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Int. J. Production Economics 102 (2006) 289–301

www.elsevier.com/locate/ijpe

A fuzzy approach for supplier evaluation and selectionin supply chain management

Chen-Tung Chena,�, Ching-Torng Linb, Sue-Fn Huangb

aDepartment of Information Management, National United University, 1, Lien Da, Kung-Ching Li, Miao-Li, TaiwanbDepartment of Information Management, Da-Yeh University, 112, Shan-Jiau Road, Da-Tsuen, Changhua, Taiwan

Received 14 April 2003; accepted 29 March 2005

Available online 25 May 2005

Abstract

This paper is aimed to present a fuzzy decision-making approach to deal with the supplier selection problem in supply

chain system. During recent years, how to determine suitable suppliers in the supply chain has become a key strategic

consideration. However, the nature of these decisions usually is complex and unstructured. In general, many

quantitative and qualitative factors such as quality, price, and flexibility and delivery performance must be considered

to determine suitable suppliers. In this paper, linguistic values are used to assess the ratings and weights for these

factors. These linguistic ratings can be expressed in trapezoidal or triangular fuzzy numbers. Then, a hierarchy multiple

criteria decision-making (MCDM) model based on fuzzy-sets theory is proposed to deal with the supplier selection

problems in the supply chain system. According to the concept of the TOPSIS, a closeness coefficient is defined to

determine the ranking order of all suppliers by calculating the distances to the both fuzzy positive-ideal solution (FPIS)

and fuzzy negative-ideal solution (FNIS) simultaneously. Finally, an example is shown to highlight the procedure of the

proposed method at the end of this paper. This paper shows that the proposed model is very well suited as a decision-

making tool for supplier selection decisions.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Supplier selection; Supply chain; Linguistic variables; Fuzzy set theory; MCDM

1. Introduction

Recently, supply chain management and thesupplier (vendor) selection process have receivedconsiderable attention in the business-management

e front matter r 2005 Elsevier B.V. All rights reserve

e.2005.03.009

ng author. Tel.: +886 37 381833;

0776.

ss: ctchen@nuu.edu.tw (C.-T. Chen).

literature. During the 1990s, many manufacturersseek to collaborate with their suppliers in order toupgrade their management performance and com-petitiveness (Ittner et al., 1999; Shin et al., 2000). Thematerial flow in a supply chain is shown in Fig. 1.The purchasing function is increasingly seen as astrategic issue in organizations. Buyer and supplierrelationships in manufacturing enterprises havereceived a great deal of attention. When it is built

d.

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Manufacturer /Company

Distributor

Direct

Supplier

Direct Supplier

Direct Supplier

Distributor Consumer

Consumer

Fig. 1. Material flow in supply chain.

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301290

on long-term relationships, a company’s supplychain creates one of the strongest barriers to entryfor competitors (Briggs, 1994; Choi and Hartley,1996). In other words, once a supplier becomespart of a well-managed and established supplychain, this relationship will have a lasting effect onthe competitiveness of the entire supply chain.Therefore, the supplier selection problem hasbecome one of the most important issues forestablishing an effective supply chain system. Theoverall objective of supplier selection process is toreduce purchase risk, maximize overall value tothe purchaser, and build the closeness and long-term relationships between buyers and suppliers(Monczka et al., 1998).

In supply chains, coordination between amanufacturer and suppliers is typically a difficultand important link in the channel of distribu-tion. Many models have been developed forsupplier selection decisions are based on rathersimplistic perceptions of decision-making process(Boer et al., 1998; Lee et al., 2001). Most of thesemethods do not seem to address the complex andunstructured nature and context of many present-day purchasing decisions (Boer et al., 1998). Infact, many existing decision models only quantitiescriteria are considered for supplier selection.However, several influence factors are often nottaken into account in the decision making process,such as incomplete information, additional quali-tative criteria and imprecision preferences. Ac-cording to the vast literature on supplier selection(Boer et al., 1998; Choi and Hartley, 1996; Weberet al., 1991), we conclude that some properties areworth considering when solving the decision-making problem for supplier selection. First, thecriteria may consider quantitative as well as

qualitative dimensions (Choi and Hartley, 1996;Dowlatshahi, 2000; Verma and Pullman, 1998;Weber et al., 1991, 1998). In general, theseobjectives among these criteria are conflicted. Astrategic approach towards supplier selection mayfurther emphasize the need to consider multiplecriteria (Donaldson, 1994; Ellram, 1992; Swift,1995). Second, several decision-makers are veryoften involved in the decision process for supplierselection (Boer et al., 1998). Third, decision-making is often influenced by uncertainty inpractice. An increasing number of supplier deci-sions can be characterized as dynamic andunstructured. Situations are changing rapidly orare uncertain and decision variables are difficult orimpossible to quantify (Cook, 1992). Fourth, thetypes of decision models can be divided intocompensatory and non-compensatory methods(Boer et al., 1998; Ghodsypour and O’Brien,1998; Roodhooft and Konings, 1996). The com-pensatory decision models leading to an optimalsolution for dealing with supplier selection pro-blems. The non-compensatory methods are thatuse a score of an alternative on a particularcriterion can be compensated by high scores onother criteria. From the literature it can beconcluded that in supplier selection the classicconcept of ‘‘optimality’’ may not always be themost appropriate model (Boer et al., 1998). Over-all speaking, we can conclude that supplierselection may involve several and different typesof criteria, combination of different decisionmodels, group decision-making and various formsof uncertainty. It is difficult to find the best way toevaluate and select supplier, and companies use avariety of different methods to deal with it.Therefore, the most important issue in the process

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of supplier selection is to develop a suitablemethod to select the right supplier.

