Available online at ScienceDirect. Procedia Computer Science 91 (2016 )

Transcription

1 Avalable onlne at ScenceDrect Proceda Computer Scence 91 (2016 ) Informaton Technology and Quanttatve Management (ITQM 2016) Fuzzy TOPSIS: A General Vew Sorn Nădăban a, Smona Dztac b,c, Ioan Dztac a,c, a Aurel Vlacu Unversty of Arad, Department of Mathematcs and Computer Scence, Elena Drago 2, RO Arad, Romana b Unversty of Oradea, Department of Energetcs, Unverstat 1, RO Oradea, Romana c Agora Unversty of Oradea, Department of Socal Scences, Pata Tneretulu 8, RO Oradea, Romana Abstract The am of ths survey paper s to offer a general vew of the developments of fuzzy TOPSIS methods. We begn wth a lterature revew an we explore dfferent fuzzy models that have been appled to the decson mang feld. Fnally, we present some applcatons of fuzzy TOPSIS. c 2016 The Authors. Publshed by Elsever B.V. Ths s an open access artcle under the CC BY-NC-ND lcense ( Selecton and/or peer-revew under responsblty of ITQM2016. Peer-revew under responsblty of the Organzng Commttee of ITQM 2016 Keywords: Mult-crtera decson mang, fuzzy sets, fuzzy MCDM, fuzzy numbers, lngustc varable, ntutonstc fuzzy sets, neutrosophc set, hestant fuzzy set, fuzzy AHP, fuzzy TOPSIS 1. Introducton and lterature revew The problems of Mult-Crtera Decson Mang (MCDM) appear and are ntensely appled n many domans, such as Economcs, Socal Scences, Medcal Scences etc. Sometmes, MCDM problems are mentoned as Multple-Crtera Decson Analyss (MCDA) or Mult-Attrbute Decson-Mang (MADM) (see [22, 27, 47, 60]). In spte of ther dversty, the MCDM have as common characterstc multple obectves and multple crtera whch usually are n conflct wth each other. The decson maers have to select, assess or ran these alternatves accordng to the weghts of the crtera. In the last decades the MCDM technques have become an mportant branch of operatons research (see [23, 46, 65]). In many real-world stuatons, the problems of decson mang are subected to some constrants, obectves and consequences that are not accurately nown. After Bellman and Zadeh [8] ntroduced for the frst tme fuzzy sets wthn MCDM, many researchers have been preoccuped by decson mang n fuzzy envronments. The fuson between MCDM and fuzzy set theory has led to a new decson theory, nown today as fuzzy mult-crtera decson mang (FMCDM), where we have decson-maer models that can deal wth ncomplete and uncertan nowledge and nformaton. The most mportant thng s that, when we want to assess, udge or decde we usually use a natural language n whch the words do not have a clear, defnte meanng. As a result, we need fuzzy numbers to express lngustc varables, to descrbe the subectve udgement of a decson maer n a quanttatve manner. Fuzzy numbers (FN) most often used are trangular FN, trapezodal FN and Gaussan FN. We hghlght that the concept of lngustc varable ntroduced by Zadeh n 1975 (see [61]) allows computaton wth words nstead of numbers and thus lngustc terms defned by fuzzy sets are ntensely used n problems of decson theory for modelng uncertan nformaton. There are very good monographs (see for nstance [17]) and surveys papers [1, 12, 25, 33, 38] on FMCDM. Recently, some new methods have been explored [3, 53, 67]. Correspondng author. Tel.: ; fax: E-mal address: oan.dztac@uav.ro The Authors. Publshed by Elsever B.V. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the Organzng Commttee of ITQM 2016 do: /.procs

