a fuzzy multi-objective programming model for supplier selection with volume discount and risk...

ORIGINAL ARTICLE

A fuzzy multi-objective programming model for supplierselection with volume discount and risk criteria

Shima Aghai & Naser Mollaverdi &Mohammad Saeed Sabbagh

Received: 5 October 2010 /Accepted: 16 December 2013 /Published online: 12 January 2014# Springer-Verlag London 2014

Abstract In contemporary supply chain management, theperformance of potential suppliers is evaluated against multi-ple criteria. In this paper, a fuzzy multi-objective program-ming model is outlined to propose supplier selection takingquantitative, qualitative, and risk factors into consideration.Also quantity discount has been considered to determine thebest suppliers and to place the optimal order quantities amongthem. The mixed integer derivative nonlinear programming isobtained from fuzzy multi-objective programming model bychance-constrained method. To solve this problem, an inno-vative method is proposed. In addition, several “what if”scenarios are facilitated. Finally, a real-life sample is used tovalidate the proposed model.

Keywords Supplier selection . Risk . Fuzzymulti-objectiveprogramming . Chance-constrained . Discount quantity

1 Introduction

The task of supplier selection has always been considered as akey factor within purchasing and supply management [1].Furthermore, suppliers have a direct and significant impacton the quality, cost, and lead time of new products andtechnologies which are needed to meet new market demands[2]. The task of selecting the right suppliers is also an ex-tremely demanding. One key reason is that today’s businessenvironments are typically seen as becoming inherently moreunstable due to fast-changing market conditions, customerdemands, actions of competitors, and so on [3]. Extensive

multi-criteria decision making approaches such as TOPSIS,analytic hierarchy process (AHP), analytic network process,case-based reasoning, data envelopment analysis, fuzzy settheory, genetic algorithm, mathematical programming, simplemulti-attribute rating technique, and their hybrids have beenproposed for supplier selection [4]. Supplier selection is one ofthe critical activities for firms to gain competitive advantageand achieve the objectives of the whole supply chain [5]. Inpractice, decision making in supplier selection problem in-cludes a high degree of fuzziness and uncertainties. Fuzzy settheory is one of the effective tools to handle uncertainty andvagueness [5]. Supplier selection is the process by whichsuppliers are reviewed, evaluated, and chosen to become apart of the company’s supply chain [6]. Supplier selection andevaluation is one of the most vital actions of companies in asupply chain. Selecting the wrong supplier could be enough todeteriorate the whole supply chain’s financial and operationalposition [7]. This paper develops a fuzzy multi-objectiveprogramming (FMOP) supplier selection model for supplierselection and risk management. Both quantitative andqualitative supplier selection factors have been considered.Quantitative factors include cost. Qualitative factors arerejecting rate, items late rate, and risk factors such as econom-ic environmental factors and supplier satisfactory ratings. Allfactors are considered fuzzy data. Possibility multi-objectiveprogramming models have been yield by applying possibilitymeasures of fuzzy events in fuzzy multi-objective program-ming models. The projected method allows decision makersto perform tradeoff analysis among costs, quality acceptancelevels, on-time delivery, and risk factors. Also in this paper,we consider quantity discount for cost. The rest of thepaper is ordered as follows. Section 2 reviews the existentliterature on supplier selection and present methods tosolve supplier selection problem. Section 3 develops oursolution approach. Section 4 presents results and analysisof applied model in real life example, and Sect. 5 concludesthe paper results and suggestions for future.

S. Aghai (*)Tehran Payamnoor University (TPNU), Esfahan, Irane-mail: [email protected]

N. Mollaverdi :M. S. SabbaghIsfahan University of Technology (IUT), Esfahan, Iran

Int J Adv Manuf Technol (2014) 71:1483–1492DOI 10.1007/s00170-013-5562-0

2 Literature review

In this stage, literature review is described in three parts asfollows:

2.1 Supplier evaluation and selection importance

Maloni and Benton concluded that the adoption of a supplychain perspective necessitates a significant shift from theconventional adversarial relationships to openness and trustbetween purchasers and vendors [1].