In essential, the supplier selection problem insupply chain system is a group decision-makingunder multiple criteria. The degree of uncertainty,the number of decision makers and the nature ofthe criteria those have to be taken into account insolving this problem. In classical MCDM methods,the ratings and the weights of the criteria are knownprecisely (Delgado et al., 1992; Hwang and Yoon,1981; Kaufmann and Gupta, 1991). A survey of themethods has been presented in Hwang and Yoon(1981). Technique for Order Performance bySimilarity to Ideal Solution (TOPSIS), one of theknown classical MCDM methods, may provide thebasis for developing supplier selection models thatcan effectively deal with these properties. It basesupon the concept that the chosen alternative shouldhave the shortest distance from the Positive IdealSolution (PIS) and the farthest from the NegativeIdeal Solution (NIS).

Under many conditions, crisp data are inadequateto model real-life situations. Since human judge-ments including preferences are often vague andcannot estimate his preference with an exactnumerical value. A more realistic approach maybe to use linguistic assessments instead of numericalvalues. In other words, the ratings and weights ofthe criteria in the problem are assessed by means oflinguistic variables (Bellman and Zadeh, 1970;Chen, 2000; Delgado et al., 1992; Herrera et al.,1996; Herrera and Herrera-Viedma, 2000). In thispaper, we further extended to the concept ofTOPSIS to develop a methodology for solvingsupplier selection problems in fuzzy environment(Chen, 2000). Considering the fuzziness in thedecision data and group decision-making process,linguistic variables are used to assess the weights ofall criteria and the ratings of each alternative withrespect to each criterion. We can convert thedecision matrix into a fuzzy decision matrix andconstruct a weighted-normalized fuzzy decisionmatrix once the decision-makers’ fuzzy ratings havebeen pooled. According to the concept of TOPSIS,we define the fuzzy positive ideal solution (FPIS)and the fuzzy negative ideal solution (FNIS). Andthen, a vertex method is applied in this paper tocalculate the distance between two fuzzy ratings.

Using the vertex method, we can calculate thedistance of each alternative from FPIS and FNIS,respectively. Finally, a closeness coefficient of eachalternative is defined to determine the ranking orderof all alternatives. The higher value of closenesscoefficient indicates that an alternative is closer toFPIS and farther from FNIS simultaneously.The paper is organized as follows. Next section

introduces the basic definitions and notations ofthe fuzzy numbers and linguistic variables. InSection 3, we present a fuzzy decision-makingmethod to cope with the supplier selectionproblem. And then, the proposed method isillustrated with an example. Finally, some conclu-sions are pointed out at the end of this paper.

2. Fuzzy numbers and linguistic variables

In this section, some basic definitions of fuzzysets, fuzzy numbers and linguistic variables arereviewed from Buckley (1985), Kaufmann andGupta (1991), Negi (1989), Zadeh (1975). Thebasic definitions and notations below will be usedthroughout this paper until otherwise stated.

Definition 2.1. A fuzzy set ~A in a universe ofdiscourse X is characterized by a membershipfunction m ~AðxÞ which associates with each elementx in X a real number in the interval [0,1]. Thefunction value m ~AðxÞ is termed the grade of member-ship of x in ~A (Kaufmann and Gupta, 1991).

Definition 2.2. A fuzzy set ~A in the universe ofdiscourse X is convex if and only if

m ~Aðlx1 þ ð1� lÞx2ÞXminðm ~Aðx1Þ; m ~Aðx2ÞÞ (1)

for all x1; x2 in X and all l 2 ½0; 1�, where mindenotes the minimum operator (Klir and Yuan,1995).

Definition 2.3. The height of a fuzzy set is thelargest membership grade attained by any elementin that set. A fuzzy set ~A in the universe ofdiscourse X is called normalized when the heightof ~A is equal to 1 (Klir and Yuan, 1995).

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n1 n2 n3 n4x 0

1

)(n~ xµ

Fig. 3. Trapezoidal fuzzy number ~n.

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301292

Definition 2.4. A fuzzy number is a fuzzy subset inthe universe of discourse X that is both convex andnormal. Fig. 2 shows a fuzzy number ~n in theuniverse of discourse X that conforms to thisdefinition (Kaufmann and Gupta, 1991).

Definition 2.5. The a-cut of fuzzy number ~n isdefined as

~na ¼ fxi : m ~nðxiÞXa;xi 2 X g, (2)

where a 2 ½0; 1�.The symbol ~na represents a non-empty bounded

interval contained in X, which can be denoted by~na ¼ ½na

l ; nau�, na

l and nau are the lower and upper

bounds of the closed interval, respectively (Kauf-mann and Gupta, 1991; Zimmermann, 1991). Fora fuzzy number ~n, if na

l 40 and naup1 for all

a 2 ½0; 1�, then ~n is called a standardized (normal-ized) positive fuzzy number (Negi, 1989).