2 824 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) After Atanassov [4] ntroduced the concept of ntutonstc fuzzy sets, where each element s characterzed by a membershp functon, as n fuzzy sets, as well as by a non-membershp functon, the nterest n the study of the problems of decson mang theory wth the help of ntutonstc fuzzy sets (see [11, 26, 29, 31, 32, 62]) has ncreased. As a generalzaton of the concept of the classc set, fuzzy set, ntutonstc fuzzy set etc., Smarandache [42] frstly proposed the concept of neutrosophc set. In paper [49] there are proposed set-theoretc operators on an nstance of neutrosophc set called nterval neutrosophc set. Recently, neutrosophc sets have been appled n MCDM (see [36, 56, 57, 58, 63]). Torra and Naraawa [45] and Torra [44] ntroduced the concept of hestant fuzzy set, whch undergoes a much more flexble approach for decson maers when they provde ther decsons. Therefore, hestant fuzzy sets have become useful n MCDM problems [37, 39, 48, 52]. The am of ths survey paper s to offer a general vew of the developments of fuzzy TOPSIS methods. We begn wth a lterature revew an we explore dfferent fuzzy models that have been appled to the decson mang feld. Fnally, we present some applcatons of fuzzy TOPSIS. 2. Basc concepts and defntons Defnton 2.1. [21] A fuzzy number (FN) s a fuzzy set n R, namely a mappng x : R [0, 1], wth the followng propertes: 1. x s convex,.e. x(t) mn{x(s), x(r)}, for s t r; 2. x s normal,.e. ( )t 0 R : x(t 0 ) = 1; 3. x s upper semcontnuous,.e. ( )t R, ( )α (0, 1] : x(t) <α,( )δ >0 such that s t <δ x(s) <α. (1) Remar 2.2. Among the varous types of FNs, trangular FNs and trapezodal FNs are the most popular. A trangular FN s defned by ts membershp functon 0 f t < a t a x(t) = b a f a t < b c t, where a b c, (2) c b f b t < c 0 f t > c and t s denoted x = (a, b, c). A trapezodal FN s defned by ts membershp functon x(t) = and t can be expressed as x = (a, b, c, d). 0 f t < a t a b a f a t b 1 f b < t < c d t d c f c t d 0 f t > d, where a b c d, (3) Remar 2.3. [14, 22, 24] Let x = (a 1, b 1, c 1 ), ỹ = (a 2, b 2, c 2 ) be two non negatve trangular FNs and α R +. Accordng to the extenson prncple, the arthmetc operatons are defned as follows: 1. x + ỹ = (a 1 + a 2, b 1 + b 2, c 1 + c 2 ); 2. x ỹ = (a 1 c 2, b 1 b 2, c 1 a 2 ); 3. α x = (αa 1,αb 1,αc 1 ); 4. x 1 (1/c 1, 1/b 1, 1/a 1 ); 5. x ỹ (a 1 a 2, b 1 b 2, c 1 c 2 ); 6. x/ỹ (a 1 /c 2, b 1 /b 2, c 1 /a 2 ). We note that the results of (4) (6) are not trangular FNs, but they can be approxmated by trangular FNs. Remar 2.4. [17, 28, 59] Let x = (a 1, b 1, c 1, d 1 ), ỹ = (a 2, b 2, c 2, d 2 ) be two non negatve trapezodal FNs and α R +. The arthmetc operatons are defned as follows: 1. x + ỹ = (a 1 + a 2, b 1 + b 2, c 1 + c 2, d 1 + d 2 ); 2. x ỹ = (a 1 d 2, b 1 c 2, c 1 b 2, d 1 a 2 ); 3. α x = (αa 1,αb 1,αc 1,αd 1 ); 4. x 1 (1/d 1, 1/c 1, 1/b 1, 1/a 1 );

3 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) x ỹ (a 1 a 2, b 1 b 2, c 1 c 2, d 1 d 2 ); 6. x/ỹ (a 1 /d 2, b 1 /c 2, c 1 /b 2, d 1 /a 2 ). We menton that the results of (4) (6) are not trapezodal FNs, but they can be approxmated by trapezodal FNs. Defnton 2.5. [4] An ntutonstc fuzzy set (IFS) A n X s gven by A = {< x,μ A (x),ν A (x) > : x X}, (4) where μ A,ν A : X [0, 1] such that 0 μ A (x) + ν A (x) 1 and represent the degree of membershp and the degree of non-membershp of an element x to A. Defnton 2.6. [5] Let D([0, 1]) be the set of all closed subnterval of [0, 1]. An nterval valued ntutonstc fuzzy set (IVIFS) n X s gven by à = {< x,μã(x),νã(x) > : x X}, (5) where μ A,ν A : X D([0, 1]) such that 0 sup x X μã(x) + sup νã(x) 1. x X Remar 2.7. If we denote μã(x) = [a, b],νã(x) = [c, d], then à can be wrtten à = ([a, b], [c, d]). Let à 1 = ([a 1, b 1 ], [c 1, d 1 ]), à 2 = ([a 2, b 2 ], [c 2, d 2 ]) and α R +. The arthmetc operatons are defned by Xu [54]: 1. à 1 + à 2 = ([a 1 + a 2 a 1 a 2, b 1 + b 2 b 1 b 2 ], [c 1 c 2, d 1 d 2 ]); 2. αã 1 = ([1 (1 a 1 ) α, 1 (1 b 1 ) α ], [c α 1, dα 1 ]); 3. à 1 à 2 = ([a 1 a 2, b 1 b 2 ], [c 1 + c 2 c 1 c 2, d 1 + d 2 d 1 d 2 ]). 