Selection Of Best Alternative For An Automotive Company By Intuitionistic Fuzzy TOPSIS Method

Transcription

1 Selecton Of Best lternatve For n utomotve Company By Intutonstc Fuzzy TOPSIS Method Zulqarnan M., Dayan F. bstract: Mult Crtera Decson Makng MCDM uses dfferent technques to fnd a best alternatve from mult-alternatve and mult-crtera condtons. Classcal TOPSIS uses crsp technques for the lngustc assessments, but due to mprecse and fuzzness nature of the lngustc assessments, there must be some tools to deal ths vague nformaton. In ths paper, e dscuss about Intutonstc Fuzzy TOPSIS IF-TOPSIS and use ths method for the selecton of best alternatve for an automotve company. Index Terms: MCDM, IF-TOPSIS, Intutonstc fuzzy postve deal soluton IFPIS, Intutonstc fuzzy negatve deal soluton IFNIS, Intutonstc Fuzzy Sets IFS, nterval valued ntutonstc fuzzy numbers IVIFN. Mult ttrbute Decson Makng MDM. 1 INTRODUCTION Hang and Yoon nvented the Technque for Order Preference by Smlarty to Ideal Soluton TOPSIS n order to solve MCDM Problem th many alternatves [1]. om crsp to fuzzy data, Chen & Hang remodeled TOPSIS [2]. Furthermore, Chen dened the TOPSIS for oup Decson Makng n fuzzy atmosphere [3]. asth et al. used fuzzy TOPSIS for the evaluaton and selecton of the best locaton plannng for urban dstrbuton centers [4]. Chu [5] and Yong [6] appled fuzzy TOPSIS for choosng plant locaton th mnmum costs and maxmum use of resources. The relatve closeness RC coeffcents ere obtaned as fuzzy numbers and after defuzzfcaton, alternatves ere ranked [7]. Wang and Elhang found that ther method s much closer to the fuzzy eghted approach presented by Dong and Wong [8]. Mahmoodzadeh et al. ncorporatng fuzzy HP and TOPSIS method gave a ne procedure for the project selecton problem. Improved fuzzy HP as used to compute the eghts of each crteron at frst and then TOPSIS algorthm as engaged for rankng the projects to be selected [9]. Yong tao et al. utlzed fuzzy TOPSIS for supportng contractors to choose sutable project for bddng th the help of MGDM. Trangular fuzzy numbers TFNs ere assgned to each lngustc varable for alternatve ratngs and crtera eghts [10]. Zadeh proposed Type-2 fuzzy sets to encompass uncertanty about the membershp functon n fuzzy set theory FST [11]. Sarem and Montazer, to cope th the decson-makng problems havng data th large uncertanty, used a Type-2 fuzzy TOPSIS based method. Study of TOPSIS n Classcal, Fuzzy, Intutonstc Fuzzy and Neutrosophc Envronments Interval-valued fuzzy IVF set s a specal type of Type-2 fuzzy set. Sarem and Montazer appled IVF sets havng loer and upper trangular membershp functons; they ranked the alternatves [12]. Chen and Tsao broadened IVF-TOPSIS to solve MDM problem [13]. Zulqarnan. M and Saeed. M., proposed and proved the credblty of nterval valued fuzzy soft matrx IVFSM n decson makng also dscussed ts dfferent propertes [14]. They also studed fuzzy soft matrx FSM and IVFSM and redefned the product of IVFSM, they used IVFSM and FSM n decson makng problem th