1 (multiobjective optimization problems (MOP))

Min 119865 (119909) = 1198911 (

119909) 1198912 (

119909) 119891119896 (

119909)

Subject to 119892119894 (

119909) le 0 119894 = 1 2 119898

(20)

A general MOP consists of 119899 decision variables 119896 objec-tive functions and 119898 constrains In Definition 1 119909 refers tothe decision space and 119892

119894(119909) are constrains of MOP

Definition 2 (Pareto optimal solution) The optimal solutionofMOP is often referred to as the Pareto optimal solution Letvector 119886 belong to 119909 and suppose that 119909

lowast is a subset of 119909 Ifthere does not exist any vector in 119909

lowast that is better than 119886 then119886 is called noninferior solution (or Pareto optimal solution)of 119909lowast Moreover if vector 119886 is the noninferior solution of 119909

then vector 119886 is the Pareto optimal of the MOP


Step 1 For each 119909119894isin 119868 initialize distance 119863

(119894)= 0

Step 2 Calculate the crowding distance of 119909119894

Step 21 Sorting 119868 for each objective 119898 119868 = sort(119868 119898)

119863(119868(1)

) = 119863(119868(1)

) + [2(119868(2)sdot119898

minus 119868(1)sdot119898

)]2

119863(119868119868) = 119863(119868

119868) + [2(119868

119868sdot119898minus 119868119868minus1sdot119898

)]2

For 119894 = 2 |119868| minus 1

119863(119868(119894)

) = 119863(119868(119894)

) + [(119868(119894+1)sdot119898

minus 119868(119894minus1)sdot119898

)]2

Step 22119863(119894)

= 119904119902119903119905(119863(119894)

)

where |119868| is the number of individuals contained in 119868 119868(119894)sdot119898

is the objective value of individual 119894

Algorithm 1 Steps of calculating crowding distance

422 HDE-Based Procedures for MSJRD Using MOEAApproach There exist many difficulties when applying DEto solve an MOP compared with single objective problemThe main challenges for solving MOP are as follows howto generate offspring and how to keep Pareto solutionsuniformly distributed The classical DE is not suitable foran MOP since many good solutions may be abandoned dueto its one-to-one competing mechanism This will also beconfirmed by a numerical example

Therefore we also use an HDE which uses truncationselection to choose next generation based on front rank andcrowding distance adopted by Qian and li [27] The steps ofcalculating crowding distance are presented in Algorithm 1

In this algorithm the low front rank corresponds to thehigh quality of a solution As to the those individuals with thesame front rank the larger crowding distance means betterdistributionTherefore individuals with lower front rank andlarger crowding distance are selected to the next generation

The first target can be divided into an inventory problemand a delivery problem When all 119896

119894are determined the

optimal delivery cost can be calculated by solving a TSPIn addition for a stochastic JRP with normal distributeddemand when 119896

119894 119911119894 and 119879 are known the stochastic JRP

can then be solved With the same value of 119896119894 119911119894 and 119879 in

the second target we can obtain the corresponding value ofthe second targetThen change 119896

119894 119911119894 and119879with the following

steps until the termination condition is satisfied The steps ofHDE-based approach are described as follows

Step 1 Initialization set related parameters (CR F and NP)for the HDE Set the lower bound and the upper bound of 119896

119894

respectively that is 119896LB119894

= 1 and 119896UB119894119895

= 100 119911119894is randomly

generated in the range of 0 and 3 which can cover 997 ofthe demand 119879 is randomly generated in the range of 0 and1 Combining 119896

119894 119911119894 and 119879 we get the 119905th individual 119909

119905=

(119896119894 119911119894 119879) Create initial population randomly

Step 2 Calculate the objective function that is the total costand the total shortage quantity of all items

Step 3 Calculate the Pareto front and crowding distance ofeach individual

Step 4 Differential operations while stopping criterion is notmet implement mutation and crossover for each individual

Table 1 Supply relationship between items and suppliers

Supplier 1 Supplier 2 Supplier 3Item 1 1 0 0Item 2 0 1 0Item 3 0 0 1Item 4 0 0 1119865119901

40 50 60

Table 2 Parameters of items

Item 1 Item 2 Item 3 Item 4119863 (unityear) 600 900 1200 10001205902 (unit2year) 800 600 700 500

119871 (year) 002 002 002 002ℎ ($unityear) 56 21 42 15119904 ($order) 25 14 20 30120587 ($unit) 28 35 40 30The other two parameters are as follows 119878 = 100 and 119888 = 05

After that the number of population is two times the originalone

Step 5 Genetic operations select the individuals accordingto the front rank and crowding distance Then the number ofpopulation is the same as the original one


5 Contrastive Example and Results Analysis

51 Basic Data of Numerical Example The data come fromQu et al [9] Table 1 describes the items to be replenished andthe center warehouse correspondingly Tables 2 and 3 are therelated parameters of items and distances between suppliersand warehouse respectively

In the following two approaches named LP and MOEAare compared The comparison contains two aspects thePareto solutions and some specific solutions obtained by eachmethod In the meanwhile three algorithms used in each


Table 3 Distances between suppliers and warehouse

Warehouse Supplier 1 Supplier 2 Supplier 3Warehouse 0 11 9 7Supplier 1 11 0 5 8Supplier 2 9 5 0 10Supplier 3 7 8 10 0

Table 4 Parameters of the algorithms

Parameter Value AlgorithmsNumber of population NP 200 HDE DE and GAMaximum number of iteration GenM 300 HDE DE and GAMutation factor 119865 06 HDE and DECrossover rate CR 03 HDE and DEProbability of mutation 119875

11989801 GA

Probability of crossover 119875119888

09 GA

Table 5 Results for LP with HDE DE and GA (1199081

= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536

method are comparedwith each other Table 4 reports relatedparameters of HDE DE and GA

For LP-based approach we directly set 119865max1

(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2

(119879 119896119894 119911119894) = 0 according to the advice of the decision

makers This approach is also widely used by other scholars(Wee et al [18])

52 Comparisons for LP-Based and MOEA-Based SolutionsIn this section the above numerical example is handled usingLP and MOEA For LP the weight of each objective mustbe assigned firstly In order to compare with MOEA theobjectives can be converted to single index by setting thetotal cost and total shortage quantity with the same weightsfor MOEA when the Pareto solutions are obtained The bestresults for LP when 119908

1= 056 are presented in Table 5 As

to MOEA the highest index after converting is shown inTable 6

Table 5 shows thatHDEandDEare better thanGA for LPTable 6 implies that HDE is better than GA and GA is betterthan DE forMOEA In order to further verify the conclusionwe obtained for different 119908