3. Fuzzy MCDM problem formulaton A MCDM problem wth m alternatves {A 1, A 2,, A m } whch should be assessed by applyng n crtera (or attrbutes) {C 1, C 2,, C n } can be expressed by the decson matrx x 11 x 12 x 1n x X = 21 x 22 x 2n, (6) x m1 x m2 x mn where x s a numerc data whch represents the value of the th alternatve wth respect to the th crteron. The mportance (or weght) of the crteron C to the decson s denoted by w. Let w be the vector w = [w 1, w 2,, w n ]. (7) Generally, the weghts are determned on a subectve bass by a sngle decson maer or by a group of experts. Some remars must be made: Remar 3.1. In fuzzy MCDM, n order to assgn the mportance degree to the crtera, t can be used an emprcal method descrbed n [55], where an equvalence between the mportance of an attrbute and a trangular FN s presented. Table 1. Trangular FNs for the mportance of crtera Ran Attrbute grade Trangular FN Very low 1 (0.00, 0.10, 0.30) Low 2 (0.10, 0.30, 0.50) Medum 3 (0.30, 0.50, 0.75) Hgh 4 (0.50, 0.75, 0.90) Very hgh 5 (0.75, 0.90, 1.00) Smlarly, alternatves can be evaluated by lngustc terms whch can be represented by trangular FNs [15]. Remar 3.2. If the performance ratngs of alternatves on qualtatve crtera s expressed by lngustc term, these lngustc terms can be represented by trapezodal FNs or IVIFS as n the followng tables: Remar 3.3. It s often dffcult for decson maers to assgn a precse value to an alternatve for the crtera consdered. In ths stuaton the fuzzy MCDM problem can be expressed by the decson matrx X = x 11 x 12 x 1n x 21 x 22 x 2n x m1 x m2 x mn, (8) where x are fuzzy value (trangular FN, trapezodal FN, IFS, IVIFS, trapezodal hestant fuzzy element etc.)

4 826 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) Table 2. Lngustc terms for alternatves ratngs Lngustc terms for alternatves ratngs Trangular FN Very good (9,10,10) Good (7,9,10) Medum (3,5,7) Poor (1,3,5) Very poor (1,1,3) Table 3. Lngustc values of trapezodal FNs for lngustc terms Lngustc term Trapezodal FN Very low (0.00, 0.00, 0.00, 0.10) Low (0.10, 0.20, 0.25, 0.30) Medum low (0.30, 0.40, 0.45, 0.50) Medum (0.50, 0.60, 0.65, 0.70) Medum hgh (0.70, 0.80, 0.85, 0.90) Hgh (0.90, 0.95, 1.00, 1.00) Very hgh (1.00, 1.00, 1.00, 1.00) 4. Fuzzy AHP There exsts a number of methods n order to obtan crtera weghts [64, 66], but the AHP (Analytc Herarchy Process) developed by Saaty [40] s the most used. Bucley [9] ncorporated the fuzzy theory nto AHP and obtaned n ths way fuzzy AHP. We note that a new method for fndng fuzzy weghts, based on a drect fuzzfcaton of method proposed by Saaty s presented n paper [10]. The procedure of fuzzy AHP (see for nstance [22, 51] etc.) s: Step 1: Construct fuzzy parwse comparson matrces. Each decson maer assgns lngustc term represented by trangular FN to the parwse comparson among all crtera. Let P = [ã ]bean n matrx, where ã s the mportance of crteron C wth respect to crteron C, accordng to the fuzzy preference scale shown n the next table: We note that P = (1, 1, 1) ã 12 ã 1n ã 21 (1, 1, 1) ã 2n ã n1 ã n2 (1, 1, 1) = Step 2: Compute the fuzzy weghts by normalzaton. The fuzzy weght of crteron C, denoted w, s obtaned by where r = [ã 1 ã 2 ã n ] 1/n. (1, 1, 1) ã 12 ã 1n (1, 1, 1)/ã 12 (1, 1, 1) ã 2n. (9) (1, 1, 1)/ã 1n (1, 1, 1)/ã 2n (1, 1, 1) w = r ( r 1 + r r n ) 1, (10) 5. Fuzzy TOPSIS Technque for Order Performance by Smlarty to Ideal Soluton (TOPSIS) was proposed by Hwang and Yoon [27] and t s the most nown technque for solvng MCDM problems. Ths method s based on the concept that the chosen alternatve should have the shortest dstance to Postve Ideal Soluton (PIS) (the soluton whch mnmzes the cost crtera and maxmzes the beneft crtera) and the farthest dstance to Negatve Ideal Soluton (NIS). Chen [15] extended TOPSIS wth trangular FNs. Chen ntroduced a vertex method to calculate the dstance between two trangular FNs. If x = (a 1, b 1, c 1 ), ỹ = (a 2, b 2, c 2 ) are two trangular FNs then 1 [ d( x, ỹ) := (a1 a 2 ) (b 1 b 2 ) 2 + (c 1 c 2 ) 2]. (11) The procedure of fuzzy TOPSIS (see for nstance [6, 15] etc.) s: Step 1. Assgnment ratng to the crtera and to the alternatves. We assume that we have a decson group wth K members. The fuzzy ratng of the th decson maer about alternatve A w.r.t. crteron C s denoted x = (a, b, c ) and the weght of crteron C s denoted w = (w 1, w 2, w 3 ). Step 2. Compute the aggregated fuzzy ratngs for alternatves and the aggregated fuzzy weghts for crtera.

