An Intuitionistic Fuzzy Multi-Objective Vendor. Selection Problem

Transcription

1 Applied Mathematical Scieces, vol. 8, 2014, o. 149, HIKARI Ltd, A Ituitioistic Fuzzy Multi-Obective Vedor Selectio Problem Prabot Kaur * Applied Mathematics Birla Istitute of Techology Mesra, Rachi, Jharkhad, Idia *Correspodig Author Copyright 2014 Prabot Kaur. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract A vedor selectio problem is a multiobective decisio makig problem ivolvig the optimizatio of cost, quality ad delivery performace o the basis of evaluatio of criteria like cost, quality, service etc. The problem of vedor selectio ivolves its selectio ad allocatio of order. The selectios of appropriate vedors eable firms to improve performace related to customer's eeds, requiremets ad satisfactio. I real world most of the iput iformatio related to criteria is ot kow precisely due to several coflictig factors. Imprecisio is best hadled by fuzzy set theory. A more advaced form of fuzzy set theory with a additioal degree of freedom is ituitioistic fuzzy sets. We choose triagular ituitioistic fuzzy umber to represet the criteria because of its ability to hadle ucertaity i data ad its simple arithmetic operatios to solve ay liear programmig problem. This paper develops a ituitioistic fuzzy multiobective liear model (IFMOLM) to vedor selectio problem with the coefficiet of the obectives as triagular ituitioistic fuzzy umbers ad rest of the data i the costraits as crisp. The multiple obectives are miimizatio of et price, maximizatio of quality ad maximizatio of o time delivery for vedors. Specifically the paper icorporates the LPP is formulated i ituitioistic fuzzy form ad a rakig fuctio coverts the IFMOLM to equivalet crisp liear form (Dubey ad Mehra(2011)). A umerical example illustrates the applicatio of the methodology.

2 7444 Prabot Kaur The results of the umerical example show that the results obtaied by this methods is better tha fuzzy approach. Keywords: Vedor selectio, Triagular Ituitioistic Fuzzy Number (TIFN), Multicriteria decisio Makig (MCDM) 1.0 Itroductio Vedor selectio problem is a multiobective decisio makig problem where decisios may be drive by more tha oe obective.the obectives may be of low cost or good quality ad good service facility.the vedor selectio problem is a selectio ad allocatio of order to various vedors selected. The first study of vedor selectio problem was doe by Dickso i the 1960's. Sice the various multi-criteria decisio makig approaches have bee proposed for vedor selectio, such as the aalytic hierarchy process(narasimha (1983)), case-based reasoig(choy ad Lee (2002)), data evelopmet aalysis (Baker ad Talluri (1997)), fuzzy set theory (Che et al. (2006),Sarkar ad Mohapatra (2006)), geetic algorithm (Dig et al. (2005)), mathematical programmig(weber & Curret(1993), Zydiak, & Chaudhry, 1995) ad multiple obective programmig (Buffa & Jackso, 1983; Feg; Ghoudsypour & O Brie, 1998; Weber & Ellram,1992). Sigificat aalytical methods are lackig i VSP as discussed by Weber (1991) where he listed oly te articles (Gaballa (1974), Athoy ad Buffa (1977), Pa (1989) etc) correspodig to mathematical programmig. The disadvatage of such models is that they are applicatio specific or ot readily geeralizable or limited i terms of the scope of assumptios. The ifluece of the criteria ad its impact o alteratives provided by decisio makers are difficult to exactly express by crisp data i the selectio of vedors. Fuzzy sets was itroduced by Zadeh (1965) to express impreciseess or vagueess i data.a further improvemet i fuzzy sets was the cocept of ituitioistic fuzzy set (IFS) itroduced by Ataassov (1986). It is defied by two fuctios expressig the degree of membership ad the degree of omembership, respectively. Accordigly, IFS is a appropriate tool to describe the impreciseess or ucertaiity i data with a additioal degree of freedom. IFS has foud applicatio i various areas ad oe area beig the supply chai. Guo,Qi ad Zhao(2010) gave a approach based o ituitioistic fuzzy topsis to deal with supplier selectio problem. Bora et al (2009) gave a multicriteria itutioistic fuzzy group decisio makig for supplier selectio with Topsis method. Shahrokhi et al. (2011) gave a itegrated method usig ituitioistic fuzzy set ad liear programmig for supplier selectio problem.i this paper we set a multiobective optimizatio problem i a ituitioistic fuzzy eviromet ad results obtaied by ituitioistic approach are better compared to fuzzy approach.