2.2 Supplier selection criteria in literature

Dickson et al. examined the importance of supplier evaluationcriteria and presented 23 supplier attributes that managersconsider in such an evaluation. He identified quality, cost,and delivery performance history as the three most importantcriteria in supplier selection [3]. Ellram proposed three prin-cipal criteria as follows: (1) the financial statement of thesupplier, (2) the organizational culture and strategy of thesupplier, and (3) the technological state of the supplier [8].Thomas and Janet studied purchasing managers of US auto-motive companies and concluded that the quality and deliveryof stock remained the most important criteria among all levels[9]. Barbarosoglu and Yazgac further distinguished three dif-ferent primary criteria as follows: (1) the performance of thesupplier, (2) the technical capabilities and financial situationof the supplier, and (3) the quality system of the supplier [8].Wang et al. adopted 12 performance metrics as the standardcriteria for evaluating a company’s performance [8].Nassimbeni surveyed 78 Italian enterprises concerning theirinternational sourcing, finding that quality and technologicalcontent were the highest ranked criteria for vendor selection,with cost ranked only the fifth [10]. Sarkar and Mohapatrasuggested that the performance and capability of the supplierwere the two major measures in the supplier evaluation andselection problem. The authors used the fuzzy set approach toaccount for the imprecision involved in the numerous subjec-tive characteristics of suppliers. A hypothetical case wasadopted to illustrate how the two best suppliers were selectedwith respect to four performance-based and ten capability-based factors [4]. Florez-Lopez selected 14 most importantevaluating factors from 84 potential added-value attributes,which were based on the questionnaire response from USpurchasing managers. To obtain a better representation ofsuppliers’ ability to create value for the customers, atwo-tuple fuzzy linguistic model was illustrated to combineboth numerical and linguistic information [4]. Another studyconducted by Sung, Ramayya in a survey showed that thefollowing factors were important to supplier selection in liter-ature up to 2008: price, quality, delivery, warranties, andclaims, after sales service, technical support, training aids

attitude, performance history, financial position, geographicallocation, management and organization, labor relations,communication system, response to customer, JIT capability,request e-commerce capability, technical capability, productionfacilities and capacity, packaging ability, operational controls,ease-of-use, maintainability, reputation and position in industry,reciprocal arrangements, impression environmentally friendlyproducts, product appearance, and catalog technology [11].Ozturkoglu and Turker also studied supplier (stakeholder) se-lection problem and their criteria are as follows: power andinterest of the supplier [19].

2.3 Methods to solve supplier selection problem in literature

Moore and Fearon, Anthony and Buffa, Buffa and Jackson,Kingsman, Turner, Sevkli, Koh, Zaim, Demirbag, and Tatogluand Razmi, Songhori, and Khakbaz apply LP to solve supplierselection problem [12]. Talluri developed a binary integerlinear programming model to evaluate alternative supplierbids based on ideal targets forbid attributes which is set bythe buyer to select an optimal set of bids by matching demandand capacity constraints. Based on four variations of model,effective negotiation strategies were proposed for unselectedbids [4]. Lin and Chen highlighted the importance of formu-lating criteria while constructing a fuzzy decision makingframework. A system based on generous criteria was pro-posed which can be modified for some industries. However,their focus was on the performance of the supply chain; theypay little attention to the individual supplier [1]. Narasimhanet al. introduced a multi-objective programming model toselect the optimal suppliers and determine the optimal orderquantity. Five criteria were proposed to evaluate the perfor-mance of suppliers. Before solving the model to optimality,the relative importance weightings of five criteria werederived in advance. It was suggested by the authors thatAHP could be one of the possible ways for generatingthe weightings [4]. Amid et al. developed a fuzzy multi-objective linear programming model for supplier selection.The model could handle the vagueness and imprecision ofinput data, and help decision makers to find out the optimalorder quantity from each supplier. Three objective functionswith different weights were included in the model. An algo-rithm was developed to solve the model [4]. Kokangul et al.solved the model considering capacity, cost, and discountquantity with the hybrid method that concluded AHP, NLP,and multi-objective programming [9]. Tai et al. classified thevast majority of quantitative methodologies into three catego-ries as follows: (1) multi-attribute decision making, (2) math-ematical programming models, and (3) intelligent approaches[3]. Fatih et al. constructed a multi-criteria intuitionist fuzzygroup decision making for supplier selection with TOPSISmethod [13]. Wu et al. developed a FMOP vendor selectionmodel for supply chain outsourcing risk management. Both