5 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) Table 4. Lngustc values of IVIFS for lngustc terms Lngustc term IVIFS Very low ([0.00,0.10], [0.85,0.90]) Low ([0.10,0.15], [0.75,0.80]) Medum low ([0.25,0.30], [0.60,0.65]) Medum ([0.35,0.40], [0.45,0.50]) Medum hgh ([0.50,0.60], [0.30,0.35]) Hgh ([0.65,0.70], [0.15,0.20]) Very hgh ([0.75,0.80], [0.10,0.15]) Very very hgh ([0.90,0.95], [0.01,0.02]) Extremely hgh ([1.00,1.00], [0.00,0.00]) Table 5. Fuzzy preference scale Lngustc value Trangular FN (ã ) Absolutely mportant (7,9,9) Very strongly extreme mportant (6,8,9) Very strongly mportant (5,7,9) Strongly mportant (4,6,8) Moderately strong mportant (3,5,7) Moderate mportant (2,4,6) Wealy mportant (1,3,5) Equally moderate mportant (1,2,4) Equally mportant (1,1,3) The aggregated fuzzy ratng x = (a, b, c )of th alternatve w.r.t. th crteron s obtaned as follows: a = mn{a }, b = 1 K =1 b, c = max{c }. (12) The aggregated fuzzy weght w = (w 1, w 2, w 3 ) for the crteron C are calculated by formulas: or w 1 = mn{w 1 }, w 2 = 1 K =1 w 2, w 3 = max{w 3 }. (13) Step 3. Compute the normalzed fuzzy decson matrx. The normalzed fuzzy decson matrx s R = [ r ], where a r = c, b c, c c and c = max {c } (beneft crtera) (14) r = a c, a b, a a and c = mn{a } (cost crtera). (15) Step 4. Compute the weghted normalzed fuzzy decson matrx. The weghted normalzed fuzzy decson matrx s Ṽ = (ṽ ), where ṽ = r w. Step 5. Compute the Fuzzy Postve Ideal Soluton (FPIS) and Fuzzy Negatve Ideal Soluton (FNIS). The FPIS and FNIS are calculated as follows: A = (ṽ 1, ṽ 2,, ṽ n), where ṽ = max {v 3 } ; (16) A = (ṽ 1, ṽ 2,, ṽ n ), where ṽ = mn{v 1 }. (17) Step 6. Compute the dstance from each alternatve to the FPIS and to the FNIS. Let d = d(ṽ, ṽ ), d = d(ṽ, ṽ ) (18) be the dstance from each alternatve A to the FPIS and to the FNIS, respectvely. Step 7. Compute the closeness coeffcent CC for each alternatve. For each alternatve A we calculate the closeness coeffcent CC as follows: =1 CC = Step 8. Ran the alternatves. The alternatve wth hghest closeness coeffcent represents the best alternatve. d d + d =1. (19)

6 828 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) Remar 5.1. If for the evaluaton ratngs of alternatves and the weghts of crtera there are used lngustc values represented by trapezodal FNs, then the fuzzy TOPSIS approach s slghtly modfed (see [16]): If x = (a 1, b 1, c 1, d 1 ), ỹ = (a 2, b 2, c 2, d 2 ) are two trapezodal FNs then 1 [ d( x, ỹ) := (a1 a 2 ) (b 1 b 2 ) 2 + (c 1 c 2 ) 2 + (d 1 d 2 ) 2]. (20) 1. The aggregated fuzzy ratngs x = (a, b, c, d ) are defned as: a = mn{a }, b = 1 K b, c = 1 K =1 =1 c, d = max{d }, (21) where x = (a, b, c, d ) are the fuzzy ratngs of th decson maer. Let w = (w 1, w 2, w 3, w 4 ) be the mportance weght of th decson maer. The aggregated fuzzy weght w = (w 1, w 2, w 3, w 4 ) for the crteron C can be calculated as: w 1 = mn{w 1 }, w 2 = 1 K w 2, w 3 = 1 K =1 =1 w 3, w 4 = max{w 4 }. (22) 2. The normalzed fuzzy decson matrx s R = [ r ], where a r = d, b d, c d, d d and d = max {d } for Beneft crtera (23) 3. r = a d, a c, a b, a a and a = mn{a } for Cost crtera. (24) A = (ṽ 1, ṽ 2,, ṽ n), where ṽ = max {v 4 } ; (25) A = (ṽ 1, ṽ 2,, ṽ n ), where ṽ = mn {v 1 }. (26) Remar 5.2. In 2008, Chen and Tsao [19] extended TOPSIS method based on nterval-valued fuzzy sets. In 2010, Chen and Lee [18] presented a fuzzy TOPSIS technque based on nterval type-2 fuzzy sets. Remar 5.3. In 2010, L developed n paper [30] a methodology that s based on TOPSIS to solve MCDM problems wth both ratngs of alternatves w.r.t. crtera and weghts of