1 1199081is set from 01 to 09 and the

results are reported in Table 7Set the total cost and total shortage quantitywith the same

weights 1199081and 119908

2 respectively for MOEA Then solutions

with the highest weighted objective from the obtained Paretosolutions are shown in Table 8

Table 6 Results for MOEA with HDE DE and GA (1199081

= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543

Table 7 shows that values of each target 1198651and 119865

2change

correspondently when 1199081varies which means that different

settings of the weight will result in different decisionsMoreover when the weight of the first objective (total cost)equals 01 the solution is the best When the differencebetween 119908

1and 119908

2becomes smaller the solutions become

worse Table 8 implies a similar conclusion for HDE DE andGA

From the comparisons for specific solutions the follow-ing conclusion can be easily drawn (1) HDE is better thanDEor GA no matter whether LP or MOEA is adopted HDE andDE aremore suitable than GAwhen LP is used HDE andGAare better than DEwhenMOEA is used (2) Different weightsfor objectives will influence the solutions for the conflictedobjectives and the assigned weights with large ratios (ie1199081

1199082

ge 3 1) may result in better solutions

53 Nondominated Solution Analysis ofMOEA For theMOPthere have several metrics to evaluate the quality of thenondominated solutions (Robic and Filipic [38]) Howeverthe implementation ofmostmetrics needs a prerequisite thatis the true Pareto front must be known In this study it isimpossible to find the true Pareto front because theMSJRD isa practical problem Sowe adopt themetric (Spacing SP) usedby Esparcia-Alcazar et al [39 40] to measure the distributionof solutions on the Pareto front by evaluating the varianceof neighboring solutions The lower value of SP means thatbetter nondominated solution is obtained

SP measures the relative distances between the membersof Pareto front as

SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)

where 119899 is the number of the first nondominated solutionsfound The distance 119889

119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896

119873(sdot) is the fitness of point 119896 on objective 119873 and 119889 is

the mean of all 119889119894 Table 9 shows the mean and variance of SP

by 10 runs using MOEATable 9 shows that the mean and variance of SP obtained

by HDE is the lowest and corresponding values obtained byDE are biggest That is to say HDE is better than DE and


Table 7 Results for LP with HDE DE and GA (1199081varies)

1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439

Table 8 Results for MOEA with HDE DE and GA (1199081varies)

1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389

Table 9 Statistical analysis of SP by 10 runs

Mean VarianceHDE 875 193DE 1748 1048GA 1215 396

GA for theMOEAmethodThe conclusion is consistent withSection 52 At the same time it verifies that the conversionusing weights for the MSJRD is feasible

In order to have a better understanding of the solutionsrsquodistribution of the last generation the entire nondominatedfronts found by HDE DE and GA are presented fromFigure 2 to Figure 4

Figures 2 3 and 4 show that HDE and GA are capa-ble of obtaining Pareto solutions while the effectivenessand distribution of solutions are much worse for DE Themain reason is the one-to-one competing of DE Whentrial individual dominates target individual the trial vector(otherwise the target vector) remains to the next generationThis mechanism is different from the selection operation ofthe HDE where all the target and trial individuals are kept tobe chosen according to the front rank and crowding distanceSo DE is not suitable for solving this MSJRD

Results obtained by HDE for MOEA

Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120

Total shortage quantity

Figure 2 Nondominated solutions of the final population obtainedby HDE

6 Conclusions and Future Research

In this study a newmultiobjective JRDmodel with stochasticdemand is proposed which takes into account the servicelevel while making the replenishment and delivery decisionsThen two approaches to solve this complex optimizationproblem are designed using an improved HDE The maincontributions are as follows

(1) Considering the difficulty to estimate the shortagecost in reality the shortage quantity is utilized as


Results obtained by DE for MOEA

Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35

Figure 3 Nondominated solutions of the final population obtainedby DE

0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500

Results obtained by GA for MOEA

Figure 4 Nondominated solutions of the final population obtainedby GA

another standard to evaluate the rationality of deci-sions besides the total cost To our best knowledgethis is the first time to propose a practical multiobjec-tive stochastic JRD model

(2) TheMOEA is adopted to solve the proposedmultiob-jective JRDmodelThe results of the numerical exam-ple and Pareto solution analysis show the feasibility oftheMOEA tohandle the proposedMSJRD It enrichesthe application field of the MOEA

(3) The comparison of two approaches for the MSJRDverifies that LP and MOEA are suitable for solvingthis MSJRD problem Furthermore results show thatthe proposed HDE is more effective than DE and GAwhatever LPorMOEAmethod is usedThis illustratesthat DE is likely to combine with other algorithms soas to provide a more effective way to solve complexproblems

The future research on the multiobjective JRD problemshould considermore realistic assumptions such as uncertaincosts freight consolidation and budget constraint

Abbreviations

119863119894 Annual demand rate of item 119894

119878 Major ordering cost119904119894 Minor ordering cost of item 119894

ℎ119894 Annual inventory holding cost of item 119894

119879 Basic cycle time (decision variable)119896119894 Multiplier of 119879 (decision variable)

119879119894 Cycle time of item 119894 (decision variable)

119877119894 Maximum inventory level of item 119894 during

replenishment interval119911119894 Safety stock factor of item 119894 (decision

variable)119871 Lead time of items120590119894 Standard deviation of item 119894

119872 Least common multiple of 119896119894

120587119894 Stock-out cost per unit for item 119894

119888 Distribution cost per unit distance119865119901 Stopover cost at supplier 119901

119889(119895) Shortest path needed for distribution in119895th basic cycle

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (70801030 71101060 7113100471371080) andHumanities and Social Sciences Foundation ofthe Chinese Ministry of Education (no 11YJC630275)

References

[1] S K Goyal ldquoDetermination of optimum packaging frequencyof items jointly replenishedrdquoManagement Science vol 21 no 4pp 436ndash443 1974

[2] 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 9ndash12 pp 1907ndash1920 2013

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

[4] A Narayanan P Robinson and F Sahin ldquoCoordinated deter-ministic dynamic demand lot-sizing problem a review ofmodels and algorithmsrdquo Omega vol 37 no 1 pp 3ndash15 2009

[5] J-M Chen and T-H Chen ldquoThe multi-item replenishmentproblem in a two-echelon supply chain the effect of centraliza-tion versus decentralizationrdquoComputersampOperations Researchvol 32 no 12 pp 3191ndash3207 2005

[6] S Axsater J Marklund and E A Silver ldquoHeuristic methodsfor centralized control of one-warehouse N-retailer inventorysystemsrdquo Manufacturing amp Service Operations Managementvol 4 no 1 pp 75ndash97 2002

