Expert Systems with Applications

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1 Expert Systems wth Applcatons 37 (2010) Contents lsts avalable at ScenceDrect Expert Systems wth Applcatons ournal homepage: Group decson mang process for suppler selecton wth VIKOR under fuzzy envronment Amr Sanaye a, *, S. Fard Mousav b, A. Yazdanhah c a Industral and Manufacturng Engneerng Department, Wayne State Unversty, Detrot, MI 48202, USA b Department of management, Faculty of Humantes, Tarbyat Modares Unversty, Tehran, Iran c Department of Industral Eng., Bonab Islamc Azad Unversty, Bonab, Iran artcle Keywords: Suppler selecton Fuzzy set VIKOR nfo abstract Durng recent years, how to determne sutable supplers n the supply chan has become a ey strategc consderaton. However, the nature of suppler selecton s a complex mult-crtera problem ncludng both quanttatve and qualtatve factors whch may be n conflct and may also be uncertan. The VIKOR method was developed to solve multple crtera decson mang (MCDM) problems wth conflctng and non-commensurable (dfferent unts) crtera, assumng that compromsng s acceptable for conflct resoluton, the decson maer wants a soluton that s the closest to the deal, and the alternatves are evaluated accordng to all establshed crtera. In ths paper, lngustc values are used to assess the ratngs and weghts for these factors. These lngustc ratngs can be expressed n trapezodal or trangular fuzzy numbers. Then, a herarchy MCDM model based on fuzzy sets theory and VIKOR method s proposed to deal wth the suppler selecton problems n the supply chan system. A numercal example s proposed to llustrate an applcaton of the proposed model. Ó 2009 Publshed by Elsever Ltd. 1. Introducton In today s ferce compettve envronment characterzed by thn proft margns, hgh consumer expectatons for qualty products and short lead-tmes, companes are forced to tae the advantage of any opportunty to optmze ther busness processes. To reach ths am, academcs and practtoners have come to the same concluson: for a company to reman compettve, t has to wor wth ts supply chan partners to mprove the chan s total performance. Thus, beng the man process n the upstream chan and affectng all areas of an organzaton, the purchasng functon s tang an ncreasng mportance. Thus supply chan management and the suppler (vendor) selecton process s an ssue that receved relatvely large amount of attenton n both academa and ndustry. Suppler selecton s a fundamental ssue of supply chan area whch heavly contrbutes to the overall supply chan performance. Partcularly for companes who spend a hgh percentage of ther sales revenue on parts and materal supples, and whose materal costs represent a larger porton of total costs, savngs from supples are of partcular mportance. These, strongly urge for a more systematc and transparent approach to purchasng decson mang, especally regardng the area of suppler selecton. Selectng the supplers sgnfcantly reduces the purchasng cost and mproves corporate compettveness, and that s why many experts beleve * Correspondng author. E-mal addresses: mousavfard@yahoo.com, sanaye@wayne.edu (A. Sanaye). that the suppler selecton s the most mportant actvty of a purchasng department. Suppler selecton s the process by whch supplers are revewed, evaluated, and chosen to become part of the company s supply chan. The overall obectve of suppler selecton process s to reduce purchase rs, maxmze overall value to the purchaser, and buld the closeness and long term relatonshps between buyers and supplers (Chena, Ln, & Huangb, 2006). Several factors affect a suppler s performance. Dcson (1966), Ellram (1990), Roa and Kser (1980), Stamm and Golhar (1993) dentfed, respectvely 60, 18, 13 and 23 crtera for