Definition 2.6. A positive trapezoidal fuzzy num-ber (PTFN) ~n can be defined as ðn1; n2; n3; n4Þ,shown in Fig. 3. The membership function, m ~nðxÞ isdefined as (Kaufmann and Gupta, 1991)

m ~nðxÞ ¼

0; xon1;x�n1n2�n1

; n1pxpn2;

1; n2pxpn3;x�n4n3�n4

; n3pxpn4;

0; x4n4:

8>>>>>><>>>>>>:

(3)

For a trapezoidal fuzzy number ~n ¼ ðn1; n2; n3; n4Þ,if n2 ¼ n3, then ~n is called a triangular fuzzynumber. A non-fuzzy number r can be expressed

0

1

x

(x)n~µ

Fig. 2. Fuzzy number ~n.

as (r, r, r, r). By the extension principle (Duboisand Prade, 1980), the fuzzy sum � and fuzzysubtraction � of any two trapezoidal fuzzynumbers are also trapezoidal fuzzy numbers; butthe multiplication � of any two trapezoidal fuzzynumbers is only an approximate trapezoidal fuzzynumber. Given any two positive trapezoidal fuzzynumbers, ~m ¼ ðm1;m2;m3;m4Þ, ~n ¼ ðn1; n2; n3; n4Þ

and a positive real number r, some main opera-tions of fuzzy numbers ~m and ~n can be expressedas follows:

~m� ~n ¼ ½m1 þ n1;m2 þ n2;m3 þ n3;m4 þ n4�, (4)

~m� ~n ¼ ½m1 � n4;m2 � n3;m3 � n2;m4 � n1�; (5)

~m� r ¼ ½m1r;m2r;m3r;m4r�, (6)

~m� ~nffi ½m1n1;m2n2;m3n3;m4n4�. (7)

Definition 2.7. A matrix ~D is called a fuzzy matrixif at least one element is a fuzzy number (Buckley,1985).

Definition 2.8. A linguistic variable is a variablewhose values are expressed in linguistic terms(Zimmermann, 1991).The concept of a linguistic variable is very

useful in dealing with situations, which are toocomplex or not well defined to be reasonablydescribed in conventional quantitative expressions

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

Very Low

Low (L)

Medium Low

Medium (M)

Medium High

High (H)

Very High

Fig. 4. Linguistic variables for importance weight of each

criterion.

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301 293

(Zimmermann, 1991). For example, ‘‘weight’’ is alinguistic variable whose values are very low, low,medium, high, very high, etc. Fuzzy numbers canalso represent these linguistic values.

Let ~m ¼ ðm1;m2;m3;m4Þ and ~n ¼ ðn1; n2; n3; n4Þ

be two trapezoidal fuzzy numbers. Then thedistance between them can be calculated by usingthe vertex method as (Chen, 2000)

dvð ~m; ~nÞ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4½ðm1 � n1Þ

2þ ðm2 � n2Þ

2þ ðm3 � n3Þ

2þ ðm4 � n4Þ

2�

r.

ð8Þ

Let ~m ¼ ðm1;m2;m3Þ and ~n ¼ ðn1; n2; n3Þ be twotriangular fuzzy numbers. Then the distancebetween them can be calculated by using thevertex method as (Chen, 2000)

dvð ~m; ~nÞ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

3½ðm1 � n1Þ

2þ ðm2 � n2Þ

2þ ðm3 � n3Þ

2�

r.

ð9Þ

The vertex method is an effective and simplemethod to calculate the distance between twotrapezoidal fuzzy numbers. According to thevertex method, two trapezoidal fuzzy numbers ~mand ~n are identical if and only if dvð ~m; ~nÞ ¼ 0. Let~m, ~n and ~p be three trapezoidal fuzzy numbers.Fuzzy number ~n is closer to fuzzy number ~m thanthe other fuzzy number ~p if and only ifdvð ~m; ~nÞodvð ~m; ~pÞ (Chen, 2000).

1

0 1 2 3 4 5 6 7 8 9 10

Very Poor

Poor (P)

Medium Poor

Fair (F)

Medium Good

Good (G)

Very Good

Fig. 5. Linguistic variables for ratings.

3. Proposed method for supplier selection

A systematic approach to extend the TOPSIS isproposed to solve the supplier-selection problemunder a fuzzy environment in this section. In thispaper the importance weights of various criteriaand the ratings of qualitative criteria are consid-ered as linguistic variables. Because linguisticassessments merely approximate the subjectivejudgment of decision-makers, we can considerlinear trapezoidal membership functions to beadequate for capturing the vagueness of these

linguistic assessments (Delgado et al., 1998;Herrera et al., 1996; Herrera and Herrera-Viedma,2000). These linguistic variables can be expressedin positive trapezoidal fuzzy numbers, as in Figs. 4and 5. The importance weight of each criterion canbe by either directly assigning or indirectly usingpairwise comparison (Cook, 1992). It is suggestedin this paper that the decision-makers use thelinguistic variables shown in Figs. 4 and 5 toevaluate the importance of the criteria and theratings of alternatives with respect to qualitativecriteria. For example, the linguistic variable‘‘Medium High (MH)’’ can be represented asð0:5; 0:6; 0:7; 0:8Þ, the membership function of

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which is

mMedium HighðxÞ ¼

0; xo0:5;x�0:50:6�0:5 ; 0:5pxp0:6;

1; 0:6pxp0:7;x�0:80:7�0:8 ; 0:7pxp0:8;

0; x40:8:

8>>>>>><>>>>>>:

(10)

The linguistic variable ‘‘Very Good (VG)’’ can berepresented as (8,9,9,10), the membership functionof which is

mVery GoodðxÞ ¼

0; xo8;x�89�8

; 8pxp9;

1; 9pxp10:

8><>: (11)

In fact, supplier selection in supply chain systemis a group multiple-criteria decision-making(GMCDM) problem, which may be described bymeans of the following sets:

(i)

a set of K decision-makers called E ¼ fD1;D2; . . . ;DKg;

(ii)

a set of m possible suppliers called A ¼ fA1;A2; . . . ;Amg;

(iii)

a set of n criteria, C ¼ fC1;C2; . . . ;Cng, withwhich supplier performances are measured;

(iv)

a set of performance ratings of Aiði ¼ 1; 2; . . . ;mÞ with respect to criteria Cjðj ¼ 1; 2; . . . ; nÞ,called X ¼ fxij ; i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; ng.

