HESITANT TRIANGULAR FUZZY TOPSIS APPROACH FOR MULTIPLE ATTRIBUTES GROUP DECISION MAKING

Transcription

1 GESJ: Computer Scence and Telecommuncatons 07 No.(5) UDC 004.8, 004.9, 005. HESITNT TRINGULR FUZZY TOPSIS PPROCH FOR MULTIPLE TTRIBUTES GROUP DECISION MKING Irna Khutsshvl, Ga Srbladze, Iral Gotsrdze, Sopho Cobulashvl, Goderdz Jobadze and Davt Sprashvl Department of Computer Scences, Ivane Javahshvl Tbls State Unversty, Unversty str., 086 Tbls, Georga bstract In ths artcle a decson support methodology for multple attrbutes group decson mang (MGDM) problem s developed. The proposed methodology s based on the hestant fuzzy TOPSIS (Technque for Order Performance by Smlarty to Ideal Soluton) method. The case s consdered, when both the values and weghts of the attrbutes are expressed n lngustc terms, gven by all decson maers. Then, these lngustc terms are descrbed by trangular fuzzy numbers. Followng the TOPSIS method s algorthm, a relatve closeness coeffcent s defned to determne the ranng order of all alternatves by calculatng the dstances to the fuzzy postve-deal soluton (FPIS), as well as to the fuzzy negatve-deal soluton (FNIS). n example s shown to explan the procedure of the proposed methodology. Keywords: Multple attrbutes group decson mang, lngustc varable, hestant trangular fuzzy set, TOPSIS method, ranng of alternatves.. Introducton The man obectve of the multple attrbutes group decson mang (MGDM) problem s to choose one best of the feasble alternatves or to ran all the alternatves that are evaluated by a group of decson-maers (DMs) based on multple, often conflctng attrbutes. From ths perspectve, many practcal applcatons (such as medcal dagnostcs, proect management, personnel evaluaton, busness and fnance management, supply chan management, etc.) represent a MGDM problem. To solve a MGDM problem varous methods have been proposed, among them one of the most popular s the TOPSIS approach. It was frst developed by Hwang and Yoon []. In classcal TOPSIS the weght and values of each attrbute are presented by crsp numbers. However, the attrbute and weght values cannot be expressed always wth exact numbers due to uncertanty of decson maers preferences, as well as the vagueness and complexty of attrbutes. Ignorng the fuzzness and uncertanty of evaluated obects leads to nadequate and non-acceptable decsons. Processng fuzzy data n decson-mang models s based on the concept of fuzzy sets ntroduced by Zadeh [] and researched by Bellman and Zadeh []. s a generalzaton of a fuzzy set, Torra proposed noton of a hestant fuzzy set and ts applcaton n decson-mang [4, 5]. In ths connecton, many well-nown MGDM methods have been extended to tae nto account fuzzy types of values of attrbutes and ther weghts. The latter led to a great number of researches, n whch evaluatons of attrbutes nvolved n the decson mang problems most often are expressed n fuzzy numbers, trangular fuzzy numbers, confdence ntervals, hestant fuzzy elements, nterval-valued hestant fuzzy elements and so on [6-4 and others]. However, a more natural representaton of decson maers' assessments may be lngual expressons (lngustc varables) [7]. 5