examples and compare the results and sa that FSM method s more approprate for decson makng [15]. They proposed a ne decson makng method on IVFSM named as nterval valued fuzzy soft max-mn decson makng method th the help of nterval valued fuzzy soft max-mn decson makng functon and used ths method for decson makng [16]. shtan et al also establshed IVF- TOPSIS, n hch alternatve ratngs and eghts of the crtera ere treated as lngustc varables represented by Trangular Interval Valued Fuzzy Numbers TIVFN. They used t to solve MCDM problems [17]. By developng an Interval Valued Intutonstc Fuzzy set IVIFS TOPSIS technque, a faclty locaton problem as tackled by Verma et al. [18]. By usng Intutonstc Fuzzy sets, Hung and Chen ntroduced a novel Fuzzy TOPSIS method nvolvng Entropy Weghts for Decson Makng [19]. L and Nan used the technque proposed by Hung and Chen to enhance TOPSIS for solvng MDM problems under IFS condtons [20]. Nurandah and Lazm modelled IFS- TOPSIS to fnd a soluton of a decson problem under IFS condtons [21]. In ther ork, eghts of the decson makers DMs ere computed; an IF-Decson Matrx s computed accordng to DMs judgements. The eghts of the crtera and Weghted Intutonstc Fuzzy Decson Matrx ere computed. IFPIS and IFNIS ere determned. Dstances and RC of the alternatves from IFPIS and IFNIS are calculated for rankng the alternatves. Boran utlzed the theory of IFS to supply chan management SCM [22]. F. Ye extended the dea of TOPSIS method th IVIFN for the selecton of vrtual enterprse partner [23]. In ths paper, e dscuss about Intutonstc Fuzzy TOPSIS IF-TOPSIS and use ths method for the selecton of best alternatve for an automotve company. 2 PRELIMINRIES M. Zulqarnan receved the M.S. degree n mathematcs from Unversty of Management and Technology, Lahore n E-mal: ranazulqarnan7777@gmal.com Fazal Dayan receved the M.S. degree n mathematcs from Unversty of Management and Technology, Lahore n E-mal: fazaldayan1@gmal.com Defnton 1 [22] IFS n a fnte set X, can be rtten as = *x, µ x x: xεx+ Where µ x x: X [0, 1] are membershp and nonmembershp functons respectvely, such that 0 µ x + v x 1 tanassov [24] ntroduced IFS, as an extenson of classcal FST. The purpose of IFS as to deal the vagueness..jstr.org 126

2 Intutonstc Fuzzy TOPSIS lgorthm [25] Consder a set of m lternatves = {,,,... } a set of n Evaluaton Crtera C = {C, C, C,... C }a set of Decson Makers DMs = {DM 1, DM 2,... DM } Step 1: Selecton of Intutonstc Fuzzy Ratngs Scale for Lngustc Varables Snce mportance of crtera, DMs and alternatve ratngs all are n the form of lngustc varables. Step 2: Determne the eghts of the DMs The Intutonstc Fuzzy Number IFN for ratng the k th DM s gven belo D = µ, π The formula to fnd the eght of the k th DM s represented by the follong equaton λ = : λ = 1 Step 3: ggregated Intutonstc Fuzzy Decson Matrx IFDM accordng to the ratngs of DMs The ggregated Intutonstc Fuzzy Decson Matrx IFDM s r r r r r r D = [ ] = [r ] m n r r r Where r can be defned as r = µ x x, π x = 1,2,3,,m and j = 1,2,3,.,n So the