Assume that a decision group has K decisionmakers, and the fuzzy rating of each decision-maker Dkðk ¼ 1; 2; . . . ;KÞ can be represented as apositive trapezoidal fuzzy number ~Rkðk ¼ 1; 2; . . . ;KÞ with membership function m ~Rk

ðxÞ. A goodaggregation method should be considered therange of fuzzy rating of each decision-maker. Itmeans that the range of aggregated fuzzy ratingmust include the ranges of all decision-makers’fuzzy ratings. Let the fuzzy ratings of all decision-makers be trapezoidal fuzzy numbers ~Rk ¼ ðak; bk;ck; dkÞ, k ¼ 1; 2; . . . ;K . Then the aggregated fuzzyrating can be defined as

~R ¼ ða; b; c; dÞ; k ¼ 1; 2; . . . ;K (12)

where

a ¼ minkfakg; b ¼

1

K

XK

k¼1

bk,

c ¼1

K

XK

k¼1

ck; d ¼ maxkfdkg.

Let the fuzzy rating and importance weight ofthe kth decision maker be ~xijk ¼ ðaijk; bijk; cijk; dijkÞ

and ~wjk ¼ ðwjk1;wjk2;wjk3;wjk4Þ; i ¼ 1; 2; . . . ;m,j ¼ 1; 2; . . . ; n, respectively. Hence, the aggregatedfuzzy ratings ( ~xij) of alternatives with respect toeach criterion can be calculated as

~xij ¼ ðaij ; bij ; cij ; dijÞ (13)

where

aij ¼ minkfaijkg; bij ¼

1

K

XK

k¼1

bijk,

cij ¼1

K

XK

k¼1

cijk; dij ¼ maxkfdijkg.

The aggregated fuzzy weights ( ~wj) of eachcriterion can be calculated as

~wj ¼ ðwj1;wj2;wj3;wj4Þ, (14)

where

wj1 ¼ minkfwjk1g; wj2 ¼

1

K

XK

k¼1

wjk2,

wj3 ¼1

K

XK

k¼1

wjk3; wj4 ¼ maxkfwjk4g.

As stated above, a supplier-selection problemcan be concisely expressed in matrix format asfollows:

~D ¼

~x11 ~x12 ::::: ~x1n

~x21 ~x22 ::::: ~x2n

..

. ...

::::: ...

~xm1 ~xm2 ::::: ~xmn

26666664

37777775,

~W ¼ ½ ~w1; ~w2; . . . ~wn�,

where ~xij ¼ ðaij ; bij ; cij ; dijÞ and ~wj ¼ ðwj1;wj2;wj3;wj4Þ; i ¼ 1; 2; . . . ;m, j ¼ 1; 2; . . . ; n can be approxi-mated by positive trapezoidal fuzzy numbers.

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To avoid complexity of mathematical opera-tions in a decision process, the linear scaletransformation is used here to transform thevarious criteria scales into comparable scales.The set of criteria can be divided into benefitcriteria (the larger the rating, the greaterthe preference) and cost criteria (the smaller therating, the greater the preference). Therefore, thenormalized fuzzy-decision matrix can be repre-sented as

~R ¼ ½~rij �m�n, (15)

where B and C are the sets of benefit criteria andcost criteria, respectively, and

~rij ¼aij

d�j;bij

d�j;cij

d�j;dij

d�j

!; j 2 B;

~rij ¼a�j

dij

;a�j

cij

;a�j

bij

;a�j

aij

� �; j 2 C;

d�j ¼ maxi

dij ; j 2 B;

a�j ¼ mini

aij ; j 2 C:

The normalization method mentioned above isdesigned to preserve the property in which theelements ~rij ;8i; j are standardized (normalized)trapezoidal fuzzy numbers.

Considering the different importance of eachcriterion, the weighted normalized fuzzy-decisionmatrix is constructed as

~V ¼ ½~vij �m�n; i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; n,

(16)

where

~vij ¼ ~rijðÞ ~wj.

According to the weighted normalized fuzzy-decision matrix, normalized positive trapezoidalfuzzy numbers can also approximate the elements~vij ;8i; j. Then, the fuzzy positive-ideal solution(FPIS, A�) and fuzzy negative-ideal solution(FNIS, A�) can be defined as

A� ¼ ð~v�1; ~v�2; . . . ; ~v

�nÞ, (17)

A� ¼ ð~v�1 ; ~v�2 ; . . . ; ~v

�n Þ, (18)

where

~v�j ¼ maxifvij4g and ~v�j ¼ min

ifvij1g,

i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; n.

The distance of each alternative (supplier) fromA� and A� can be currently calculated as

d�i ¼Xn

j¼1

dvð~vij ; ~v�j Þ; i ¼ 1; 2; . . . ;m, (19)

d�i ¼Xn

j¼1

dvð~vij ; ~v�j Þ; i ¼ 1; 2; . . . ;m, (20)

where dvð; Þ is the distance measurement betweentwo fuzzy numbers.A closeness coefficient is defined to determine

the ranking order of all possible suppliers once d�iand d�i of each supplier Aiði ¼ 1; 2; . . . ;mÞ hasbeen calculated. The closeness coefficient repre-sents the distances to the fuzzy positive-idealsolution (A�) and the fuzzy negative-ideal solution(A�) simultaneously by taking the relative close-ness to the fuzzy positive-ideal solution. Thecloseness coefficient (CCi) of each alternative(supplier) is calculated as

CCi ¼d�i

d�i þ d�i; i ¼ 1; 2; . . . ;m. (21)

It is clear that CCi ¼ 1 if Ai ¼ A� and CCi ¼ 0 ifAi ¼ A�. In other words, supplier Ai is closer tothe FPIS (A�) and farther from FNIS (A�) as CCi

approaches to 1. According to the descendingorder of CCi, we can determine the ranking orderof all suppliers and select the best one from amonga set of feasible suppliers. Although we candetermine the ranking order of all feasiblesuppliers, a more realistic approach may be touse a linguistic variable to describe the currentassessment status of each supplier in accordancewith its closeness coefficient. In order to describethe assessment status of each supplier, we dividethe interval [0,1] into five sub-intervals. Fivelinguistic variables with respect to the sub-intervalsare defined to divide the assessment status ofsupplier into five classes. The decision rules of the

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Table 1

Approval status

Closeness coefficient (CCi) Assessment status

CCi 2 ½0; 0:2Þ Do not recommend

CCi 2 ½0:2; 0:4Þ Recommend with high risk

CCi 2 ½0:4; 0:6Þ Recommend with low risk

CCi 2 ½0:6; 0:8Þ Approved

CCi 2 ½0:8; 1:0� Approved and preferred

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301296

five classes are shown in Table 1. According to theTable 1, it means that

If CCi 2 ½0; 0:2Þ, then supplier Ai belongs toClass I and the assessment status of supplier Ai

is ‘‘not recommend’’;If CCi 2 ½0:2; 0:4Þ, then supplier Ai belongs toClass II and the assessment status of supplier Ai

is ‘‘recommend with high risk’’;If CCi 2 ½0:4; 0:6Þ, then supplier Ai belongs toClass III and the assessment status of supplierAi is ‘‘recommend with low risk’’;If CCi 2 ½0:6; 0:8Þ, then supplier Ai belongs toClass IV and the assessment status of supplierAi is ‘‘approved’’;If CCi 2 ½0:8; 1:0�, then supplier Ai belongs toClass V and the assessment status of supplier Ai

is ‘‘approved and preferred to recommend.’’

According to Table 1 and decision rules, we canuse a linguistic variable to describe the currentassessment status of each supplier. In addition, ifany two suppliers belong to the same class ofassessment status, the closeness coefficient value isused to determine their rank.

In summation, an algorithm of the fuzzydecision-making method for dealing with thesupplier selection is given as follows.

Step 1: Form a committee of decision-makers,and then identify the evaluation criteria.

Step 2: Choose the appropriate linguistic vari-ables for the importance weight of the criteria andthe linguistic ratings for suppliers.

Step 3: Aggregate the weight of criteria to getthe aggregated fuzzy weight ~wj of criterion Cj, andpool the decision-makers’ ratings to get theaggregated fuzzy rating ~xij of supplier Ai undercriterion Cj.

Step 4: Construct the fuzzy-decision matrix andthe normalized fuzzy-decision matrix.

Step 5: Construct weighted normalized fuzzy-decision matrix.

Step 6: Determine FPIS and FNIS.Step 7: Calculate the distance of each supplier

from FPIS and FNIS, respectively.Step 8: Calculate the closeness coefficient of

each supplier.Step 9: According to the closeness coefficient, we

can understand the assessment status of eachsupplier and determine the ranking order of allsuppliers.

4. Numerical example

A high-technology manufacturing companydesires to select a suitable material supplier topurchase the key components of new products.After preliminary screening, five candidates (A1,A2, A3, A4, A5) remain for further evaluation. Acommittee of three decision-makers, D1; D2 andD3, has been formed to select the most suitablesupplier. Five benefit criteria are considered:

(1)

profitability of supplier (C1), (2) relationship closeness (C2), (3) technological capability (C3), (4) conformance quality (C4), (5) conflict resolution (C5).

The hierarchical structure of this decision problemis shown in Fig. 6. The proposed method iscurrently applied to solve this problem, thecomputational procedure of which is summarizedas follows:

Step 1: Three decision-makers use the linguisticweighting variables shown in Fig. 4 to assess theimportance of the criteria. The importance weightsof the criteria determined by these three decision-makers are shown in Table 2.

Step 2: Three decision-makers use the linguisticrating variables shown in Fig. 5 to evaluate theratings of candidates with respect to each criterion.The ratings of the five candidates by the decision-makers under the various criteria are shown inTable 3.