2 GESJ: Computer Scence and Telecommuncatons 07 No.(5) In the proposed methodology both the values and weghts of the attrbutes are expressed n lngustc terms, gven by all decson maers. Then, these assessments are converted nto the trangular fuzzy numbers. Decsons are made usng TOPSIS method. Hence, proposed approach s based on hestant trangular fuzzy TOPSIS decson mang model. The dea of the TOPSIS method conssts n fndng as optmal the alternatve wth the nearest dstance from the fuzzy postve deal soluton (FPIS) and the farthest dstance from the fuzzy negatve deal soluton (FNIS). Followng the TOPSIS method s algorthm, a relatve closeness coeffcent s defned to determne the ranng order of all alternatves by calculatng the dstances to the FPIS, as well as to the FNIS. The developed approach s appled to evaluaton of nvestment proects wth the am of ther ranng and dentfcaton of hgh-qualty proects for nvestment. The artcle provdes an nvestment decson-mang example clearly llustratng the wor of the proposed methodology.. Prelmnares Ths Secton presents some basc defntons and notatons on the trangular fuzzy numbers, hestant fuzzy sets and hestant trangular fuzzy sets... On the trangular fuzzy numbers Defnton []. Let X be a reference set. fuzzy set of X s defned by a membershp functon µ ( x) [0,], where µ ( x), x X, ndcates the possble membershp degree of x to. Defnton [5]. trangular fuzzy number a can be determned by a trplet ( a, a a ) membershp functon µ a ( x) s defned as 0, x a a a µ a ( x) = a x a a 0 5 f f x < a, a x a, f a x a, otherwse., Let a and b be two trangular fuzzy numbers (TFNs) gven by the ( a, a a ) and ( b, b b ), respectvely. Some arthmetc operatons are determned on these two numbers as follows: ;. a ( ) b = ( a, a, a )( )( b, b, b ) = ( a b, a b, a b ). a ( ) b = ( a, a, a )( ) ( b, b, b ) = ( a b, a b, a b ) ;. a ( ) b = ( a, a, a )( )( b, b, b ) = ( ab, ab, a b ), > 0 4. a ( ) b = ( a, a, a )( )( b, b, b ) = ( a / b, a / b, a / b ) 5. λ a = ( λa, λa, λ a ), λ > 0 ; 6. ( a ) = ( / a,/ a,/ a ), a > 0;. Its, a, b > 0 ; (), a > 0, b > 0 ; 7. a > b f a > b ; and f a b then a = > b f a ; a > b b otherwse a = b. Defnton [6]. The dstance between two TFNs a = ( a, a a ) and b = ( b, b b ) by formula:,, s calculated

3 GESJ: Computer Scence and Telecommuncatons 07 No.(5) ( a b ) ( a b ) ( a ) ) d( a, b ) =. () b The followng propertes that wll be used later are vald:. Two TFNs a and b are dentcal ff d ( a, b ) = 0;. Let a, b and c be three TFNs. The TFN b s closer to TFN a that the other TFN c ff d ( a, b ) < d( a, c ). lngustc varable s a varable wth values expressed n lngustc terms [7]. For nstance, the lngustc varable weght may have the followng values - very low, low, medum, hgh, very hgh and so on. These lngustc values can be represented by fuzzy numbers... On the hestant fuzzy sets and hestant trangular fuzzy sets Hestant fuzzy set (HFS) was ntroduced by Torra and Naruawa n [4] and Torra n [5] as a generalzaton of a fuzzy set. In HFS the degree of membershp of an element to a reference set s presented by several possble fuzzy values. Ths allows descrbng stuatons when DMs have hestancy n provdng ther preferences over alternatves. The HFS s defned as follows: Defnton 4 [4, 5]. Let X be a reference set, a hestant fuzzy set H on X s defned n terms of a functon h H (x), whch when appled to X returns a subset of [0,]: { < x h ( x) > x X } H =, H, () where h H (x) s a set of some dfferent values n [0,], representng