Supplier evaluation with fuzzy similarity based fuzzy TOPSIS with new fuzzy similarity measure

Transcription

1 Suppler evaluaton wth fuzzy smlarty based fuzzy TOPSIS wth new fuzzy smlarty measure Leonce Nygena Laboratory of Appled Mathematcs Lappeenranta Unversty of Technology Lappeenranta, Fnland. Emal: Pas Luukka Laboratory of Appled Mathematcs Lappeenranta Unversty of Technology Lappeenranta, Fnland. Emal: Mkael Collan School of Busness Lappeenranta Unversty of Technology Lappeenranta, Fnland. Emal: Abstract In ths paper we present how a recently publshed new fuzzy smlarty measure can be used wthn a fuzzy TOPSIS method and appled to a suppler evaluaton problem. Wth ths new smlarty measure we are able to consder the area of the fuzzy numbers under comparson. Ths allows us to nclude more relevant nformaton n the rankng of supplers and n ths way mprove the qualty of the rankng result reached wth smlarty measure based fuzzy TOPSIS. We compare the new fuzzy smlarty measure wth the prevously used smlarty measures and apply t to two dfferent suppler evaluaton problems. We show that by ncludng more relevant nformaton n the rankng can result n a new rankng, ths s not nsgnfcant from the decson-makng pont-of-vew. I. INTRODUCTION Many manufacturng frms want to develop a strong collaboraton wth ther supplers, n order to optmze performance and compettveness of ther supply chans. For ths reason, supply chan management and the suppler selecton process has been the subect of much research n management lterature. Determnaton of relevant supplers and rankng supplers has often been done by consderng multple crtera [1] [2]. To llustrate the pont, n Klr et al [3], t was noted: Each crteron from a number of crtera helps to evaluate a relevant number of supplers, so that a procedure s needed by whch to construct an overall preference orderng. The overall obectve of the suppler rankng and selecton process s to reduce purchase rsk, to maxmze overall value to the purchaser, and to buld the closeness and long-term relatonshps between buyers and supplers []. In ths paper, the Technque for Order Performance by Smlarty to Ideal Soluton (TOPSIS) method s appled to the suppler rankng and selecton problem. Here, nstead of usng a dstance measure, as s done n the classcal TOPSIS, we test two smlarty measures for fuzzy numbers and nfact we are usng a fuzzy TOPSIS method. The fuzzy TOPSIS uses fuzzy numbers as nputs and s thus able to ncorporate naccurate and mprecse nformaton n the analyss (not havng to smplfy realty by usng crsp numbers). The man dfference n usng the smlarty measures and the dstance measure n the TOPSIS envronment wth fuzzy numbers s that smlarty measures can take nto consderaton the permeter and the area of fuzzy number more comprehensvely, whle the dstance measure requres a defuzzfcaton of the fuzzy number n order to calculate a dstance between the resultng crsp numbers. Usng the dstance based measure may cause more loss of relevant nformaton. The frst smlarty measure used here was ntroduced by We and Chen n [5], ths smlarty measure takes n account the permeter of fuzzy numbers. The second smlarty measure used takes addtonally nto account the area of the consdered fuzzy numbers, ths smlarty measure s more recent and was presented n [6]. Ths s the frst tme the new smlarty measure of [6] s studed n the context of fuzzy smlarty based TOPSIS method [7]. We also consder three dfferent deal soluton schemes. II. FUZZY SIMILARITY MEASURES The concept of smlarty s of fundamental mportance n ths work. In the same way as the noton of a fuzzy subset generalzes that of the classcal subset, the concept of smlarty can be consdered as a many-valued generalzaton of the classcal noton of equvalence [8]. As an equvalence relaton s a famlar way to classfy smlar obects, fuzzy smlarty s an equvalence relaton that can be used to classfy mult-valued obects. Knowng the dstance between two obects s dfferent from knowng how smlar those obects are. Let us focus on uncertan obects lke n fuzzy sets or fuzzy numbers, when the dstance between two fuzzy sets s zero, t means that the membershp functons of these two fuzzy sets are the same almost everywhere and not smlar [9]; as two fuzzy sets can be smlar but at a certan dstance one from another. Defnton 1: For any fuzzy subset F of R n, and for any elements A, B F the smlarty measure functon s defned as [10]: s(a, B) : F F [0, 1] In the early 1987, a unversal law was proposed [10], sayng that dstance and smlarty are exponentally related as shows the followng functon: s 1 (A, B) = e d(a,b) (1)

2 The defned smlarty measures s has a certan number of propertes to satsfy for any x, y, z F, s(x, x) = s(y, y), x, y F ( Reflexvty ) s(x, y) s(y, y), x, y F ( Mnmalty ) s(x, y) = s(y, x) ( Symmetry ) If s(x, y) = s(x, z) t mples that s(x, y) = s(x, z) = s(y, z) (Transtvty) [10] Smlarty measures for generalzed fuzzy numbers come from smlarty measures for fuzzy sets. Ths s n a sense obvous snce fuzzy numbers can be consdered to be certan type of restrcted fuzzy sets. Chen [11] represented a generalzed trapezodal fuzzy number à as à = (a, b, c, d; w), where a,b,c and d are real values and 0 < w 1. The membershp functon µã satsfes the followng condtons [11]: 1. µã s a contnuous mappng from the unverse of dscourse X to the closed nterval n [0, 1] 2. µã = 0, where < x a 3. µã s monotoncally ncreasng n [a, b]. µã = w, where b x c 5. µã s monotoncally decreasng n [c, d] 6. µã = 0, where d x < There s a close lnk between the noton of smlarty and that of dstance (see for example [12] and [13]). One relatonshp formula between these two measures has been establshed n [1], as follows: Let us have two generalzed fuzzy numbers M = (m 1, m 2, m 3, m ) and N = (n 1, n 2, n 3, n ) wth ther correspondng membershp functons M(x ) and N(x ) wth {1, 2, 3, } for generalzed trapezodal fuzzy numbers, d(m, N), the dstance between M and N, and for F = {x 1, x 2,..., x k }, ther smlarty measures s(m, N) are calculated as follow: s 2 (M, N) = d(m, N) Generally, the fuzzy smlarty measures between two fuzzy numbers have been developed from the concept of the center of gravty ponts and the dstance of ponts between those fuzzy numbers [15]. Snce the measure of smlarty s very mportant n decson makng, varous smlarty measures have been proposed to calculate the degree of smlarty between fuzzy numbers. New smlarty measures have been ntroduced throughout the years to account for dfferent stuatons. Here we use two smlarty measures. The frst s a smlarty measure s 3 (M, N) ntroduced by We and Chen n [5] that consders the permeter of fuzzy numbers. The measure s defned n equaton (3). The second measure s (M, N) s based on the frst measure. The second measure was ntroduced by Heaz et al n [6] and n addton to consderng the permeter of fuzzy numbers t also consders ther area, the second measure s defned n (). Both of these smlarty measures nvolve fuzzy numbers M = (m 1, m 2, m 3, m ; ω m ) and N = (n 1, n 2, n 3, n ; ω n ) wth 0 m 1 m 2 m 3 m 1, 0 n 1 n 2 n 3 n 1, and M(x ) and N(x ) (2) ther correspondng membershp functons, where ω m and ω n are ther correspondng heghts. The frst fuzzy smlarty measure by [5] s gven as =1 s 3 (M, N) = (1 m n ) mn(p (m), P (n)) + mn(ω m, ω n ) max(p (m), P (n)) + max(ω m, ω n ) (3) where the two values P (m) and P (n) represent the permeters of the trapezodal fuzzy numbers, and are defned as: P (m) = (m 1 m 2 ) 2 + ωm 2 + (m 3 m ) 2 + ωm 2 + (m 3 m 2 ) + (m m 1 ) and P (n) = (n 1 n 2 ) 2 + ωn 2 + (n 3 n ) 2 + ωn 2 + (n 3 n 2 ) + (n n 1 ) The second smlarty measure ntroduced n [6] s defned as: s (M, N) = =1 (1 n ) mn(p (m), P (n)) max(p (m), P (n)) mn(a(m), A(n)) + mn(ω m, ω n ) max(a(m), A(n)) + max(ω m, ω n ) () Where the values A(m) and A(n) represent the areas of the trapezodal fuzzy numbers, and are defned as: and A(m) = 1 2 ω m(m 3 m 2 + m m 1 ), A(m) = 1 2 ω n(n 3 n 2 + n n 1 ). In addton, s 3 (M, N) and s (M, N) belong to the unt nterval [0, 1] and the larger the value of smlarty measure s, the stronger the smlarty s between the fuzzy numbers M and N [5]. III. FUZZY SIMILARITY BASED FUZZY TOPSIS METHOD FOR SUPPLIER SELECTION PROBLEM Fuzzy extenson to the Technque for Order Performance by Smlarty to Ideal Soluton (TOPSIS) was been proposed by Chen [1] and t has been extended to solve problems nvolvng trapezodal fuzzy numbers and appled, e.g., to solvng the suppler selecton problems [2]. It s a Multple Crtera Decson Makng (MCDM) method [2] [16] usable n rankng obects based on the smlarty of the obect characterstcs to the characterstcs of an deal obect (deal soluton). The method s based on the dea that obects are ranked hgher the shorter ther dstance s from the Fuzzy Postve Ideal Soluton (FPIS) and the further they are from the Fuzzy Negatve Ideal Soluton (FNIS). One advantage of havng extended the TOPSIS method to the fuzzy envronment s that lngustc assessment can be properly used nstead of beng constraned

3 to usng only numercal values, because lngustc varables can be mapped to correspondng fuzzy numbers [1], [2]. Soluton to the suppler selecton problem (SSP), when usng the T OP SIS approach, can be suggested by consderng a stuaton of a fnte set of supplers A = {A = 1, 2,..., m} whch need to be evaluated by a commttee of decson-makers D = {D l l = 1, 2,..., k}, by consderng a fnte set of gven crtera C = {C = 1, 2,..., n}. Let us consder a decson matrx representng a set of performance ratngs of each alternatve suppler A, = 1, 2,..., m wth respect to each crteron C, = 1, 2,..., n [17], as follow: X = x 11 x x 1n x 21 x x 2n x m1 x m2... x mn Let us also assume the weght w of the th crteron C, such that the weght vector s represented as follows: W = [ w 1, w 2,..., w n ] Where m lnes represent m possble alternatves, n columns represent n relevant crtera, and x represent the performance ratng of the th alternatve A wth respect to the th crteron C. The above fuzzy ratngs for each decson-maker D l, l = 1, 2,..., k are represented by postve trapezodal fuzzy numbers ˆR l = (a l, b l, c l, d l ), l = 1, 2,..., k wth the respectve membershp functon µ ˆRl (x). As the ratng ˆR l = (a l, b l, c l, d l ) s for the l th decson-maker, the aggregated fuzzy number whch can stand for all decson-makers ratng s: ˆR = (a, b, c, d) wth: a = mn l {a l }, b = 1 k k l=1 b l, c = 1 k k l=1 c l, d = max l {d l } The fuzzy ratng and mportance weght of the l th decson-maker can respectvely be represented by x l = (a l, b l, c l, d l ) and ŵ = (w l1, w l2, w l3, w l ) wth = 1, 2,..., m; = 1, 2,..., n. Then, the aggregated fuzzy ratngs x of alternatves wth respect to each crteron are: x = (a, b, c, d ), calculated as: a = mn l {a l }, b = 1 k k l=1 b l, c = 1 k k l=1 c l, d = max l {d l } The aggregated fuzzy weght ŵ of each crteron can be calculated as : ŵ = (w 1, w 2, w 3, w ) wth w 1 = mn l {w l1 }, w 2 = 1 k k l=1 w l2, w 3 = 1 k k l=1 w l3, w = max l {w l }. From the fact that the SSP s elements are matrces of the followng form: X = {x } mn and W = {w } 1n, where = 1, 2,..., m and = 1, 2,..., n. These matrces elements are gven by postve trapezodal fuzzy numbers as : x = (a, b, c, d ) and w = (w 1, w 2, w 3, w ). The lnear scale transformaton s used to transform the varous crtera scales nto comparable scales n order to avert overly complex mathematcal operatons n a decson process. By dvdng the set of crtera nto beneft crtera B, where the larger the ratng, the greater the preference and cost crtera C, where the smaller the ratng, the greater the preference. A normalzaton method desgned to preserve the property n whch the elements are normalzed trapezodal fuzzy numbers s used. The normalzed value of x s r, and the normalzed fuzzy decson matrx s then represented as: wth r = ( a d + R = [r ] mn (5), b d +, c d +, d d + ), B r = ( a d, a c, a b, a a ), C where d + = max {d }, B and a = mn {a }, C The weghted normalzed value of r s v, and by consderng the mportance of each crteron, the weghted normalzed fuzzy decson matrx s represented as: V = [v ] mn (6) where v = r w. For all,, the elements v are now normalzed postve trapezodal fuzzy numbers. In the SSP, the deal solutons must be determned and taken from the gven crtera whch are lngustcally expressed, that s why they are named Fuzzy Postve Ideal Soluton (FPIS) and Fuzzy Negatve Ideal Soluton (FNIS), whch means that they are fuzzy numbers. By consderng a fnte set of gven crtera C = {C = 1, 2,..., n}, the ways to select the FPIS(A + ) and the FNIS(A ) come from the weghted normalzed decson matrx V = (v ) mn, where the obtaned weghted normalzed values v are fuzzy numbers expressed as: v = (v 1, v 2, v 3, v ) The fuzzy postve-deal soluton A + and the fuzzy negatvedeal soluton A, respectvely are: A + = [v + 1, v+ 2,..., v+ n ] (7) A = [v 1, v 2,..., v n ] (8) In Chen et al [2], one way for choosng the FPIS (A + ) and the FNIS (A ) have been gven as for A +, the maxma, and A, the mnma of the weghted normalzed values: v + v = max v (9) = mn v 1 (10) In addton, two other ways of selectng the FPIS (A + ) and the FNIS (A ) have been explaned n [7], as follow:

4 A + as a ones vector, and A as a zeros vector: v + v = (1, 1, 1, 1) (11) = (0, 0, 0, 0) (12) Every element of A + s the maxmum for all weghted normalzed value, and every element of A s the mnmum for all weghted normalzed value: v + v = (max = (mn v 1, max v 1, mn v 2, max v 3, max v ) (13) v 2, mn v 3, mn v ) (1) The separaton dstance of each alternatve from the above deal solutons A + and A, or the smlarty measure between each alternatve and the deal solutons A + and A wll be needed later when calculatng the closeness coeffcents to determne the rankng order of all possble alternatve supplers. The dstances d + or smlartes s + from the postve deal soluton are calculated as: d + = n n d(v, v + ) = (v v + )2 (15) =1 =1 or by usng the two smlartes under nvestgaton n ths paper as: s + 3 = s 3(v, v + t=1 ) = (1 t v + t ) mn(p (v ), P (v + )) + mn(ωv, ω v + ) max(p (v ), P (v + )) + max(ωv, ω v + ) (16) or s + = s (v, v + t=1 ) = (1 t v + t ) mn(p (v ), P (v + )) max(p (v ), P (v + )) mn(a(v ), A(v + )) + mn(ω v, ω v + ) max(a(v ), A(v + )) + max(ω v, ω v + ) (17) n and average smlartes by s + =1 3 = s 3(v, v + ) or s + = n n =1 s (v, v + ) n The dstances d or smlartes s from the negatve deal soluton are calculated as: n n d = d(v, v ) = (v v )2 (18) =1 =1 s 3 = s 3 (v, v t=1 ) = (1 t v t ) mn(p (v ), P (v )) + mn(ω v, ω v ) max(p (v ), P (v )) + max(ω v, ω v ) s = s (v, v t=1 ) = (1 t v t ) mn(p (v ), P (v )) max(p (v ), P (v )) mn(a(v ), A(v )) + mn(ωv, ω v ) max(a(v ), A(v )) + max(ωv, ω v ) (20) n and average smlartes by s =1 3 = s 3(v, v ) or n n s = =1 s (v, v ) n The closeness coeffcent s calculated usng the dstance measures from the postve and the negatve deal soluton [18] [2]. The closeness coeffcent can be appled to smlarty measures n a straghtforward way. The closeness coeffcents of the alternatve suppler A wth respect to the postve deal soluton by usng the dstance matrx (CC + ) or by usng the smlarty measures matrx (CCS + ) are defned as: CC + = CCS + = d + s + d + d s + + s, = 1, 2,..., m (21), = 1, 2,..., m (22) For all = 1, 2,..., m and = 1, 2,..., n, steps for the T OP SIS algorthm are presented as follow [7]: Step1: Form a decson-makers commttee, and dentfy the evaluaton crtera. Step2: Choose the approprate lngustc varables for the mportance weght of the crtera and the lngustc ratngs for alternatve supplers. Step3: Aggregate the weght of crtera to get the aggregated fuzzy weght ŵ of the crteron C and on the decson-makers ratngs to get the aggregated fuzzy ratng x of the suppler A n consderaton of the crteron C. Step: Construct the fuzzy decson matrx and the normalzed fuzzy decson matrx. Step5: Construct the weghted normalzed fuzzy decson matrx. Step6: Determne the fuzzy postve (or negatve) deal soluton FPIS (or FNIS). Step7: Construct the smlarty matrx by calculatng the smlarty measure of each suppler from the FPIS (or FNIS), and then average each suppler s smlarty value Step8: Calculate smlarty based closeness coeffcent to each suppler n order to determne all supplers rankng order. IV. EXPERIMENTAL RESULTS In ths secton, we present two numercal examples n whch the fuzzy TOPSIS method s appled by usng the above two fuzzy smlarty measures from equatons (3) and (). We shall

5 also demonstrate how the three FPIS and FNIS crtera affects the results. A. Numercal example 1 Our frst numercal example has been worked on by Chen et al (see [2]), the problem s to evaluate and to rank fve canddate supplers (A 1,A 2,A 3,A, and A 5 ) by consderng fve crtera (C 1,C 2,C 3,C, and C 5 ), and nputs from three decson-makers (D 1, D 2, and D 3 ). The consdered fve crtera are: 1. Proftablty of suppler (C 1 ) 2. Relatonshp closeness (C 2 ) 3. Technologcal capablty (C 3 ). Conformance qualty (C ) 5. Conflct resoluton (C 5 ) Ths knd of attrbutes are called beneft attrbutes. The decson-makers frst use the lngustc weghtng varable shown n Fgure 1 to assess the mportance of each crteron. Importance weghts of the crtera can be seen n Table I. The lngustc varables used for the ratng of supplers, by The fuzzy TOPSIS method s appled to solvng ths problem, and as we have prevously seen the TOPSIS general algorthm, the problem s computatonal procedure s summarzed as follow: 1: Three decson-makers use the lngustc weghtng varables shown n Fgure 1 to assess the mportance of the crtera. The mportance weghts of each crteron can be seen n Table I. TABLE I IMPORTANCE WEIGHT OF CRITERIA FROM THREE DECISION-MAKERS. Crtera Decson-maker D 1 D 2 D 3 C 1 H H H C 2 VH VH VH C 3 VH VH H C H H H C 5 H H H 2: Three decson-makers use the lngustc ratng varables shown n Fgure 2 to evaluate the ratngs of supplers wth regards to each crteron. The resultng ratngs are shown n Table II. TABLE II RATINGS OF THE FIVE SUPPLIERS BY THREE DECISION-MAKERS UNDER FIVE CRITERIA. Fg. 1. Lngustc varables for mportance weghts of each crteron. consderng all gven crtera, are shown n Fgure 2. The Crtera Suppler Decson-makers D 1 D 2 D 3 C 1 A 1 MG MG MG C 1 A 2 G G G C 1 A 3 VG VG G C 1 A G G G C 1 A 5 MG MG MG C 2 A 1 MG MG VG C 2 A 2 VG VG VG C 2 A 3 VG G G C 2 A G G MG C 2 A 5 MG G G C 3 A 1 G G G C 3 A 2 VG VG VG C 3 A 3 VG VG G C 3 A MG MG G C 3 A 5 MG MG MG C A 1 G G G C A 2 G VG VG C A 3 VG VG VG C A G G G C A 5 MG MG G C 5 A 1 G G G C 5 A 2 VG VG VG C 5 A 3 G VG G C 5 A G G VG C 5 A 5 MG MG MG Fg. 2. Lngustc varables for ratngs. lngustc varables shown n Fgures 1 and 2 are used to evaluate the mportance of the crtera and n the ratng of the alternatves wth respect to the qualtatve crtera. For example, the lngustc varable Medum Good can be represented as (5, 6, 7, 8) and Hgh as (0.7, 0.8, 0.8, 0.9). 3: The lngustc evaluatons n Tables I-II are converted nto trapezodal fuzzy numbers and the fuzzy weght of each crteron s determned. : The fuzzy-decson matrx s normalzed. 5: The weghted normalzed fuzzy-decson matrx as shown n Table III s bult. 6: The FPIS and FNIS are determned. Accordng to the three dfferent crtera we get three dfferent FPIS and FNIS that are gven n Table IV. 7: The smlarty of each suppler s calculated, by usng the weghted normalzed fuzzy decson matrx and the FPIS (and the FNIS) by calculatng the smlarty matrx

6 and then the average smlarty values for each suppler. Results are demonstrated n Table V and Table VI for two smlarty measures wth FPIS. In the tables Average1 s calculated from (3) and Average2 s calculated from (). 8: Accordng to the closeness coeffcent we next rank the supplers to descendng order. Ths way we get the rankng order A 2 > A 3 > A > A 1 > A 5 wth frst two deal solutons. Wth the thrd deal soluton (F P IS d3 ) and the new smlarty measure a permutaton between the the supplers A 2 and A 3 occurred and the rankng order becomes A 3 > A 2 > A > A 1 > A 5 (see Table VII). Ths ndcates that when we consder also the area of fuzzy numbers wthn the smlarty measure the new nformaton ncluded affects the rankng. Shortly, we can say that the rankng order s the same n the two frst cases (A 2 > A 3 > A > A 1 > A 5 ), but not n the thrd case (A 3 > A 2 > A > A 1 > A 5 ). The dfference s caused by the fact that n the frst two cases the deal solutons are crsp numbers, whle n the thrd case the deal solutons are fuzzy numbers. B. Numercal example 2 The second numercal example s somewhat dfferent from the frst one. The dfferences are: the number of supplers TABLE III WEIGHTED NORMALIZED FUZZY DECISION MATRIX. C 1 C 2 C 3 C C 5 A 1 (0.,0.5,0.6,0.7) (0.,0.6,0.8,1) (0.5,0.7,0.7,0.9) (0.5,0.6,0.6,0.8) (0.5,0.6,0.6,0.8) A 2 (0.5,0.6,0.6,0.8) (0.6,0.8,1,1) (0.6,0.8,0.9,1) (0.5,0.7,0.7,0.9) (0.6,0.7,0.8,0.9) A 3 (0.5,0.7,0.8,0.9) (0.6,0.8,0.9,1) (0.5,0.8,0.9,1) (0.6,0.7,0.8,0.9) (0.5,0.7,0.7,0.9) A (0.5,0.6,0.6,0.8) (0.,0.7,0.8,0.9) (0.,0.6,0.7,0.9) (0.5,0.6,0.6,0.8) (0.5,0.7,0.7,0.9) A 5 (0.,0.5,0.6,0.7) (0.,0.7,0.8,0.9) (0.,0.5,0.7,0.8) (0.,0.5,0.6,0.8) (0.,0.5,0.6,0.7) TABLE IV FUZZY POSITIVE IDEAL SOLUTIONS AND FUZZY NEGATIVE IDEAL SOLUTIONS ACCORDING TO DIFFERENT CRITERIA. A + d1 A d1 A + d2 A d2 A + d3 A d3 (0.9,0.9,0.9,0.9) (1,1,1,1) (1,1,1,1) (0.9,0.9,0.9,0.9) (0.9,0.9,0.9,0.9) (0.,0.,0.,0.) (0.,0.,0.,0.) (0.,0.,0.,0.) (0.,0.,0.,0.) (0.,0.,0.,0.) (1,1,1,1) (1,1,1,1) (1,1,1,1) (1,1,1,1) (1,1,1,1) (0,0,0,0) (0,0,0,0) (0,0,0,0) (0,0,0,0) (0,0,0,0) (0.5,0.7,0.8,0.9) (0.6,0.8,1,1) (0.6,0.8,0.9,1) (0.6,0.7,0.8,0.9) (0.6,0.7,0.8,0.9) (0.,0.5,0.6,0.7) (0.,0.6,0.8,0.9) (0.,0.5,0.7,0.8) (0.,0.5,0.6,0.8) (0.,0.5,0.6,0.7) TABLE V SIMILARITIES BETWEEN FPIS AND WEIGHTED NORMALIZED FUZZY DECISION MATRIX FROM EQUATION3. C 1 C 2 C 3 C C 5 Average1: S 3(A 1, A + d1 ) S 3 (A 2, A + d1 ) S 3 (A 3, A + d1 ) S 3 (A, A + d1 ) S 3 (A 5, A + d1 ) S 3 (A 1, A + d2 ) S 3(A 2, A + d2 ) S 3 (A 3, A + d2 ) S 3 (A, A + d2 ) S 3 (A 5, A + d2 ) S 3(A 1, A + d3 ) S 3 (A 2, A + d3 ) S 3(A 3, A + d3 ) S 3 (A, A + d3 ) S 3 (A 5, A + d3 ) TABLE VI SIMILARITIES BETWEEN FPIS AND WEIGHTED NORMALIZED FUZZY DECISION MATRIX FROM EQUATION. C 1 C 2 C 3 C C 5 Average2: S (A 1, A + d1 ) S (A 2, A + d1 ) S (A 3, A + d1 ) S (A, A + d1 ) S (A 5, A + d1 ) S (A 1, A + d2 ) S (A 2, A + d2 ) S (A 3, A + d2 ) S (A, A + d2 ) S (A 5, A + d2 ) S (A 1, A + d3 ) S (A 2, A + d3 ) S (A 3, A + d3 ) S (A, A + d3 ) S (A 5, A + d3 ) TABLE VII SIMILARITIES BETWEEN FPIS AND FNIS AND CLOSENESS COEFFICIENTS BY SIMILARITIES (3) AND (). SOLUTIONS ARE GIVEN FOR THREE DIFFERENT WAYS OF COMPUTING IDEAL SOLUTIONS GIVEN IN THE SAME ORDER AS INTRODUCED IN THE EQUATIONS (9) TO (1). Suppler s + 3 s 3 s + s CCS + 3 CCS + A A A A A A A A A A A A A A A to be evaluated, the crtera to be consdered (we have a cost crteron nvolved), and the number of decson-makers. As decson-makers act ndvdually, the two examples also dffer n the way the decson-makers attrbute the mportance weghts to the crtera and n ther ratngs of the supplers. Here we gve an example of results from usng the fuzzy TOPSIS method and by applyng the two (above presented) fuzzy smlarty measures, we see how four decson-makers (D 1, D 2, D 3 and D ) tryng to evaluate sx canddate supplers (A 1,A 2,A 3,A, A 5 and A 6 ) whle consderng four crtera (lsted below). 1. Product qualty (C 1 ) 2. Servce qualty (C 2 ) 3. Delvery tme (C 3 ). Prce (C ) In ths case, the frst three crtera are beneft crtera, and the last one (prce) s a cost crteron. The lngustc varables prevously shown n Fgures 1 and 2 are also used to evaluate the mportance of the crtera and the ratngs of alternatves wth respect to qualtatve crtera. The mportance weghts of the crtera can be seen n Table VIII. The resultng ratngs are shown n Table IX. Ths problem has been solved n the

7 exact same way as the frst one, and by followng the TOPSIS general algorthm, the problem s computatonal results are shown n Tables X and XI. From the rankng order for three dfferent deal soluton wth ths data set (see Table XII), we can see that the rankng order s the same (A 1 > A 2 > A 3 > A > A 5 > A 6 ) for both smlarty measures n case of the TABLE VIII IMPORTANCE WEIGHT OF CRITERIA FROM THREE DECISION-MAKERS. Crtera Decson-maker D 1 D 2 D 3 D C 1 VH H H H C 2 H VH VH H C 3 MH H H MH C M M MH MH TABLE IX RATINGS OF THE SIX SUPPLIERS BY FOUR DECISION-MAKERS UNDER FOUR CRITERIA. Crtera Suppler Decson-makers D 1 D 2 D 3 D C 1 A 1 G MG G G C 1 A 2 MG G F MG C 1 A 3 F F MG G C 1 A F MG MG F C 1 A 5 MG F F MG C 1 A 6 G MG MG MG C 2 A 1 G G MG MG C 2 A 2 G MG MG G C 2 A 3 F F P F C 2 A P MP MP P C 2 A 5 MP MP MP MP C 2 A 6 MP P P MP C 3 A 1 G MG MG G C 3 A 2 MG G G G C 3 A 3 G G F MG C 3 A G MG MG G C 3 A 5 MG F F MG C 3 A 6 F F MG F C A 1 F G G G C A 2 G G F MG C A 3 VG VG G G C A G MG G G C A 5 MG MG G MG C A 6 G VG VG G TABLE X SIMILARITIES BETWEEN FPIS AND WEIGHTED NORMALIZED FUZZY DECISION MATRIX FROM EQUATION3. C 1 C 2 C 3 C Average1: S 3 (A 1, A + d1 ) S 3(A 2, A + d1 ) S 3 (A 3, A + d1 ) S 3(A, A + d1 ) S 3 (A 5, A + d1 ) S 3 (A 6, A + d1 ) S 3 (A 1, A + d2 ) S 3(A 2, A + d2 ) S 3 (A 3, A + d2 ) S 3(A, A + d2 ) S 3 (A 5, A + d2 ) S 3 (A 6, A + d1 ) S 3 (A 1, A + d3 ) S 3(A 2, A + d3 ) S 3 (A 3, A + d3 ) S 3(A, A + d3 ) S 3 (A 5, A + d3 ) S 3 (A 6, A + d1 ) TABLE XI SIMILARITIES BETWEEN FPIS AND WEIGHTED NORMALIZED FUZZY DECISION MATRIX FROM EQUATION. C 1 C 2 C 3 C Average2: S (A 1, A + d1 ) S (A 2, A + d1 ) S (A 3, A + d1 ) S (A, A + d1 ) S (A 5, A + d1 ) S (A 6, A + d1 ) S (A 1, A + d2 ) S (A 2, A + d2 ) S (A 3, A + d2 ) S (A, A + d2 ) S (A 5, A + d2 ) S (A 6, A + d2 ) S (A 1, A + d3 ) S (A 2, A + d3 ) S (A 3, A + d3 ) S (A, A + d3 ) S (A 5, A + d3 ) S (A 6, A + d1 ) TABLE XII SIMILARITIES BETWEEN FPIS AND FNIS AND CLOSENESS COEFFICIENTS BY SIMILARITIES (3) AND (). SOLUTIONS ARE GIVEN FOR THREE DIFFERENT WAY OF COMPUTING IDEAL SOLUTIONS GIVEN IN SAME ORDER AS INTRODUCED IN THE EQUATIONS (9) TO (1). Suppler s + 3 s 3 s + s CCS + 3 CCS + A A A A A A A A A A A A A A A A A A frst deal solutons (F P IS d1, F NIS d1 ). Wth the second deal solutons (F P IS d2, F NIS d2 ), we get the orderng (A 1 > A 2 > A 3 > A > A 5 > A 6 ) usng smlarty from (3), but the orderng (A 1 > A 2 > A > A 3 > A 5 > A 6 ) by usng smlarty from (). Wth the thrd set of deal solutons (F P IS d3, F NIS d3 ) we get order nterchanges for both smlarty measures; for the permeters fuzzy smlarty measures from (3) gves (A 1 > A 2 > A > A 3 > A 5 > A 6 ), and for the areas fuzzy smlarty measures from () we get orderng (A 1 > A 2 > A > A 3 > A 5 > A 6 ). Clearly from ths experment we can see that both, selecton of proper deal solutons and selecton of proper fuzzy smlarty measure affects the orderng. Two ssues that nfluence the orderng are that second smlarty measure can take nto consderaton the area of fuzzy numbers somethng that the frst one cannot do. In the thrd alternatve of creatng the deal solutons changes may also be caused by the fact that the deal solutons are trapezodal fuzzy numbers nstead of crsp numbers.