suppler selecton. The suppler selecton process s often n nfluenced by uncertanty n practce (de Boer, van der Wegen, & Telgen, 1998; Mn, 1994). Due to strategc mportance and nvolvement of varous uncertantes and rss assocated wth the suppler selecton process, several decson maers from departments other than purchasng such as producton, fnance, and maretng are very often nvolved n the decson mang process for suppler selecton process. Therefore, some scholars emphaszed the need for a ratonal and systematc group decson mang process for suppler selecton (de Boer et al., 1998). In essental; the suppler selecton problem n supply chan system s a group decson mang combnaton of several and dfferent crtera wth dfferent forms of uncertanty (Chena et al., 2006). Hence ths problem s a nd of multple crtera decson mang problem (MCDM) whch requres MCDM methods for an effectve problem-solvng. Due to nature of the problem, the technques of MCDM are coherently derved to /$ - see front matter Ó 2009 Publshed by Elsever Ltd. do: /.eswa

2 A. Sanaye et al. / Expert Systems wth Applcatons 37 (2010) Table 1 Suppler selecton methods and examples. Category Approach Example MADM AHP Narasmhan (1983), Barbarosoglu and Yazgaç (1997); Nydc and Hll (1992), Tam (2001), Lee et al. (2001), Lu and Ha (2005) ANP Sars and Tallur (2002) MAUT Mn (1994) Outranng de Boer et al. (1998) method TOPSIS Chena et al. (2006) Mathematcal LP Pan (1989) programmng GP Bufa and Jacson (1983), Karpa et al. (1999) MIP Weber and Ellram (1993), Chaudhry et al. (1993) DEA Weber (1996) Statstcal/ probablstc Artfcal ntellgence Hybrd and nnovatve approaches Hnle et al. (1969) Ronen and Tretsch (1998) Souup (1987) Neural Albno and Garavell (1998), Choy et al. (2002) Networs Case-based Coo (1997) reasonng Expert Voura et al. (1996) System AHP LP Ghodsypour and O Bren (1998) ANP MIP Demrtas and Üstün (n.d) ANP Shyur and Shh (2006) TOPSIS Fuzzy QFD Bevlacqua et al. (2006) manage t. de Boer, Labro, and Morlacch (2001) and Assaou, Haouar, and Hassn (2007) gave a good revew and classfcaton of the methods for supportng suppler selecton. We can roughly dvde these methods nto sx man categores: mult-attrbute decson mang (MADM), mult-obectve decson mang and mathematcal programmng (MP), statstcs/probablstc approaches, ntellgent approaches, hybrd approaches and others. Sx categores, each wth ther own related approaches and examples, are lsted n Table 1. Methods of the frst category concentrate on selecton actvtes. They select a lmted and countable number of predetermned alternatves through multple attrbutes or crtera. These methods nvolves mult attrbute utlty theory (MAUT), outranng methods, analytcal herarchy process (AHP) and ts sophstcated verson, analytcal networ process (ANP) and technque for order performance by smlarty to deal soluton (TOPSIS). Among these methods, t s dffcult to obtan a mathematcal representaton of the decson maer s utlty functon for usng MAUT (Oprcovc & Tzeng, 2007). The outranng methods are normally not used for the actual selecton of alternatves, but they are very sutable for the ntal screenng process (to categorze alternatves nto acceptable or unacceptable). After the screenng process another method must be used to get a full ranng or actual recommendatons among the alternatves (Loen, 2007). Also AHP and ANP have ther own problems: ran reversal and dffculty n accommodatng a great many canddates. The other method n ths category, TOPSIS, s dscussed n Secton 2. The methods n the second category optmze the nteractons and tradeoffs among dfferent factors of nterest by consderng constrants and dfferent ssues le dscount, sngle or multple sourcng and logstc costs; whch allow the buyer to mae an effectve decson usually by