the possble membershp degrees of the element x X to H ; h H (x) s called a hestant fuzzy element (HFE). In [8] Yu ntroduced the hestant trangular fuzzy set (HTFS), where the membershp degrees of an element to a gven set are expressed by TFN. Defnton 5 [8]. For a reference set X, a hestant trangular fuzzy set T on X s defned n terms of a functon (x) as follows: f T { < x f ( x) > x X } T =, T, (4) where f T (x) s a set of several trangular fuzzy numbers, representng the possble membershp degrees of the element x X to the HTFS T ; f T (x) s called a hestant trangular fuzzy element (HTFE).. MGDM problem n hestant fuzzy envronment Consder a multple attrbutes problem for group decson mang. ssume that there are m decson mang alternatves = {,,, m }, and the group E = { e, e,, e } of t DMs (experts) evaluates them wth respect to an n attrbutes X = { x, x,, x n }. Consderng that the attrbutes have dfferent mportance degrees, the weghtng vector of all attrbutes, gven by the DMs, s defned by w = ( w, w,, w ) T n, where w s the mportance degree of th attrbute. DMs provde evaluatons of the attrbutes and of ther weghts n the form of lngual 54

4 GESJ: Computer Scence and Telecommuncatons 07 No.(5) assessments lngustc terms. Then, these assessments are expressed n trangular fuzzy numbers as shown n the followng tables: Table. Lngustc scale for the mportance of the weght of each attrbute Lngustc term Correspondng TFNs Very low (VL) (0, 0., 0.) Low (L) (0., 0., 0.5) Medum (M) (0., 0.5, 0.7) Hgh (H) (0.5, 0.7, 0.9) Very hgh (VH) (0.7, 0.9,.0) Table. Lngustc terms for ratng of alternatves Lngustc term Correspondng TFNs Very poor (VP) (0, 0, ) Poor (P) (,, 5) Far (F) (, 5, 7) Good (G) (6, 8, 9) Very good (VG) (8, 0, 0) Therefore, DMs ont assessments concernng each alternatve represent HTFSs. HTFS of the th alternatve on X s gven by = { < x, f ( x ) > x X }, where f ( ) x, =,,, m; =,,, n ndcates the possble membershp degrees of the th alternatve under the th attrbute x, and t can be expressed as a HTFE t. Then a MGDM problem n our case can be expressed n matrx format as follows T = x x x n t t t n t t tn, m t m t m tmn w = w, w,,, ( w ) T n { e, e, } E =,, e t where T s the fuzzy decson matrx (fuzzy hestant trangular decson matrx), each element of whch represents a HTFE t. 4. The hestant trangular fuzzy TOPSIS approach for MGDM problem The dea of TOPSIS method as appled to the problem of MGDM s to choose an alternatve wth the nearest dstance from the so-called fuzzy postve deal soluton (FPIS) and the farthest dstance from the fuzzy negatve deal soluton (FNIS). The algorthm of practcal solvng an MGDM problem can be formulated as follows: 55

5 GESJ: Computer Scence and Telecommuncatons 07 No.(5) Step : Determne the attrbutes weghng vector w = ( w, w,, w ) T n (see Table ) usng followng rules. If w = ( w, w, w ) s the mportance weght of the x attrbute gven by th DM, then the aggregated fuzzy weght of each attrbute are calculated as w = ( w, w, w ) where w = mn { w }, w = w, w = max{ w }. (5) Step : Based on the DMs evaluatons (see Table ) construct the fuzzy decson matrx = ( ). T t m n = Step : Compute the score matrx - aggregate fuzzy decson matrx - as follows. If t = ( t, t, t ) s an evaluaton of the alternatve wth respect to x attrbute gven t of alternatves wth