1484 Int J Adv Manuf Technol (2014) 71:1483–1492

quantitative and qualitative supplier selection risk factors wereconsidered. Quantitative risk factors include cost, quality, andlogistics [14]. According to literature, it is necessary to payattention to a model that considers all real-life circumstances.In brief, before models considered both qualitative and quan-titative factors. Also some of them combined these factorswith risk. Another group considered for example qualitativeand quantitative factors with quantity discount. But there is nomodel constructed considering all of these items. In thisarticle, a model is considered which concludes qualitativeand quantitative factors, risk and quantity discount to selectthe suppliers, and define order quantity for each customer. Inaddition, the quality of the resulting models is evaluated onreal-life data. We propose the innovative method to solve themodel with fuzzy and mathematical analysis.

3 Basic definitions and proposed method

Actions taken in this stage are described as follows:

3.1 Problem definition and modeling

In our model, both qualitative and quantitative factors wereconsidered. Late items late, rejected ones, and risk factorsincluding environment conditions and vendor rate aremodeled as qualitative factors. Only product unit cost isconsidered as a quantitative factor. This kind of classificationthat we considered was chosen according to our real casestudy. But model can become easier, for example the numberof rejected items or the number of late items can be defined asa quantitative factor if they are calculated with one definedand exact number. Our classification for factors is shown inTable 1. Vendor rate is defined according to vendor satisfac-tory, but because we minimize all objectives, we considerdissatisfactory for each supplier. Qualitative variables suchas those evaluated by customers are typically fuzzy in nature.For example, it is easy to say whether the environment is goodor not, but hard to give a specific value for it [14]. Also, in thispaper, product unit cost is calculated considering quantitydiscount. Maximum and minimum amount of business givento supplier j by the ith customer and order quantity from

supplier j by the ith customer are defined as fuzzy parametersin this model. The proposed approach allows decision makersto perform trade-off analysis among costs, quality acceptancelevels, on-time delivery, and risk factors [14]. Also our modelcalculates product amount purchased from supplier j by the ithcustomer.

Nomenclature

n The number of candidate suppliers desired bythe ith customer

m The number of customersxij Decision variable, product amount that is

purchased from supplier j by the ith customerzij Binary variable, equal to unity if supplier j is

selected by customer i otherwise 0cij Unit cost from supplier j by the ith customer

λijPercentage of items late from supplier j to theith customer

βij Percentage of rejected items from supplier j

Di Demand of ith customer from all suppliersuuij Maximum amount of business for item to be

given to supplier j by the ith customer(capacity of supplier j )

ulij Minimum amount of business given tosupplier j by the ith customer

wuij Maximum order quantity from supplier j

by the ith customerwlij Minimum order quantity from supplier j

by the ith customerObj1 Minimize the purchased costObj2 Minimize the items lateObj3 Minimize the items rejectedObj4 Minimize the environment riskObj5 Minimize the vendor rateConst 6 Ensure that all of customer’s demand is metConst 7 Ensure that quantity purchased is in defined

limits from suppliers and customers

First model

min f 1 ¼Xi¼1

m Xj¼1

n

cijxij ð1Þ

Table 1 Classification for factors

ID Factor name Quantitative Qualitative(Fuzzy)

1 Unit cost √2 Late items √3 Rejected items √4 Environment conditions √5 Vendor rate √

µ

a1 a2 a3 a4

Fig. 1 A trapezoidal fuzzy number

Int J Adv Manuf Technol (2014) 71:1483–1492 1485

min f 2 ¼Xi¼1

m Xj¼1

n

λijxij ð2Þ

min f 3 ¼Xi¼1

m Xj¼1

n

βijxij ð3Þ

min f 4 ¼Xi¼1

m Xj¼1

n

ϕijxij ð4Þ

min f 5 ¼Xi¼1

m Xj¼1

n

εijxij ð5Þ

Subject to

Xj¼1

n

xij≥Di;∀i; ð6Þ

xij≥ziju lij

xij≥zijw lij

xij≤zijuuij

xij≤zijwuij

9>>>>>>>=>>>>>>>;