IFDM can be rtten as µ x x, π x µ x x, π x µ x x, π x µ x x, π x µ x x, π x µ x x, π x D= [µ x x, π x µ x x, π x µ x x, π x ] Let r denotes the ratng for the th alternatve.r.t. the j th crteron by the k th DM r = µ r can be calculated by usng Intutonstc Fuzzy Weghted veragng Operator IFWO gven n belo equaton. [1-1 µ, v, 1 µ v ] Step 4: Computaton of eghts for the crtera Let eght assgned to the crteron X j by the k th DM s represented as = µ, π IFWO, to compute eghts of the crtera s defned as = [1-1 µ, v, 1 µ v ] The aggregated eght for the crteron X j s represented as = µ, π, j = 1,2,3,...,n W = [ 1, 2, 3,, n ] Transpose Step 5: Compute ggregated Weghted Intutonstc Fuzzy Decson Matrx WIFDM fter fndng the eght matrx and IFDM the WIFDM s calculated and s represented as r r r r r r R = [ ] = [r ] m n r r r Where r = µ, π = µ x x, π x = 1,2,3,,m and j = 1,2,3,.,n Therefore R can be rtten as R = µ x x, π x µ x x, π x µ x x, π x µ x x, π x µ x x, π x µ x x, π x [ µ x x, π x µ x x, π x µ x x, π x ] To fnd the values of µ x x, π x e used ths product rule R W = {< x, µ x x > / xϵ X} µ x = µ x. µ x V x = v x + v x - v x. v x In addton, the hestaton degree n x can be calculated as π x = 1 - µ x. µ x - v x - v x + v x. v x Step 6: Determne IFPIS and IFNIS For fndng the IFPIS and IFNIS. j = beneft crtera j =cost crtera * = IFPIS = IFNIS.jstr.org * = {µ x x } = {µ x x } µ x = max µ x / jϵ j 1, mn µ x / jϵ j 2 v x = mn v x / jϵ j 1, max v x / jϵ j 2 µ x = mn µ x / jϵ j 1, max µ x / jϵ j 2 v x = max v x / jϵ j 1, mn v x / jϵ j 2 Step 7: Computaton of Separaton Measures To fnd the separaton measures d * and d, Normalzed 127

3 Eucldean Dstance s used as,µ x µ x + v x v x,,π x π x ] 0.5,µ x µ x + v x v x,,π x π x ] 0.5 Step 8: Computaton of Relatve Closeness Coeffcent RCC The RCC of an alternatve. r. t the IFPIS can computed as RCC = : 0 RCC 1 Step 9: Rankng alternatves fter computaton of RCC for each alternatve, the alternatves are ranked n descendng order of RCC s. 3 PPLICTION OF INTUITIONISTIC FUZZY TOPSIS PROBLEM SCENRIO Company: utomotve Problem: Suppler Selecton Consder three Decson Makers represented by D = {DM 1, DM 2, DM 3 } Fve lternatves m = 5 represented by = { : = 1, 2, 3, 4, 5} Four Evaluaton Crtera n = 4 represented by Benefte crtera C = { cost crtera : : : * : { Soluton by Intutonstc Fuzzy TOPSIS Intutonstc fuzzy ratngs scale for mportance of crtera and DMs s gven by the Table LVs Table 1: Lngustc varables for ratng the mportance of crtera and decson makers IFNs Very mportant 0.90, 0.10 Important 0.75, 0.20 Medum 0.50, 0.45 Unmportant 0.35, 0.60 Very unmportant 0.10, 0.90 Table 3: The mportance and eghts of decson makers DM 1 DM 2 DM 3 Lngustc VI0.90, 0.10, M0.50,0.45, I 0.75, 0.20, Varables µ, π µ, π µ, π Weghts λ = λ = λ = No by usng the alternatve ratngs r and the DM eghts λ k, the aggregated ntutonstc fuzzy ratngs for the alternatves are calculated belo λ r µ, v, 1 µ v Where = 1, 2, 3, 4, 5. : j = 1, 2, 3, 4, and = 3 For = j = 1 and l= 3 λ r 1 1 µ, v, 1 µ v 1.1 µ /.1 µ /.1 µ /,.v /.v /.v /,.1 µ /.v / 1 µ v 1 µ v , 1, , 0.170, For the sake of brevty, remanng 19 r matrx gven by the table 5. s are tabulated n The alternatve ratngs are gven n the belo table Table 2a: Lngustc varables for ratng the alternatves LVs Extremely good EG/extremely hgh EH 1.00, 0.00 IFNs 1.00, 0.00 Very very good VVG/very very hgh VVH 0.90, 0.10 Very good VG/very hgh VH 0.80, 0.10 Good G/hgh H 0.70, 0.20 Medum good MG/medum hgh MH 0.60, 0.30 Far F/medum M 0.50, 0.40 Table 2b: Lngustc varables for ratng the alternatves LVs IFNs Medum bad MB/medum lo ML 0.40, 0.50 Bad B/lo L 0.25, 0.60 Very bad VB/very lo VL 0.10, 0.75 Very very bad VVB/very very lo VVL 0.10, 0.90.jstr.org 128

4 Table 4: lternatve ratngs Cr. lts. Decson Makers DM 1 DM 2 DM 3 X 1 1 = µ = 0.7, 0.2 V = µ = 0.8, 0.1 = µ = 0.7, 0.2 = µ = 0.6, 0.3 = µ = 0.7, 0.2 = µ = 0.5, 0.4 VV = µ = 0.9, 0.1 V = µ = 0.8, 0.1 V = µ = 0.8, = µ = 0.6, 0.3 = µ = 0.7, 0.2 = µ = 0.6, = µ = 0.5, 0.4 M = µ = 0.6, 0.3 M = µ = 0.6, 0.3 X 2 1 = µ = 0.6, 0.3 = µ = 0.7, 0.2 M = µ = 0.6, 0.3 = µ = 0.5, 0.4 M = µ = 0.6, 0.3 = µ = 0.7, 0.2 V = µ = 0.8, 0.1 = µ = 0.7, 0.2 V = µ = 0.8, = µ = 0.5, 0.4 = µ = 0.5, 0.4 = µ = 0.6, 0.3 X 3 5 MBr = µ = 0.4, 0.5 = µ = 0.5, 0.4 = µ = 0.5, V = µ = 0.8, 0.1 = µ = 0.7, 0.2 V = µ = 0.8, 0.1 = µ = 0.7, 0.2 M = µ = 0.6, 0.3 M = µ = 0.6, 0.3 V = µ = 0.8, 0.1 V = µ = 0.8, 0.1 = µ = 0.7, V = µ = 0.8, 0.1 = µ = 0.7, 0.2 = µ = 0.7, = µ = 0.7, 0.2 = µ = 0.7, 0.2 = µ = 0.6, 0.3 X 3 1 = µ = 0.7, 0.2 = µ = 0.7, 0.2 = µ = 0.7, 0.2 M = µ = 0.6, 0.3 = µ = 0.5, 0.4 M = µ = 0.6, 0.3 V = µ = 0.8, 0.1 V = µ = 0.8, 0.1 = µ = 0.7, = µ = 0.7, 0.2 M = µ = 0.6, 0.3 M = µ = 0.6, = µ = 0.5, 0.4 = µ = 0.6, 0.3 = µ = 0.5, 0.4 Table 5: ggregated Intutonstc Fuzzy Decson Matrx D =,r - X X X X 0.728, 0.626, 0.780, 0.700, 0.170, , , , , 0.605, 0.644, 0.578, 0.302, , , , , 0.780, 0.769, 0.769, 0.100, , , , , 0.538, 0.746, 0.644, 0.236, , , , , 0.462, 0.526, 0.526, 0.337, , , , Step 4: Computaton of the eghts of the Crtera The ndvdual eghts gven by each DM are lsted n the Table 6 Table 6: Weghts of Crtera determned by the DMs = µ, π Crtera DM 1 DM 2 DM 3 X 1 X 2 X 3 X 4 VI0.90, =µ I 0.75, 0.20, 0.05 =µ I 0.75, 0.20, 0.05 =µ M0.50,0.45, 0.05 =µ VI0.90,0.10, 0.00 =µ I 0.75, 0.20, 0.05 =µ I 0.75, 0.20, 0.05 =µ I 0.75, 0.20, 0.05 =µ I 0.90, 0.10, 0.00 =µ I 0.75, 0.20, 0.05 =µ M0.50,0.45, 0.05 =µ M0.50,0.45, 0.05 =µ By usng the table 6, the aggregated crtera eghts are computed by usng the gven operator = λ +λ +λ +.+λ = 1 1 µ, v, 1 µ - v Where = 1, 2, 3, 4, 5. : j = 1, 2, 3, 4. nd l = 3 For j = 1 and l= 3 = λ +λ +λ = 1 1 µ, v, 1 µ v =1 1 µ 1 µ 1 µ v v 1 µ v, 1 µ v 1 µ v.jstr.org 129