crtera are expressed n nterval-valued ntutonstc fuzzy sets. In ths case the decson matrx s X = [ x ], where x = ([a, b ], [c, d ]) s nterpreted as follows: - the nterval [a, b ] represents the performance ratng of the alternatve A w.r.t. the crteron C, namely the degree that the alternatve A satsfy the crteron C may tae any value between a and b ; - the nterval [c, d ] represents the degree that the alternatve A does not satsfes the crteron C, whch means that the non-membershp degree of alternatve A w.r.t. crteron C may tae value between c and d. The weght of crteron C s denoted w = ([a, b ], [c, d ]). The nterval [a, b ] means that the membershp degree of crteron C may tae any value between a and b. The nterval [c, d ] shows the non-membershp degree of crteron C. For each alternatve A the closeness coeffcent CC s defned as follows: {(w μ ) 2 + [ρ (1 ν )] 2 } CC ((μ ), (ν ), (w ), (ρ )) = =1, (27) {(w μ ) 2 + [ρ (1 ν )] 2 } + {[w (1 μ )] 2 + (ρ ν ) 2 } =1 where (μ ) and (ν ) are m n matrces wth elements μ [a, b ] and ν [c, d ] and (w ) and (ρ ) are n-dmensonal vectors wth elements w [a, b ] and ρ [c, d ]. We note that the values of CC are closed and bounded subnterval of [0, 1]. =1

7 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) Remar 5.4. In 2011, another TOPSIS method to solve MCDM problems n nterval-valued ntutonstc fuzzy envronment s proposed n paper [35]. Step 1. Let X () = [ x ] be the nterval-valued ntutonstc fuzzy (IVIF) decson matrx of decson-maer D ( = 1, K), where x = ([a, b ], [c, d ]). Step 2. Aggregate all ndvdual IVIF decson matrx X () nto a collectve IVIF decson matrx X = [ x ] usng IIFHG operator. Step 3. Compute the weghted IVIF decson matrx. Step 4. Compute the nterval-valued ntutonstc postve deal soluton (IVIPIS) and nterval-valued ntutonstc negatve deal soluton (IVINIS). Step 5. Compute the dstance from each alternatve to the IVIPIS and to the IVINIS usng Hammng dstance or Eucldean dstance. Step 6. Compute the closeness coeffcent CC for each alternatve as follows: CC = S S + S. (28) Step 7. Ran the alternatves. The alternatve wth hghest closeness coeffcent represents the best alternatve. 6. Some applcatons Frst of all we must note that there are very good surveys concernng fuzzy MCDM applcatons (see for example [1, 2, 7, 33, 34]. We menton n ths paper some fuzzy TOPSIS applcatons: 6.1. Locaton problem A fuzzy TOPSIS approach for selectng plant locaton s frstly proposed by Chu n 2002, n paper [20], where the ratng of alternatves and the weghts of crtera are assessed n lngustc terms represented by trangular FN. In paper [6] fuzzy TOPSIS method s also used. A logstc company s nterested n mplementng a new urban dstrbuton center and there are three alternatves (A 1, A 2, A 3 ). Frstly a commttee of three decson-maers s formed. The crtera are: accessblty (C1), securty (C2), connectvty to multmodal transport (C3), costs (C4), envronmental mpact (C5), proxmty to customers (C6), proxmty to supplers (C7), resource avalablty (C8), conformance to sustanable freght regulatons (C9), possblty of expanson (C10), qualty of servce (C11). Smlar problems are consdered by varous authors [41, 50] Suppler selecton In paper [16] a fuzzy TOPSIS approach based on trapezodal FNs s used to solve the suppler-selecton problem. Fve beneft crtera are consdered: proftablty of suppler, relatonshp closeness, technologcal capablty, conformance qualty, conflct resoluton Sustanable and renewable energy In paper [13], a fuzzy TOPSIS methodology s used to compare dfferent heat transfer fluds. An mportant step, n problem formulaton, s choosng the