∀i; j; ð7Þ

Table 2 Trapezoidal fuzzy data

Supplier number c R L V E D W min W max U min U max

1 97,500,000 0.06 0 0.1 0.05 4 1 5 3 4

0.08 0.001 0.15 0.1 5 2 6 4 5

96,000,000 0.1 0.002 0.2 0.15 6 3 7 5 6

0.12 0.003 0.25 0.2 7 4 8 6 7

2 116,000,000 0.3 0.15 0.3 0.06 4 1 5 1 1.5

0.4 0.2 0.4 0.08 5 2 6 2 2.5

115,000,000 0.5 0.25 0.5 0.1 6 3 7 3 3.5

0.6 0.3 0.6 0.12 7 4 8 4 4.5

3 120,200,000 0.06 0.01 0.1 0.1 4 1 5 3 4

0.08 0.025 0.15 0.2 5 2 6 4 5

120,000,000 0.1 0.04 0.2 0.3 6 3 7 5 6

0.12 0.055 0.25 0.4 7 4 8 6 7

4 115,000,000 0 0.1 0.25 0.3 4 1 5 1 1.5

0.05 0.2 0.3 0.4 5 2 6 2 2.5

115,000,000 0.1 0.3 0.35 0.5 6 3 7 3 3.5

0.15 0.4 0.4 0.6 7 4 8 4 4.5

5 114,600,000 0.01 0.02 0.02 0.05 4 1 5 3 4

0.02 0.03 0.05 0.08 5 2 6 4 5

114,000,000 0.03 0.04 0.08 0.11 6 3 7 5 6

0.04 0.05 0.11 0.14 7 4 8 6 7

6 120,400,000 0.25 0.1 0.25 0.15 4 1 5 1 2

0.3 0.2 0.3 0.2 5 2 6 1.5 3

120,000,000 0.35 0.3 0.35 0.25 6 3 7 2 4

0.4 0.4 0.4 0.3 7 4 8 2.5 5

50

150

250

350

450

550

0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

0.55 0.6

0.65 0.7

0.75 0.8

0.85 0.9

0.95 1

alpha value

ob

j val

ue

Fig. 2 Sensitivity graph


xij≥0 ∀i; j; ð8Þ

zij∈ 0; 1f g ∀i; j; ð9Þ

The sign “¯” to each decision variable shows that it is afuzzy variable.

The above model is completed with quantity discount that is:

Cij ¼Cij1 for 0 < xij≤ qij1Cij2 for qij1 < xij≤ qij2⋮Cijr for qij r−1ð Þ < xij≤qijr

8>><>>:

ð10Þ

q (i, j, r) denotes discount level r from jth supplier by ithcustomer. r is number of discount levels. According to litera-ture review, none of the models consider all factors to select.But this model organizes all factors such as quantitative,qualitative, and risk factors. Also, for the first time in thispaper, in addition to the above factors, discount quantity levelsand unit cost in each level are added to the model. This paperdevelops a FMOP vendor selection model.

3.2 Proposed method to solve model

To solve mentioned model, first of all, according to belowformula, weighting method is used to change multi-objectiveproblem to single-objective problem [15].