3 A ituitioistic fuzzy multi-obective vedor selectio problem 7445 The orgaizatio of the paper is as follows: Sectio 2 we explai about basics of ituitioistic fuzzy sets. I sectio 3 model formulatio i a ituitioistic fuzzy eviromet ad its solutio is discussed. Sectio 4 takes a umerical example for illustratio of the methodology.i sectio 5 we coclude the results. 2.0 Prelimiaries Defiitio 1 : Give a fixed set X= {x1,x2,..,x}, a ituitioistic fuzzy set (IFS) is defied as A = (<xi, ta(xi), fa(xi)>/xi X ) which assigs to each elemet xi, a membership degree ta(xi) ad a o-membership degree fa(xi) uder the coditio 0 ta( xi ) fa( xi ) 1, for all xi X. Defiitio 2: A (TIFN) A = (a 1, a 2, a 3 ; w a )(a 1, a 2, a 3 ; u a )} is a IFS i R with the followig membership fuctio μ A (x) ad o- membership θ A (x). (x a 1 )w a a 2 a 1, a 1 x a 2 μ A (x) = w a x = a 2 (a 3 x )w a a 3 a 2, a 2 x a 3 { 0, otherwise a 2 x + u a ( x a 1 ) a 2 a, a 1 x a 2 1 ad θ A (x) = u a, x = a 2 x a 2 +u a (a 3 x) a 3 a 2, a 2 x a 3 { 1, otherwise The values w a ad u a respectively represet the maximum degree of membership ad the o-membership such that 0 wa 1,0 ua 1.+ Defiitio 3: Arithmetic Operatios of TIFN is give by: If A = (a 1, a 2, a 3 ; w a )(a 1, a 2, a 3 ; u a )} ad B = (b 1, b 2, b 3 ; w b )(b 1, b 2, b 3 ; u b )} are two TIFNs, ad k be a real umber the A + B = {(a 1 + b 1,a 2 + b 2, a 3 + b 3 ; mi {w a, w b )(a 1 + b 1, a 2 + b 2, a 3 + b 3 ; max {u a, u b} )} is also a TIFN. ka =is a TIFN {(ka1, ka2, ka3; wa)(ka1 ',ka2,ka3 ' ; ua)}.k>0.

4 7446 Prabot Kaur Defiitio 4: Let A = (a 1, a 2, a 3 ; w a )(a 1, a 2, a 3 ; u a )} be a TIFN. The value ad the ambiguity of A is give as follows: 1. The value of the membership fuctio of A is V μ (A ) = (a 1+4 a 2 +a 3 )w a 6 (1) 2. The value of the o-membership fuctio is V θ (A ) = (a 1 +4 a 2 +a 3 )(1 u a ) 6 (2) 3. The ambiguity of the membership fuctio of A is A μ (A ) = (a 3 a 1 )w a 6 (3) 4.The ambiguity of o-membership fuctio A is A θ (A ) = (a 3 a 1 )(1 u a ) 6 (4) Also A μ (A ) A θ (A ). Defiitio 5: Let A = (a 1, a 2, a 3 ; w a )(a 1, a 2, a 3 ; u a )} be a TIFN.The the value idex ad ambiguity idex of A is defied as follows: V (A, λ) = V μ (A ) + λ(v ν (A ) V μ (A )) (5) ad A(A, λ) = A ν (A ) λ(a ν (A ) A μ (A )) (6) where λ ɛ[0,1] is a weight which represets the decisio maker's preferece iformatio. Defiitio 6: Rakig relatio F (A, λ) = V (A, λ) A (A, λ) (7) 3.0 Methodology 3.1 Crisp Multi-Obective liear model The classical liear programmig problem is to fid a optimum value of a liear fuctio subect to costraits represeted by liear iequalities or equatios. The formulatio of liear model ca be expressed as:

5 A ituitioistic fuzzy multi-obective vedor selectio problem 7447 Mi Z= cx Subect to Ax b (8) x 0 where x=[x1,x2,..x] T is a vector of decisio variables ad z is the obective fuctio Ituitioistic Fuzzy Multiobective liear model Cosider the ituitioistic liear programmig problem (IFMOLM) i which coefficiet of the obective fuctio are cosidered as itutioistic fuzzy umbers. Mi z = c =1 x Subect to: (9) =1 a i x 0, =1, 2,, x b i, i=1,2,,m where c = {(c 1, c 2, c 3 ; w c )(c 1, c 2, c 3 ; u c )} are TIFN's. 3.3 Solutio to Ituitioistic Fuzzy Multiobective liear model The elemets i the obective fuctio are ituitioistic fuzzy umbers ad the costraits are i the crisp form. To covert the ituitioistic fuzzy obectio fuctio to crisp form usig the rakig fuctio f, for predefied λ ɛ [0,1], IFLP is equivalet to the followig crisp optimizatio problem. Max F ( =1 c, λ) Subect to (10) =1 a i x b i, i=1,2,,m x 0, =1,2,. The liear equivalet is as follows: max(1 λ) mi {w c V μ (c ) } =1 x (1 λ) mi {1 u c } w c λ mi {1 u c } =1 x λ mi V ν (c ) 1 u c {w c A υ (c )x A υ (c )x =1 + 1 u c } =1 (11) w c

6 7448 Prabot Kaur Subect to: =1 a i x b i,i=1,2,,m. x 0, =1,2,. For optimistic attitude λ=1, equatio (*) reduces to max mi {1 u c } =1 V υ (c )x 1 u c mi {w c } =1 A µ (c )x w c Subect to: (12) =1 a i x b i,i=1,2,,m x 0, =1,2, For pessimistic λ=0 (*) reduces to max mi {w c } V μ(c ) =1 w c x (1 λ) mi {1 u c } A υ(c )x 1 u c =1 + mi {w c V µ (c ) } =1 x mi {1 u c } w c =1 A υ (c )x 1 u c Subect to: (13) =1 a i x b i,i=1,2,,m x 0, =1,2, The above crisp LPP is solved by optimizatio software Tora 2.0.We obtai solutio of MOLPP for the various obectives. 4.0 Numerical Example A textile compay desires to select suitable suppliers to purchase yar for a ew product (Yucel ad Gueri (2011)). A committee of decisio makers, D1, D2 ad D3 has bee costituted ad the committee selected et price, quality ad otime delivery as selectio criteria ad demad as determiistic i the costrait. The data for vedor selectio problem is give i Table1.I Table 2 data as TIFNS is preseted.

7 A ituitioistic fuzzy multi-obective vedor selectio problem 7449 Table 1: Supplier's quatitative iformatio. Supplier Net Price Quality(%) Delivery(%) Capacity A A A Table 2:TIFN data for Supplier's quatitative iformatio. Supplier Net Price Quality(%) Delivery(%) Capacity A 1 (4,5,6;.75)(3,5,7,.25) (.75,.80,.90;.75))(.75,.80,95;.25) (.85,.90,.95;1)(.65,.85,.95;0) 400 A 2 (6,7,8;1)(5,7,9;0) (.85,.90,.95;1)(.65,.85,.95;0) (.75,.80,.90;.75))(.75,.80,95;.25) 450 A 3 (3,4,5;.75)(4,5,6;.25) (.70,.85,.90;.75)(.65,.85,.95;.25) (.70,.85,.90;.75)(.65,.85,.95;.25) 450 Usig data of Table 2 i equatios 10, 11, 12, 1d 13,a IMOLPP is formulated. For λ=0, the equivalet crisp formulatio for the three obectives is as follows: Obective fuctio 1: Mi 2.75 x1+4.25x2+2.5x3 Subect to: x1+x2+x3=800 x1 400 x2 450 x3 450 x1, x2, x3 0 Obective fuctio 2: Max.56x1+.66x2+.55x3 Subect to: x1+x2+x3=800 x1 400 x2 450 x3 450 x1, x2, x3 0 Obective fuctio 3: Max.66x 1+.55x 2+.55x 3 Subect to: x1+x2+x3=800 x1 400 x2 450 x3 450 x1, x2, x3 0