respect to each attrbute are t = t, t t where by th DM, then the aggregated fuzzy ratngs ( ) gven by ( ), { t }, t = t, t { t } = mn = max t =. (6) Step 4: Compute the normalzed fuzzy decson matrx. Because varous attrbutes are usually measured n varous unts, t s necessary to transform varous measurements of attrbutes nto dmensonless attrbutes allowng to mae comparsons between attrbutes. The normalzed fuzzy decson matrx R s gven by R = [ r ], =,,..., m, =,,..., n, m n where r r r =,, * r, r * * * = max r for beneft (the more the better) attrbutes; (7) r r r r = r r r,,, r = mn r r r r for cost (the less the better) attrbutes. Step 5: Compute the weghted normalzed fuzzy decson matrx. The weghted normalzed matrx V s computed by multplyng the weghts ( w ) of evaluaton attrbutes wth the normalzed fuzzy decson matrx R : V = [ v ], =,,..., m, =,, n..., m n, where v = r ( ) w. (8) Step 6: Compute the fuzzy postve deal soluton (FPIS) and the fuzzy negatve deal soluton (FNIS). Determne the FPIS and the FNIS by formulas: = { J } ( v, v,..., v ) = v = max{ v } J ; v = mn{ v } n, (9) 56

6 GESJ: Computer Scence and Telecommuncatons 07 No.(5) = { J } ( v, v,..., v ) = v = mn{ v } J ; v = mn{ v } n, (0) =,,.., m, =,,.., n, where J s assocated wth a beneft attrbutes, and J - wth a cost attrbutes. Step 7: Compute the dstance of each alternatve from FPIS and FNIS. The dstances d and d of each weghted alternatve, =,,.., m from the FPIS and the FNIS s computed as follows: n d = = d( v, v ) ; n d = = d( v, v ), =,,.., m, () where d (.,.) s the dstance between two fuzzy numbers (see formula ()). soluton Step 8: Compute the relatve closeness coeffcent ( RC ) of each alternatve to the FPIS. The relatve closeness coeffcent and the fuzzy negatve deal soluton RC represents the dstances to the fuzzy postve deal smultaneously and s calculated as RC d ( d d ), =,,.., m. () = Step 9: Ran the alternatves. Perform the ranng of the alternatves, =,,, m accordng to the relatve closeness coeffcents RC, n decreasng order by the rule: for two alternatves α and β α β, f RC α > RC β, where s a preference relaton on. The best alternatve wll be the closest to the FPIS and farthest from FNIS. 5. pplcaton to evaluaton of nvestment proects Consder an nvestment decson-mang example adapted from [0] that clearly llustrates the wor of the proposed methodology. Suppose that n the competton for nvestment fve constructon companes are nvolved. The group of four DMs evaluates the nvestment proects tang nto account the fve attrbutes that are mportant for grantng nvestment: x busness proftablty; x pledge guaranteeng repayment of the credt; x locaton of constructon obect; x 4 wormanshp; x percent rato of the pledge to the credt monetary amount. 5 In our concrete case, all attrbutes are of a beneft type. DMs gve evaluatons n form of lngustc terms. ssume the nformaton of the mportance weghts of the attrbutes gven by all DMs wth lngustc assessments loos le: 57