Min w1 f 1 þ w2 f 2 þ w3 f 3 þ w4 f 4 þ w5 f 5 ð11Þ

Also, chance-constrained method is used to convert fuzzyvariables to deterministic variables. The concept of chance-constrained programming (CCP) introduced by Charnes andCooper [14] is adopted to solve fuzzy single-objective pro-gramming (SOP) models. CCP deals with uncertainty byspecifying the desired level so confidence with which theconstraints hold. Using the concepts of CCP and possibilityof fuzzy events, the FSOP model becomes the followingdeterministic SOP model [14]:

Subject to

1−αð ÞXi¼1

m Xj¼1

n

Cij

� �L

1xij þ α

Xi¼1

m Xj¼1

n

Cij

� �L

2xij ≤ f 1 ð12Þ

1−αð ÞXi¼1

m Xj¼1

n

λij

� �L

1xij þ α

Xi¼1

m Xj¼1

n

λij

� �L

2xij ≤ f 2 ð13Þ

1−αð ÞXi¼1

m Xj¼1

n

βij

� �L

1xij þ α

Xi¼1

m Xj¼1

n

βij

� �L

2xij ≤ f 3 ð14Þ

1−αð ÞXi¼1

m Xj¼1

n

ϕij

� �L

1xij þ α

Xi¼1

m Xj¼1

n

ϕij

� �L

2xij ≤ f 4 ð15Þ

0

100

200

300

400

500

600

0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

0.55 0.6

0.65 0.7

0.75 0.8

0.85 0.9

0.95 1

w1 value

ob

j

obj

Fig. 3 Sensitivity graph for w1

0

100

200

300

400

500

600

0.00

0.06

0.12

0.18

0.24

0.30

0.36

0.42

0.48

0.54

0.60

0.66

0.72

0.78

0.84

0.90

0.96

w2 value

ob

j

obj



1−αð ÞXi¼1

m Xj¼1

n

εij� �L

1xij þ α

Xi¼1

m Xj¼1

n

εij� �L

2xij ≤ f 5 ð16Þ

Xj¼1

n

xij ≥ 1−αð Þ Di

� �L

1þ α Di

� �L

2∀i; ð17Þ

xij ≤ 1−αð ÞZij uuij

� �u

4þ αZij uuij

� �u

3

xij ≤ 1−αð ÞZij wuij

� �u

4þ αZij wu

ij

� �u

3

xij ≤ 1−αð ÞZij ulij

� �u

4þ αZij ulij

� �u

3

xij ≤ 1−αð ÞZij wlij

� �u

4þ αZij wl

ij

� �u

3

9>>>>>>>=>>>>>>>;

∀i; j; ð18Þ

xij ≥ 0∀i; j; ð19Þ

zij ∈ 0; 1f g ∀i; j; ð20Þ

Cij ¼Cij1 for 0 < xij≤ qij1Cij2 for qij1 < xij≤ qij2⋮Cijr for qij r−1ð Þ < xij≤qijr

8>><>>:

ð21Þ

α is desired level of confidence for CCP method. TheEqs. (11) to (21) show a model that is the crisp single-objective MIP1 model because of binary variables. Also, it isDNLP2 model because of discount quantity. Thus in total, themodel isMIDNLPmodel. An innovative method is developedto change a MIDNP problem to MINLP. Equation number(21) is replaced by below formulas:

a i; j; 1ð Þ ¼ sign sign x i; jð Þð Þ þ sign q i; j; 1ð Þ−x i; jð Þð Þð ð22Þ

a i; j; 2ð Þ ¼ sign sign x i; jð Þ−q i; j; 1ð Þð Þ þ sign q i; j; 2ð Þ−x i; jð Þð Þðð23Þ

a i; j; 3ð Þ ¼ sign sign x i; jð Þ−q i; j; 2ð Þð Þ þ sign q i; j; 3ð Þ−x i; jð Þð Þðð24Þ

a i; j; rð Þ ¼ sign signð xð i; jð Þ−q i; j; r−1ð Þð Þ þ sign q i; j; rð Þ−x i; jð Þð Þð25Þ

Cij ¼ Cij1�a i; j; 1ð Þ þ Cij2

�a i; j; 2ð Þ þ…þ Cijr�a i; j; rð Þ ð26Þ

Sign function is a function that when x is positive, it returns1.When x is zero, it returns zero and finally when x is negativeit returns −1. So, in Eqs. numbers (22) to (26), when forexample X quantity is in discount level one, sign(x(i, j) is 1and sign(q(i, j, 1) – x(i, j) is 1. So sign(x(i, j))+sign(q(i, j, 1) –x(i, j) is 2 and sign(2)=1. Therefore, a(i, j, 1) is equal to 1.Other a (i, j, r) are to be equal to 0. Thus, Cij

← in thislevel is to be gained, to 1. For other cases also, theseformulas are true. With this proposed algorithm andGAMS 22.9 software, we solve this problem. The rest of thepaper shows a real-life problem that is solved with proposedmethod.

4 Case study: experimental results and verification

4.1 Problem definition

In this section, a real case study from the Airplane Com-pany is considered to demonstrate how the proposedalgorithm is applied. Thereafter, a sensitivity analysis isperformed by change in α parameter and also appliesdifferent weights for objectives in the model to see the resultsin answers.

0

100

200

300

400

500

600

0.01

0.07

0.13

0.19

0.25

0.31

0.37

0.43

0.49

0.55

0.61

0.67

0.73

0.79

0.85

0.91

0.97

w3 value

ob

j

obj


1 Mixed integer programming2 Derivative nonlinear programming


Our model needs fuzzy data input. Fuzzy data are usuallyyielded in two ways as follows [14]: (1) subjective and exper-tise estimation by decision makers and (2) using historicaldata for stating vague input data. Interested readers can refer toRommelfanger [16]. In our case study, qualitative and quan-titative data is transformed in to fuzzy numbers correspondingto the first and second means according to Desheng [14]. Themodel is applied to select vendor and calculate order quantityfor machining one of the most sensitive parts in helicopter.

There were six suppliers and one customer. The parametersare as follows:

C Product unit cost in two discount level:

1. 0<×<152. 25<×<25

R Reject rate of each supplierL Late items rate of each supplier

0

100

200

300

400

500

600

0.00

0.06

0.12

0.18

0.24

0.30

0.36

0.42

0.48

0.54

0.60

0.66

0.72

0.78

0.84

0.90

0.96

w5 value

ob

j

obj


0

100

200

300

400

500

600

0

0.06

0.12

0.18

0.24 0.3

0.36

0.42

0.48

0.54 0.6

0.66

0.72

0.78

0.84 0.9

0.96

obj

objFig. 6 Sensitivity graph for w4

Table 3 Interval dataSuppliernumber

c r l v e d Wmin

Wmax

Umin

Umax

1 97,500,000 0.07 0.0005 0.125 0.075 4.5 1.5 5.5 3.5 4.5

96,000,000 0.11 0.0025 0.225 0.175 6.5 3.5 7.5 5.5 6.5

2 116,000,000 0.35 0.175 0.35 0.07 4.5 1.5 5.5 1.5 2

115,000,000 0.55 0.275 0.55 0.11 6.5 3.5 7.5 3.5 4

3 120,200,000 0.07 0.0175 0.125 0.15 4.5 1.5 5.5 3.5 4.5

120,000,000 0.11 0.0475 0.225 0.35 6.5 3.5 7.5 5.5 6.5

4 115,000,000 0.025 0.15 0.275 0.35 4.5 1.5 5.5 1.5 2

115,000,000 0.125 0.35 0.375 0.55 6.5 3.5 7.5 3.5 4

5 114,600,000 0.015 0.025 0.035 0.065 4.5 1.5 5.5 3.5 4.5

114,000,000 0.035 0.045 0.095 0.125 6.5 3.5 7.5 5.5 6.5

6 120,400,000 0.275 0.15 0.275 0.175 4.5 1.5 6 1.25 2.5

120,000,000 0.4 0.35 0.375 0.275 6.5 3.5 7.5 2.25 4.5


V Dissatisfactory percent of each supplierE Environment risk effect rate on each supplierD Customer total demandWmin Minimum customer demandWmax Maximum customer demandUmin Minimum capacity of each supplierUmax Maximum capacity of each supplier

Trapezoidal fuzzy numbers are used in the model. Atrapezoidal membership function (see Fig. 1) is confined byfour parameters {a1, a2, a3, a4}. All fuzzy data are shown inTable 1. A trapezoidal fuzzy number a=(a1, a2, a3, a4)depicted in Fig. 1 can be transformed into an interval numberin α-cut level by use of this formula [17]:

a αð Þ ¼ a2þ �α a1−a2ð Þ; a4þ α � a3−a4ð Þ½ � ð27Þ

Numbers in Table 2 are converted to interval numbers using(27) formula and consider α is equal to 0.5 (we explain whyconsiderα equal to 0.5 as follows). Results are shown in Table 2.