determnng the best order quantty/ perod from the sutable suppler/supplers. Several optmzaton methods such as goal programmng, lnear programmng, mxed nteger and data envelopment analyss have been appled n ths area. A sgnfcant problem wth usng mathematcal programmng methods s that most of them are too complex for practcal use by operatng managers. The other fallbac of these methods s ther lac to consder qualtatve factors. Furthermore the methods n ths category are manly used n multple sourcng envronments for assgnng order quanttes between suppler/supplers. Statstcal studes ncorporate uncertanty; there are not many artcles n the lterature that utlze statstcs n the suppler selecton process. The publshed statstcal models only accommodate for uncertanty wth regard to one crteron at a tme (de Boer et al., 2001). Artfcal Intellgence (AI) based models are based on computeraded systems that n one way or another can be traned by a purchasng expert or hstorc data, however, the complexty of the system s not sutable for enterprses to solve the ssue effcently wthout hgh capablty n advanced computer programs. The ffth category s hybrd and nnovatve methods whch authors ntegrate one or more methods together to utlze ther both advantages. However the dsadvantages of combned methods affect the effectveness of hybrd models. In other way the VIKOR method, a recently ntroduced new MCDM method developed to solve multple crtera decson mang (MCDM) problems wth conflctng and non-commensurable (dfferent unts) crtera (Oprcovc & Tzeng, 2007), may provde the bass for developng suppler selecton models that can effectvely deal wth characterstcs of ths problem. In ths paper, we used the concept of fuzzy set theory and lngustc values to overcome uncertanty and qualtatve factors. Then, a herarchy MCDM model based on fuzzy sets theory and VIKOR method s proposed to deal wth the suppler selecton problems n the supply chan system. The rest of ths paper s structured as follows. In the next secton, an overvew and bacground of the VIKOR method s presented. In Secton 3, an overvew of the concepts of the fuzzy approach s gven. Secton 4 wll focus on the proposed model. Then a numercal example s llustrated n Secton 5. In the fnal secton, some conclusons are drawn for the study. 2. VIKOR method Oprcovc (1998), Oprcovc and Tzeng (2002) developed VIKOR, the Serban name: VlseKrterumsa Optmzaca I Kompromsno Resene, means mult-crtera optmzaton and compromse soluton (Chu, Shyu, Tzeng, & Khosla, 2007). The VIKOR method was developed for mult-crtera optmzaton of complex systems (Oprcovc & Tzeng, 2004). Ths method focuses on ranng and selectng from a set of alternatves, and determnes compromse solutons for a problem wth conflctng crtera, whch can help the decson maers to reach a fnal decson. Here, the compromse soluton s a feasble soluton whch s the closest to the deal, and a compromse means an agreement establshed by mutual concessons (Oprcovc & Tzeng, 2007). It ntroduces the mult-crtera ranng ndex based on the partcular measure of closeness to the deal soluton (Oprcovc, 1998). Accordng to (Oprcovc & Tzeng, 2004) the mult-crtera measure for compromse ranng s developed from the PL p -metrc used as an aggregatng functon n a compromse programmng method (Yu, 1973). The varous J alternatves are denoted as a 1 ; a 2 ;...; a J. For alternatve a, the ratng of the th aspect s denoted by f,.e. f s the value of th crteron functon for the alternatve a ; n s the number of crtera. Development of the VIKOR method started wth the followng form of L p -metrc: ( ) 1=p L p; ¼ Xn ½w ðf f Þ=ðf f ÞŠ p ; ð1þ ¼1 1 6 p 6 1; ¼ 1; 2;...; J: Wthn the VIKOR method L 1; (as S n Eq. (15)) and L 1; (as R n Eq. (16)) are used to formulate ranng measure. L 1; s nterpreted as