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

[8] B Abdul-Jalbar A Segerstedt J Sicilia and A Nilsson ldquoAnew heuristic to solve the one-warehouse N-retailer problemrdquoComputers amp Operations Research vol 37 no 2 pp 265ndash2722010

[9] W W Qu J H Bookbinder and P Iyogun ldquoAn integratedinventory-transportation system with modified periodic pol-icy for multiple productsrdquo European Journal of OperationalResearch vol 115 no 2 pp 254ndash269 1999


[10] 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

[11] S Sindhuchao H E Romeijn E Akcali and R Boondiskul-chok ldquoAn integrated inventory-routing system for multi-itemjoint replenishment with limited vehicle capacityrdquo Journal ofGlobal Optimization vol 32 no 1 pp 93ndash118 2005

[12] C K Chan L Y-O Li C T Ng B K-S Cheung and ALangevin ldquoScheduling of multi-buyer joint replenishmentsrdquoInternational Journal of Production Economics vol 102 no 1 pp132ndash142 2006

[13] B C Cha I K Moon and J H Park ldquoThe joint replenish-ment and delivery scheduling of the one-warehouse n-retailersystemrdquo Transportation Research E vol 44 no 5 pp 720ndash7302008

[14] I K Moon B C Cha and C U Lee ldquoThe joint replenishmentand freight consolidation of a warehouse in a supply chainrdquoInternational Journal of Production Economics vol 133 no 1 pp344ndash350 2011

[15] L Wang H Qu Y H Li and J He ldquoModeling and optimiza-tion of stochastic joint replenishment and delivery schedulingproblemwith uncertain costsrdquoDiscrete Dynamics in Nature andSociety vol 2013 Article ID 657465 12 pages 2013

[16] T K Roy and M Maiti ldquoMulti-objective inventory models ofdeteriorating items with some constraints in a fuzzy environ-mentrdquo Computers amp Operations Research vol 25 no 12 pp1085ndash1095 1998

[17] M Rong N K Mahapatra and M Maiti ldquoA multi-objectivewholesaler-retailers inventory-distributionmodel with control-lable lead-time based on probabilistic fuzzy set and triangularfuzzy numberrdquo Applied Mathematical Modelling vol 32 no 12pp 2670ndash2685 2008

[18] S Islam ldquoMulti-objectivemarketing planning inventorymodela geometric programming approachrdquo Applied Mathematics andComputation vol 205 no 1 pp 238ndash246 2008

[19] H-M Wee C-C Lo and P-H Hsu ldquoA multi-objective jointreplenishment inventory model of deteriorated items in a fuzzyenvironmentrdquo European Journal of Operational Research vol197 no 2 pp 620ndash631 2009

[20] E Arkin D Joneja and R Roundy ldquoComputational complexityof uncapacitatedmulti-echelonproduction planning problemsrdquoOperations Research Letters vol 8 no 2 pp 61ndash66 1989

[21] G Aiello G La Scalia and M Enea ldquoA non dominatedranking multi-objective genetic algorithm and electre methodfor unequal area facility layout problemsrdquo Expert Systems withApplications vol 40 no 12 pp 4812ndash4819 2013

[22] P C Lin ldquoPortfolio optimization and risk measurement basedon non-dominated sorting genetic algorithmrdquo Journal of Indus-trial and Management Optimization vol 8 no 3 pp 549ndash5642012

[23] E Zitzler and L Thiele ldquoMultiobjective evolutionary algo-rithms a comparative case study and the strength Paretoapproachrdquo IEEETransactions onEvolutionaryComputation vol3 no 4 pp 257ndash271 1999

[24] W X Sheng Y M Liu X L Meng and T S ZhangldquoAn improved strength pareto evolutionary algorithm 2 withapplication to the optimization of distributed generationsrdquoComputers amp Mathematics with Applications vol 64 no 5 pp944ndash955 2012

[25] L V Santana-Quintero and C A C Coello ldquoAn algorithmbased on differential evolution for multi-objective problemsrdquo

International Journal of Computational Intelligence Researchvol 1 no 2 pp 151ndash169 2005

[26] B Qian L Wang D-X Huang and X Wang ldquoSchedulingmulti-objective job shops using a memetic algorithm basedon differential evolutionrdquo International Journal of AdvancedManufacturing Technology vol 35 no 9-10 pp 1014ndash1027 2008

[27] W Qian and A li ldquoAdaptive differential evolution algorithm formulti-objective optimization problemsrdquo Applied Mathematicsand Computation vol 201 no 1-2 pp 431ndash440 2008

[28] L Wang J He D S 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

[29] 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

[30] L Wang Q L Fu C G Lee and Y R Zeng ldquoModeland algorithm of fuzzy joint replenishment problem undercredibility measure on fuzzy goalrdquo Knowledge-Based Systemsvol 39 pp 57ndash66 2013

[31] A Eynan and D H Kropp ldquoPeriodic review and joint replen-ishment in stochastic demand environmentsrdquo IIE Transactionsvol 30 no 11 pp 1025ndash1033 1998

[32] A Eynan and D H Kropp ldquoEffective and simple EOQ-like solutions for stochastic demand periodic review systemsrdquoEuropean Journal of Operational Research vol 180 no 3 pp1135ndash1143 2007

[33] 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

[34] L G Cui L Wang and J Deng ldquoRFID technology investmentevaluation model for the stochastic joint replenishment anddelivery problemrdquo Expert Systems with Applications 2013

[35] O Hrstka and A Kucerova ldquoImprovements of real codedgenetic algorithms based on differential operators preventingpremature convergencerdquo Advances in Engineering Software vol35 no 3-4 pp 237ndash246 2004

[36] D He F Wang and Z Mao ldquoA hybrid genetic algorithmapproach based on differential evolution for economic dispatchwith valve-point effectrdquo International Journal of Electrical Powerand Energy Systems vol 30 no 1 pp 31ndash38 2008

[37] W-Y Lin ldquoA GA-DE hybrid evolutionary algorithm for pathsynthesis of four-bar linkagerdquoMechanism and Machine Theoryvol 45 no 8 pp 1096ndash1107 2010

[38] T Robic and B Filipic ldquoDEMO differential evolution for mul-tiobjective optimizationrdquo Lecture Notes in Computer Sciencepp 520ndash533

[39] A I Esparcia-Alcazar J J Merelo A Martınez-Garcıa PGarcıa-Sanchez E Alfaro-Cid and K Sharman ldquoCompar-ing single andmultiobjective evolutionary approaches to theinventory and transportation problemrdquo CoRR httparxivorgabs09093384