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Table 2

Importance weight of criteria from three decision-makers

Criteria Decision-makers

D1 D2 D3

C1 H H H

C2 VH VH VH

C3 VH VH H

C4 H H H

C5 H H H

Goal

C11

Profitability

of supplier

C2

Relationship

closeness

C3

Technological

capability

C4

Conformance

quality

C5

Conflict

resolution

A2 A3A1 A4 A5

Fig. 6. Hierarchical structure of decision problem.

Table 3

Ratings of the five candidates by decision-makers under various

criteria

Criteria Suppliers Decision-makers

D1 D2 D3

C1 A1 MG MG MG

A2 G G G

A3 VG VG G

A4 G G G

A5 MG MG MG

C2 A1 MG MG VG

A2 VG VG VG

A3 VG G G

A4 G G MG

A5 MG G G

C3 A1 G G G

A2 VG VG VG

A3 VG VG G

A4 MG MG G

A5 MG MG MG

C4 A1 G G G

A2 G VG VG

A3 VG VG VG

A4 G G G

A5 MG MG G

C5 A1 G G G

A2 VG VG VG

A3 G VG G

A4 G G VG

A5 MG MG MG

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301 297

Step 3: Then the linguistic evaluations shown inTables 2 and 3 are converted into trapezoidal fuzzynumbers to construct the fuzzy-decision matrixand determine the fuzzy weight of each criterion,as in Table 4.

Step 4: The normalized fuzzy-decision matrix isconstructed as in Table 5.

Step 5: Weighted normalized fuzzy-decisionmatrix is constructed as in Table 6.

Step 6: Determine FPIS and FNIS asA�¼ ½ð0:9; 0:9; 0:9; 0:9Þ; ð1; 1; 1; 1Þ; ð1; 1; 1; 1Þ; ð0:9;

0:9; 0:9; 0:9Þ; ð0:9; 0:9; 0:9; 0:9Þ�,A� ¼ ½ð0:35; 0:35; 0:35; 0:35Þ; ð0:4; 0:4; 0:4; 0:4Þ;ð0:35; 0:35; 0:35; 0:35Þ; ð0:35; 0:35; 0:35; 0:35Þ; ð0:35;0:35; 0:35; 0:35Þ�.

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Table 4

Fuzzy-decision matrix and fuzzy weights of five candidates

C1 C2 C3 C4 C5

A1 (5,6,7,8) (5,7,8,10) (7,8,8,9) (7,8,8,9) (7,8,8,9)

A2 (7,8,8,9) (8,9,10,10) (8,9,10,10) (7,8.7,9.3,10) (8,9,10,10)

A3 (7,8.7,9.3,10) (7,8.3,8.7,10) (7,8.7,9.3,10) (8,9,10,10) (7,8.3,8.7,10)

A4 (7,8,8,9) (5,7.3,7.7,9) (5,6.7,7.3,9) (7,8,8,9) (7,8.3,8.7,10)

A5 (5,6,7,8) (5,7.3,7.7,9) (5,6,7,8) (5,6.7,7.3,9) (5,6,7,8)

Weight (0.7,0.8,0.8,0.9) (0.8,0.9,1.0,1.0) (0.7,0.87,0.93,1.0) (0.7,0.8,0.8,0.9) (0.7,0.8,0.8,0.9)

Table 5

Normalized fuzzy-decision matrix

C1 C2 C3 C4 C5

A1 (0.5,0.6,0.7,0.8) (0.5,0.7,0.8,1) (0.7,0.8,0.8,0.9) (0.7,0.8,0.8,0.9) (0.7,0.8,0.8,0.9)

A2 (0.7,0.8,0.8,0.9) (0.8,0.9,1,1) (0.8,0.9,1,1) (0.7,0.87,0.93,1) (0.8,0.9,1,1)

A3 (0.7,0.87,0.93,1) (0.7,0.83,0.87,1) (0.7,0.87,0.93,1) (0.8,0.9,1,1) (0.7,0.83,0.87,1)

A4 (0.7,0.8,0.8,0.9) (0.5,0.73,0.77,0.9) (0.5,0.67,0.73,0.9) (0.7,0.8,0.8,0.9) (0.7,0.83,0.87,1)

A5 (0.5,0.6,0.7,0.8) (0.5,0.73,0.77,0.9) (0.5,0.6,0.7,0.8) (0.5,0.67,0.73,0.9) (0.5,0.6,0.7,0.8)

Table 6

Weighted normalized fuzzy-decision matrix

C1 C2 C3 C4 C5

A1 (0.35,0.48,0.56,0.72) (0.4,0.63,0.8,1) (0.49,0.7,0.74,0.9) (0.49,0.64,0.64,0.81) (0.49,0.64,0.64,0.81)

A2 (0.49,0.64,0.64,0.81) (0.64,0.81,1,1) (0.56,0.78,0.93,1) (0.49,0.7,0.74,0.9) (0.56,0.72,0.8,0.9)

A3 (0.49,0.7,0.74,0.9) (0.56,0.75,0.87,1) (0.49,0.76,0.86,1) (0.56,0.72,0.8,0.9) (0.49,0.66,0.7,0.9)

A4 (0.49,0.64,0.64,0.81) (0.4,0.66,0.77,0.9) (0.35,0.58,0.68,0.9) (0.49,0.64,0.64,0.81) (0.49,0.66,0.7,0.9)

A5 (0.35,0.48,0.56,0.72) (0.4,0.66,0.77,0.9) (0.35,0.52,0.65,0.8) (0.35,0.54,0.58,0.81) (0.35,0.48,0.56,0.72)

Table 7

Distances between Aiði ¼ 1; 2; . . . ; 5Þ and A� with respect to

each criterion

C1 C2 C3 C4 C5

d(A1,A*) 0.4 0.37 0.33 0.28 0.28

d(A2,A*) 0.28 0.2 0.25 0.24 0.2

d(A3,A*) 0.24 0.26 0.29 0.2 0.26

d(A4,A*) 0.28 0.37 0.42 0.28 0.26

d(A5,A*) 0.4 0.37 0.45 0.37 0.4

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301298

Step 7: Calculate the distance of each supplierfrom FPIS and FNIS with respect to eachcriterion, respectively, as Tables 7 and 8.