7 GESJ: Computer Scence and Telecommuncatons 07 No.(5) Wegh t Table. The mportance weghts of the attrbutes n lngustc form ttrbutes x x x x 4 x 5 {H,VH,VH,H} {L,M,L,H} {M,H,VH,H} {M,L,M,L} {H,H,M,M} Lngustc terms then wll be transformed to the correspondng TFN shown n the Table. The HTFS obtaned for the weght of each attrbute are gven below n Table 4. Weght Table 4. Weghts of attrbutes n form of HTFS ttrbutes x x x x 4 x 5 { (0.5,0.7,0.9), (0.7,0.9,.0), (0.7,0.9,.0), (0.5,0.7,0.9) } { (0.,0.,0.5), (0.,0.5,0.7), (0.,0.,0.5), (0.5, 0.7, 0.9) } { (0., 0.5, 0.7), (0.5, 0.7, 0.9), (0.7, 0.9,.0), (0.5, 0.7, 0.9) } { (0., 0.5, 0.7), (0., 0., 0.5), (0., 0.5, 0.7), (0., 0., 0.5) } { (0.5, 0.7, 0.9), (0.5, 0.7, 0.9), (0., 0.5, 0.7), (0., 0.5, 0.7) } Usng the formula (5) we determne the attrbutes weghng vector w as = w {(0.5,0.8,.0),(0.,0.45,0.9),(0.,0.7,.0),(0.,0.4,0.7),(0.,0.6,0.9)} To evaluate the ratng of alternatves wth respect to each attrbute DMs use the lngustc ratng terms as gven n Table. ggregated results are presented n Table 5 as the lngustc fuzzy decson matrx.. lternatves Table 5. Ratngs of alternatves gven by DMs ttrbutes x x x x 4 x 5 {G,VG,VP,VG} {G,G,F,VG} {VG,VG,F,VG} {VP,P,VP,VP} {F,F,VP,VP} {P,P,VP,VP} {P,F,VP,F} {VP,P,VP,VP} {P,VP,P,P} {F,F,P,P} {F,VG,G,G} {P,G,F,G} {F,VG,VG,F} {F,G,F,F} {F,G,G,F} 4 {G,VP,F,F} {P,VP,G,P} {G,G,G,G} {VG,G,VG,VG} {G, F,G, VG} 5 {F,P,VP,VP} {P,P,F,P} {F,F,F,P} {F,G,G,G} {F,G,F,G} These lngustc evaluatons are transformed nto HTF matrx by assgnng the approprate TFN as gven n Table. Thus, we obtaned the followng hestant trangular fuzzy decson matrx. lternatves 4 Table 6. The hestant trangular fuzzy decson matrx T ttrbutes x x x x 4 x 5 {(6,8,9), (6,8,9), {(8,0,0),(8,0,0), {(0,0,), (,,5), (,5,7),(8,0,0)} (,5,7), (8,0,0)} (0,0,), (0,0,)} {(6,8,9), (8,0,0), (0,0,), (8,0,0)} {(,,5), (,,5), (0,0,), (0,0,)} {(,5,7), (8,0,0), (6,8,9), (6,8,9)} {(6,8,9), (0,0,), (,5,7), (,5,7)} {(,,5), (,5,7), (0,0,), (,5,7)} {(,,5), (6,8,9), (,5,7), (6,8,9)} {(,,5), (0,0,), (6,8,9), (,,5)} {(0,0,), (,,5), (0,0,), (0,0,)} {(,5,7), (8,0,0), (8,0,0), (,5,7)} {(6,8,9), (6,8,9), (6,8,9), (6,8,9)} {(,,5), (0,0,), (,,5), (,,5)} {(,5,7), (6,8,9), (,5,7), (,5,7)} {(8,0,0), (6,8,9), (8,0,0),(8,0,0)} {(,5,7), (,5,7), (0,0,), (0,0,)} {(,5,7), (,5,7), (,,5), (,,5)} {(,5,7), (6,8,9), (6,8,9), (,5,7)} {(6,8,9), (,5,7), (6,8,9), (8,0,0)} 58

8 GESJ: Computer Scence and Telecommuncatons 07 No.(5) 5 {(,5,7), (,,5), (0,0,), (0,0,)} {(,,5), (,,5), (,5,7), (,,5)} {(,5,7), (,5,7), (,5,7), (,,5)} {(,5,7), (6,8,9), (6,8,9), (6,8,9)} {(,5,7), (6,8,9), (,5,7), (6,8,9)} To avod computatonal complexty n the decson mang process the aggregate fuzzy decson matrx s developed usng formula (6) from Step. lternatves Table 7. The aggregate fuzzy decson matrx ttrbutes x x x x 4 x 5 (0, 7, 0) (, 7.75, 0) (, 8.75, 0) (0, 0.75, 5) (0,.5, 7) (0,.5, 5) (0,.5, 7) (0, 0.75, 5) (0,.5, 5) (, 4, 7) (, 7.75, 0) (, 6, 9) (, 7.5, 0) (, 5.75, 9) (, 6.5, 9) 4 (0, 4.5, 9) (0,.5, 9) (6, 8, 9) (6, 9.5, 0) (, 7.75, 0) 5 (0,, 7) (,.5, 7) (, 4.5, 7) (, 7.5, 9) (, 6.5, 9) Then, by formula (7) from Step 4, the ratngs of each alternatve attrbutes are normalzed, and the result s lsted n Table 8. lternatves Table 8. The normalzed fuzzy decson matrx R ttrbutes x x x x 4 x 5 (0, 0.7,.0) (0., 0.775,.0) (0., 0.875,.0) (0, 0.075, 0.5) (0, 0.5, 0.7) (0, 0.5, 0.5) (0, 0.5, 0.7) (0, 0.075, 0.5) (0, 0.5, 0.5) (0., 0.4, 0.7) (0., 0.775,.0) (0., 0.6, 0.9) (0., 0.75,.0) (0., 0.575, 0.9) (0., 0.65, 0.9) 4 (0, 0.45, 0.9) (0, 0.5, 0.9) (0.6, 0.8, 0.9) (0.6, 0.95,.0) (0., 0.775,.0) 5 (0, 0., 0.7) (0., 0.5, 0.7) (0., 0.45, 0.7) (0., 0.75, 0.9) (0., 0.65, 0.9) Usng