8 V. CONCLUSIONS We have studed the use of the fuzzy TOPSIS method wth fuzzy smlarty measures. A recently publshed, new fuzzy smlarty measure has been tested for the frst tme n connecton wth the fuzzy TOPSIS to solve a suppler evaluaton problem as a Mult-Crtera Decson-Makng problem. The use of the fuzzy TOPSIS allows us to account for the fact that decson-makers udgements are sometmes mprecse, makng the suppler rankng problem becomes more complcated. The fuzzy TOPSIS also allows the use of lngustc varables that can be used to deal wth the vagueness of the supplers ratngs and the mprecse weghts for the crtera to be consdered. We consdered also dfferent versons of the deal soluton used n the analyss. For fuzzy postve and negatve deal solutons (FPIS, FNIS) that are fuzzy trapezodal numbers, nstead of crsp numbers, the fuzzy smlarty based TOPSIS s able to fnd a dfferent rankng than the dstance based method for the same numbers defuzzfed. Ths ponts to the effect from ncreased nformaton. There s also a dfference between the rankngs done wth the (new) smlarty measure that s able to take also the fuzzy numbers area nto account () and the rankng done wth the smlarty measure that only consders the permeter (3), ths ndcates new results wth the ncluson of more nformaton. The thrd deal soluton type consdered s to represent the deal soluton by truly fuzzy trapezodal numbers nstead of crsp numbers (the two frst types of deal solutons are crsp). Usng such a fuzzy deal soluton may be relevant and benefcal n some cases and as was shown n the examples, may affect the rankng obtaned. Consderng the two numercal examples, reasonable results have been obtaned from the fuzzy smlarty measures between each suppler and the deal solutons. We llustrate wth a numercal example that under relevant crcumstances the use of the dfferent smlarty measures result n dfferent rankngs of supplers - dfferent from the dstance based classcal TOPSIS and dfferent from each other. Includng more relevant nformaton n the rankng wll cause the results to be dfferent, thus makng the testng and use of more sophstcated fuzzy smlarty measures a topc that has relevance for the real world decson-makng wth regards to the selecton of supplers. [6] S.R. Heaz, A.Doostparast, and S.M. Hossen (2011), An mproved fuzzy rsk analyss based on a new smlarty measures of generalzed fuzzy numbers, Expert Systems wth Applcatons, Vol. 38, Issue 8, 2011, p [7] P. Luukka (2011), Fuzzy Smlarty n Multcrtera Decson-Makng Problem Appled to Suppler Evaluaton and Selecton n Supply Chan Management, Hndaw Publshng Corporaton, Advances n Artfcal Intellgence, 2011, p [8] Zadeh,L.(1971). Smlarty Relatons and Fuzzy Orderngs. Inform Sc, 3, pp [9] J. Bednář(2005), Fuzzy dstances, Kybernetka, Vol.1, No.3, 2005, p [10] Shepard, R. N. (1987). Toward a unversal law of generalzaton for psychologcal scence. Scence, 237, [11] Chen, S.H. (1999). Rankng generalzed fuzzy number wth graded mean ntegraton. In Proceedngs of the eghth nternatonal fuzzy systems assocaton world congress, Vol , pp Tape, Tawan, Republc of Chna. [12] Formato, F., Gerla, G. & Scarpat, L.(1999), Fuzzy Subgroups and Smlartes. Soft Computng, 3, pp [13] Valverde, L., On the Structure of F-Indstngushablty Operators, Fuzzy Sets and Systems 17. (1981) [1] C.H.Hseh and S.H.Chen (1999), Smlarty of generalzed fuzzy numbers wth graded mean ntegraton representaton, Fuzzy Systems Assocaton World Congr., Vol.2, Tawan, Chna, 1999, p [15] B. Srdev and R. Nadaraan (2009), Fuzzy smlarty measure for generalzed fuzzy numbers, Int. J. Open Compt. Math., Vol. 2, No. 2, 2009, p [16] M. Socorro García-Cascales, M. Teresa Lamata (2007), Solvng a decson problem wth lngustc nformaton, Pattern Recognton letters, Elsever, Vol. 28, No. 16, p [17] Zhu Xan Cu, Han Ku Yoo, Jun Yeol Cho, and Hee Yong Youn (2011), Mult-crtera Group Decson Makng wth Fuzzy Logc and Entropy based Weghtng, Proceedngs of the 5th ICUIMC 11, Feb. 2011, p [18] S. Mahmoodzadeh, J. Shahrab, M. Parazar, and M.S. Zaer (2007), Proect selecton by usng Fuzzy AHP and TOPSIS Technque, Internatonal Journal of Human and Socal Scences, Vol. 1, No. 3, 2007, p REFERENCES [1] Chen C. T. (2000), Extensons of the TOPSIS for group decson-makng under fuzzy envronment, Fuzzy Sets and Systems, Vol. 11, 2000, p [2] Chen-Tung Chen, Chng-Torng Ln, and Sue-Fn Huang (2006), A fuzzy approach for suppler evaluaton and selecton n supply chan management, Internatonal ournal of producton economcs, Vol. 102, Issue 2, p [3] George J. Klr, Bo Yuan (1995), Fuzzy Sets and Fuzzy Logc, Theory and Applcatons, Prentce Hall, Upper Saddle Rver, New Jersey. [] Monczka, R. Trent, R., Handfeld, R. (1998), Purchasng and Supply Chan Management, South-Western college Publshng, New York, 1998, p [5] S.-H. We and S.-M. Chen (2009), A new approach for fuzzy rsk analyss based on smlarty measures of generalzed fuzzy numbers, Expert Systems wth Applcatons 36. p