Step 8: Calculate d�i and d�i of five possiblesuppliers Aiði ¼ 1; 2; . . . ; 5Þ as Table 9.

Step 9: Calculate the closeness coefficient ofeach supplier as

CC1 ¼ 0:50; CC2 ¼ 0:64; CC3 ¼ 0:62,

CC4 ¼ 0:51; CC5 ¼ 0:40.

Step 10: According to the closeness coefficientsof five suppliers and the approval status level, weknow that suppliers A2 and A3 belong to Class IV,

the assessment status of which are ‘‘Approved’’.Suppliers A1, A4 and A5 belong to Class III. Thismeans that their assessment status is ‘‘Recommend

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Table 8

Distances between Aiði ¼ 1; 2; . . . ; 5Þ and A� with respect to

each criterion

C1 C2 C3 C4 C5

d(A1,A�) 0.22 0.38 0.39 0.32 0.32

d(A2,A�) 0.32 0.49 0.5 0.39 0.41

d(A3,A�) 0.39 0.43 0.47 0.41 0.37

d(A4,A�) 0.32 0.34 0.34 0.32 0.37

d(A5,A�) 0.22 0.34 0.28 0.27 0.22

Table 9

Computations of d�i , d�i and CCi

d�i d�i d�i þ d�i CCi

A1 1.66 1.63 3.29 0.5

A2 1.17 2.11 3.28 0.64

A3 1.25 2.07 3.32 0.62

A4 1.61 1.69 3.3 0.51

A5 1.99 1.33 3.32 0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

Very Low

Low (L)

Medium Low

Medium (M)

Medium High

High (H)

Very High

Fig. 7. Alterative linguistic variables for importance weight of

each criterion.

0 1 2 3 4 5 6 7 8 9 10

1

Very Poor

Poor (P)

Medium Poor

Fair (F)

Medium Good

Good (G)

Very Good

Fig. 8. Alterative linguistic variables for ratings.

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301 299

with low risk’’. However, according to the close-ness coefficients, supplier A2 is preferred to A3

because CC24CC3 in Class IV. In Class III, thepreferred order is A44A14A5 because CC44CC14CC5. Finally, the ranking order of fivesuppliers is fA24A3g4fA44A14A5g.

In the other case, the different membershipfunctions of linguistic variables are used tocompare the result of the ranking order. Themembership functions of linguistic variables aretriangular fuzzy numbers. Three decision-makersuse the linguistic variables shown in Figs. 7 and 8to assess the importance of criteria and evaluatethe ratings of candidates with respect to eachcriterion. Applying the computational procedureof proposed method, the closeness coefficient ofeach supplier can be calculated as

CC1 ¼ 0:50; CC2 ¼ 0:64; CC3 ¼ 0:63,

CC4 ¼ 0:51; CC5 ¼ 0:40.

According to the closeness coefficients, we knowthat suppliers A2 and A3 belong to Class IV.Suppliers A1, A4 and A5 belong to Class III. Theranking order of five suppliers is fA24A3g4fA44A14A5g. Obviously, the results of ranking

order are identical when the different membershipfunctions of linguistic variables are used in theproposed method. Therefore, it can confirm thatthis proposed method is very effective to deal withthe problem of supplier selection.

5. Conclusions

Many practitioners and researchers have pre-sented the advantages of supply chain manage-ment. In order to increase the competitiveadvantage, many companies consider that a well-designed and implemented supply chain system isan important tool. Under this condition, buildingon the closeness and long-term relationshipsbetween buyers and suppliers is critical success

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C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301300

factor to establish the supply chain system.Therefore, supplier selection problem becomesthe most important issue to implement a successfulsupply chain system.

In general, supplier selection problems adhere touncertain and imprecise data, and fuzzy-set theoryis adequate to deal with them. In a decision-making process, the use of linguistic variables indecision problems is highly beneficial when per-formance values cannot be expressed by means ofnumerical values. In other words, very often, inassessing of possible suppliers with respect tocriteria and importance weights, it is appropriateto use linguistic variables instead of numericalvalues. Due to the decision-makers’ experience,feel and subjective estimates often appear in theprocess of supplier selection problem, an extensionversion of TOPSIS in a fuzzy environment isproposed in this paper. The fuzzy TOPSIS methodcan deal with the ratings of both quantitative aswell as qualitative criteria and select the suitablesupplier effectively. It appears from the foregoingsections that fuzzy TOPSIS method may be auseful additional tool for the problem of supplierselection in supply chain system.

In fact, the fuzzy TOPSIS method is veryflexible. According to the closeness coefficient,we can determine not only the ranking order butalso the assessment status of all possible suppliers.Significantly, the proposed method provides moreobjective information for supplier selection andevaluation in supply chain system. The systematicframework for supplier selection in a fuzzyenvironment presented in this paper can be easilyextended to the analysis of other managementdecision problems. However, improving the ap-proach for solving supplier selection problemsmore efficiently and developing a group decision-support system in a fuzzy environment can beconsidered as a topic for future research.