4.2 Sensitivity analysis

4.2.1 α sensitivity analysis

Random numbers are used in uniform distribution to create α-cut level in 100 simulations run. This process was done usingC++ codes and link with GAMS22.9. Objective trend of thesechanges is shown in Fig. 2. The trend of objective values withrespect to different α quantities shows that α-cut level doesnot change objective. Therefore, the α-cut level value of 0.5 isused in our case computation.

4.2.2 Weight sensitivity analysis

Weights attached to five objectives are assumed to beindependently identically distributed with uniform distributionin the interval (0, 1). Using C++ codes, the algorithm is imple-mented with the assumed distributions. One hundred simulationruns were done; 5 series of simulation has been done for 5weights of objectives. Results are shown in Figs. 3, 4, 5, 6, and7. They show that different weight values can affect objectivevalues. Therefore, in our case study, weights of objective havebeen computed with AHP3 method and by using Expert choicesoftware [18]. Results are shown in Table 3.

4.3 Solving process and evaluation

The model with numbers in Tables 3 and 4 is solved byGAMS 22.9.

If X being considered as a positive variable, the objectivequantity would be 81.865 and if X being considered as an

integer variable, the objective quantity would be 115.825.Because the aim is to select suppliers for machining of part,X can be positive. It can be seen that with this assumption, theobjective quantity is better and results with first assumptionare better for company. Therefore, first supplier is selected andall demands can be met by this supplier.

To evaluate the model, some examples with some specialscenarios were solved as follows:

1. Only late rate and cost are considered; according toTable 5, it is clear that supplier 5 should be selected. Ourmodel is solved with this specific condition and resultsshow that model is well.

2. Another example was considered when according toTable 6 only reject rate, cost, and risk about vendor rateare considered. According to Table 6, it is clear thatsupplier 3 should be selected. Our model is solved withthis specific condition and show model results are well.

3. Another example is consideredwhen according to Table 7.Reject rate, late rate, risk factor about economicenvironment, and risk about vendor rate are considered.According to Table 7, it is clear that supplier 1 should beselected. Our model was solved with this specific conditionand results show model is well.

Our scenarios showed that our model and our method forsolution work correct. So our model has been evaluated and itcan be used for other situations.

5 Conclusions and future research

This paper develops a fuzzy multi-objective programming(FMOP) supplier selection model for supply chainoutsourcing risk management. Both quantitative, qualitative,and risk supplier selection factors were considered. Also, weconsidered quantity discount for cost. It is an integrated modelthat is similar to real cases. All factors except unit cost areconsidered as fuzzy factors. These factors are integrated withdiscount quantity and constructed a FMIDNLP4 model. First,by using chance-constrained method, this model was changed