crtera. Ten crtera are selected both techncal-economc and envronmental. Three of them are qualtatve and expressed n lngustc terms: state of nowledge of nnovatve technology, envronmental rs and safety freezng pont. A fuzzy TOPSIS method s appled n paper [43] for ranng renewable energy supply systems n Turey. There are fve crtera wth postve mpact: value of CO 2 emsson (envronmental), ob creaton (socal), effcency, nstalled capacty, amount of energy produced (techncal) and four crtera wth negatve mpact: nvestment cost, operaton and mantenance cost, paybac perod (economc), land use (envronmental). Concluson Technque for Order of Preference by Smlarty to Ideal Soluton (TOPSIS) for solvng problems n decson mang ssues s a very popular method. In ths paper we presented a general overvew about the development of fuzzy TOPSIS methods. Fnally we have mentoned several wors that presents some applcatons of fuzzy TOPSIS, such as: locaton problem, suppler selecton and, sustanable and renewable energy. We beleve that ths survey wll be a support for future research n ths feld.

8 830 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) References [1] Abdullah L. Fuzzy mult crtera decson mang and ts applcatons: A bref revew of category. Proceda - Socal and Behavoral Scences 2013; 97: [2] Aruldoss M, Lashm TM, Venatesan VP. A survey on mult crtera decson mang methods and ts applcatons. Amercan Journal of Informaton Systems 2013; 1(1): [3] Ashraf S, Rehman A, Kerre EE. Group decson mang wth ncomplete nterval-valued fuzzy preference relatons based on the mnmum operator. Internatonal Journal of Computers, Communcatons & Control 2015; 10(6): [4] Atanassov KT. Intutonstc fuzzy sets. Fuzzy Sets and Systems 1986; 20: [5] Atanassov KT, Gargov G. Interval valued ntutonstc fuzzy sets. Fuzzy Sets and Systems 1989; 31(3): [6] Awasth A, Chauhan SS, Goyal SK. A mult-crtera decson mang approach for locaton plannng for urban dstrbuton centers under uncertanty. Mathematcal and Computer Modellng 2011; 53: [7] Behzadan M, Otaghsara SK, Yazdan M, Ignatus I. A state-of the-art survey of TOPSIS applcatons. Expert Systems wth Applcatons 2012; 39: [8] Bellman RE, Zadeh LA. Decson-mang n a fuzzy envronment. Management Scence 1970; 17(4): [9] Bucley JJ. Fuzzy herarchcal analyss. Fuzzy Sets and Systems 1985; 17(3): [10] Bucley JJ, Feurng T, Hayash Y. Fuzzy herarchcal analyss revsted. European Journal of Operatonal Research 2001; 129(1): [11] Bustnce H. Handlng multcrtera fuzzy decson-mang problems based on ntutonstc fuzzy sets. HIFS 1995; 1: [12] Carlsson C, Fullér R. Fuzzy multple crtera decson mang: Recent developments. Fuzzy Sets and Systems 1996; 78(2): [13] Cavallaro F. Fuzzy TOPSIS approach for assessng thermal-energy storage n concentrated solar power (CSP) systems. Appled Energy 2010; 87: [14] Chang TH, Wang TC. Usng the fuzzy mult-crtera decson mang approach for measurng the possblty of successful nowledge management. Informaton Scences 2009; 179: [15] Chen CT. Extenson of the TOPSIS for group decson-mang under fuzzy envronment. Fuzzy Sets and Systems 2000; 114: 1 9. [16] Chen CT, Ln CT, Huang SF. A fuzzy approach for suppler evaluaton and selecton n supply chan management. Internatonal Journal of Producton Economcs 2006; 102: [17] Chen SJ, Hwang CL. Fuzzy Multple Attrbute Decson-Mang, Methods and Applcatons. Lecture Notes n Economcs and Mathematcal Systems, vol. 375, Sprnger-Verlag Berln Hedelberg; [18] Chen SM, Lee LW. Fuzzy multple attrbutes group decson-mang based on the nterval type-2 TOPSIS method. Expert Systems wth Applcatons 2010; 37(4): [19] Chen TY, Tsao CY. The