5 = , 1 1, = 0.861, 0.128, In ths ay, the remanng 3 S are calculated and are tabulated n the follong matrx = 0.750, 0.200, = 0.680, 0.267, = 0.576, 0.371, Therefore 861, 1 8, 11 W * 75,, 5,,, + = [ ] 68, 67, , 371, 53 Step 5: Constructon of WIFDM fter fndng the eghts of the crtera and the alternatve ratngs the aggregated eghted Intutonstc fuzzy ratngs for the alternatves are calculated. µ, π = µ x. µ x,v x + v x- v x. v x,1-µ x. µ x - v x- v x + v x. v x For = j = 1 µ, π = µ x. µ x x + v x- v x. v x, 1-µ x. µ x- v x - v x+ v x. v x , , 1- µ v 0.627, 0.276, , 0.276, Smlarly e fnd 0.470, 0.418, , 0.353, , 0.497, , 0.391, , 0.434, , 0.453, , 0.573, , 0.215, , 0.294, , 0.361, , 0.452, , 0.334, , 0.489, , 0.378, , 0.531, , 0.422, , 0.550, , 0.436, , 0.606, ll the above values are gven n the table 7 as WIFDM Table 7: ggregated Weghted Intutonstc Fuzzy Decson Matrx X 1 X 2 X 3 X , 0.470, 0.530, 0.403, 0.276, , , , , 0.454, 0.438, 0.333, 0.391, , , , , 0.585, 0.523, 0.443, 0.215, , , , , 0.404, 0.507, 0.371, 0.334, , , , , 0.347, 0.454, 0.303, 0.422, , , , Step 6: Computaton of IFPIS and IFNIS Snce Product Qualty, Relatonshp closeness and Delvery Performance are beneft crtera that s hy they are n the set j 1 = {X 1, X 2, X 3 } hereas Prce beng the cost crtera therefore t s n the set j 2 = {X 4 } The IFPIS s calculated as = {0.731, 0.215, 0.054, 0.585, 0.294, 0.121, 0.530, 0.353, 0.117, 0.303, 0.606, 0.091} The IFNIS s calculated as = {0.484, 0.422, 0.094, 0.347, 0.550, 0.103, 0.438, 0.453, 0.109, 0.443, 0.452, 0.105} Step 7: Computaton of Separaton Measures By usng Normalzed Eucldean Dstance Measure the negatve and postve separaton measures d and d respectvely are calculated as follos [.µ x µ x / +.v x v x / +.π x π x / ] Fo1 and n=4 1 [.µ x µ x / +.v x v x / +.π x π x / ] 1 8.µ x µ x / +.v x v x / +.π x π x / +.µ x µ x / +.v x v x / +.π x π x / +.µ x µ x / +.v x v x / +.π x π x / + [.µ x µ x / +.v x v x / +.π x π x / ].jstr.org 130

6 [ ], For the sake of brevty thout fndng all values, the remanng values are gven n the table 8. Table 8: Separaton measures and the RCC for each lternatve. lternatves d d Step 8: Computaton of Relatve Closeness Coeffcent RCC The relatve closeness coeffcents are calculated as follos RCC = : = 1,2,3,4,5. RCC 1 = = RCC 2 = = RCC 3 = = RCC 4 = = RCC 5 = = Step 9: Rankng lternatves om above calculatons the RCC are ranked as follos RCC 3 RCC 1 RCC 2 RCC 4 RCC 5 Hence s the best alternatve. 4 CONCLUSION st of all, n ths paper e dscuss about IFS th some defntons and study about IF-TOPSIS to deal those problems hch have uncertanty. Secondly, e dscuss about IF- TOPSIS algorthm and fnally, e apply ths method for decson makng and used ths paper for the selecton of best suppler for automotve company and conclude that s best alternatve for automotve company. REFERENCES [1] C. Hang and K. Yoon, Multple ttrbute Decson Makng: Methods and pplcatons, State of the rt Surveyol [2] S.J. J. Chen and C.