3 26 A. Sanaye et al. / Expert Systems wth Applcatons 37 (2010) concordance and can provde decson maers wth nformaton about the maxmum group utlty or maorty. Smlarly, L 1; s nterpreted as dscordance and provdes decson maers wth nformaton about the mnmum ndvdual regret of the opponent. Also TOPSIS, another MCDM method, s based on aggregatng functon representng closeness to deal. In TOPSIS the chosen alternatve should have the shortest dstance from the deal soluton and the farthest dstance from the negatve-deal. The TOPSIS method ntroduces two reference ponts, but t does not consder the relatve mportance of the dstances from these ponts. These two MCDM methods use dfferent nds of normalzaton to elmnate the unts of the crteron functons, whereas the VIKOR method uses lnear normalzaton, the TOPSIS method uses vector normalzaton. The normalzed value n the VIKOR method does not depend on the evaluaton unt of crteron functon, whereas the normalzed values by vector normalzaton n the TOPSIS method may depend on the evaluaton unt (Chu et al., 2007). 3. Fuzzy approach In dealng wth a decson process, the decson maer s often faced wth doubts, problems and uncertantes. In other words natural language to express percepton or udgment s always subectve, uncertan or vague. To resolve the vagueness, ambguty and subectvty of human udgment, fuzzy sets theory (Zadeh, 1965) was ntroduced to express the lngustc terms n decson mang (DM) process. Bellman and Zadeh (1970) developed fuzzy multcrtera decson mang (FMCDM) methodology to resolve the lac of precson n assgnng mportance weghts of crtera and the ratngs of alternatves regardng evaluaton crtera. The logcal tools that people can rely on are generally consdered the outcome of a bvalent logc (yes/no, true/false), but the problems posed by real-lfe stuatons and human thought processes and approaches to problem-solvng are by no means bvalent (Tong & Bonssone, 1980). Just as conventonal, bvalent logc s based on classc sets, fuzzy logc s based on fuzzy sets. A fuzzy set s a set of obects n whch there s no clear-cut or predefned boundary between the obects that are or are not members of the set. The ey concept behnd ths defnton s that of membershp : any obect may be a member of a set to some degree ; and a logcal proposton may hold true to some degree. Each element n a set s assocated wth a value ndcatng to what degree the element s a member of the set. Ths value comes wthn the range [0,1], where 0 and 1, respectvely, ndcate the mnmum and maxmum degree of membershp, whle all the ntermedate values ndcate degrees of partal membershp (Bevlacqua, Carapca, & Gacchetta, 2006). Ths approach helps decson maers solve complex decson mang problems n a systematc, consstent and productve way (Carlsson & Fuller, 1996) and has been wdely appled to tacle DM problems wth multple crtera and alternatves (Wang & Chang, 2007). In short, fuzzy set theory offers a mathematcally precse way of modelng vague preferences for example when t comes to settng weghts of performance scores on crtera. Smply stated, fuzzy set theory maes t possble to mathematcally descrbe a statement le: crteron X should have a weght of around 0.8 (de Boer et al., 2001). Fuzzy set theory was also looed at as a tool for suppler selecton because of the vagueness of the nformaton related to parameters. Narasmhan (1983), Nydc and Hll (1992) handled mprecse nformaton and uncertanty n suppler selecton models for fndng the best overall ratng suppler. Amd, Ghodsypour, and O Bren (2006) developed a fuzzy mult-obectve lnear model to overcome the vagueness of the nformaton. Chena et al. (2006) developed a model that combnes the use of fuzzy set theory and TOPSIS. In the followng, for the purpose