nterval-valued fuzzy TOPSIS method and expermental analyss. Fuzzy Sets and Systems 2008; 159(11): [20] Chu TC. Selectng plant locaton va a fuzzy TOPSIS approach. The Internatonal Journal of Advanced Manufacturng Technology 2002; 20(11): [21] Dztac I. The fuzzfcaton of classcal structures: a general vew. Internatonal Journal of Computers, Communcatons & Control 2015; 10(6): [22] Elomda BM, Hefny HA, Hassan HA. An extenson of fuzzy decson maps for mult-crtera decson-mang. Egyptan Informatcs Journal 2013; 14: [23] Fguera J, Greco S, Ehrgott M. Multple Crtera Decson Analyss: State of the Art Surveys. New Yor: Sprnger; [24] Hanss M. Appled fuzzy arthmetc: an ntroducton wth engneerng applcatons. Berln Hedelberg: Sprnger-Verlag; [25] Herrera F, Alonso S, Chclana F, Herrera-Vedma E. Computng wth word n decson mang: Fundaton, trens and prospects. Fuzzy Optmzaton and Decson Mang 2009; 8(4): [26] Husan S, Ahmad Y, Alam MA. A study on the role of ntutonstc fuzzy set n decson mang problems. Internatonal Journal of Computer Applcatons 2012; 48(9): [27] Hwang CL, Yoon KP. Multple Attrbute Decson Mang: Methods and Applcatons, A State-of-the-Art Survey. Berln: Sprnger- Verlang; [28] Keshavarz Ghorabaee M, Zavadsas EK, Amr M, Turss Z. Extended EDAS method for fuzzy mult-crtera decson mang: an applcaton to suppler selecton. Internatonal Journal of Computers, Communcatons & Control 2016; 11(3): [29] L DF. Multattrbute decson mang models and methods usng ntutonstc fuzzy sets. Journal of Computer and System Scences 2005; 70: [30] L DF. TOPSIS-based nonlnear-programmng methodology for multattrbute decson mang wth nterval-valued ntutonstc fuzzy sets. IEEE Transactons on Fuzzy Systems 2010; 18(2): [31] Ln L, Yuan XH, Xa ZQ. Multcrtera fuzzy decson-mang methods based on ntutonstc fuzzy sets. Journal of Computer and System Scences 2007; 73: [32] Lu HW, Wang GJ. Mult-crtera decson-mang based on ntutonstc fuzzy sets. European Journal of Operatonal Research 2007; 179(1): [33] Mardan A, Jusoh A, Zavadsas EK. Fuzzy multple crtera decson-mang technques and applcatons - Two decades revew from 1994 to Expert Systems wth Applcatons 2015; 42: [34] Mardan A, Jusoh A, Zavadsas EK, Cavallaro F, Khalfah Z. Sustanable and Renewable Energy: An Overvew of the Applcaton of Multple Crtera Decson Mang Technques and Approaches. Sustanablty 2015; 7(10): [35] Par JH, Par IY, Kwun YC, Tan X. Extenson of the TOPSIS method for decson mang problems under nterval-valued ntutonstc fuzzy envronment. Appled Mathematcal Modellng 2011; 35: [36] Peng JJ, Wang JQ, Zhang HY, Chen XH. An outranng approach for mult-crtera decson-mang problems wth smplfed neutrosophc sets. Appled Soft Computng 2014; 25: [37] Qn J, Lu X, Pedrycz W. Fran aggregaton operators and ther applcaton to hestant fuzzy multple attrbute decson mang. Appled Soft Computng 2016; 41: [38] Rbero RA. Fuzzy multple attrbute decson mang: A revew and new preference elctaton technques. Fuzzy Sets and Systems 1996; 78(2): [39] Rodríguez RM, Martínez L, Herrera F. Hestant fuzzy lngustc term sets for decson mang. IEEE Transacton on Fuzzy System 2012; 20(1): [40] Saaty TL. The Analytc Herarchy Process: Plannng, Prorty Settng, Resource Allocaton. Mcgraw-Hll; [41] Safar H, Faghh A, Fath MR. Fuzzy mult-crtera decson mang method for faclty locaton selecton. Afrcan Journal of Busness Management 2012; 6(1):