-L. Hang, Fuzzy Multple ttrbute Decson Makng Methodsol [3] C.T. Chen, Extensons of the TOPSIS for group decsonmakng under fuzzy envronment, Fuzzy Sets Syst.ol. 114, no. 1, pp. 1 9, [4]. asth, S.S. Chauhan, and S.K. Goyal, multcrtera decson makng approach for locaton plannng for urban dstrbuton centers under uncertanty, Math. Comput. Model.ol. 53, no. 1 2, pp , [5] T.C. Chu, Selectng plant locaton va a fuzzy TOPSIS approach, Int. J. dv. Manuf. Technol.ol. 20, no. 11, pp , [6] D. Yong, Plant locaton selecton based on fuzzy TOPSIS, Int. J. dv. Manuf. Technol.ol. 28, no. 7 8, pp , [7] Y.M. Wang and T.M.S. Elhag, Fuzzy TOPSIS method based on alpha level sets th an applcaton to brdge rsk assessment, Expert Syst. ppl.ol. 31, no. 2, pp , [8] W.M. Dong and F.S. Wong, Fuzzy Weghted verages and Implementaton of the Extenson Prncple, vol. 21, pp , [9] S. Mahmoodzadeh, J. Shahrab, M. Parazar, and M. S. Zaer, Project selecton by usng fuzzy HP and TOPSIS technque, Int. J. Soc. Manag. Econ. Bus. Eng.ol. 1, no. 3, pp , [10] T. Yong tao, L. Shen, C. Langston, and Y. Lu, Constructon project selecton usng fuzzy TOPSIS approach, J. Model. Manag.ol. 5, no. 3, pp , [11] L.. Zadeh, The concept of a lngustc varable and ts applcaton to approxmate reasonng-i, Inf. Sc. Ny.ol. 8, no. 3, pp , [12] H. Q. Sarem and G.. Montazer, n applcaton of type- 2 fuzzy notons n ebste structures selecton: Utlzng extended TOPSIS method, WSES Trans. Comput.ol. 7, no. 1, pp. 8 15, [13] T. Y. Chen and C. Y. Tsao, The nterval-valued fuzzy TOPSIS method and expermental analyss, Fuzzy Sets Syst.ol. 159, no. 11, pp , [14] Zulqarnan M., and Saeed M., n applcaton of Interval valued fuzzy soft matrx IVFSM n decson makng, Sc. Int. Lahoreol. 28, no. 3, , [15] Zulqarnan, M., and Saeed. M., Comparson beteen fuzzy soft matrx FSM and nterval valued fuzzy soft matrx IVFSM n decson makng, Sc. Int. Lahoreol. 28, no. 5, pp , jstr.org 131

7 [16] Zulqarnan, M., and Saeed, M., Ne Decson Makng Method on Interval Valued Fuzzy Soft Matrx IVFSM, Brtsh Journal of Mathematcs & Computer Scenceol. 20, no. 5, pp. 1-17, [17] B. shtan, F. Haghghrad,. Maku, and G. l Montazer, Extenson of fuzzy TOPSIS method based on nterval-valued fuzzy sets, ppl. Soft Comput. J.ol. 9, no. 2, pp , [18].K. Verma, R. Verma, and N. C. Mahant, Faclty Locaton Selecton: n Interval Valued Intutonstc Fuzzy TOPSIS pproach, Journal of Modern Mathematcs and Statstcsol. 4, no. 2. pp , [19] C.C. Hung and L. H. Chen, Fuzzy TOPSIS Decson Makng Model th Entropy Weght under Intutonstc Fuzzy Envronment, Proc. Int. MultConference Eng. Comput. Sc.ol. I, pp , [20] D.F. L and J.X. Nan, Extenson of the TOPSIS for Mult- ttrbute oup Decson Makng under tanassov IFS Envronments, Int. J. Fuzzy Syst. ppl.ol. 1, no. 4, pp , [21] Z. Nurandah and. Lazm, TOPSIS Method th ntutonstc fuzzy group decson makng, Proc. 10th Int. nnu. Symp. UMTS, no. July, pp , [22] F.E. Boran, S. Genç, M. Kurt, and D. kay, multcrtera ntutonstc fuzzy group decson makng for suppler selecton th TOPSIS method, Expert Syst. ppl.ol. 36, no. 8, pp , [23] F. Ye, n extended TOPSIS method th nterval-valued ntutonstc fuzzy numbers for vrtual enterprse partner selecton, Expert Syst. ppl.ol. 37, no. 10, pp , [24] K.T. tanassov, Intutonstc fuzzy sets, Fuzzy Sets Syst.ol. 20, no. 1, pp , ug [25] F.E. Boran, S. Genç, M. Kurt, and D. kay, multcrtera ntutonstc fuzzy group decson makng for suppler selecton th TOPSIS method, Expert Syst. ppl.ol. 36, no. 8, pp , jstr.org 132