of reference, some mportant defntons and notatons of fuzzy sets theory from (Kaufmann & Gupta, 1991; Dubos & Prade, 1980; Zadeh, 1975; Chena et al., 2006) wll be revewed. Let X be the unverse of dscourse, X ¼fx 1 ; x 2 ;...; x n g. A fuzzy set A e of X s a set of order pars, fðx 1 ; f ea ðx 1 ÞÞ; ðx 2 ; f ea ðx 2 ÞÞ;...; ðx n ; f ea ðx n ÞÞg; f ea : X! ½0; 1Š s the membershp functon of e A, and f ea ðx Þ stands for the membershp degree of x n A. e The value f ea s closer to 0, the degree s low. The value f ea s closer to 1, the degree s hgh. A fuzzy set A e of the unverse of dscourse X s convex f and only f for all x 1 ; x 2 n X, f ea ðx 1 þð1 Þx 2 Þ P mn½f ea ðx 1 Þ; f ea ðx 2 ÞŠ; where 2½0; 1Š; x 1 ; x 2 2 X. The heght of a fuzzy set s the largest membershp grade attaned by any element n that set. A fuzzy set A e n the unverse of dscourse X s called normalzed when the heght of A e s equal to 1. A fuzzy number s a fuzzy subset n the unverse of dscourse X that s both convex and normal. Fuzzy membershp functon has more types. Ths paper adopts the type of a trapezodal fuzzy number. A postve trapezodal fuzzy number (PTFN) c can be defned as ða 1 ; a 2 ; a 3 ; a 4 Þ, shown n Fg. 1. The membershp functon l ea ðxþ s defned as: 8 0; x < a 1 ; x a 1 >< a 2 a 1 ; a 1 6 x 6 a 2 ; l ea ðxþ ¼ 1; a 2 6 x 6 a 3 ; ð2þ x a 4 a 3 a 4 ; a 3 6 x 6 a 4 ; >: 0; x > a 3 : A non-fuzzy number r can be expressed as ðr; r; r; rþ. By the extenson prncple, the fuzzy sum and fuzzy subtracton of any two trapezodal fuzzy numbers are also trapezodal fuzzy numbers; but the multplcaton of any two trapezodal fuzzy numbers s only an approxmate trapezodal fuzzy number. Gven any two postve trapezodal fuzzy numbers, ~a ¼ða 1 ; a 2 ; a 3 ; a 4 Þ; b ~ ¼ ðb 1 ; b 2 ; b 3 ; b 4 Þ and a postve real number r, some man operatons of fuzzy numbers A e and B e can be expressed as follows: ea B e ¼½a 1 þ b 1 ; a 2 þ b 2 ; a 3 þ b 3 ; a 4 þ b 4 Š; ð3þ ea B e ¼½a1 b 1 ; a 2 b 2 ; a 3 b 3 ; a 4 b 4 Š; ð4þ ea e B ff½a 1 b 1 ; a 2 b 2 ; a 3 b 3 ; a 4 b 4 Š; ea r ff½a 1 r; a 2 r; a 3 r; a 4 rš: The operatons of _ðmaxþ and ^ðmnþ are defned as follow: eað_þb e ¼ða 1 _ b 1 ; a 2 _ b 2 ; a 3 _ b 3 Þ; eað^þ e B ¼ða 1 ^ b 1 ; a 2 ^ b 2 ; a 3 ^ b 3 Þ: Fg. 1. Trapezodal fuzzy number e A. ð5þ ð6þ ð7þ ð8þ

4 A. Sanaye et al. / Expert Systems wth Applcatons 37 (2010) Also the crsp value of the fuzzy number A e based on Center of Area (COA) method can be expressed by followng relaton: R defuzzðaþ¼ e x lðxþdx R lðxþdx ¼ R a2 x a 1 a 1 a 2 a 1 R a2 a 1 x a 1 a 2 a 1 xdxþ R a 3 a 2 xdxþ R a 4 a 3 dx þ R a 3 a 3 dx þ R a 4 a 3 a 4 x a 4 a 3 a 4 x a 4 a 3 dx xdx ¼ a 1a 2 þ a 3 a 4 þ 1 3 ða 4 a 3 Þ ða 2 a 1 Þ 2 a 1 a 2 þ a 3 þ a 4 : ð9þ 4. Proposed method for suppler selecton A systematc approach to extend the VIKOR s proposed to solve the suppler selecton problem under a fuzzy envronment n ths secton. In ths paper the mportance weghts of varous crtera and the ratngs of qualtatve crtera are consdered as lngustc varables. Because lngustc assessments merely approxmate the subectve udgment of decson maers, we can consder lnear trapezodal membershp functons to be adequate for capturng the vagueness of these lngustc assessments. In fact, suppler selecton n supply chan system s a group multple crtera decson mang (GMCDM) problem, whch may be descrbed by means of the followng sets (Chena et al., 2006): 1. a set of K decson maers called E ¼fD 1 ; D 2 ;...; D K g; 2. a set of m possble supplers called A ¼fA 1 ; A 2 ;...; A m g; 3. a set of n crtera, C ¼fC 1 ; C 2 ;...; C n g, wth whch suppler performances are measured; 4. a set of performance ratngs of A ð ¼ 1; 2;...; mþ wth respect to crtera C ð ¼ 1; 2;...; nþ, called X ¼fx ; ¼ 1; 2;...; m; ¼ 1; 