9 Sorn N ad aban et al. / Proceda Computer Scence 91 ( 2016 ) [42] Smarandache F. A unfyng feld n logcs. Neutrosophy: Neutrosophc probablty, set and logc. Amercan Research Press Rehoboth; [43] Şengul U, Eren M, Shraz SE, Gezder V, Şengul AB. Fuzzy TOPSIS method for ranng renewable energy supply systems n Turey. Renewable Energy 2015; 75: [44] Torra V. Hestant fuzzy sets. Internatonal Journal of Intellgent Systems 2010; 25(6): [45] Torra V, Naruawa Y. On hestant fuzzy sets and decson. In Proceedng of the 18th IEEE Internatonal Conference on Fuzzy Systems, Jeu Island, Korea, 2009; [46] Trantaphyllou E. Mult-Crtera Decson Mang Methods: A Comparatve Study. London: Kluwer Academc; [47] Tzeng GH, Huang JJ. Multple Attrbute Decson Mang: Methods and Applcatons. CRC Press Taylor & Francs Group; [48] Wang J-g, Wu J-h, Wang J, Zhang H-y. Mult-crtera decson mang methods based on the Hausdorff dstance of hestant fuzzy lngustc numbers. Soft Computng 2016; 20(4): [49] Wang H, Smarandache F, Zhang YQ, Sunderraman R. Interval neutrosophc sets and logc: Theory and applcatons n computng. Arzona: Hexs; [50] Wang Y, Ma X-l, Wang Y-h, Mao H-, Zhang Y. Locaton optmzaton of multple dstrbuton centers under fuzzy envronment. Journal of Zheang Unversty SCIENCE A 2012; 13(10): [51] Wu HY, Tzeng GH, Chen YH. A fuzzy MCDM approach for evaluatng banng performance based on Balanced Scorecard. Expert Systems wth Applcatons 2009; 36: [52] Xa M, Xu Z. Hestant fuzzy nformaton aggregaton n decson mang. Internatonal Journal of Approxmate Reasonng 2011; 52: [53] Xa M, Xu Z. A novel method for fuzzy mult-crtera decson mang. Internatonal Journal of Informaton Technology & Decson Mang 2014; 13(2): [54] Xu ZS. Methods for aggregatng nterval-valued ntutonstc fuzzy nformaton and ther applcaton to decson mang. Control and Decson 2007; 22(2): [55] Yang T, Hung C. Multple-attrbute decson mang methods for plant layout desng problem. Robotcs and Computer-Integrated Manufacturng 2007; 23: [56] Ye J. Smlarty measures between nterval neutrosophc sets and ther applcatons n multcrtera decson-mang. Journal of Intellgent & Fuzzy Systems 2014; 26: [57] Ye J. Multcrtera decson-mang method usng the correlaton coeffcent under sngle-valued neutrosophc envronment. Internatonal Journal of General System 2013; 42(4): [58] Ye J. A multcrtera decson-mang method usng aggregaton operators for smplfed neutrosophc sets. Journal of Intellgent & Fuzzy Systems 2014; 26: [59] Ye J. Multcrtera decson-mang method usng expected values n trapezodal hestant fuzzy settng. Journal of Convergence Informaton Technology 2013; 8(11): [60] Yoon KP, Hwang CL. Multple Attrbute Decson Mang: An Introducton. Thousand Oas (CA): Sage Unversty Paper; [61] Zadeh LA. The concept of a lngustv varable and ts applcaton to approxmate reasonng - I. Informaton Scences 1970; 8: [62] Zavadsas EK, Antuchevcene J, Haagha SHR, Hashem SS. Extenson of weghted aggregated sum product assessment wth ntervalvalued ntutonstc fuzzy numbers (WASPAS-IVIF). Appled Soft Computng 2014; 24: [63] Zavadsas EK, Baušys R, Lazausas M. Sustanable Assessment of Alternatve Stes for the Constructon of a Waste Incneraton Plant by Applyng WASPAS Method wth Sngle-Valued Neutrosophc Set. Sustanablty 2015; 7: [64] Zavadsas EK, Podvezo V. Integrated Determnaton of Obectve Crtera Weghts n MCDM. Internatonal Journal of Informaton Technology & Decson Mang 2016; 15(2): [65] Zavadsas EK, Turss Z. Multple crtera decson mang (MCDM) methods n economcs: an overvew. Technologcal and Economc Development of Economy 2011; 17(2): [66] Zavadsas EK, Turss Z, Bagočus V. Mult-crtera selecton of a deep-water port n the Eastern Baltc Sea. Appled Soft Computng 2015; 15: [67] Zeng S, Chen J, L X. A hybrd method for pythagorean fuzzy multple-crtera decson mang. Internatonal Journal of Informaton Technology & Decson Mang 2016; 15(2):