8 7450 Prabot Kaur 5.0 Results: Usig the optimizatio software Tora 2.0 for solutio of the multiobective problem, obtai the optimal solutio for the model as follows: Z 1= , x 1=350, x 2 = 0, x 3 = 450; Z 2=493, x 1=350, x 2=450, x 3=0; Z 3=488, x 1=400, x 2=400, x 3=0. The allocatio of order for the three obectives varies accordig to situatios. The first obective was miimize the cost so maximum allocatio wet to vedor 3 whose cost was lowest ad vedor 2 got o allocatio of order because of its high cost. The secod obective was maximize quality,so allocatio wet to first ad secod vedor accordig to percetage quality.though maximum allocatio wet to vedor 2 which had high quality ad high price too. The third obective was miimizig delivery time so equal allocatio of order to vedor 1 ad vedor 2.Though delivery of goods was 90% ad 80% for vedors 1 ad 2.Other factors were also take ito cosideratio for order allocatio. 6.0 Coclusios I this study based o MOLPP i a ituitioistic eviromet for selectio of vedor ad order allocatio to vedor is doe. Compared to the results of Yucel ad Gueri (2011), the results obtaied by TIFN approach are better.i this study we use ituitioistic fuzzy umbers because of its ease of use ad its ability to represet vagueess i data with a additioal degree of freedom(o-membership fuctio). The future scope of the work icludes represetig the whole MOLPP i terms of itutioistic fuzzy sets. Refereces 1. M. Kumar, P. Vrat, R. Shakar, A fuzzy programmig approach for vedor selectio problem i a supply chai, Iteratioal Joural of Productio Ecoomics 101 (2006) Y. Crama, R. Pascual J., A. Torres, Optimal procuremet decisios i the presece of total quatity discouts ad alterative product recipes, EuropeaJoural of Operatioal Research 159 (2004) M. Díaz-Madroñero et al. / Computers ad Mathematics with Applicatios 60 (2010) R.A.D. Carvalho, H.G. Costa, Applicatio of a itegrated decisio support process for supplier selectio, Eterprise Iformatio Systems 1 (2007)

9 A ituitioistic fuzzy multi-obective vedor selectio problem G. Dickso, A aalysis of vedor selectio: systems ad decisios, Joural of Purchasig 2 (1966) Narsimha,R. (1983) A aalytical approach to supplier selectio, J. of Purchasig ad Materials Maagemet, witer, Choy, K.L., Lee, W.B., Lo, V., Developmet of a case based itelliget customer Supplier relatioship maagemet system. Expert Systems with Applicatios, 23 (3), Baker, R.C., Talluri, S., A closer look at the use of DEA for techology selectio. Computers ad Idustrial Egieerig 32 (1), Che, C.T., Li, C.T., Huag, S.F., A fuzzy approach for supplier evaluatio ad selectio i supply chai maagemet. Iteratioal Joural of Productio Ecoomics (102), Sarkar, A., Mohapatra, P.K.J., Evaluatio of supplier capability ad performace: A method for supply base reductio. Joural of Purchasig ad Supply Maagemet 12 (3), Dig, H., Beyoucef, L., Xie, X., A simulatio optimizatio methodology for supplier selectio problem. Iteratioal Joural Computer Itegrated Maufacturig 18 (2 3), Weber, C.A. ad Curret, J.R. (1993) A multiobective approach to vedor selectio, Europea J. of Operatioal Research, vol 68, Ghodsypour,S.H. ad Brie,C.O. (1998) A DSS for supplier selectio usig a itegrated AHP ad Liear Programmig, Iteratioal J. of Productio Ecoomics, vol 56-57, Buffa, F.P. ad Jackso, W.M. (1983) A goal programmig model for purchase plaig, Joural of Purchasig ad Materials Maagemet, Weber, C.A. ad Ellram, L.M. (1992) Supplier selectio usig multiobective programmig: A decisio support systems approach, Iteratioal J. of Physical Distributio ad Logistics Maagemet, vol 23 (2), 3-14.

10 7452 Prabot Kaur 15 Zixue Guo,Meria Qi ad Xi Zhao,"A ew approach based o itutioistic fuzzy set for selectio of suppliers",2010 sixth iteratioal coferece o Natural Computatio(ICNC 2010). 16 Fatih Emre Bora,Serka Gee,Mustafa Kurt ad Diyar Akay, "A multicriteria itutioistic fuzzy group decisio makig for supplier selectio with Topsis method",expert system with applicatios,vol.36,pp , Ataassov,K.(1986).Ituitioistic fuzzy sets.fuzzy Sets ad Systems,vol.79,pp M.Shahrokhi,A.Berard ad H.Shidpour,"A itegrated method usig ituitioistic fuzzy set ad liear programmig for supplier selectio problem",18 th IFAC World Cogress, Milao(Italy)August28-September 2, Dipti Dubey ad Apara Mehra (2011).Liear Programmig with triagular itutioistic fuzzy umbers,eusflat-lfa 2011,Aix-les- Bais, Frace. 20 Ataka Yucel ad Ali Fuat Gueri (2011).A weighted additive fuzzy programmig approach for multi-criteria supplier selectio,expert Systems with Applicatios,38, Received: March 10, 2014; Published: October 23, 2014