Acknowledgements

The authors gratefully acknowledge the finan-cial support of the National Science Council,Taiwan, under project number NSC 92-2213-E-212-008.

References

Bellman, B.E., Zadeh, L.A., 1970. Decision-making in a fuzzy

environment. Management Science 17 (4), 141–164.

Briggs, P., 1994. Vendor assessment for partners in supply.

European Journal of Purchasing & Supply Management 1,

49–59.

Buckley, J.J., 1985. Fuzzy hierarchical analysis. Fuzzy Sets and

Systems 17, 233–247.

Chen, C.T., 2000. Extensions of the TOPSIS for group

decision-making under fuzzy environment. Fuzzy Sets and

Systems 114, 1–9.

Choi, T.Y., Hartley, J.L., 1996. An exploration of supplier

selection practices across the supply chain. Journal of

Operations Management 14, 333–343.

Cook, R.L., 1992. Expert systems in purchasing applications

and development. International Journal of Purchasing and

Management 18, 20–27.

de Boer, L., van der Wegen, L., Telgen, J., 1998. Outranking

methods in support of supplier selection. European Journal

of Purchasing & Supply Management 4, 109–118.

Delgado, M., Verdegay, J.L., Vila, M.A., 1992. Linguistic

decision-making models. International Journal of Intelligent

Systems 7, 479–492.

Delgado, M., Herrera, F., Herrera-Viedma, E., Martinez, L.,

1998. Combining numerical and linguistic information in

group decision making. Journal of Information Sciences

107, 177–194.

Donaldson, B., 1994. Supplier selection criteria on the service

dimension. European Journal of Purchasing & Supply

Management 1, 209–217.

Dowlatshahi, S., 2000. Design–buyer–supplier interface:

Theory versus practice. International Journal of Production

Economics 63, 111–130.

Dubois, D., Prade, H., 1980. Fuzzy Sets and Systems: Theory

and Applications. Academic Press Inc., New York.

Ellram, L.M., 1992. The supplier selection decision in strategic

partnerships. International Journal of Purchasing and

Management 18, 8–14.

Ghodsypour, S.H., O’Brien, C., 1998. A decision support

system for supplier selection using an integrated analytic

hierarchy process and linear programming. International

Journal of Production Economics 56–57, 199–212.

Herrera, F., Herrera-Viedma, E., 2000. Linguistic decision

analysis: Steps for solving decision problems under linguis-

tic information. Fuzzy Sets and Systems 115, 67–82.

Herrera, F., Herrera-Viedma, E., Verdegay, J.L., 1996. A model

of consensus in group decision making under linguistic

assessments. Fuzzy Sets and Systems 78, 73–87.

Hwang, C.L., Yoon, K., 1981. Multiple Attributes Decision

Making Methods and Applications. Springer, New York.

Ittner, C.D., Larcker, D.F., Nagar, V., Rajan, M.V., 1999.

Supplier selection, monitoring practices, and firm perfor-

mance. Journal of Accounting and Public Policy 18, 253–281.

Kaufmann, A., Gupta, M.M., 1991. Introduction to Fuzzy

Arithmetic: Theory and Applications. Van Nostrand Rein-

hold, New York.

ARTICLE IN PRESS

C.-T. Chen et al. / Int. J. Production Economics 102 (2006) 289–301 301

Klir, G.J., Yuan, B., 1995. Fuzzy Sets and Fuzzy Logic: Theory

and Applications. Prentice-Hall Inc., USA.

Lee, E.K., Ha, S., Kim, S.K., Delgado, M., 2001. Supplier

selection and management system considering relationships

in supply chain management. IEEE Transactions on

Engineering Management 48 (3), 307–318.

Monczka, R., Trent, R., Handfield, R., 1998. Purchasing and

Supply Chain Management. South-Western College Pub-

lishing, New York.

Negi, D.S., 1989. Fuzzy analysis and optimization. Ph.D.

Dissertation, Department of Industrial Engineering, Kansas

State University.

Roodhooft, F., Konings, J., 1996. Vendor selection and

evaluation—an activity based costing approach. European

Journal of Operational Research 96, 97–102.

Shin, H., Collier, D.A., Wilson, D.D., 2000. Supply manage-

ment orientation and supplier/buyer performance. Journal

of Operations Management 18, 317–333.

Swift, C.O., 1995. Preference for single sourcing and sup-

plier selection criteria. Journal of Business Research 32,

105–111.

Verma, R., Pullman, M.E., 1998. An analysis of the supplier

selection process. Omega 26, 739–750.

Weber, C.A., Current, J.R., Benton, W.C., 1991. Vendor

selection criteria and methods. European Journal of

Operational Research 50, 2–18.

Weber, C.A., Current, J.R., Desai, A., 1998. Non-cooperative

negotiation strategies for vendor selection. European

Journal of Operational Research 108, 208–223.

Zadeh, L.A., 1975. The concept of a linguistic variable and its

application to approximate reasoning. Information Sciences

8, 199–249(I) 301–357(II).

Zimmermann, H.J., 1991. Fuzzy Set Theory and its Applica-

tions, second ed. Kluwer Academic Publishers, Boston,

Dordrecht, London.