[40] A I Esparcia-Alcazar E Alfaro-Cid P Garcıa-Sanchez AMartınez-Garcıa J JMerelo andK Sharman ldquoAn evolutionaryapproach to integrated inventory and routing management ina real world caserdquo in Proceedings of the IEEE Congress onEvolutionary Computation (CEC rsquo10) pp 1ndash7 Barcelona SpainJuly 2010

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However managers are usually faced with complex mul-tiobjective optimization problems (MOPs) in reality For theJRD policy it is necessary to decrease the total cost whileimproving the service level Although there are several papersthat studied multiobjective inventory models (Roy and Maiti[16] Rong et al [17] Islam [18] Wee et al [19]) no study onthe multiobjective JRD can be found

ForMOPs direct comparison among the solutions is verydifficult because of the different measurements between eachcontradicted target In this study total cost and service levelare obviously two contradictory targets reducing total costmay result in the decline of service level and vice verse Sowe should coordinate two targets Different from a single-objective optimization problem which has unique optimalsolution an MOP has a set of optimal solutions called Paretooptimal solutions Due to the characteristics of MOPs theyare much more complex and the key is to find an effectivemethod to obtain Pareto optimal solutions Unfortunatelythe classical JRPs and JRDs are already NP hard problems(Arkin et al [20]) and the multiobjective makes the JRDsbecome much more difficult to handle

Many linear or nonlinear weighted methods (Rong et al[17] Islam [18] Wee et al [19] Roy and Maiti [16]) were usedto convert the multiobjective to a single one in the existingstudies These methods undoubtedly provide one easy wayto deal with the multiobjective JRD model However theseapproaches do not solve the MOPs intrinsically since thesolutions for MOPs are multiple rather than one On theother hand multiobjective optimization methods based onPareto-basedMOEAs are widely used such as multiobjectivegenetic algorithm (MOGA) (Aiello et al [21]) nondominatedsorting genetic algorithm (NSGA) (Lin [22]) and strengthPareto evolutionary algorithm (SPEA) (Zitzler and Thiele[23] Sheng et al [24])

In recent years several MOEAs based on Pareto differ-ential evolution (DE) were utilized to solve MOPs Santana-Quintero and Coello [25] presented a DE-based multiob-jective algorithm using a secondary population and theconcept of 120598-dominance The performance of the proposedalgorithm was also compared with NSGA-II and 120598-MOEAQian et al [26] proposed a memetic algorithm based onDE (MODEMA) for multiobjective job shop schedulingproblems Qian and li [27] proposed an adaptive DE (ADEA)and the results of five test functions showed that the ADEAwas very efficient to find out the true Pareto front Howeverthe majority of the above studies focused on the effectivenessverification of the algorithms and always analyzed standardtesting functions and ignored the practical applications ofthem in MOPs Due to the existing unpredictable and uncer-tain factors of inventorymanagement it is difficult to convertshortage ratequantity to shortage cost Therefore discussingshortage ratequantity independently is meaningful

The aim of this study is to propose a new multiobjectivestochastic JRD (MSJRD) model including two minimumobjectives that is the total cost and shortage quantity Themain difference between this study and [10] is that twoobjectives are handled simultaneously Moreover effectiveapproaches are provided to handle this MSJRD Having thesuccessful applications in engineering management as well

as the effectiveness of DE in solving MOPs and JRPsJRDs(Wang et al [28ndash30]) linear programming (LP) and MOEA-based approaches using DE are provided Results of anexample show that the proposed hybrid DE is more effectivethan the original DE and GA whatever LP or MOEAmethodis used

The rest of this paper is organized as follows Section 2describes the proposed multiobjective stochastic JRD modelSection 3 introduces the hybrid DE Section 4 presents twoapproaches to solve theMSJRD Section 5 contains numericalexamples and results Section 6 discusses conclusions andprovides future research directions

2 Formulation of the Proposed MSJRD Model

Consider an enterprise has a center warehouse in a properplace and several suppliers in decentralized locations Thecenter warehouse jointly replenishes items from its suppliersaccording to the market demand or historical data Thenthe center warehouse will collect replenished items fromsuppliers In this situation the center warehouse can jointlydetermine the replenishment and distribution policy toobtain the optimal decision

Refering to the model of Qu et al [9] the differencesbetween the JRD policy and typical JRP can be concludedas follows (a) the JRP supposes deterministic demandwhile our model considers stochastic demand and allowsfor shortage (b) the JRP just discusses the replenishmentpolicy while our model analyses not only replenishment butalso transportation decision (c) the JRP is a single objectivemodel while our model is a multiobjective model

For the MSJRD reducing the related total cost as well asdecreasing shortage quantity should be considered simulta-neously Therefore the first target is to minimize total costwhich consists of replenishment cost inventory holding costand distribution costMinimizing the shortage quantity is thesecond target

21The First Target Total Cost The total cost includes inven-tory holding cost replenishment cost and distribution costDistribution cost involves the stopover cost at suppliersand cost related to distance The infinite capacity of vehicleassumption makes the distribution problem a traveling sales-man problem (TSP)

Inventory cost can be given as

119862119867

=

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)] (1)

In (1) the first term is deterministic inventory and thesecond one is safety stock The following replenishment costis the same as the classical JRP

119862119878

=

119878

119879

+

119899

sum

119894=1

119904119894

119896119894119879

=

1

119879

(119878 +

119899

sum

119894=1

119904119894

119896119894

) (2)

When 119896119894are given taking the least common multiple of

119896119894we can get integer 119872 It means that the replenishment and






119862119879

=

sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901

+ 119888119889 (119895))

119872119879

(3)

where 119909119901119895



1198651

(119879 119896119894 119911119894) = 119862119867

+ 119862119878

+ 119862119879

=

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

(119878 +

119899

sum

119894=1

119904119894

119896119894

)

+

sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901

+ 119888119889 (119895))

119872119879

(4)




the expectation 119864 = 119863119894(119871 + 119896

119894119879) variance Var = 120590

2

119894(119871 + 119896

119894119879)


119894119879)


119894 The total


1198652

(119879 119896119894 119911119894) =

sum119899

119894=1(int

infin

119877119894

(119909119894minus 119877119894) 119891 (119909

119894 119871 + 119896

119894119879) 119889119909

119894)

119896119894119879

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 int

infin

119911119894

(119910 minus 119911119894) 119891 (119910) 119889119910

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871

times (int

infin

119911119894

119910119891 (119910) 119889119910 minus 119911119894int

infin

119911119894

119891 (119910) 119889119910)

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)])

(5)

where 119877119894= 119863119894(119896119894119879 + 119871) + 119911

119894120590119894radic(119896119894119879 + 119871)