2;...; ng The man steps of the algorthms are: 4.1. Identfy the obectves of the decson mang process and defne the problem scope Decson mang s the process of defnng the decson goals, gatherng relevant nformaton and selectng the optmal alternatve (Hess & Sclano, 1996). Thus, the frst step s defnng the decson goal that here s to evaluate and select a favorable suppler/s. Mang precse statement of the problem wll help to narrow t. Gvng clear and careful thought to ths frst step s very vtal to selectng process. The way n whch the process s defned wll determnstc the character of all the other steps. In ths step, the scope of the problem s defned n terms of the product/servce to be outsourced, tme frame for outsourcng, ustfcaton of decson, constrants n the suppler selecton process, avalable alternatve sources to choose from etc. Then the obectves of suppler selecton s derved from varous areas of organzaton mpacted by the decson, e.g. assembly lne, suppler qualty assurance department, fnance group, logstcs department etc. and algnng them wth the overall organzatonal goals Arrange the decson mang group and defne and descrbe a fnte set of relevant attrbutes As mentoned prevously, n suppler evaluaton and selecton process several people and experts from dfferent functonal areas wthn the company are nvolved. So wth consderng the problem scope defned n prevous secton and ts entre dmenson, we must form a group of decson maers. Suppler selecton frst requres dentfcaton of decson attrbutes (crtera) then evaluaton scales/metrcs are determned n order to measure apposteness of suppler. These crtera must be defned accordng to the corporate strateges, company s compettve stuaton, the level of buyer suppler ntegraton (Ghodsypour & O Bren, 1998) and type of product whch be outsourced. Then wth consderng sub-crtera for each man crteron, herarchcal form called value tree s structured Identfy the approprate lngustc varables In ths step we must defne the approprate lngustc varables for the mportance weght of crtera, and the fuzzy ratng for alternatves wth regard to each crteron these lngustc varables can be expressed n postve trapezodal fuzzy numbers, as n Fgs. 1 and 2. It s suggested n ths paper that the decson maers use the lngustc varables shown n Fgs. 1 and 2 to evaluate the mportance of the crtera and the ratngs of alternatves wth respect to qualtatve crtera. For example, the lngustc varable Medum Hgh (MH) can be represented as (0.5; 0.6; 0.7; 0.8), the membershp functon of whch s: 8 0; x < 0:5; x 0:5 >< ; 0:5 6 x 6 0:6; 0:6 0:5 l Medum Hgh ðxþ ¼ 1; 0:6 6 x 6 0:7; ð10þ x 0:8 ; 0:7 6 x 6 0:8; 0:7 0:8 >: 0; x > 0:8: 4.4. Pull the decson maers opnons to get the aggregated fuzzy weght of crtera, and aggregated fuzzy ratng of alternatves and construct a fuzzy decson matrx Let the fuzzy ratng and mportance weght of the th decson maer be ~x ¼ðx 1 ; x 2 ; x 3 ; x 4 Þ and ~w ¼ð~w 1 ; ~w 2 ; ~w 3 ; ~w 4 Þ; ¼ 1; 2;...; m; ¼ 1; 2;...; n, respectvely. Hence, the aggregated fuzzy ratngs ð~x Þ of alternatves wth respect to each crteron can be calculated as: ~x ¼ðx 1 ; x 2 ; x 3 ; x 4 Þ; ð11þ where Fg. 2. Lngustc varables for mportance weght of each crtera. x 1 ¼ mnfx 1 g; x 2 ¼ 1 K ¼ maxfx 4 g: x 2 ; x 3 ¼ 1 K x 3 ; x 4

5 28 A. Sanaye et al. / Expert Systems wth Applcatons 37 (2010) The aggregated fuzzy weghts ð ~w Þ of each crteron can be calculated as: ~w ¼ðw 1 ; w 2 ; w 3 ; w 4 Þ; ð12þ where w 1 ¼ mnfw 1 g; w 2 ¼ 1 K w 4 ¼ maxfw 4 g: w 2 ; w 3 ¼ 1 K w 3 ; A suppler selecton problem can be concsely expressed n matrx format as follows: 2 3 ~x 11 ~x 12 ~x 1n ~x 21 ~x 22 ~x 2n ed ¼ ; W f ¼ ½ ~w1 ~w 2 ~w n Š; ~x m1 ~x m2 ~x mn where ~x the ratng of alternatve A wth respect to C ; ~w the mportance weght of the th crteron holds, ~x ¼ðx 1 ; x 2 ; x 3 ; x 4 Þ and ~w ¼ðw 1 ; w 2 ; w 3 ; w 4 Þ; ¼ 1; 2;...; m; ¼ 1; 2;...; n are lngustc varables can be approxmated by postve trapezodal fuzzy numbers Defuzzfy the fuzzy decson matrx and fuzzy weght of each crteron nto crsp values Deffuzzfy fuzzy decson matrx and fuzzy weght of each crteron nto crsp values usng COA defuzzfcaton relaton proposed n Secton 3 (Relaton 9) Determne the best f ratngs, ¼ 1; 2;...; n f f ¼ max x ; ¼ mn x : and the worst f 4.7. Compute the values S and R by the relatons S ¼ Xn ¼1 R ¼ max w ðf w ðf values of all crteron ð13þ ð14þ f Þ=ðf f Þ; ð15þ f Þ=ðf f Þ: ð16þ 4.8. Compute the values Q by the relatons Q ¼ vðs S Þ=ðS S Þþð1 vþðr R Þ=ðR R Þ; ð17þ where S ¼ mn S ; S ¼ max S ; R ¼ mn R ; R ¼ max R and v s ntroduced as a weght for the strategy of maxmum group utlty, whereas 1 vs the weght of the ndvdual regret Ran the alternatves, sortng by the values S; R and Q n ascendng order Propose as a compromse soluton the alternatve ða ð1þ Þ whch s the best raned by the measure Q (mnmum) f the followng two condtons are satsfed C1. Acceptable advantage: QðA ð2þ Þ QðA ð1þ Þ P DQ; ð18þ where A ð2þ s the alternatve wth second poston n the ranng lst by Q; DQ ¼ 1=ðJ 1Þ. C2. Acceptable stablty n decson mang: The alternatve A ð1þ must also be the best raned by S or/and R. Ths compromse soluton s stable wthn a decson mang process, whch could be the strategy of maxmum group utlty (when v > 0:5 s needed), or by consensus v 0:5, or wth veto ðv < 0:5Þ. Here, v s the weght of decson mang strategy of maxmum group utlty. If one of the condtons s not satsfed, then a set of compromse solutons s proposed, whch conssts of Alternatves A ð1þ and A ð2þ f only the condton C2 s not satsfed, or Alternatves A ð1þ ; A ð2þ ;...; A ðmþ f the condton C1 s not satsfed; A ðmþ s determned by the relaton QðA ðmþ Þ QðA ð1þ Þ < DQ for maxmum M (the postons of these alternatves are n closeness ). 5. Numercal example The proposed model has been appled to a suppler selecton process of a frm worng n the feld of automoble part manufacturng n the followng steps: Step 1: The Company desres to select a sutable suppler to purchase the ey components of ts new product. After prelmnary screenng, fve canddate supplers (S1, S2, S3, S4, and S5) reman for further evaluaton. Step 2: A commttee of three decson maers, D1; D2 and D3, has been formed to select the most sutable suppler. The followng crtera have been defned: Product qualty On-tme delvery Prce/cost Suppler s technologcal level Flexblty Step 3: Three decson maers use the lngustc weghtng varables shown n Fg. 2 to assess the mportance of the crtera. The mportance weghts of the crtera determned by these three decson maers are shown n Table 2. Also the decson maers use the lngustc ratng varables shown n Fg. 3 to evaluate the ratngs of canddates wth respect to each crteron. The ratngs of the fve supplers by the decson maers under the varous crtera are shown n Table 3. Step 4: The lngustc evaluatons shown n Tables 2 and 3 are converted nto trapezodal fuzzy numbers. Then the aggregated weght of crtera and aggregated fuzzy ratng of alternatves s calculated to construct the fuzzy decson matrx and determne the fuzzy weght of each crteron, as n Tables 4 and 5. Table 2 Importance weght of crtera from three decson maers. Crtera Decson maers D1 D2 D3 C1 H H H C2 VH VH H C3 VH VH VH C4 H H MH C5 H H H

6 A. Sanaye et al. / Expert Systems wth Applcatons 37 (2010) Table 5 Crsp values for decson matrx and weght of each crteron. Crtera C1 C2 C3 C4 C5 Weght S S S S S Table 6 The values of S, R and Q for all supplers. Supplers S1 S2 S3 S4 S5 Fg. 3. Lngustc varables for ratngs. S R Q Table 3 Ratngs of the fve supplers by the decson maers under the varous crtera. Supplers Crtera C1 C2 C3 C4 C5 Decson maer D 1 S1 G MG G G G S2 G VG MP G VG S3 VG MG F VG G S4 G G MG G G S5 MG MG MG MG MG D 2 S1 G MG G G G S2 G VG F VG MG S3 VG G F VG VG S4 G G MG G G S5 MG G MG MG MG D 3 S1 VG VG G G G S2 G VG MP VG VG S3 G G F VG G S4 G MG G G VG S5 MG G MG G MG Step 5: The crsp values for decson matrx and weght of each crteron are computed as shown n Table 6. Step 6: The best and the worst values of all crteron ratngs are determned as follows: f 1 ¼ 0:87; f 2 ¼ 0:92; f 3 ¼ 0:80; f 4 ¼ 0:92; f 5 ¼ 0:85; f 1 ¼ 0:65; f 2 ¼ 0:72; f 3 ¼ 0:40; f 4 ¼ 0:70; f 5 ¼ 0:65: Steps 7 and 8: The values of S, R and Q are calculated for all supplers as Table 6. Step 9: The ranng of the supplers by S, R and Q n decreasng order s shown n Table 7. Step 9: As we see n Table 6, the suppler S 3 s the best raned by Q. Also the condtons C1 and C2 are Table 7 The ranng of the supplers by S, R and Q n decreasng order. Ranng supplers By S S1 S3 S2 S4 S5 By R S1 S3 S4 S5 S2 By Q S1 S3 S2 S4 S5 6. Concluson satsfed (Q S1 Q S3 P and S 3 s best raned by R and S). So s S 3 s the best choce. Many practtoners and academcs have emphaszed the advantages of supply chan management. In order to ncrease the compettve advantage, many companes consder that a welldesgned and mplemented supply chan system s an mportant tool. Therefore beng the man process n the upstream chan and affectng all areas of an organzaton, the suppler selecton problem becomes the most mportant ssue to mplement a successful supply chan system. The suppler selecton problem s often nfluenced by uncertanty n practce, and n such stuaton fuzzy set theory s an approprate tool to deal wth ths nd of problems. In real decson mang process, the decson maer s unable (or unwllng) to express hs preferences precsely n numercal values and the evaluatons are very often expressed n lngustc terms. In ths paper an extenson of the VIKOR, a recently ntroduced MCDM method, n Table 4 Aggregated fuzzy weght of crtera and aggregated fuzzy ratng of alternatves. Crtera C1 C2 C3 C4 C5 Weght (0.70,0.80,0.80,0.90) (0.70,0.87,0.93,1.00) (0.80,0.90,1.00,1.00) (0.50,0.73,0.77,0.90) (0.70,0.80,0.80,0.90) S1 (0.70,0.83,0.87,1.00) (0.50,0.70,0.80,1.00) (0.70,0.80,0.80,0.90) (0.70,0.80,0.80,0.90) (0.70,0.80,0.80,0.90) S2 (0.70,0.80,0.80,0.90) (0.80,0.90,1.00,1.00) (0.20,0.37,0.43,0.60) (0.70,0.87,0.93,1.00) (0.50,0.80,0.90,1.00) S3 (0.70,0.87,0.93,1.00) (0.50,0.73,0.77,0.90) (0.40,0.50,0.50,0.60) (0.80,0.90,1.00,1.00) (0.70,0.83,0.87,1.00) S4 (0.70,0.80,0.80,0.90) (0.50,0.73,0.77,0.90) (0.50,0.67,0.73,0.90) (0.70,0.80,0.80,0.90) (0.70,0.83,0.87,1.00) S5 (0.50,0.60,0.70,0.80) (0.50,0.73,0.77,0.90) (0.50,0.60,0.70,0.80) (0.50,0.67,0.73,0.90) (0.50,0.60,0.70,0.80)

7 30 A. Sanaye et al. / Expert Systems wth Applcatons 37 (2010) fuzzy envronment s proposed to deal wth the both qualtatve and quanttatve crtera and select the sutable suppler effectvely. It appears ths method has some advantages whch may be useful n dealng wth suppler selecton problem. The proposed method s very flexble. Usng ths method not only enables us to determne the outranng order of supplers, but also assess and rate the supplers. These ratng can be used n combnaton wth mathematcal programmng and other methods to deal wth suppler selecton n multple sourcng envronments. Also the proposed method for suppler selecton n fuzzy envronment provdes a systematc approach whch can be easly extend to deal wth other management decson mang problems. References Assaou, N., Haouar, M., & Hassn, E. (2007). Suppler selecton and order lot szng modelng: A revew. Computers and Operatons Research, 34(12), Albno, V., & Garavell, A. (1998). 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