formula (8) from Step 5 and tem of formula () the weghted normalzed matrx V s computed (see Table 9). lternatves Table 9. The weghted normalzed fuzzy decson matrx V ttrbutes x x x x 4 x 5 (0, 0.56,.0) (0.0, 0.5, 0.9) (0.09, 0.6,.0) (0, 0.0, 0.5) (0, 0.5, 0.6) (0, 0., 0.5) (0, 0.5, 0.6) (0, 0.05, 0.5) (0, 0.09, 0.5) (0.0, 0.4, 0.5) (0.5, 0.6,.0) (0.0, 0.7, 0.8) (0.09, 0.5,.0) (0.0, 0., 0.6) (0.09, 0.9, 0.8) 4 (0, 0.6, 0.9) (0, 0.6, 0.8) (0.8, 0.56, 0.9) (0.06, 0.8, 0.7) (0.09, 0.47, 0.9) 5 (0, 0.6, 0.7) (0.0, 0.6, 0.6) (0.0, 0., 0.7) (0.0, 0.9, 0.6) (0.09, 0.9, 0.8) Followng the fuzzy TOPSIS method s algorthm, we determne the FPIS by formulas (9)-(0) and tem 7 of formula (), respectvely: and the FNIS 59

9 GESJ: Computer Scence and Telecommuncatons 07 No.(5) the FNIS { = (0.5, 0.6,.0), (0.0, 0.5, 0.9), (0.09, 0.6,.0), (0.06, 0.8, 0.7), (0.09, 0.47, 0.9)}; = { (0.0, 0., 0.5), (0.0, 0.5, 0.6), (0.0, 0.05, 0.5, ), (0.0, 0.5, 0.6) }. Then we calculate the dstances d and (0.0, 0.0, 0.5), d of each alternatve from the FPIS by formulas () from Step 7 and formula (), respectvely: d = 0.667, =. d 504, = 0. d 8567, d 4 = , d 5 = ; d =.0708, d = , =. d 06, 4 =. d 976, d 5 = and Usng formula () from Step 8 to calculate the relatve closeness coeffcent alternatve to the FPIS we obtan: RC of each RC = 0.689, RC = , RC = , RC 4 = , RC 5 = Fnally, we perform the ranng of the alternatves, =,,, 5 accordng to the relatve closeness coeffcents RC, =,,, 5 (see Step 9). Usng the values of RC, the alternatves are raned as:. 4 5 From the obtaned ranng of proects, t s possble to mae a concluson that the proect wll be the most preferable, whle the proect wll be the least preferable choce of the decson. That means that when nvestng the captal only n one proect, DMs prefer the nvestment proect,.e. the proect receves nvestment. 6. Conclusons In the present wor a novel approach for solvng MGDM problem s developed. The proposed methodology s the extenson of TOPSIS method n the fuzzy envronment usng HTFS. Unle other fuzzy TOPSIS methods, n ths study HTFS are appled to processng the lngustc expressons used by decson maers. If decson maers gve lngustc evaluatons of the mportance of the attrbutes' weghts and ratng of the alternatves on the bass of these attrbutes, they only roughly descrbe these values because of the uncertanty and vagueness of the nformaton and also due to hestance of DMs preferences. In these cases, HTFS may be the best tool for solvng decson-mang problems wth lngustc assessments. The use of HTFS wll gve an adequate converson of lngustc terms used by the decson maers nto an MGDM problem. The new aspects n the hestant trangular TOPSIS approach have been used:. To dentfy both FPIS and FNIS the new formulas are appled, where the hghest and lowest values of attrbutes are found by comparng the trangular numbers. These formulas also tae nto account both types of attrbutes - a beneft attrbutes, as well a cost attrbutes;. The developed approach was appled n the problem of nvestment decson mang. Based on proposed approach the software pacage has been developed and used to ran nvestment proects n the real nvestment decson mang problem. The applcaton and testng of the software was carred out based on the data provded by the Ban of Georga. The results are llustrated n the example.