4 Fuzzy mixed integer derivative nonlinear programming3 Analytical hierarchy process

Table 4 Final weight of objectives

Objective number Objective description Weight

1 Reject rate of supplier 0.284

2 Late items rate of supplier 0.19

3 Product unit cost 0.186

4 Environment risk factor 0.122

5 Dissatisfactory rate of supplier 0.218



c r l v e d Wmin

Wmax

Umin

Umax

1 97,500,000 0.07 0.0005 0.125 0.075 4.5 1.5 5.5 3.5 4.5

96,000,000 0.11 0.0025 0.225 0.175 6.5 3.5 7.5 5.5 6.5

2 116,000,000 0.35 0.175 0.35 0.07 4.5 1.5 5.5 1.5 2

115,000,000 0.55 0.275 0.55 0.11 6.5 3.5 7.5 3.5 4

3 80,200,000 0.01 0.0175 0.1 0.15 4.5 1.5 5.5 3.5 4.5

80,000,000 0.015 0.0475 0.2 0.35 6.5 3.5 7.5 5.5 6.5

4 115,000,000 0.025 0.15 0.275 0.35 4.5 1.5 5.5 1.5 2

115,000,000 0.125 0.35 0.375 0.55 6.5 3.5 7.5 3.5 4

5 90,500,000 0.015 0.005 0.035 0.065 4.5 1.5 5.5 3.5 4.5

90,000,000 0.035 0.006 0.095 0.125 6.5 3.5 7.5 5.5 6.5

6 120,400,000 0.275 0.15 0.275 0.175 4.5 1.5 6 1.25 2.5

120,000,000 0.4 0.35 0.375 0.275 6.5 3.5 7.5 2.25 4.5


c r l v e d Wmin

Wmax

Umin

Umax

1 97,500,000 0.01 0.0005 0.1 0.05 4.5 1.5 5.5 3.5 4.5

96,000,000 0.03 0.002 0.22 0.1 6.5 3.5 7.5 5.5 6.5

2 116,000,000 0.35 0.175 0.35 0.07 4.5 1.5 5.5 1.5 2

115,000,000 0.55 0.275 0.55 0.11 6.5 3.5 7.5 3.5 4

3 120,200,000 0.07 0.0175 0.125 0.15 4.5 1.5 5.5 3.5 4.5

120,000,000 0.11 0.0475 0.225 0.35 6.5 3.5 7.5 5.5 6.5

4 115,000,000 0.025 0.15 0.275 0.35 4.5 1.5 5.5 1.5 2

115,000,000 0.125 0.35 0.375 0.55 6.5 3.5 7.5 3.5 4

5 114,600,000 0.015 0.025 0.035 0.065 4.5 1.5 5.5 3.5 4.5

114,000,000 0.035 0.045 0.095 0.125 6.5 3.5 7.5 5.5 6.5

6 120,400,000 0.275 0.15 0.275 0.175 4.5 1.5 6 1.25 2.5

120,000,000 0.4 0.35 0.375 0.275 6.5 3.5 7.5 2.25 4.5


c r l v e d Wmin

Wmax

Umin

Umax

1 97,500,000 0.07 0.0005 0.125 0.075 4.5 1.5 5.5 3.5 4.5

96,000,000 0.11 0.0025 0.225 0.175 6.5 3.5 7.5 5.5 6.5

2 116,000,000 0.35 0.175 0.35 0.07 4.5 1.5 5.5 1.5 2

115,000,000 0.55 0.275 0.55 0.11 6.5 3.5 7.5 3.5 4

3 120,200,000 0.07 0.0175 0.125 0.15 4.5 1.5 5.5 3.5 4.5

120,000,000 0.11 0.0475 0.225 0.35 6.5 3.5 7.5 5.5 6.5

4 115,000,000 0.025 0.15 0.275 0.35 4.5 1.5 5.5 1.5 2

115,000,000 0.125 0.35 0.375 0.55 6.5 3.5 7.5 3.5 4

5 90,500,000 0.015 0.005 0.035 0.065 4.5 1.5 5.5 3.5 4.5

90,000,000 0.035 0.006 0.095 0.125 6.5 3.5 7.5 5.5 6.5

6 120,400,000 0.275 0.15 0.275 0.175 4.5 1.5 6 1.25 2.5

120,000,000 0.4 0.35 0.375 0.275 6.5 3.5 7.5 2.25 4.5


to MIDNLP model. After that, an innovative method wascreated and MIDNLP model was changed to MINLP andfinally was solved by GAMS22.9. To validate our model, itwas applied in a real-life case and results show that the modelis applicable for supplier selection problems.

For future research, other methods can be applied to changefuzzy variables to deterministic variable; also, other methodsto change DNLP model to NLP. Also in this model, multi-objective problem is changed to single-objective. Others maynot change this and can solve multi-objective problems. Wehope our comprehensive and systematic model is applied inall companies.

Acknowledgments The final part of this research was done in AirplaneCompany. Thanks a lot for their outsourcing unit because of theircooperation and constructive suggestions. The helpful comments of theanonymous reviewers are gratefully acknowledged.

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a fuzzy multi-objective programming model for supplier selection with volume discount and risk...

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