Min 1198651

(119879 119896119894 119911119894) =

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

(119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

)

Min 1198652

(119879 119896119894 119911119894) =

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871


119894)])

(6)

where 119892(119896119894) = sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901









119866

119905(119905 = 1 2 NP) the


V119866+1119905

= 119909119866

1199031+ 119865 times (119909

119866

1199032minus 119909119866

1199033) (7)

In (7) 1199031 1199032 and 119903



119866

11990311199091198661199032 and119909

119866




number of iteration


119906119866+1

119905119895=

V119866+1119905119895

if 119903119886119899119889119898 (119895) le CR or 119895 = 119903119886119899119889119899 (119905)

119909119866


(8)



119866+1

119905


119909119866+1

119905=

119906119866+1

119905 if 119891 (119906

119866+1

119905) lt 119891 (119909

119866

119905)

119909119866

119905 otherwise

(9)



119866


numbers 1199031 1199032 and 119903




Vector x1 6 5 2 4 0090

sdot sdot sdot

Vector 11990923

5 4 2 1 0578

sdot sdot sdot

Vector 11990940

5 1 3 6 0745

sdot sdot sdot

Vector 119909NP 4 2 5 7 0024

(10)




Vector k1 56 34 08 04 1010

sdot sdot sdot

(11)



Vector u1 56 5 08 04 0090

sdot sdot sdot

(12)


1

Since 119891(119906119866+1

1) lt 119891(119909

119866

1) vector 119906




Vector 1 56 5 08 04 0090

sdot sdot sdot

(13)








1and119908

2 and


Max 120582 = 11990811199061

+ 11990821199062

st

1199061

=

119865max1

(119879 119896119894 119911119894) minus 1198651

(119879 119896119894 119911119894)

119865max1

(119879 119896119894 119911119894) minus 119865

min1

(119879 119896119894 119911119894)

1199062

=

119865max2

(119879 119896119894 119911119894) minus 1198652

(119879 119896119894 119911119894)

119865max2

(119879 119896119894 119911119894) minus 119865

min2

(119879 119896119894 119911119894)

1199081

+ 1199082

= 1

(14)

119865max1

(119879 119896119894 119911119894) 119865

min1

(119879 119896119894 119911119894) 119865

max2

(119879 119896119894 119911119894) and 119865

min2

(119879



Set 1199081015840

1= 1199081(119865

max1

minus 119865min1

) 11990810158402

= 1199082(119865

max2

minus 119865min2


Max 120582 = 119865119888

minus (1199081015840

11198651

+ 1199081015840

21198652) (15)

where 119865119888

= 1199081119865max1

(119865max1

minus 119865min1

) + 1199082119865max2

(119865max2

minus 119865min2

)

and

119865119908

= 1199081015840

11198651

+ 1199081015840

21198652

= 1199081015840

1(

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)]))

(16)

Let 120597119865119908

120597119911119894

= 0 that is 1199081015840

1ℎ119894120590119894radic119896119894119879 + 119871 + 119908

1015840

2(120590119894

119896119894119879)radic119896

119894119879 + 119871((119889119891(119911

119894)119889119911119894) minus [1 minus 119865(119911

119894)] + 119911119894119891(119911119894)) = 0


=

minus119911119894119891(119911119894) so that

1 minus 119865 [119911119894(119896119894119879)] =

1199081015840

1ℎ119894

1199081015840

2

119896119894119879 (17)





2119879119862(119879 119896

119894 119911119894)1205971199112

119894= (119908

1015840

2120590119894

119896119894119879)radic119896

119894119879 + 119871119891(119911


119894is


119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879)



119908for

given 119896119894and 119879 is

119865119908

= 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894))

(18)


Max 120582 (119879 119896119894)

= 119865119888

minus 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

minus 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894

))

st

119865119888

=

1199081119865max1

119865max1

minus 119865min1

+

1199082119865max2

119865max2

minus 119865min2

1199081015840

1=

1199081

119865max1

minus 119865min1

1199081015840

2=

1199082

119865max2

minus 119865min2

1199081

+ 1199082

= 1

119911lowast

119894

(119896119894119879) = 119865

minus1(1 minus

1199081015840

1ℎ119894

1199081015840

2

119896119894119879)

(19)



1





119911lowast

119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879) when 119896



obtained



(119896UB119894) of 119896



LB119894


UB119894


119894and 119879 we



randomly




119905and

119911lowast



Start



G gt GenM Yes

No




G = G + 1


Stop

NP

NP








Min 119865 (119909) = 1198911 (

119909) 1198912 (

119909) 119891119896 (

119909)

Subject to 119892119894 (

119909) le 0 119894 = 1 2 119898

(20)









(119894)= 0



119863(119868(1)

) = 119863(119868(1)

) + [2(119868(2)sdot119898

minus 119868(1)sdot119898

)]2

119863(119868119868) = 119863(119868

119868) + [2(119868


)]2

For 119894 = 2 |119868| minus 1

119863(119868(119894)

) = 119863(119868(119894)

) + [(119868(119894+1)sdot119898

minus 119868(119894minus1)sdot119898

)]2

Step 22119863(119894)

= 119904119902119903119905(119863(119894)

)
















119894


= 1 and 119896UB119894119895

= 100 119911119894is randomly



119905=







40 50 60















11989801 GA


09 GA


= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536



(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2













= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543


2change



1and 119908




1199082




SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)


119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896







1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








119862119879

=

sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901

+ 119888119889 (119895))

119872119879

(3)

where 119909119901119895



1198651

(119879 119896119894 119911119894) = 119862119867

+ 119862119878

+ 119862119879

=

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

(119878 +

119899

sum

119894=1

119904119894

119896119894

)

+

sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901

+ 119888119889 (119895))

119872119879

(4)




the expectation 119864 = 119863119894(119871 + 119896

119894119879) variance Var = 120590

2

119894(119871 + 119896

119894119879)


119894119879)


119894 The total


1198652

(119879 119896119894 119911119894) =

sum119899

119894=1(int

infin

119877119894

(119909119894minus 119877119894) 119891 (119909

119894 119871 + 119896

119894119879) 119889119909

119894)

119896119894119879

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 int

infin

119911119894

(119910 minus 119911119894) 119891 (119910) 119889119910

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871

times (int

infin

119911119894

119910119891 (119910) 119889119910 minus 119911119894int

infin

119911119894

119891 (119910) 119889119910)

=

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)])

(5)

where 119877119894= 119863119894(119896119894119879 + 119871) + 119911

119894120590119894radic(119896119894119879 + 119871)


Min 1198651

(119879 119896119894 119911119894) =

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

(119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

)

Min 1198652

(119879 119896119894 119911119894) =

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871


119894)])

(6)

where 119892(119896119894) = sum119872

119895=1(sum119875

119901=1119909119901119895

119865119901









119866

119905(119905 = 1 2 NP) the


V119866+1119905

= 119909119866

1199031+ 119865 times (119909

119866

1199032minus 119909119866

1199033) (7)

In (7) 1199031 1199032 and 119903



119866

11990311199091198661199032 and119909

119866




number of iteration


119906119866+1

119905119895=

V119866+1119905119895

if 119903119886119899119889119898 (119895) le CR or 119895 = 119903119886119899119889119899 (119905)

119909119866


(8)



119866+1

119905


119909119866+1

119905=

119906119866+1

119905 if 119891 (119906

119866+1

119905) lt 119891 (119909

119866

119905)

119909119866

119905 otherwise

(9)



119866


numbers 1199031 1199032 and 119903




Vector x1 6 5 2 4 0090

sdot sdot sdot

Vector 11990923

5 4 2 1 0578

sdot sdot sdot

Vector 11990940

5 1 3 6 0745

sdot sdot sdot

Vector 119909NP 4 2 5 7 0024

(10)




Vector k1 56 34 08 04 1010

sdot sdot sdot

(11)



Vector u1 56 5 08 04 0090

sdot sdot sdot

(12)


1

Since 119891(119906119866+1

1) lt 119891(119909

119866

1) vector 119906




Vector 1 56 5 08 04 0090

sdot sdot sdot

(13)








1and119908

2 and


Max 120582 = 11990811199061

+ 11990821199062

st

1199061

=

119865max1

(119879 119896119894 119911119894) minus 1198651

(119879 119896119894 119911119894)

119865max1

(119879 119896119894 119911119894) minus 119865

min1

(119879 119896119894 119911119894)

1199062

=

119865max2

(119879 119896119894 119911119894) minus 1198652

(119879 119896119894 119911119894)

119865max2

(119879 119896119894 119911119894) minus 119865

min2

(119879 119896119894 119911119894)

1199081

+ 1199082

= 1

(14)

119865max1

(119879 119896119894 119911119894) 119865

min1

(119879 119896119894 119911119894) 119865

max2

(119879 119896119894 119911119894) and 119865

min2

(119879



Set 1199081015840

1= 1199081(119865

max1

minus 119865min1

) 11990810158402

= 1199082(119865

max2

minus 119865min2


Max 120582 = 119865119888

minus (1199081015840

11198651

+ 1199081015840

21198652) (15)

where 119865119888

= 1199081119865max1

(119865max1

minus 119865min1

) + 1199082119865max2

(119865max2

minus 119865min2

)

and

119865119908

= 1199081015840

11198651

+ 1199081015840

21198652

= 1199081015840

1(

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)]))

(16)

Let 120597119865119908

120597119911119894

= 0 that is 1199081015840

1ℎ119894120590119894radic119896119894119879 + 119871 + 119908

1015840

2(120590119894

119896119894119879)radic119896

119894119879 + 119871((119889119891(119911

119894)119889119911119894) minus [1 minus 119865(119911

119894)] + 119911119894119891(119911119894)) = 0


=

minus119911119894119891(119911119894) so that

1 minus 119865 [119911119894(119896119894119879)] =

1199081015840

1ℎ119894

1199081015840

2

119896119894119879 (17)





2119879119862(119879 119896

119894 119911119894)1205971199112

119894= (119908

1015840

2120590119894

119896119894119879)radic119896

119894119879 + 119871119891(119911


119894is


119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879)



119908for

given 119896119894and 119879 is

119865119908

= 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894))

(18)


Max 120582 (119879 119896119894)

= 119865119888

minus 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

minus 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894

))

st

119865119888

=

1199081119865max1

119865max1

minus 119865min1

+

1199082119865max2

119865max2

minus 119865min2

1199081015840

1=

1199081

119865max1

minus 119865min1

1199081015840

2=

1199082

119865max2

minus 119865min2

1199081

+ 1199082

= 1

119911lowast

119894

(119896119894119879) = 119865

minus1(1 minus

1199081015840

1ℎ119894

1199081015840

2

119896119894119879)

(19)



1





119911lowast

119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879) when 119896



obtained



(119896UB119894) of 119896



LB119894


UB119894


119894and 119879 we



randomly




119905and

119911lowast



Start



G gt GenM Yes

No




G = G + 1


Stop

NP

NP








Min 119865 (119909) = 1198911 (

119909) 1198912 (

119909) 119891119896 (

119909)

Subject to 119892119894 (

119909) le 0 119894 = 1 2 119898

(20)









(119894)= 0



119863(119868(1)

) = 119863(119868(1)

) + [2(119868(2)sdot119898

minus 119868(1)sdot119898

)]2

119863(119868119868) = 119863(119868

119868) + [2(119868


)]2

For 119894 = 2 |119868| minus 1

119863(119868(119894)

) = 119863(119868(119894)

) + [(119868(119894+1)sdot119898

minus 119868(119894minus1)sdot119898

)]2

Step 22119863(119894)

= 119904119902119903119905(119863(119894)

)
















119894


= 1 and 119896UB119894119895

= 100 119911119894is randomly



119905=







40 50 60















11989801 GA


09 GA


= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536



(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2













= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543


2change



1and 119908




1199082




SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)


119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896







1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119866

119905(119905 = 1 2 NP) the


V119866+1119905

= 119909119866

1199031+ 119865 times (119909

119866

1199032minus 119909119866

1199033) (7)

In (7) 1199031 1199032 and 119903



119866

11990311199091198661199032 and119909

119866




number of iteration


119906119866+1

119905119895=

V119866+1119905119895

if 119903119886119899119889119898 (119895) le CR or 119895 = 119903119886119899119889119899 (119905)

119909119866


(8)



119866+1

119905


119909119866+1

119905=

119906119866+1

119905 if 119891 (119906

119866+1

119905) lt 119891 (119909

119866

119905)

119909119866

119905 otherwise

(9)



119866


numbers 1199031 1199032 and 119903




Vector x1 6 5 2 4 0090

sdot sdot sdot

Vector 11990923

5 4 2 1 0578

sdot sdot sdot

Vector 11990940

5 1 3 6 0745

sdot sdot sdot

Vector 119909NP 4 2 5 7 0024

(10)




Vector k1 56 34 08 04 1010

sdot sdot sdot

(11)



Vector u1 56 5 08 04 0090

sdot sdot sdot

(12)


1

Since 119891(119906119866+1

1) lt 119891(119909

119866

1) vector 119906




Vector 1 56 5 08 04 0090

sdot sdot sdot

(13)








1and119908

2 and


Max 120582 = 11990811199061

+ 11990821199062

st

1199061

=

119865max1

(119879 119896119894 119911119894) minus 1198651

(119879 119896119894 119911119894)

119865max1

(119879 119896119894 119911119894) minus 119865

min1

(119879 119896119894 119911119894)

1199062

=

119865max2

(119879 119896119894 119911119894) minus 1198652

(119879 119896119894 119911119894)

119865max2

(119879 119896119894 119911119894) minus 119865

min2

(119879 119896119894 119911119894)

1199081

+ 1199082

= 1

(14)

119865max1

(119879 119896119894 119911119894) 119865

min1

(119879 119896119894 119911119894) 119865

max2

(119879 119896119894 119911119894) and 119865

min2

(119879



Set 1199081015840

1= 1199081(119865

max1

minus 119865min1

) 11990810158402

= 1199082(119865

max2

minus 119865min2


Max 120582 = 119865119888

minus (1199081015840

11198651

+ 1199081015840

21198652) (15)

where 119865119888

= 1199081119865max1

(119865max1

minus 119865min1

) + 1199082119865max2

(119865max2

minus 119865min2

)

and

119865119908

= 1199081015840

11198651

+ 1199081015840

21198652

= 1199081015840

1(

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)]))

(16)

Let 120597119865119908

120597119911119894

= 0 that is 1199081015840

1ℎ119894120590119894radic119896119894119879 + 119871 + 119908

1015840

2(120590119894

119896119894119879)radic119896

119894119879 + 119871((119889119891(119911

119894)119889119911119894) minus [1 minus 119865(119911

119894)] + 119911119894119891(119911119894)) = 0


=

minus119911119894119891(119911119894) so that

1 minus 119865 [119911119894(119896119894119879)] =

1199081015840

1ℎ119894

1199081015840

2

119896119894119879 (17)





2119879119862(119879 119896

119894 119911119894)1205971199112

119894= (119908

1015840

2120590119894

119896119894119879)radic119896

119894119879 + 119871119891(119911


119894is


119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879)



119908for

given 119896119894and 119879 is

119865119908

= 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894))

(18)


Max 120582 (119879 119896119894)

= 119865119888

minus 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

minus 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894

))

st

119865119888

=

1199081119865max1

119865max1

minus 119865min1

+

1199082119865max2

119865max2

minus 119865min2

1199081015840

1=

1199081

119865max1

minus 119865min1

1199081015840

2=

1199082

119865max2

minus 119865min2

1199081

+ 1199082

= 1

119911lowast

119894

(119896119894119879) = 119865

minus1(1 minus

1199081015840

1ℎ119894

1199081015840

2

119896119894119879)

(19)



1





119911lowast

119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879) when 119896



obtained



(119896UB119894) of 119896



LB119894


UB119894


119894and 119879 we



randomly




119905and

119911lowast



Start



G gt GenM Yes

No




G = G + 1


Stop

NP

NP








Min 119865 (119909) = 1198911 (

119909) 1198912 (

119909) 119891119896 (

119909)

Subject to 119892119894 (

119909) le 0 119894 = 1 2 119898

(20)









(119894)= 0



119863(119868(1)

) = 119863(119868(1)

) + [2(119868(2)sdot119898

minus 119868(1)sdot119898

)]2

119863(119868119868) = 119863(119868

119868) + [2(119868


)]2

For 119894 = 2 |119868| minus 1

119863(119868(119894)

) = 119863(119868(119894)

) + [(119868(119894+1)sdot119898

minus 119868(119894minus1)sdot119898

)]2

Step 22119863(119894)

= 119904119902119903119905(119863(119894)

)
















119894


= 1 and 119896UB119894119895

= 100 119911119894is randomly



119905=







40 50 60















11989801 GA


09 GA


= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536



(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2













= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543


2change



1and 119908




1199082




SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)


119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896







1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







1and119908

2 and


Max 120582 = 11990811199061

+ 11990821199062

st

1199061

=

119865max1

(119879 119896119894 119911119894) minus 1198651

(119879 119896119894 119911119894)

119865max1

(119879 119896119894 119911119894) minus 119865

min1

(119879 119896119894 119911119894)

1199062

=

119865max2

(119879 119896119894 119911119894) minus 1198652

(119879 119896119894 119911119894)

119865max2

(119879 119896119894 119911119894) minus 119865

min2

(119879 119896119894 119911119894)

1199081

+ 1199082

= 1

(14)

119865max1

(119879 119896119894 119911119894) 119865

min1

(119879 119896119894 119911119894) 119865

max2

(119879 119896119894 119911119894) and 119865

min2

(119879



Set 1199081015840

1= 1199081(119865

max1

minus 119865min1

) 11990810158402

= 1199082(119865

max2

minus 119865min2


Max 120582 = 119865119888

minus (1199081015840

11198651

+ 1199081015840

21198652) (15)

where 119865119888

= 1199081119865max1

(119865max1

minus 119865min1

) + 1199082119865max2

(119865max2

minus 119865min2

)

and

119865119908

= 1199081015840

11198651

+ 1199081015840

21198652

= 1199081015840

1(

119899

sum

119894=1

ℎ119894[

1

2

119863119894119896119894119879 + 119911119894120590119894radic(119871 + 119896

119894119879)]

+

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871 (119891 (119911

119894) minus 119911119894[1 minus 119865 (119911

119894)]))

(16)

Let 120597119865119908

120597119911119894

= 0 that is 1199081015840

1ℎ119894120590119894radic119896119894119879 + 119871 + 119908

1015840

2(120590119894

119896119894119879)radic119896

119894119879 + 119871((119889119891(119911

119894)119889119911119894) minus [1 minus 119865(119911

119894)] + 119911119894119891(119911119894)) = 0


=

minus119911119894119891(119911119894) so that

1 minus 119865 [119911119894(119896119894119879)] =

1199081015840

1ℎ119894

1199081015840

2

119896119894119879 (17)





2119879119862(119879 119896

119894 119911119894)1205971199112

119894= (119908

1015840

2120590119894

119896119894119879)radic119896

119894119879 + 119871119891(119911


119894is


119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879)



119908for

given 119896119894and 119879 is

119865119908

= 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

+ 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894))

(18)


Max 120582 (119879 119896119894)

= 119865119888

minus 1199081015840

1(

119899

sum

119894=1

119863119894ℎ119894119896119894

2

119879 +

1

119879

[119878 + 119892 (119896119894) +

119899

sum

119894=1

119904119894

119896119894

])

minus 1199081015840

2(

119899

sum

119894=1

120590119894

119896119894119879

radic119896119894119879 + 119871119891 (119911

lowast

119894

))

st

119865119888

=

1199081119865max1

119865max1

minus 119865min1

+

1199082119865max2

119865max2

minus 119865min2

1199081015840

1=

1199081

119865max1

minus 119865min1

1199081015840

2=

1199082

119865max2

minus 119865min2

1199081

+ 1199082

= 1

119911lowast

119894

(119896119894119879) = 119865

minus1(1 minus

1199081015840

1ℎ119894

1199081015840

2

119896119894119879)

(19)



1





119911lowast

119894(119896119894119879) = 119865


1015840

1ℎ1198941199081015840

2)119896119894119879) when 119896



obtained



(119896UB119894) of 119896



LB119894


UB119894


119894and 119879 we



randomly




119905and

119911lowast



Start



G gt GenM Yes

No




G = G + 1


Stop

NP

NP








Min 119865 (119909) = 1198911 (

119909) 1198912 (

119909) 119891119896 (

119909)

Subject to 119892119894 (

119909) le 0 119894 = 1 2 119898

(20)









(119894)= 0



119863(119868(1)

) = 119863(119868(1)

) + [2(119868(2)sdot119898

minus 119868(1)sdot119898

)]2

119863(119868119868) = 119863(119868

119868) + [2(119868


)]2

For 119894 = 2 |119868| minus 1

119863(119868(119894)

) = 119863(119868(119894)

) + [(119868(119894+1)sdot119898

minus 119868(119894minus1)sdot119898

)]2

Step 22119863(119894)

= 119904119902119903119905(119863(119894)

)
















119894


= 1 and 119896UB119894119895

= 100 119911119894is randomly



119905=







40 50 60















11989801 GA


09 GA


= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536



(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2













= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543


2change



1and 119908




1199082




SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)


119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896







1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Start



G gt GenM Yes

No




G = G + 1


Stop

NP

NP








Min 119865 (119909) = 1198911 (

119909) 1198912 (

119909) 119891119896 (

119909)

Subject to 119892119894 (

119909) le 0 119894 = 1 2 119898

(20)









(119894)= 0



119863(119868(1)

) = 119863(119868(1)

) + [2(119868(2)sdot119898

minus 119868(1)sdot119898

)]2

119863(119868119868) = 119863(119868

119868) + [2(119868


)]2

For 119894 = 2 |119868| minus 1

119863(119868(119894)

) = 119863(119868(119894)

) + [(119868(119894+1)sdot119898

minus 119868(119894minus1)sdot119898

)]2

Step 22119863(119894)

= 119904119902119903119905(119863(119894)

)
















119894


= 1 and 119896UB119894119895

= 100 119911119894is randomly



119905=







40 50 60















11989801 GA


09 GA


= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536



(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2













= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543


2change



1and 119908




1199082




SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)


119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896







1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





(119894)= 0



119863(119868(1)

) = 119863(119868(1)

) + [2(119868(2)sdot119898

minus 119868(1)sdot119898

)]2

119863(119868119868) = 119863(119868

119868) + [2(119868


)]2

For 119894 = 2 |119868| minus 1

119863(119868(119894)

) = 119863(119868(119894)

) + [(119868(119894+1)sdot119898

minus 119868(119894minus1)sdot119898

)]2

Step 22119863(119894)

= 119904119902119903119905(119863(119894)

)
















119894


= 1 and 119896UB119894119895

= 100 119911119894is randomly



119905=







40 50 60















11989801 GA


09 GA


= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536



(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2













= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543


2change



1and 119908




1199082




SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)


119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896







1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








11989801 GA


09 GA


= 056)

HDE DE GA119896 2 1 1 1 2 1 1 1 2 1 1 1119911 167 135 093 153 167 135 093 153 170 138 096 156119879 00824 00824 007851198651

846821 846821 8487521198652

1744 1744 1691120582 07553 07553 07536



(119879 119896119894 119911119894) =

10500 119865min1

(119879 119896119894 119911119894) = 7500 119865

max2

(119879 119896119894 119911119894) = 120 and

119865min2













= 056)

HDE DE GA119896 2 1 1 1 3 1 1 1 2 1 1 1119911 159 135 096 149 149 164 111 167 155 137 099 146119879 00721 00736 007211198651

846657 858738 8475241198652

1770 1295 1736120582 07547 07495 07543


2change



1and 119908




1199082




SP =radic

sum119899

119894=1(119889 minus 119889

119894)

2

(119899 minus 1)

(21)


119894is given by

119889119894= min

119895(

10038161003816100381610038161003816119891119894

1(119909) minus 119891

119895

1(119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891119894

2(119909) minus 119891

119895

2(119909)

10038161003816100381610038161003816)

119894 119895 = 1 119899

(22)

where 119891119896







1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925301 102 09339 925699 101 09339 925298 102 0933902 903190 255 08809 904017 248 08808 911739 240 0876203 886745 474 08356 886769 476 08354 887375 467 0835404 871856 796 07977 871680 806 07975 876001 759 0794105 856708 1296 07682 857414 1277 07678 859170 1251 0765906 839613 2143 07493 840040 2134 07488 840968 2093 0748307 817659 3811 07468 818303 3786 07460 817224 3864 0746508 789098 7237 07751 789254 7287 07739 789430 7240 0774209 767480 12383 08444 767480 12383 08444 765665 13097 08439


1199081

HDE DE GA1198651

1198652

120582 1198651

1198652

120582 1198651

1198652

120582

01 925373 103 09338 928577 089 09337 922535 117 0933702 904673 242 08807 901773 272 08806 903113 259 0880703 888603 448 08352 887841 477 08343 884834 517 0835004 872004 802 07972 876847 700 07959 873946 755 0797005 854597 1394 07676 858738 1295 07648 861766 1116 0767206 842372 2000 07486 858738 1295 07394 835945 2393 0748307 820726 3576 07456 834816 3222 07215 822971 3376 0745308 791356 6971 07735 834816 3222 07201 789298 7303 0773509 768528 12157 08431 834816 3222 07187 771152 11717 08389







Tota

l cos

t

10000

9500

9000

8500

8000

75000 20 40 60 80 100 120








Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





Tota

l cos

t


10500

10000

9500

9000

8500

80000 5 10 15 20 25 30 35


0 20 40 60 80 100 120


Tota

l cos

t

10500

10000

9500

9000

8500

8000

7500







Abbreviations













Acknowledgments


References


















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article modeling and optimization of the multiobjective...

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