10 GESJ: Computer Scence and Telecommuncatons 07 No.(5) References. Hwang C.L. and Yoon K., Multple ttrbute Decson Mang: Methods and pplcatons, New Yor: Sprnger-Verlag, 98.. Zadeh L.., Fuzzy sets, Informaton and control, 8(), 995, 8-5, 85, -9.. Bellman, R.E. and Zadeh L.., Decson-Mang n a Fuzzy Envronment, Management Scence, 7(4), 970, B-4-B Torra V., Hestant fuzzy sets, Internatonal Journal of Intellgent Systems, 5, 00, pp Torra V. and Naruawa Y., On hestant fuzzy sets and decson, n: The 8th IEEE Internatonal Conference on Fuzzy Systems, Jeu Island, Korea, 009, pp L L. and Fan G., Fuzzy MDM wth trangular numbers for proect nvestment model based on left and rght scores, Research Journal of ppled Scences, Engneerng and Technology, 7, 04, Chen X.H. and Yang X., Multple attrbutve group decson mang method based on trangular fuzzy numbers, Syst. Eng. Electron., 0(), 008, Srbladze G., Khutsshvl I. and Ghvaberdze B., Multstage decson-mang fuzzy methodology for optmal nvestments based on experts evaluatons, European Journal of Operatonal Research, vol., ssue, 04, pp We G., Zhao X. and Ln R., Some hestant nterval-valued fuzzy aggregaton operators and ther applcatons to multple attrbute decson mang, Knowledge-Based Systems, 46, 0, Khutsshvl I. and Srbladze G., Multple-ttrbutes Decson Mang n Hestant Fuzzy Envronment: pplcaton to Evaluaton of Investment Proects, Georgan Electronc Scentfc Journal: Computer Scence and Telecommuncatons, No.(4), 04, Saghafan S. and Heaz S.R., Mult-crtera group decson mang usng a modfed fuzzy TOPSIS procedure, n Proceedngs - Internatonal Conference on Computatonal Intellgence for Modellng, Control and utomaton, CIMC 005 and Internatonal Conference on Intellgent gents, Web Technologes and Internet, vol., Venna, ustra, 005, pp Wang Y.J. and Lee H.S., Generalzng TOPSIS for fuzzy multcrtera group decson mang, Computers and Mathematcs wth pplcatons, 5, 007, Srbladze G., New Fuzzy ggregaton Operators Based on the Fnte Choquet Integral pplcaton n the MDM Problem, Internatonal Journal of Informaton Technology & Decson Mang, vol. 5, no. 0, 06, pp Srbladze G. and Badagadze O., Intutonstc Fuzzy Probablstc ggregaton Operators Based on the Choquet Integral: pplcaton n Multcrtera Decson-Mang, Internatonal Journal of Informaton Technology & Decson Mang, vol. 6, no. 0, 07, pp Kaufmann. and Gupta M.M., Introducton to Fuzzy rthmetc: Theory and pplcatons, New Yor: Van Nostrand Renhold, Chen C.T., Extensons of the TOPSIS for group decson-mang under fuzzy envronment, Fuzzy Sets and Systems, 4, 000, Zadeh L.., The concept of a lngustc varable and ts applcaton to approxmate reasonng, Inform. Sc,. 8, 975, 99-49(I), 0-57(II). 8. Yu D., Trangular hestant fuzzy set and ts applcaton to teachng qualty evaluaton, Journal of Informaton & Computatonal Scence, vol. 7, 0, pp rtcle receved: