Research Article Multidemand Multisource Order Quantity ...downloads.hindawi.com/journals/mpe/2015/363056.pdf · allocation [ ]: total purchase cost minimization, total delay risk

Research ArticleMultidemand Multisource Order Quantity Allocation withMultiple Transportation Alternatives

Jun Gang12 Yan Tu23 and Jiuping Xu2

1Sichuan Institute of Building Research Chengdu 610081 China2Business School Sichuan University Chengdu 610064 China3School of Management Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Jiuping Xu xujiupingscueducn

Received 25 July 2015 Revised 26 September 2015 Accepted 29 September 2015

Academic Editor Young Hae Lee

Copyright copy 2015 Jun Gang et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper focuses on a multidemand multisource order quantity allocation problem with multiple transportation alternatives Tosolve this problem a bilevel multiobjective programming model under a mixed uncertain environment is proposed Two levels ofdecisionmakers are considered in themodel On the upper level the purchaser aims to allocate order quantity to multiple suppliersfor each demand node with the consideration of three objectives total purchase cost minimization total delay risk minimizationand total defect risk minimization On the lower level each supplier attempts to optimize the transportation alternatives with totaltransportation and penalty costsminimization as the objective In contrast to prior studies considering the information asymmetryin the bilevel decision random and fuzzy random variables are used to model uncertain parameters of the construction companyand the suppliers To solve the bilevel model a solution method based on Kuhn-Tucker conditions sectional genetic algorithmand fuzzy random simulation is proposed Finally the applicability of the proposed model and algorithm is evaluated througha practical case from a large scale construction project The results show that the proposed model and algorithm are efficient indealing with practical order quantity allocation problems

1 Introduction

In todayrsquos ever-changing competitive market environmentpurchasing strategy is extremely important for eachmanufac-turing company because of the tremendous impact of mate-rial costs on profits [1] This situation is also appropriate forconstruction companies which are manufacturers of con-struction projects In purchasing management two decisionsare mainly involved supplier selection and order allocationSupplier selection is a critical task to achieve the differentobjectives of supply chain because it can help the companyto maintain long relationship with few reliable suppliers [2]To deal with supplier selection many methodologies havebeen proposed including multiattribute decision makingtechniques mathematical programming and artificial intel-ligence methods [3] In material purchasing process afterchoosing suitable suppliers order allocation is the nextimportant stage to determine the optimal order quantity allo-cation scheme especially in the case of a multiple suppliers

environment Many models and methods have been pro-posed to deal with order quantity allocation problems such aslinear programming model [4] mixed integer programmingmodel [5 6] nonlinear programming [7 8] and artificialintelligence technique [9 10]

In the above literatures almost all consider multiplesuppliers (ie multisource) while only one demand nodeis included However in some large scale companies orconstruction projects we often face multiple demand nodessuch as multiple production factories or multiple construc-tion sites at the same time Moreover it is possible that asupplier provides materials for the multiple demand nodesat the same time In this situation we have to allocate orderquantity under a multidemand multisource environmentThis is a more difficult problem where each supplier has totransport materials to multiple demand nodes As we knowfor most suppliers there are multiple optional transportationalternatives defined by vehicles routes and modes for a typeof specific material [11 12] However each alternative often

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 363056 18 pageshttpdxdoiorg1011552015363056

2 Mathematical Problems in Engineering

has its own limited capacity Hence in themultidemandmul-tisource order allocation problem once the order quantityis determined for each supplier they have to optimize theirtransportation alternatives In the existing researches fewconsider the vital role of the transportation alternatives in theorder allocation process although it is significantly importantto the supplier evaluation and order allocation decisionsGhodsypour and OrsquoBrien [5] proposed a mixed integernonlinear programming model to solve the multisourceproblem which took into account the total cost of logisticsincluding net price storage transportation and orderingcosts Kawtummachai and Hop [13] constructed an ordersquantity allocationmethod to the selected supplier in order tooptimize the purchase cost within the acceptable percentageof on-time delivery which is influenced by transportationalternatives to guarantee high service levels to the retailersDullaert et al [14] considered a combinatorial optimizationmodel for determining the optimal mix of transportationalternatives to minimize total logistics costs when goodsare shipped from a supplier to a customer Jafari Songhoriet al [11] first explicitly presented a decision frameworkwhich considered different transportation alternatives fromsuppliers to demander

It should be noted that in all these papers the orderallocation and transportation alternatives are determinedby the same decision maker (ie the purchaser) Howeverin practice only the price and order quantity are writteninto agreements in most cases As for how to transportmaterial or goods it is often directed by the suppliersHence two decision making processes should be consideredFirst the purchaser such as the construction company in aconstruction project decides how to allocate order quantityamong multiple suppliers for multiple demand nodes Thenbased on the allocated order quantity each supplier willdecide how to transport them to demand nodes throughmultiple transportation alternatives In this situation themultidemand multisource order allocation problem shouldbe considered as a bilevel decision making problem

Besides for the bilevel structure the uncertainty whichis also an important factor will be considered in this paperMany papers have paid attention to uncertain supplierselection and order allocation problem Xu and Nozick [6]proposed a two-stage mixed integer stochastic programmingmodel to optimize supplier selection Yang et al [15] J-LZhang and M-Y Zhang [16] and Esfandiari and Seifbarghy[17] discussed the supplier selection and order allocationproblem with stochastic demand Kumar et al [18] stud-ied a multiobjective integer programming vendor selectionproblem with fuzzy parameters Amid et al [19] and Nazari-Shirkouhi et al [20] applied fuzzy programming theory todeal with the multiobjective supplier selection and orderallocation problem In the researches mentioned abovefuzziness and randomness were often considered separateaspects But in reality we may face a hybrid uncertainenvironment where fuzziness and randomness coexist in adecision making process For example transportation costfrom one supplier to one demand node is often modeled asa stochastic parameter However in a bilevel environmentbecause of the information asymmetry between the two levels

of decision makers the upper decision maker may modifythe accuracy of the given transportation cost As a result hemay modify the uncertain transportation cost using a fuzzyvariable In this situation the cost is a hybrid of fuzzy andrandom factors and the fuzzy random variable which wasfirst proposed by Kwakernaak [21] and then extended by Puriand Ralescu [22] can be a useful tool to deal with it

Hence we face a bilevel multidemand multisource orderquantity allocation problem under a fuzzy random environ-ment In the problem two levels of decision makers mustbe considered On the upper level the purchaser attempts toallocate order quantity amongmultiple suppliers for multipledemand nodes with the objectives of total purchase cost andtotal purchase risk minimization On the lower level eachsupplier will select the transportation alternatives from thesupplier to demand nodes with the objective of total trans-portation and penalty costs minimization Simultaneouslysome uncertain parameters in the problem such as demandtransportation cost and risk coefficients must also be con-sidered To deal with this problem a bilevel multiobjectiveprogramming model with uncertain coefficients is proposedSpecifically some uncertain coefficients are modeled as fuzzyrandom variables In addition to search the optimal solutionof proposed model a solution method based on Kuhn-Tucker conditions sectional genetic algorithm (SGA) andfuzzy random simulation is also proposed Finally resultsand comparison analysis of a case study are presented todemonstrate the practicality and efficiency of the proposedmodel and algorithm

The remainder of this paper is structured as follows inSection 2 an introduction to the bilevel multiobjective mul-tidemand multisource order allocation problem is presentedalong with the motivation for employing uncertain vari-ables in the problem A bilevel multiobjective programmingmodel under a mixed uncertain environment is establishedin Section 3 In addition a solution method is illustratedin Section 4 In Section 5 a case study from a practicalexample is given to show the validity and efficiency of theproposedmodel and algorithmConclusions and a discussionregarding further research are remarked in Section 6

2 Key Problem Statement

The problem considered in this paper is a bilevel multiobjec-tive multidemandmultisource order allocation problem withmultiple transportation alternatives under a mixed uncertainenvironment In this section we explain why this problemshould be considered a bilevel multiobjective programmingmodel and why the demand transportation cost and timeare able to be modeled as random or fuzzy random variables

21 Conflict Description for Order Quantity Allocation Inlarge scale construction projects many construction mate-rials are consumed in a great deal every day these mate-rials are often purchased from multiple material suppliersIn addition because of the big construction ground it isimpossible to store these materials in a storehouse Hencethere are often multiple demand nodes which serve for

Mathematical Problems in Engineering 3

various construction activitiesThat is the purchase problemin construction engineering can be summarized as a mul-tidemand multisource problem In this problem two levelsof stakeholders should be considered On one hand theconstruction company is the purchaser who allocates orderquantity amongmultiple suppliers for each demand node Onthe other hand the material suppliers are the vendors whoare in charge of the supply and transportation of constructionmaterials They can select transportation alternatives fromsuppliers to demand nodes for each pair

As the purchaser the construction company plays aleading role in the decisionmaking which can affect the sup-pliersrsquo income with different order allocation planningThreeobjectives are often considered in supplier selection and orderallocation [23] total purchase cost minimization total delayrisk minimization and total defect risk minimization Itshould be noted that the second and third objectives on riskare related to transportation alternatives defined by vehiclesroutes and modes In other words the decision making ofupper level is influenced by lower levelrsquos decision makingIn the lower level material suppliers often aim to seek fortransportation alternatives with minimal total transportationand penalty costs when their income order quantities havebeen determined However the transportation alternativeswith low cost may lead to high delay risk and defect risk Inthis case the objectives between the construction companyand suppliers are in conflict

In the conflict situation the decision makers pursue toachieve their own goals The construction company firsthopes to reach its own targets by making a better order quan-tity allocation scheme while considering the possible reac-tions of the suppliers Then the suppliers hope to save costby selecting better transportation alternatives based on theallocated order quantity Therefore the considered problemcan be abstracted as a bilevel decision making problem Inthe bilevel problem the construction company which is theupper level decisionmaker seeks to allocate order quantity tomultiple suppliers formultiple demand nodes with lower costand lower risk while material suppliers who are the lowerlevel decision makers determine transportation alternativesfor each pair from suppliers to demand nodes In additionit should also be noted that because of the complexity ofthe decision system the complete decision making processis conducted in an uncertain environment In this papersome uncertain parameters are considered as random orfuzzy random variables Hence the considered problem isa bilevel multiobjective order quantity allocation problemunder a mixed uncertain environment and the frameworkof the bilevel decision making with conflicting objectives isillustrated in Figure 1

22 Uncertainty Description for Order Quantity AllocationIn the order quantity allocation problem of constructionprojects as well as the bilevel structure of the model it isalso important to consider the uncertainty of the parametersIn this paper our main consideration is the uncertainty ofdemand transportation cost delay risk and defect risk

In the order allocation problem the demand quantity foreach demand node is always uncertain and often considereda stochastic coefficient such as in [15ndash17] because the deci-sion maker often has enough data to obtain its probabilitydistribution although it is difficult to get the exact value Inthis paper as the decision coefficient of upper level decisionmaker it is also reasonable to assume that the decisionmakercan obtain the probability distribution function of demandquantity for each demand node Hence the demand quantityis also assumed to be stochastic in this paper

Different from the demand quantity the transportationcost from one supplier to one demand node with one specifictransportation alternative is a decision coefficient of the lowerdecision maker Essentially it can also be modeled as arandom variable just as the demand quantity Here we just aswell assume the stochastic cost with a normal distribution 119888 sim119873(120583 120590) However as described above in the bilevel problemthere are conflicting objectives between the two levels ofdecision makers Moreover the information between them isalso asymmetrical That is it is impossible that the decisionmaker on the upper level knows all information of the lowerlevel decision makers As a result the lower level decisionmakers may provide inaccurate information to the upperlevel decision maker in the trade Of course the upper leveldecision maker may also not trust the obtained informationIn this case we can only just say that ldquothe transportationcost is about 119888rdquo Here the word ldquoaboutrdquo is an ambiguousexpression which can be modeled using triangular fuzzyvariable (119888 minus 119886 119888 119888 + 119887) where 119888 minus 119886 119888 and 119888 + 119887 are theminimum value the most possible value and the maximumvalue of the transportation cost respectively It can be seenthat the fuzzy transportation cost is a modified value of therandom cost in a certain extent Finally the transportationcost can be expressed as 119888 = (119888 minus 119886 119888 119888 + 119887) with 119888 sim 119873(120583 120590)This is a triangular fuzzy random variable which has beensuccessfully applied to many areas such as inventory problem[24] vehicle routing [25] andwater resources allocation [26]In conclusion the motivation for employing fuzzy randomvariables in the bilevel order allocation problem can beillustrated in Figure 2 The cases of delay risk and defect riskare similar with the cost which are also decision coefficientsof the lower model Hence they are also modeled as fuzzyrandom variables in this paper

3 Modelling

In this section a bilevel multiobjective programming modelfor the multidemand multisource purchase problem consid-ering fuzziness and randomness is constructed The mathe-matical description for the problem is given as follows

31 Assumptions and Notations Before constructing themodel in this paper the following assumptions are adopted

(1) Only one type of commodity is considered to bepurchased in the model at each time

(2) There are multiple suppliers and a lowest orderquantity has been determined in contract for eachsupplier


Decision variableOrder quantity to supplier jfor demand node i

Order quantityallocation model

Upper level

Decision makerConstruction company

Upper leveldecision making

system

Objectives(1) Total purchase cost minimization(2) Total delay risk minimization(3) Total defect risk minimization

Randomenvironment

Randomdemand quantity

Twostakeholders

Conflictof

interests

Bileveldecisionmodel

Multiple demands

Demandnode 1

Demandnode 2

Demandnode i

Demandnode I

Transportation Transportation

Ord

er q

uant

ity

Order quantity

Multiple sources

Supplier 1 Supplier 2 Supplier j Supplier J

middot middot middot


Decision variable

Transportation alternative fromsupplier j to demand node i

Lower leveldecision making

system

ObjectiveTotal transportation and penaltycosts minimization

Lower level

Transportationalternative selectionmodel

Decision makerSuppliersj = 1 2 J

Fuzzy randomtransportation costdelay risk and defect risk

Fuzzy randomenvironment

Figure 1 The decision framework of the bilevel order quantity allocation problem

(3) There are multiple demand nodes and their demandquantities are uncertain which can be modelled asrandom variables

(4) Each supplier has multiple transportation alterna-tives and different transportation alternatives havedifferent transportation cost and risk

(5) The transportation cost percentage of late deliveredunits and percentage of rejected units for eachtransportation alternative from one supplier to one

demand node are considered fuzzy random varia-bles

The following symbols are used in this paper

IndicesThe indices are as follows

119894 demand node 119894 isin Ψ = 1 2 119868119895 supplier (source) 119895 isin Φ = 1 2 119869119896 transportation alternative 119896 isin Ω = 1 2 119870


Fuzzy random transportation cost

Upper level DMPurchaser

(construction company)Lower level DM

SuppliersConfront some uncertain parameterswhen attempting to optimize their objectives

Obj

ectiv

es

Obj

ectiv

e

(1) Total purchase costs minimization(2) Total delay risk minimization(3) Total defect risk minimization

Total transportation and penaltycosts minimization

Analysis to history statistical data

Information asymmetry

Conflict+

Queryto

It is about

Considering transportation cost

a

b

x0010

Representationof a fuzzy

random variable

10

a

b

Transportation cost Defect riskDelay risk

cc

c Modify c to cby a and b

120583c

120583

120593 c(x)120583

c

c2

c1

c3

x isin Rvalues in the probability distribution

120593c =1

radic2120587120590eminus(xminus120583)221205902

c = (c minus a c c + b) with c sim N(120583 120590)

c = (c minus a c c + b) c sim N(120583 120590)

c1 c2 and c3 just represent three possible

Figure 2 Employing fuzzy random variable to express the transportation cost for bilevel order allocation problem

Certain Variables Certain variables are as follows

119861119894 total budget for demand node 119894119878119895 supply capacity of supplier 119895119875119895 unit price in supplier 119895119862119894119895 order cost from supplier 119895 for demand node 119894

MO119895 minimal order quantity for supplier 119895

TP119895119896 transportation capacity for supplier 119895 with

transportation alternative 119896CP119895 unit defect penalty cost for supplier 119895

119897max119895

highest delay rate for supplier 119895

Random Variables

119876119894 demand quantity of demand node 119894

Fuzzy Random Variables Fuzzy random variables are asfollows

119897119895119896 percentage of late delivered units for supplier 119895

with transportation alternative 119896119903119895119896 percentage of rejected units for supplier 119895 with

transportation alternative 119896119888119894119895119896 unit transportation cost from supplier 119895 to

demand node 119894 using transportation alternative 119896


Decision Variables Decision variables are as follows

119909119894119895 order quantity from supplier 119895 to demand node 119894

119910119894119895119896 transported quantity from supplier 119895 to demand

node 119894 using transportation alternative 119896

119911119894119895 1 if at least one unit of material is provided

by supplier 119894 for demand node 1198950 otherwise

The multidemand multisource purchase problem by thepurchaser and the transportation alternatives decision mak-ing problems by the suppliers are expressed in detail in thefollowing sections and the relationship between these twodecision makers is also discussed which can be abstracted asa bilevel decision making model

32 Order Quantity Allocation Model As the upper decisionmaker the purchaser will consider how to allocate orderquantity among multiple suppliers for multiple demandnodesThe decision variables are the allocated order quantity119909119894119895for each pair of supplier and demand node

321 Objectives of Construction Company As mentionedabove in an order quantity allocation problem of a construc-tion project the purchaser (eg the construction company)often has three objectives [20 23] total purchase cost mini-mization total delay risk minimization and total defect riskminimization

(1) Total Purchase Cost Minimization The purchaser firstaims to control the total purchase cost which is made upof purchase price (ie sum

119894isinΨsum119895isinΦ119875119895119909119894119895) and order cost (ie

sum119894isinΨsum119895isinΦ119862119894119895119911119894119895) Therefore the first objective of the upper

level can be described as

min119909119894119895

TC = sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 (1)

(2) Total Delay Risk Minimization In construction projectsproject completed time is always discussed in any phase ofthe construction process For material transportation if thematerial is late delivered some construction activities may bepostponed as a result thewhole project cannot be finished ontime Hence the delay risk must be considered in the modelHere the delay risk is described using the percentage of latedelivered units and the total delay risk can be described assum119894isinΨsum119895isinΦsum119896isinΩ

119897119895119896119910119894119895119896

Since 119897119895119896

are fuzzy random variables the total delay riskcan be regarded as a special fuzzy randomvariable Because ofthe uncertainty it is difficult for decision makers to obtain anexact value for the delay risk Therefore the decision makeroften tends to obtain an expected total delay risk with anexpected value operation for the uncertain variables whichcan be denoted as 119864[sum

119894isinΨsum119895isinΦsum119896isinΩ

119897119895119896119910119894119895119896] according to the

definition of expected value in [27] The equation above can

also be transformed into sum119894isinΨsum119895isinΦsum119896isinΩ119864[119897119895119896]119910119894119895119896 There-

fore the second objective for the upper level can be expressedas

min119909119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 (2)

(3) Total Defect Risk Minimization The third objective isrelated to the quality of ordered items It is described withthe number of rejected items which is a product term ofthe percentage of rejected units 119903

119895119896and the order quantity

119910119894119895119896 For most of construction materials the percentage

of rejected units is closely related to the transportationalternative because they often have high loss (1ndash6) duringtransportation and the loss is mainly affected by the typeof material transportation mode vehicle and route Takingcement for example it can be transported by cement cannedcars specialized cement trucks and common trucks wherethe rates of loss during transportation are about 1 2 and3 respectively For the same reason as the second objectivethe decision maker also aims to obtain an expected defectrisk Hence the third objective for the upper level can bedescribed as

min119909119894119895

119864 (TR) = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896 (3)

322 Constraints on Order Quantity Allocation There arefive main types of constraints on the upper level for the orderquantity allocation problem details of them are explained inthe following

(1) Requirement Constraint The total purchase quantitysum119895isinΦ119909119894119895

must exceed the demand quantity 119876119894for each

demand node 119894 Technically it is not possible to strictly ensuresum119895isinΦ119909119894119895exceed 119876

119894because of the uncertain variables 119876

119894

In a practical problem the decision makers often ensurethe restriction is satisfied to a certain extent In these caseschance-constrained operation which was first introduced byCharnes and Cooper [28] is used to deal with this constraintIn this problem the upper level decision maker provides aconfidence level 120572 for the random event thus we have thefollowing constraint

Pr

sum

119895isinΦ

119909119894119895ge 119876119894

ge 120572 forall119894 isin Ψ (4)

(2) Supply ConstraintFor each constructionmaterial supplier119895 the total order quantity sum

119894isinΨ119909119894119895from all demand nodes

must be smaller than its supply capacity 119878119895 which can be

denoted as

sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ (5)

(3) Minimal Order Quantity Constraint In constructionprojects the demand quantity is often very big for manytypes of construction materials In order to maintain


the contract with the suppliers more effectively a minimalorder quantity is usually set up for each supplier Hence wehave the following constraint

sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ (6)

(4) Budget Constraint In order to control cost the con-struction company always determines a cost budget for eachconstruction unit Thus for each demand node 119894 the totalpurchase cost sum

119895isinΦ119875119895119909119894119895+ sum119895isinΦ119862119894119895119911119894119895must be smaller than

the cost budget 119861119894for this material purchasing thus we can

obtain the following constraints

sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119894 forall119894 isin Ψ (7)

(5) Logicality Conditions If the construction company pur-chases material from supplier 119895 for demand node 119894 (ie 119909

119894119895gt

0) the auxiliary variable 119911119894119895must take value of 1 Otherwise

(ie 119909119894119895= 0) 119911

119894119895has to take value of 0 Let 119872 be a great

number then the following constraints are obtained

119909119894119895

119872

le 119911119894119895le 119909119894119895 forall119894 isin Ψ 119895 isin Φ (8)

From the descriptions above based on the three objectivefunctions and five types of constraints we can obtain theorder quantity allocation model for the upper level decisionmaker (ie the construction company) as the following

(ULM)

min119909119894119895 119911119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895

min119909119894119895 119911119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896

min119909119894119895 119911119894119895

119864 [TR] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894

ge 120572 forall119894 isin Ψ

sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ

sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872

le 119911119894119895le 119909119894119895 forall119894 isin Ψ 119895 isin Φ

119909119894119895ge 0 119911

119894119895isin 0 1 forall119894 isin Ψ 119895 isin Φ

where 119910119894119895119896

is solved in the lower level model

(9)

It should be noted that the decision making of theconstruction company (ie the upper level decision maker)is influenced by the decisions made by suppliers (ie

the lower level decision makers) The construction companyhas to consider the possible reactions of the suppliers beforechoosing the optimal decision The transportation alterna-tives selection model (ie the lower level model) is discussedin the following

33 Transportation Alternatives Selection for Suppliers Asthe lower level decision maker each supplier can choosetransportation alternatives independently which can affectthe delay risk and defect risk on the upper level Eachsupplier has multiple transportation alternatives and eachalternative has limited transportation volume Hence thesuppliers have to distribute the allocated orders to these alter-natives Different transportation alternatives mean differenttransportation routes transportation vehicles transportationrisks and transportation costs The decision variables forsuppliers are 119910

119894119895119896

331 Transportation Alternatives Selection Objectives for Sup-pliers For suppliers (ie lower level decision makers) theyhope to save cost by selecting better transportation alterna-tives Therefore minimizing the costs (including transporta-tion cost and penalty cost) is often their objectives For eachsupplier the transportation cost is associated with the unittransportation cost 119888

119894119895119896and transportation volume 119910

119894119895119896for

each transportation alternative Because the transportationcost 119888

119894119895119896is fuzzy random variables the same as the upper

level the suppliers also use expected value operation to dealwith the uncertain parameters the expected transportationcost can be described as sum

119894isinΨsum119896isinΩ119864[119888119894119895119896]119910119894119895119896 The penalty

cost means the loss when the transported units are rejectedin the process of examination it is the product of the unitdefect penalty cost the expected rate of rejected units andthe transportation quantity that is CP

119895119864[119903119894119895119896]119910119894119895119896 Hence the

objective function for supplier 119895 can be described as

min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896

+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

(10)

332 Constraints on Transportation Alternatives SelectionThere are three types of constraints for the transportationalternatives selection on the lower level that must be consid-ered

(1) Supply Constraints For each supplier 119895 the total deliveredquantity sum

119896isinΩ119910119894119895119896

to each demand node 119894 from all thealternatives must exceed the allocated order quantity 119909

119894119895for

this demand node If the allocated order quantity 119909119894119895is 0 then

sum119896isinΩ119910119894119895119896

should also be 0 by introducing a great number119872we can have the following constraints

119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909

119894119895 forall119894 isin Ψ 119895 isin Φ (11)

(2) Transportation Capacity Constraints For each supplier119895 the total delivered quantity sum

119894isinΨ119910119894119895119896

of transportation


alternative 119896 to all demand nodes cannot exceed the trans-portation capacity of transportation alternative 119896 thus wehave the following constraints

sum

119894isinΨ

119910119894119895119896le TP119895119896 forall119895 isin Φ 119896 isin Ω (12)

(3) Delay Rate Constraints In current competitive businessworld for sustainable business each supplier will often sethighest delay rate to gain a good performance evaluationIn the considered problem the average percentage of latedelivered units from suppler 119895 to demand node 119894 cannotexceed the highest delay rate that is sum

119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895

However in this equation 119909119894119895may take the value of 0

after conversation we can get the following constraints

sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897

max119895119909119894119895 forall119894 isin Ψ 119895 isin Φ (13)

From the descriptions above based on the objective oftransportation cost minimization by synthesizing supplyconstraints transportation capacity constraints and delayrate constraints we can obtain the lower level model (LLM)for suppliers

(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896

forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909

119894119895 forall119894 isin Ψ 119895 isin Φ

sum

119894isinΨ

119910119894119895119896le TP119895119896 forall119895 isin Φ 119896 isin Ω

sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897

max119895119909119894119895 forall119894 isin Ψ 119895 isin Φ

119910119894119895119896ge 0 forall119894 isin Ψ 119895 isin Φ 119896 isin Ω

(14)

It should be noted that the suppliersrsquo decision making isbased on the decision made by the construction companyMoreover the suppliersrsquo decision making also brings influ-ences to the upper levelrsquos decision making because differenttransportation alternatives mean different risks

34 Global Model for Bilevel Order Quantity Allocation Prob-lem In the uncertain bilevel multiobjective order quantityallocation problem there are two levels of decision makersconstruction company (purchaser on the upper level) andmaterial suppliers (vendors on the lower level)The construc-tion company on the upper level will allocate order quantityamong multiple suppliers for multiple demand nodes withthe objectives of total purchase cost minimization total delayrisk minimization and total defect risk minimization underconstraints on requirement supply capacity order quantitybudget and logicality conditions The suppliers on the lowerlevel select transportation alternatives for each pair fromsuppliers to demand nodes based on their own transportationand penalty costs objectives under constraints on supplytransportation capacity and delay rate

Usually the objectives of decision makers on two levelsconflict The decision maker on the upper level attemptsto minimize the delay risk and defect risk However boththe two types of risks are related to the transportationalternatives which are determined by suppliers on the lowerlevel Unfortunately suppliers always take their ownminimalcosts (including transportation cost and penalty cost) as theirobjectives instead of considering the risk and transportationalternatives with low cost may bring high risk In thissituation the decision maker on the upper level has toconsider the possible reactions of decision makers on thelower level before choosing the optimal decision Thus abilevel model is proposed In the model the decision makeron the upper level first decides on a feasible order quantityallocation scheme and then the decisionmakers on the lowerlevel select their transportation alternatives for each pair fromsuppliers to demand nodes in turn By means of this methodthey hope to obtain an optimal Stackelberg solution Basedon the above equations (9) and (14) the following globalmodel for the bilevel multiobjective programming can nowbe formulated for the order quantity allocation problem

min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)

4 Proposed Solution Method

In order to obtain Stackelberg solutions to a bilevel pro-gramming model many methods have been proposed Theycan be classified roughly into three categories the vertexenumeration approach the Kuhn-Tucker approach and thepenalty function approach [29] Among them the Kuhn-Tucker approach ismostwidely used if the objective functionsof the lower level model are continuous and differentia-ble

41 Kuhn-Tucker Conditions In the considered problem alldecision variables on the lower level are continuous and theobjective functions are differentiable Hence we can useKuhn-Tucker conditions to transform the lower level modelinto constraint conditions of the upper level model In theKuhn-Tucker approach the upper level decision makerrsquosproblem with constraints involving the optimality conditionsof the lower level decision makersrsquo problems is solved Byintroducing auxiliary variables 1199061

119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the

Kuhn-Tucker conditions of model (15) can be shown asfollows

min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065

119894119895119896= 0 forall119894 isin Ψ 119895 isin Φ 119896 isin Ω

1199061

119894119895(sum

119896isinΩ

119910119894119895119896minus 119909119894119895) = 0 forall119894 isin Ψ 119895 isin Φ

1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909

119894119895) = 0 forall119894 isin Ψ 119895 isin Φ


1199063

119895119896(sum

119894isinΨ

119910119894119895119896minus TP119895119896) = 0 forall119895 isin Φ 119896 isin Ω

1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897

max119895119909119894119895) = 0 forall119894 isin Ψ 119895 isin Φ

1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911

119894119895isin 0 1 forall119894 isin Ψ 119895 isin Φ 119896 isin Ω

(16)

It can be seen that the bilevel multiobjective model hasbeen transformed into an equivalent single-level multiobjec-tive mathematical programming model through the Kuhn-Tucker approach

42 Satisfaction Function Although the bilevel model (15)has been transformed into a singlemodel (16) through Kuhn-Tucker conditions it is still difficult to solve the problembecause of themultiple objectives In order to handle themul-tiple objective functions and integrate the attitude of decisionmaker we propose a weight-sum satisfaction method belowAs the three objectives of the upper level of model (16) areminimum problems for any feasible solution (x y zu) isin Λthe degree of satisfaction for each objective can be calculatedusing the following equation

119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)

where 119865119905 119905 = 1 2 3 each separately represent the three

objective functions on the upper level and 119865min119905

119865max119905

repre-sent the minimal value and maximal value respectively

Once a feasible solution is produced the above equationcan be used to calculate degree of satisfaction for eachobjective Usually decision makers will determine lowestsatisfaction of each objective in advance named as 119878min

119905

Then the following constraintsmust be satisfiedwhenmakingdecision

119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)

In this situation decision makers always hope to seekfor a solution with maximal satisfaction degree At the same

time taking the weight of each objective into considerationthe goal for the model can be transformed to maximize theweight-sum satisfaction degree so the above model can beconverted into the following satisfaction model

max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)

where 119908119905is the weight of the 119905th objective which meets 119908

1+

1199082+ 1199083= 1 and Λ is the feasible region of model (16)

It can be seen that through the operation of satisfactionfunction themultiobjectivemodel (16) has been transformedinto a single-objective model

43 Sectional Genetic Algorithm Based on Uncertain Simu-lation Although model (19) is only a single-objective andsingle-level model it is still difficult to solve the problemusing a commercial solver because of too many decisionvariables and constraints For example in a decision makingprocess there are 10 demand nodes 20 suppliers and 3 trans-portation alternatives for each supplier then 1860 decisionvariables and 4443 constraintsmust be consideredMoreoversome nonlinear constraints and uncertain objectives alsoincrease the difficulty to solve the model In this situation asectional genetic algorithm based on fuzzy random simula-tion is introduced here to solve the model (19)

431 Uncertain Simulation In the proposed model theuncertain operation mainly means the expected value opera-tion of the fuzzy randomobjective which actually is a weight-sum function and can be described as the following equation

119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)


Decisionvariables forupper level

Decisionvariables forlower level Auxiliary variables

xij ge 0 yijk ge 0

x11 middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middotxIJ y111 yIJK u111 u1

IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0

Figure 3 Real number string structure of floating-point sectional representation

where 1198651= sum

119894isinΨsum119895isinΦ119875119895119909119894119895+ sum119894isinΨsum119895isinΦ119862119894119895119911119894119895 1198652=

sum119894isinΨsum119895isinΦsum119896isinΩ119864[119897119895119896]119910119894119895119896

and 1198653

=

sum119894isinΨsum119895isinΦsum119896isinΩ119864[119903119895119896]119910119894119895119896 In the second and third functions

the expected values of fuzzy random variables (119864[TL] and119864[TR]) which are very difficult to be accurately calculatedare included Hence the following fuzzy random simulationis proposed

Step 1 Let 119899 = 1 119865min = minusinfin 119891(119897119895119896119903119895119896) =

sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))

Step 2 Generate 120596 from Ω according to the probabilitymeasure Pr of the fuzzy random variables119897

119895119896119903119895119896

Step 3 Generate a determined vector 119891(119897119895119896(120596)119903119895119896(120596)) uni-

formly from the 120572-cut of fuzzy vector 119891(119897119895119896(120596)119903119895119896(120596))

Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =

119891(119897119895119896(120596)119903119895119896(120596)) Return to Step 3 and repeat119872 times

Step 5 If 119899 = 119873 set 1198731015840 = 120573119873 and return the 1198731015840th greatestelement in 119865min

1 119865min

2 119865min

119873 as the fitness value

else go to Step 2 and 119899 = 119899 + 1

44 Sectional Genetic Algorithm As mentioned above theproposed model is a nonlinear programming model It isalways with high time complexity and easy to fall into localoptimum using traditional algorithms Hence the geneticalgorithm which has good performance in global searchis suggested to solve this model Moreover according tothe characteristics of the model a sectional chromosomerepresentation and a sectional evolution operation are alsoproposed

441 Sectional Chromosome Representation The solution ofmodel (19) consists of four parts (x y z u) where z can bedetermined in accordance with x Therefore the solutioncan be expressed as (x yu) As is known the x y and uhave different meanings and value ranges So we suggestrepresenting the solution using a chromosome with threesections as Figure 3Thus each gene is expressed as a floating-point number since these variables are nonnegative realnumbers

442 Sectional Crossover and Mutation Operation For eachchromosome consists of three parts the genetic crossoveroperations are also divided as three sections Each sectionhas an independent crossover probability Let the crossoverprobabilities for three sections be 119901

1198881 1199011198882 and 119901

1198883 It is

assumed that 119875(119904)1 = (x1 y1 u1) 119875(119904)2 = (x2 y2 u2) are twoparent chromosomes and 119862(119904)1 119862(119904)2 are the correspondingoffspring chromosomes Then the crossover operation is asfollows

119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)

where 119886 119887 and 119888 are random numbers between 0 and 1which meet the inequalities 119886 lt 119875

1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883

Similarly the chromosome mutation is also carried outwith a three-stage operation Set the three mutation proba-bilities as 119901

1198981 1199011198982 and 119901

1198983 respectively From left to right

successively produce random numbers 1199031 1199032 and 119903

3(isin [0 1])

in each gene position If 1199031le 1199011198981 1199032le 1199011198982 and 119903

3le 1199011198983

then replace the gene using a new randomly generated gene

443 Framework of the Sectional Genetic Algorithm In sum-mary the main steps of proposed sectional genetic algorithmare as follows


Table 1 Parameters of demand nodes

Nodes Demand quantity Total budget Suppliers Order cost(103 ton) (105 CNY) (105 CNY)

Number 1 119873(224 012) 226 Numbers 1 2 3 18 26 22Number 2 119873(156 014) 168 Numbers 1 2 3 22 21 26Number 3 119873(142 022) 147 Numbers 4 5 6 7 22 26 16 15Number 4 119873(558 024) 588 Numbers 1 3 4 5 6 7 18 16 22 26 22 19Number 5 119873(324 018) 286 Numbers 8 9 10 11 12 18 22 22 19 21Number 6 119873(136 010) 186 Numbers 8 9 10 14 26 22Number 7 119873(124 008) 142 Numbers 10 11 12 18 26 22

Step 1 Initialize the genetic algorithm operation parametersincluding population size pop size crossover probability119901119888= (1199011198881 1199011198882 1199011198883) mutation probability 119901

119898= (1199011198981 1199011198982

and1199011198983) and maximal iterations 119878

Step 2 Set 119904 = 0 and initialize genetic population 119875(119904)

Step 21 Generate x randomly in [0 119909119894119895]

Step 22 Generate y randomly in [0min119909119894119895TP119895119896]

Step 23 Generate u randomly in [0 119864[119888119894119895119896]]

Step 24 Check whether or not 119875(119904) meets all constraints ifnot repeat the above operation until all constraints are met

Step 3 Calculate the fitness value of each chromosome usingfuzzy random simulation procedures In this paper theobjective function of model (19) is considered as the fitnessfunction of genetic algorithmThat is

Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)

Step 4 Calculate the selection probability po of ℎth chromo-some 119901

ℎ(ℎ = 1 2 pop size) and choose the chromosome

using wheel selection method to build a new population119875(119904)1015840 while 119901

ℎis calculated by the following equation

119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)

Step 5 Produce offspring chromosomes from parent chro-mosomes using the sectional crossover and mutation oper-ation Check whether or not all constraints are met If notregenerate offspring chromosomes

Step 6 If 119904 gt 119878 end the process or set 119875(119904) = 119862(119904) 119904 = 119904 + 1and return to Step 3

5 Case Study

In the following a practical example inChina is introduced todemonstrate the complete modelling and algorithm process

51 Presentation of Case Problem The Shuibuya hydropowerproject is located in Badong County in the middle reachesof the Qingjiang River It is the first cascaded project onthe Qingjiang mainstream At 233m height and containing15640000m3 of material it is the tallest concrete face rock-fill dam in the world For constructing the project hugeamounts of construction material are consumed every dayTaking cement as example more than 2 thousand tons isused per day on average Due to the very big demandquantity it is impossible to purchase the cement from onesupplier Hence in practice multiple suppliers have beenselected to supply the cement Meanwhile for convenience ofstorage and usage the construction company has built sevenstorehouses to store the cement Hence there are multipledemand nodes and multiple suppliers for cement purchasingin the construction project Now according to the demandinformation the construction company needs to purchaseabout 14 thousand tonsrsquo cement in the next week Then theproblem the construction company currently faces is how toallocate the purchased quantity among multiple suppliers formultiple demand nodes

In this section we will discuss the order quantity alloca-tion problem of cement in Shuibuya hydropower project Itis assumed that there are 7 demand nodes and 12 suppliersEach supplier has three types of trucks light truck mediumtruck and heavy truck Different types of trucks will leadto different transportation alternatives transportation costsand transportation risks The positions of the demand nodesand the related links between the demand nodes and sup-pliers are illustrated in Figure 4 The data on demand nodesare shown in Table 1 and the data on suppliers are statedin Table 2 Because of the uncertain environment of orderquantity allocation and the information asymmetry betweenthe construction company and cement suppliers the dataon transportation cost percentage of late delivered unitsand percentage of rejected units are modeled as triangularfuzzy random variables In the considered case the differenttransportation alternatives means different transportationvehicles For each pair from one supplier to one demandnode the unit transportation cost and risk are linearly depen-dent on the transportation distance Hence the coefficients119888119894119895119896

can be split into two parts 119888119895119896 unit transportation

cost of supplier 119895 using transportation alternative 119896 and 119889119894119895


Table 2 Parameters of suppliers (part 1)

Suppliers Capacity Price Penalty cost Minimal order Maximal late rate(103 ton) (103 CNYton) (103 CNYton) (103 tons) ()

Number 1 184 064 051 080 84Number 2 236 060 048 120 86Number 3 265 061 049 180 82Number 4 128 061 049 070 86Number 5 146 062 050 075 82Number 6 224 062 051 120 88Number 7 196 060 048 085 76Number 8 162 062 051 080 78Number 9 148 062 052 060 84Number 10 188 059 047 140 78Number 11 124 062 050 070 82Number 12 153 062 050 040 80

1

2

3

4

5

6

7

Supplier 1

Supplier 8

Supplier 3Supplier 7

Supplier 6Supplier 4Supplier 2

Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11

Demanded node TransportationSupplier

Figure 4 Order quantity allocation of cement in Shuibuya hydro-power project

transportation distance from supplier 119895 to demand node 119894The detailed data on 119888

119895119896are stated in Table 3

52 Computation Results In order to run the program for theproposed SGA algorithm the algorithm parameters for thecase problem were set as follows population size pop size =40 iteration number 119878 = 500 three segmental probabilitiesof crossover 119901

1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three

mutation probabilities 1199011198981= 02 119901

1198982= 01 and 119901

1198983= 005

After 4574 minutes on average the optimal solutionsof model (16) were determined with fitness value of 0734Figure 5 shows the optimal order quantity allocation schemeand selected transportation alternatives Based on this resultsit is easy to calculate the values of the three objectivesThe total purchase cost is 1261 million yuan while the totaldelay risk and total defect risk are 265 and 325 respectivelyFrom the results it can be seen that most of demand nodes

needmultiple suppliers to providematerialMeanwhile somesuppliers are also supplying material for multiple demandnodes which reflects that it is meaningful to consider thetransportation alternatives on the lower level

53 Model Analysis In this paper we study a multide-mand multisource order quantity allocation problem using abilevel multiobjective programming Traditionally the orderquantity allocation problem is considered as a single-levelmodel To test the significance of the proposed model acomparison between the bilevelmodel and single-levelmodelis conducted At the same time the interactivity between themodels and objectives are also considered The results areshown in Table 4

From the results in Table 4 it can be seen that thesolutions of the bilevel and single-level models are very nearonly if the objective of total cost minimization is consideredHowever oncewe consider the risk objectives the single-levelmodel can always obtain better objective values for the upperlevel which seems like the single-level model is better thanthe bilevel Nevertheless it is not true Let us consider the caseof 1199081= 05 119908

2= 03 and 119908

3= 02 Compared to the

bilevel model the total cost of the upper level is decreasedto 1247 from 1261 with a decrement of 015 However at thesame time the total transportation and penalty costs of thelower level are increased to 231 from 201 with an incrementof 03 Obviously relative to the bilevel model the suppliershave to expend higher costs which even exceeds the increasedincome of purchaser if the single-level model is used As aresult the suppliers have to raise the price which may reducethe stability of the supply chainMoreover the administrativecost will also be increased if the purchaser takes chargeof the selection of transportation alternatives Hence it ismeaningful to consider the order quantity allocation problemusing a bilevel model

In addition the relationship among the objectives shouldalso be discussed Figure 6 describes the trend lines of thethree objectives of upper level model and the sum of lowerlevelrsquos objectives It can be seen that the delay risk has apositive correlationwith defect risk and a negative correlation



Supplier Alternative 119888119895119896(CNYTNsdotKM)

119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)


Table 4 Comparison analysis to single-level and bilevel model with different objectives

Weight Models Upper level objectives Lower level objective(1199081 1199082 1199083) TC 119864[TL] 119864[TR] sum119864[Tc

119895]

(1 0 0) Bilevel 1053 290 343 187Single-level 1048 288 345 192





184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032

Index of demanded node 1

Purchased quantity

Index of supplier 1

Allocated order quantity for supplier 3

Transported quantity with 3 alternatives

0

0

Figure 5 Results on order quantity allocation and transportation alternatives selection

with the total transportation and penalty costs The total costdoes not specifically relate to any other single objective Theresult also shows the necessity to considermultiple objectivessince the other objectives will take very bad values if only asingle objective is considered in the model

54 Algorithm Evaluation To demonstrate the effectivenessand efficiency of the proposed sectional genetic algorithmbased on fuzzy random simulation (FRS-SGA) an analysis of

parameter selection and a performance comparison to otheralgorithms are carried out successively

541 Parameter Selection for the FRS-SGA For genetic algo-rithm three types of parameters will observably influencethe performance of the algorithm that is population sizecrossover probability and mutation probability Many schol-ars such as Zouein et al [30] Dimou and Koumousis [31]and Xu and Yao [32] have given suggestions on the proper


S single-levelB bilevel

Delay riskDefect riskTotal costTransportation and penalty costs

S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)

Figure 6 Relation among the multiple objectives

Table 5 The parameters selection of the proposed FRS-SGA

pop size 1199011198881

1199011198882

1199011198883

Computing time Fitness value

30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734

parameters In this paper the parameters are set from theresults of 20 preliminary experiments that were conductedto observe the behavior of the algorithm at different param-eter settings Table 5 shows the experiment result on theparameter of population size and crossover probability Itcan be seen that the computing time is extended with theincrease of population size At the same time compared to thesame crossover probability the proposed sectional one hasbetter performance both in the computing time and in fitnessvalue We also carried out experiments on the mutationprobability and get a similar result to the crossover probabil-ity

542 Algorithm Comparison The performances of the pro-posed FRS-SGA are also compared with the traditional

genetic algorithm based on fuzzy random simulation (FRS-GA) and particle swarm optimization based on fuzzy randomsimulation (FRS-PSO) over 50 experiments using eight differ-ent problem sizes In the experiments the three algorithmsuse the same fitness value function which are calculated byfuzzy random simulation in the evolutionThe performancesof the three algorithms in the experiment are shown inTable 6 In the table ldquoaccuracyrdquo means the frequency to findoptimal solution while ldquoACTrdquo means the average computingtime

From Table 6 it can be seen that (1) all three algorithmsare able to obtain the best solution within 50 runs for all theeight problems (2) SGAalgorithmhas a higher accuracy thanPSO and faster computing speed thanGA (3) SGA algorithmcan deal with these problems whose sizes are within 2000variables while the GA is more efficient to deal with thelarge scale problem From this it can be concluded that theproposed SGA algorithm is efficient to solve the proposedbilevel multidemand multisource order quantity allocationproblem in most cases

6 Conclusions

In this paper we consider a multidemand multisource orderquantity allocation problem with multiple transportationalternatives under uncertain environment In this problemtwo levels of decision makers should be considered thepurchaser on the upper level and the suppliers on the lowerlevel As the decisionmaker on the upper level the purchaserhopes to allocate order quantity among multiple suppliersfor multiple demand nodes To minimize the total purchasecost the total delay risk and the total defect risk are thedecision objectives As the decision makers on the lowerlevel to reduce the transportation and penalty costs thesuppliers hope to optimize the transportation alternativesIn the decision making process some uncertain parametersalso have to be considered In order to solve this problema bilevel multiobjective programming model with mixeduncertain parameters is established Then the bilevel modelis transformed into a single model by writing the lowerlevel model with Kuhn-Tucker conditions Next a sectionalgenetic algorithm based on fuzzy random simulation isproposed to deal with the transformed model Finally acase study is presented to demonstrate the practicality andefficiency of the model and algorithm The results show thatthe bilevelmultiobjective order quantity allocationmodel hasa greater ability in maintaining stability of the supply chainthan a single-level model In addition the proposed sectionalgenetic algorithm based on fuzzy random simulation is alsoefficient to deal with the order quantity allocation problemsin most cases

In this study only one type of production or material isconsidered into the model In the future we will focus on themultidemand multisource order quantity allocation problemwith multiple productions In addition the order quantityallocation problems under other uncertain environments alsoneed to be considered


Table 6 Comparison among SGA GA and PSO with different problem sizes

Number Size of tested problem FRS-SGA FRS-GA FRS-PSO119868 119869 119870 Variables Accuracy ACT Accuracy ACT Accuracy ACT

1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research was supported by the National Science Foun-dation for the Key Program of NSFC (Grant no 70831005)the National Natural Science Foundation of China (Grantno 71301109) and the Western and Frontier Region Projectof Humanity and Social Sciences Research Ministry ofEducation of China (Grant no 13XJC630018)

References

[1] L de Boer E Labro and P Morlacchi ldquoA review of methodssupporting supplier selectionrdquo European Journal of Purchasingand Supply Management vol 7 no 2 pp 75ndash89 2001

[2] M Setak S Sharifi and A Alimohammadian ldquoSupplier selec-tion and order allocation models in supply chain managementa reviewrdquo World Applied Sciences Journal vol 18 no 1 pp 55ndash72 2012

[3] Z Li W K Wong and C K Kwong ldquoAn integrated modelof material supplier selection and order allocation using fuzzyextended AHP and multiobjective programmingrdquo Mathemat-ical Problems in Engineering vol 2013 Article ID 363718 14pages 2013

[4] S H Ghodsypour and C OrsquoBrien ldquoA decision support systemfor supplier selection using an integrated analytic hierarchyprocess and linear programmingrdquo International Journal ofProduction Economics vol 56-57 pp 199ndash212 1998

[5] S H Ghodsypour and C OrsquoBrien ldquoThe total cost of logisticsin supplier selection under conditions of multiple sourcingmultiple criteria and capacity constraintrdquo International Journalof Production Economics vol 73 no 1 pp 15ndash27 2001

[6] N Xu and L Nozick ldquoModeling supplier selection and the useof option contracts for global supply chain designrdquo Computersand Operations Research vol 36 no 10 pp 2786ndash2800 2009

[7] H Fazlollahtabar I Mahdavi M T Ashoori S Kaviani andNMahdavi-Amiri ldquoAmulti-objective decision-making processof supplier selection and order allocation for multi-periodscheduling in an electronic marketrdquo International Journal ofAdvancedManufacturing Technology vol 52 no 9ndash12 pp 1039ndash1052 2011

[8] Y Tang Z Wang and J-A Fang ldquoController design forsynchronization of an array of delayed neural networks usinga controllable probabilistic PSOrdquo Information Sciences vol 181no 20 pp 4715ndash4732 2011

[9] Y Tang Z Wang and J-A Fang ldquoFeedback learning particleswarm optimizationrdquo Applied Soft Computing Journal vol 11no 8 pp 4713ndash4725 2011

[10] WZhu Y Tang J-A Fang andWZhang ldquoAdaptive populationtuning scheme for differential evolutionrdquo Information Sciencesvol 223 pp 164ndash191 2013

[11] M Jafari Songhori M Tavana A Azadeh and M H KhakbazldquoA supplier selection and order allocation model with multipletransportation alternativesrdquo International Journal of AdvancedManufacturing Technology vol 52 no 1ndash4 pp 365ndash376 2011

[12] E Kopytov and D Abramov ldquoMultiple-criteria analysis andchoice of transportation alternatives in multimodal freighttransport systemrdquoTransport and Telecommunication vol 13 no2 pp 148ndash158 2012

[13] R Kawtummachai and N V Hop ldquoOrder allocation in amultiple-supplier environmentrdquo International Journal of Pro-duction Economics vol 93-94 pp 231ndash238 2005

[14] W Dullaert B Maes B Vernimmen and F Witlox ldquoAn evo-lutionary algorithm for order splitting with multiple transportalternativesrdquo Expert Systems with Applications vol 28 no 2 pp201ndash208 2005

[15] P C Yang H M Wee S Pai and Y F Tseng ldquoSolving astochastic demandmulti-product supplier selectionmodel withservice level and budget constraints using Genetic AlgorithmrdquoExpert Systems with Applications vol 38 no 12 pp 14773ndash14777 2011

[16] J-L Zhang and M-Y Zhang ldquoSupplier selection and purchaseproblem with fixed cost and constrained order quantitiesunder stochastic demandrdquo International Journal of ProductionEconomics vol 129 no 1 pp 1ndash7 2011

[17] N Esfandiari and M Seifbarghy ldquoModeling a stochasticmulti-objective supplier quota allocation problem with price-dependent orderingrdquo Applied Mathematical Modelling vol 37no 8 pp 5790ndash5800 2013

[18] M Kumar P Vrat and R Shankar ldquoA fuzzy programmingapproach for vendor selection problem in a supply chainrdquoInternational Journal of Production Economics vol 101 no 2 pp273ndash285 2006

[19] A Amid S H Ghodsypour and C OrsquoBrien ldquoA weighted max-min model for fuzzy multi-objective supplier selection in asupply chainrdquo International Journal of Production Economicsvol 131 no 1 pp 139ndash145 2011


[20] S Nazari-Shirkouhi H Shakouri B Javadi and A KeramatildquoSupplier selection and order allocation problem using atwo-phase fuzzy multi-objective linear programmingrdquo AppliedMathematical Modelling vol 37 no 22 pp 9308ndash9323 2013

[21] H Kwakernaak ldquoFuzzy random variables-I Definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[22] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[23] C A Weber and J R Current ldquoA multiobjective approach tovendor selectionrdquo European Journal of Operational Researchvol 68 no 2 pp 173ndash184 1993

[24] J Xu and Y Liu ldquoMulti-objective decisionmakingmodel underfuzzy random environment and its application to inventoryproblemsrdquo Information Sciences vol 178 no 14 pp 2899ndash29142008

[25] J Xu F Yan and S Li ldquoVehicle routing optimization with softtime windows in a fuzzy random environmentrdquo TransportationResearch Part E Logistics and Transportation Review vol 47 no6 pp 1075ndash1091 2011

[26] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecisionMaking Springer 2011

[28] A Charnes and W W Cooper ldquoChance-constrained program-mingrdquoManagement Science vol 6 no 1 pp 73ndash79 1959

[29] M Sakawa and I Nishizaki Cooperative and NoncooperativeMulti-Level Programming Springer 2009

[30] P P Zouein H Harmanani and A Hajar ldquoGenetic algorithmfor solving site layout problem with unequal-size and con-strained facilitiesrdquo Journal of Computing in Civil Engineeringvol 16 no 2 pp 143ndash151 2002

[31] C K Dimou and V K Koumousis ldquoGenetic algorithmsin competitive environmentsrdquo Journal of Computing in CivilEngineering vol 17 no 3 pp 142ndash149 2003

[32] J Xu and L Yao Random-Like Multiple Objective DecisionMaking Springer Berlin Germany 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


has its own limited capacity Hence in themultidemandmul-tisource order allocation problem once the order quantityis determined for each supplier they have to optimize theirtransportation alternatives In the existing researches fewconsider the vital role of the transportation alternatives in theorder allocation process although it is significantly importantto the supplier evaluation and order allocation decisionsGhodsypour and OrsquoBrien [5] proposed a mixed integernonlinear programming model to solve the multisourceproblem which took into account the total cost of logisticsincluding net price storage transportation and orderingcosts Kawtummachai and Hop [13] constructed an ordersquantity allocationmethod to the selected supplier in order tooptimize the purchase cost within the acceptable percentageof on-time delivery which is influenced by transportationalternatives to guarantee high service levels to the retailersDullaert et al [14] considered a combinatorial optimizationmodel for determining the optimal mix of transportationalternatives to minimize total logistics costs when goodsare shipped from a supplier to a customer Jafari Songhoriet al [11] first explicitly presented a decision frameworkwhich considered different transportation alternatives fromsuppliers to demander

It should be noted that in all these papers the orderallocation and transportation alternatives are determinedby the same decision maker (ie the purchaser) Howeverin practice only the price and order quantity are writteninto agreements in most cases As for how to transportmaterial or goods it is often directed by the suppliersHence two decision making processes should be consideredFirst the purchaser such as the construction company in aconstruction project decides how to allocate order quantityamong multiple suppliers for multiple demand nodes Thenbased on the allocated order quantity each supplier willdecide how to transport them to demand nodes throughmultiple transportation alternatives In this situation themultidemand multisource order allocation problem shouldbe considered as a bilevel decision making problem

Besides for the bilevel structure the uncertainty whichis also an important factor will be considered in this paperMany papers have paid attention to uncertain supplierselection and order allocation problem Xu and Nozick [6]proposed a two-stage mixed integer stochastic programmingmodel to optimize supplier selection Yang et al [15] J-LZhang and M-Y Zhang [16] and Esfandiari and Seifbarghy[17] discussed the supplier selection and order allocationproblem with stochastic demand Kumar et al [18] stud-ied a multiobjective integer programming vendor selectionproblem with fuzzy parameters Amid et al [19] and Nazari-Shirkouhi et al [20] applied fuzzy programming theory todeal with the multiobjective supplier selection and orderallocation problem In the researches mentioned abovefuzziness and randomness were often considered separateaspects But in reality we may face a hybrid uncertainenvironment where fuzziness and randomness coexist in adecision making process For example transportation costfrom one supplier to one demand node is often modeled asa stochastic parameter However in a bilevel environmentbecause of the information asymmetry between the two levels

of decision makers the upper decision maker may modifythe accuracy of the given transportation cost As a result hemay modify the uncertain transportation cost using a fuzzyvariable In this situation the cost is a hybrid of fuzzy andrandom factors and the fuzzy random variable which wasfirst proposed by Kwakernaak [21] and then extended by Puriand Ralescu [22] can be a useful tool to deal with it

Hence we face a bilevel multidemand multisource orderquantity allocation problem under a fuzzy random environ-ment In the problem two levels of decision makers mustbe considered On the upper level the purchaser attempts toallocate order quantity amongmultiple suppliers for multipledemand nodes with the objectives of total purchase cost andtotal purchase risk minimization On the lower level eachsupplier will select the transportation alternatives from thesupplier to demand nodes with the objective of total trans-portation and penalty costs minimization Simultaneouslysome uncertain parameters in the problem such as demandtransportation cost and risk coefficients must also be con-sidered To deal with this problem a bilevel multiobjectiveprogramming model with uncertain coefficients is proposedSpecifically some uncertain coefficients are modeled as fuzzyrandom variables In addition to search the optimal solutionof proposed model a solution method based on Kuhn-Tucker conditions sectional genetic algorithm (SGA) andfuzzy random simulation is also proposed Finally resultsand comparison analysis of a case study are presented todemonstrate the practicality and efficiency of the proposedmodel and algorithm

The remainder of this paper is structured as follows inSection 2 an introduction to the bilevel multiobjective mul-tidemand multisource order allocation problem is presentedalong with the motivation for employing uncertain vari-ables in the problem A bilevel multiobjective programmingmodel under a mixed uncertain environment is establishedin Section 3 In addition a solution method is illustratedin Section 4 In Section 5 a case study from a practicalexample is given to show the validity and efficiency of theproposedmodel and algorithmConclusions and a discussionregarding further research are remarked in Section 6

2 Key Problem Statement

The problem considered in this paper is a bilevel multiobjec-tive multidemandmultisource order allocation problem withmultiple transportation alternatives under a mixed uncertainenvironment In this section we explain why this problemshould be considered a bilevel multiobjective programmingmodel and why the demand transportation cost and timeare able to be modeled as random or fuzzy random variables

21 Conflict Description for Order Quantity Allocation Inlarge scale construction projects many construction mate-rials are consumed in a great deal every day these mate-rials are often purchased from multiple material suppliersIn addition because of the big construction ground it isimpossible to store these materials in a storehouse Hencethere are often multiple demand nodes which serve for








3 Modelling








Upper level



system


Randomenvironment


Twostakeholders

Conflictof

interests


Multiple demands

Demandnode 1

Demandnode 2

Demandnode i

Demandnode I


Ord

er q

uant

ity

Order quantity

Multiple sources




Decision variable



system


Lower level


















Obj

ectiv

es

Obj

ectiv

e





Conflict+

Queryto

It is about


a

b

x0010


random variable

10

a

b


cc


120583c

120583

120593 c(x)120583

c

c2

c1

c3


120593c =1











119897max119895


Random Variables





















min119909119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 (1)


119897119895119896119910119894119895119896

Since 119897119895119896



119897119895119896119910119894119895119896] according to the




min119909119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 (2)





min119909119894119895

119864 (TR) = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896 (3)






119894


Pr

sum

119895isinΦ

119909119894119895ge 119876119894

ge 120572 forall119894 isin Ψ (4)




denoted as

sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ (5)




sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ (6)





sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119894 forall119894 isin Ψ (7)


119894119895gt


(ie 119909119894119895= 0) 119911



119909119894119895

119872



(ULM)

min119909119894119895 119911119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895

min119909119894119895 119911119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896

min119909119894119895 119911119894119895

119864 [TR] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119909119894119895ge 0 119911


where 119910119894119895119896


(9)




119894119895119896



119894119895119896for






119895119864[119903119894119895119896]119910119894119895119896 Hence the


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896

+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

(10)



119896isinΩ119910119894119895119896


119894119895for


sum119896isinΩ119910119894119895119896


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909



119894isinΨ119910119894119895119896

of transportation



sum

119894isinΨ



119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895



sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896


st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(14)




min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















3 Modelling








Upper level



system


Randomenvironment


Twostakeholders

Conflictof

interests


Multiple demands

Demandnode 1

Demandnode 2

Demandnode i

Demandnode I


Ord

er q

uant

ity

Order quantity

Multiple sources




Decision variable



system


Lower level


















Obj

ectiv

es

Obj

ectiv

e





Conflict+

Queryto

It is about


a

b

x0010


random variable

10

a

b


cc


120583c

120583

120593 c(x)120583

c

c2

c1

c3


120593c =1











119897max119895


Random Variables





















min119909119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 (1)


119897119895119896119910119894119895119896

Since 119897119895119896



119897119895119896119910119894119895119896] according to the




min119909119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 (2)





min119909119894119895

119864 (TR) = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896 (3)






119894


Pr

sum

119895isinΦ

119909119894119895ge 119876119894

ge 120572 forall119894 isin Ψ (4)




denoted as

sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ (5)




sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ (6)





sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119894 forall119894 isin Ψ (7)


119894119895gt


(ie 119909119894119895= 0) 119911



119909119894119895

119872



(ULM)

min119909119894119895 119911119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895

min119909119894119895 119911119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896

min119909119894119895 119911119894119895

119864 [TR] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119909119894119895ge 0 119911


where 119910119894119895119896


(9)




119894119895119896



119894119895119896for






119895119864[119903119894119895119896]119910119894119895119896 Hence the


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896

+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

(10)



119896isinΩ119910119894119895119896


119894119895for


sum119896isinΩ119910119894119895119896


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909



119894isinΨ119910119894119895119896

of transportation



sum

119894isinΨ



119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895



sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896


st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(14)




min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Upper level



system


Randomenvironment


Twostakeholders

Conflictof

interests


Multiple demands

Demandnode 1

Demandnode 2

Demandnode i

Demandnode I


Ord

er q

uant

ity

Order quantity

Multiple sources




Decision variable



system


Lower level


















Obj

ectiv

es

Obj

ectiv

e





Conflict+

Queryto

It is about


a

b

x0010


random variable

10

a

b


cc


120583c

120583

120593 c(x)120583

c

c2

c1

c3


120593c =1











119897max119895


Random Variables





















min119909119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 (1)


119897119895119896119910119894119895119896

Since 119897119895119896



119897119895119896119910119894119895119896] according to the




min119909119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 (2)





min119909119894119895

119864 (TR) = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896 (3)






119894


Pr

sum

119895isinΦ

119909119894119895ge 119876119894

ge 120572 forall119894 isin Ψ (4)




denoted as

sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ (5)




sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ (6)





sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119894 forall119894 isin Ψ (7)


119894119895gt


(ie 119909119894119895= 0) 119911



119909119894119895

119872



(ULM)

min119909119894119895 119911119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895

min119909119894119895 119911119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896

min119909119894119895 119911119894119895

119864 [TR] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119909119894119895ge 0 119911


where 119910119894119895119896


(9)




119894119895119896



119894119895119896for






119895119864[119903119894119895119896]119910119894119895119896 Hence the


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896

+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

(10)



119896isinΩ119910119894119895119896


119894119895for


sum119896isinΩ119910119894119895119896


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909



119894isinΨ119910119894119895119896

of transportation



sum

119894isinΨ



119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895



sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896


st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(14)




min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Obj

ectiv

es

Obj

ectiv

e





Conflict+

Queryto

It is about


a

b

x0010


random variable

10

a

b


cc


120583c

120583

120593 c(x)120583

c

c2

c1

c3


120593c =1











119897max119895


Random Variables





















min119909119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 (1)


119897119895119896119910119894119895119896

Since 119897119895119896



119897119895119896119910119894119895119896] according to the




min119909119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 (2)





min119909119894119895

119864 (TR) = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896 (3)






119894


Pr

sum

119895isinΦ

119909119894119895ge 119876119894

ge 120572 forall119894 isin Ψ (4)




denoted as

sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ (5)




sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ (6)





sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119894 forall119894 isin Ψ (7)


119894119895gt


(ie 119909119894119895= 0) 119911



119909119894119895

119872



(ULM)

min119909119894119895 119911119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895

min119909119894119895 119911119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896

min119909119894119895 119911119894119895

119864 [TR] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119909119894119895ge 0 119911


where 119910119894119895119896


(9)




119894119895119896



119894119895119896for






119895119864[119903119894119895119896]119910119894119895119896 Hence the


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896

+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

(10)



119896isinΩ119910119894119895119896


119894119895for


sum119896isinΩ119910119894119895119896


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909



119894isinΨ119910119894119895119896

of transportation



sum

119894isinΨ



119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895



sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896


st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(14)




min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























min119909119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 (1)


119897119895119896119910119894119895119896

Since 119897119895119896



119897119895119896119910119894119895119896] according to the




min119909119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 (2)





min119909119894119895

119864 (TR) = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896 (3)






119894


Pr

sum

119895isinΦ

119909119894119895ge 119876119894

ge 120572 forall119894 isin Ψ (4)




denoted as

sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ (5)




sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ (6)





sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119894 forall119894 isin Ψ (7)


119894119895gt


(ie 119909119894119895= 0) 119911



119909119894119895

119872



(ULM)

min119909119894119895 119911119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895

min119909119894119895 119911119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896

min119909119894119895 119911119894119895

119864 [TR] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119909119894119895ge 0 119911


where 119910119894119895119896


(9)




119894119895119896



119894119895119896for






119895119864[119903119894119895119896]119910119894119895119896 Hence the


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896

+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

(10)



119896isinΩ119910119894119895119896


119894119895for


sum119896isinΩ119910119894119895119896


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909



119894isinΨ119910119894119895119896

of transportation



sum

119894isinΨ



119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895



sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896


st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(14)




min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

119894isinΨ

119909119894119895ge MO

119895 forall119895 isin Φ (6)





sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119894 forall119894 isin Ψ (7)


119894119895gt


(ie 119909119894119895= 0) 119911



119909119894119895

119872



(ULM)

min119909119894119895 119911119894119895


sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895

min119909119894119895 119911119894119895

119864 [TL] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896

min119909119894119895 119911119894119895

119864 [TR] = sum119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119909119894119895ge 0 119911


where 119910119894119895119896


(9)




119894119895119896



119894119895119896for






119895119864[119903119894119895119896]119910119894119895119896 Hence the


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896

+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

(10)



119896isinΩ119910119894119895119896


119894119895for


sum119896isinΩ119910119894119895119896


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909



119894isinΨ119910119894119895119896

of transportation



sum

119894isinΨ



119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895



sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896


st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(14)




min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

119894isinΨ



119896isinΩ119864[119897119894119895119896]119910119894119895119896119909119894119895le

119897max119895



sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(LLM)

min119910119894119895119896

119864 [119879119888119895]

= sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896


st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(14)




min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ


119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119909119894119895

119872


119909119894119895ge 0 119911


min119910119894119895119896

119864 [119879119888119895] = sum

119894isinΨ

sum

119896isinΩ

119864 [119888119894119895119896] 119910119894119895119896+ sum

119894isinΨ

sum

119896isinΩ

CP119895119864 [119903119894119895119896] 119910119894119895119896 forall119895 isin Φ

st 119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897



(15)




119894119895 1199062119894119895 1199063119895119896 1199064119894119895 and 1199065

119894119895119896 the


min119909119894119895 119911119894119895

(TC 119864 [TL] 119864 [TR]) = (sum119894isinΨ

sum

119895isinΦ

119875119895119909119894119895+ sum

119894isinΨ

sum

119895isinΦ

119862119894119895119911119894119895 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119897119895119896] 119910119894119895119896 sum

119894isinΨ

sum

119895isinΦ

sum

119896isinΩ

119864 [119903119895119896] 119910119894119895119896)

st Pr

sum

119895isinΦ

119909119894119895ge 119876119894


sum

119894isinΨ

119909119894119895le 119878119895 forall119895 isin Φ

sum

119894isinΨ

119909119894119895ge MO


sum

119895isinΦ

119875119895119909119894119895+ sum

119895isinΦ

119862119894119895119911119894119895le 119861119895 forall119894 isin Ψ

119909119894119895

119872


119864 [119888119894119895119896] + CP

119895119864 [119903119894119895119896] minus 1199061

119894119895+ 1199062

119894119895+ 1199063

119895119896+ 1199064

119894119895minus 1199065


1199061

119894119895(sum

119896isinΩ


1199062

119894119895(sum

119896isinΩ

119910119894119895119896minus119872119909



1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1199063

119895119896(sum

119894isinΨ


1199064

119894119895(sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896minus 119897


1199065


119909119894119895le sum

119896isinΩ

119910119894119895119896le 119872119909


sum

119894isinΨ


sum

119896isinΩ

119864 [119897119894119895119896] 119910119894119895119896le 119897


119909119894119895ge 0 119910

119894119895119896ge 0 119906

1

119894119895 1199062

119894119895 1199063

119895119896 1199064

119894119895 1199065

119894119895119896ge 0 119911


(16)



119878119905(x y z u)

=

1 119865119905(x y z u) lt 119865min

119905

119865119905(x y z u) minus 119865max

119905

119865min119905minus 119865

max119905

119865min119905le 119865119905(x y zu) lt 119865max

119905

0 119865119905(x y z u) ge 119865max

119905

(17)



119865max119905



119905


119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3 (18)



max(xyzu)

119865 (x y z u) =3

sum

119905=1

119908119905119878119905(119865119905(x y z u))

st 119878119905(119865119905(x y z u)) ge 119878min

119905 forall119905 = 1 2 3

(x y z u) isin Λ

(19)


1+





119865 (x y zu) =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(20)




xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












xij ge 0 yijk ge 0


IJ u211 u2

11 u2IJ u3

11 u3JK u4

11 u4IJ u5

111 u5IJK

u1ij u2

ij u3jk u4

ij u5ijk ge 0


where 1198651= sum



and 1198653

=




sum3

119905=1(119908119905(119865119905(x y z u) minus 119865max

119905)(119865

min119905minus 119865

max119905))


119895119896119903119895119896



Step 4 If 119891(119897119895119896(120596)119903119895119896(120596)) ge 119865min

119899 then let 119865min =



1 119865min

2 119865min






1198881 1199011198882 and 119901

1198883 It is


119862 (119904)1= 119886

x1

0

0

+ (1 minus 119886)

x2

0

0

+ 119887

0

y1

0

+ (1 minus 119887)

0

y2

0

+ 119888

0

0

u1

+ (1 minus 119888)

0

0

u2

119862 (119904)2= (1 minus 119886)

x1

0

0

+ 119886

x2

0

0

+ (1 minus 119887)

0

y1

0

+ 119887

0

y2

0

+ (1 minus 119888)

0

0

u1

+ 119888

0

0

u2

(21)


1198881 119887 lt 119875

1198882 and 119888 lt 119875

1198883


1198981 1199011198982 and 119901



3(isin [0 1])


3le 1199011198983








119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119898= (1199011198981 1199011198982








Fitness =3

sum

119905=1

119908119905(119865119905(x y z u) minus 119865max

119905)

119865min119905minus 119865

max119905

(22)





119901ℎ=

Fitness (119875ℎ (119904))

sumpop sizeℎ=1

fitness (119875ℎ (119904))

(23)



5 Case Study










1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1

2

3

4

5

6

7

Supplier 1

Supplier 8



Supplier 5

Supplier 12

Supplier 10

Supplier 9

Supplier 11






1198881= 07 119901

1198882= 05 and 119901

1198883= 03 and three


1198982= 01 and 119901

1198983= 005





2= 03 and 119908







119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119897119895119896()

119903119895119896()

1

1 (375 119888 422) (779 119897 847) (205 119903 234)119888 sim N(385 005) 119897 sim N(804 012) 119903 sim N(226 002)

2 (482 119888 516) (733 119897 819) (284 119903 308)119888 sim N(496 004) 119897 sim N(775 010) 119903 sim N(296 002)

3 (275 119888 346) (807 119897 929) (245 119903 272)119888 sim N(322 002) 119897 sim N(854 010) 119903 sim N(261 002)

2

1 (376 119888 459) (841 119897 957) (200 119903 232)119888 sim N(412 004) 119897 sim N(904 010) 119903 sim N(218 001)

2 (398 119888 464) (843 119897 858) (258 119903 265)119888 sim N(426 002) 119897 sim N(858 001) 119903 sim N(260 001)

3 (450 119888 499) (720 119897 779) (345 119903 366)119888 sim N(482 002) 119897 sim N(748 005) 119903 sim N(357 002)

3

1 (428 119888 475) (616 119897 673) (284 119903 308)119888 sim N(442 002) 119897 sim N(653 004) 119903 sim N(292 002)

2 (435 119888 516) (825 119897 854) (314 119903 336)119888 sim N(471 004) 119897 sim N(839 002) 119903 sim N(327 002)

3 (362 119888 411) (750 119897 808) (281 119903 310)119888 sim N(380 003) 119897 sim N(788 006) 119903 sim N(294 003)

4

1 (465 119888 534) (763 119897 847) (265 119903 302)119888 sim N(497 005) 119897 sim N(806 01) 119903 sim N(288 004)

2 (381 119888 431) (923 119897 1002) (304 119903 318)119888 sim N(396 003) 119897 sim N(958 008) 119903 sim N(310 002)

3 (302 119888 376) (839 119897 916) (265 119903 277)119888 sim N(338 004) 119897 sim N(868 006) 119903 sim N(271 001)

5

1 (305 119888 390) (692 119897 762) (211 119903 256)119888 sim N(342 008) 119897 sim N(730 005) 119903 sim N(233 003)

2 (365 119888 404) (666 119897 783) (199 119903 240)119888 sim N(368 003) 119897 sim N(724 01) 119903 sim N(213 002)

3 (347 119888 406) (825 119897 917) (236 119903 250)119888 sim N(366 004) 119897 sim N(874 008) 119903 sim N(243 001)

6

1 (450 119888 486) (688 119897 782) (307 119903 348)119888 sim N(473 002) 119897 sim N(742 010) 119903 sim N(325 003)

2 (429 119888 470) (616 119897 631) (303 119903 347)119888 sim N(453 003) 119897 sim N(621 001) 119903 sim N(318 003)

3 (415 119888 446) (889 119897 968) (203 119903 239)119888 sim N(430 003) 119897 sim N(912 003) 119903 sim N(224 003)

7

1 (397 119888 464) (707 119897 783) (226 119903 265)119888 sim N(415 005) 119897 sim N(738 003) 119903 sim N(239 002)

2 (425 119888 491) (738 119897 809) (271 119903 294)119888 sim N(464 006) 119897 sim N(769 005) 119903 sim N(282 002)

3 (338 119888 371) (692 119897 768) (309 119903 336)119888 sim N(344 003) 119897 sim N(726 006) 119903 sim N(317 002)

8

1 (352 119888 421) (809 119897 938) (226 119903 243)119888 sim N(384 006) 119897 sim N(878 010) 119903 sim N(228 002)

2 (377 119888 441) (594 119897 687) (234 119903 268)119888 sim N(414 002) 119897 sim N(639 006) 119903 sim N(248 002)

3 (361 119888 399) (582 119897 671) (250 119903 280)119888 sim N(380 002) 119897 sim N(626 008) 119903 sim N(265 003)




119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119895]






184

0

080

065

039

0

0

064

2481

1184

3064

0

0

133

0

052185

2 2185

0

0 0

021

10065

0

1863

6121

7065

115

086064

0

0

09

041

0

06408

0

066

6064

3201

4128

5146

7131

023060

0

057

0

10083

12057

7140

0

070

08065022

12096

9087

11070

0

027

5360

8107

0060

036

0

10105

8055

06

156

057

048

0

0230

032


Purchased quantity

Index of supplier 1



0

0









S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












S(0

10)

S(0

01)

B(0

10)

S(0

33

03

30

33)

S(0

50

30

2)

B(0

01)

B(0

33

03

30

33)

B(0

50

30

2)

B(1

00)

S(1

00)



pop size 1199011198881

1199011198882

1199011198883


30

07 07 07 4973 066305 05 05 3592 067903 03 03 5473 063207 05 03 3106 0690

40

07 07 07 5933 067105 05 05 4605 071603 03 03 4998 066107 05 03 4574 0734

50

07 07 07 6432 064805 05 05 5627 071303 03 03 8283 065907 05 03 4956 0734





6 Conclusions






1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 5 10 2 370 098 174 098 189 094 0982 5 10 3 480 096 271 096 338 090 1743 5 20 2 740 092 403 090 574 076 2654 5 20 3 960 080 457 082 764 056 3875 10 20 2 1440 076 796 076 1512 040 6386 10 20 3 1860 064 1326 068 2824 028 10187 20 50 2 7100 022 4994 044 10534 012 23348 20 50 3 9150 018 5268 034 14102 008 2825



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Multidemand Multisource Order Quantity ...downloads.hindawi.com/journals/mpe/2015/363056.pdf · allocation [ ]: total purchase cost minimization, total delay risk