185 Informática 35 (2011 ) 185-196 P. Liu An Extended TOPSIS Method for Multiple Attribute Group Decision Making Based on Generalized Interval-valued Trapezoidal Fuzzy Numbers Peide Liu Shandong Economic University, Information Management School Jinan Shandong 250014, China E-mail:Peide.liu@gmail.com Keywords: interval-valued fuzzy number, relative closeness coefficient, multiple attribute group decision making Received: May 29, 2009 An Extended TOPSIS Method deals with multiple attribute group decision making problems in which the attribute values and weights take the form of the generalized interval-valued trapezoidal fuzzy numbers (GIVTFN). First some properties are defined, such as the concept and the relational calculation rules of GIVTFN, the distance and its characteristics of GIVTFN, and the method which can transform the linguistic terms into GIVTFN. Second, the normalization method of GIVTFN is illustrated, and an extended TOPSIS method based on the GIVTFN is presented in detail. The order of the alternatives is ranked based on the relative closeness coefficient of TOPSIS. Finally, an illustrate example is given to show the effectiveness of this method and this decision making steps. Povzetek: Z izboljšano metodo TOPSIS so dosegli boljše rezultate pri odločanju z mnogoterimi atributi. 1 Introduction Multiple attribute decision making (MADM) is an important part of modern decision science. It has been extensively applied to various areas, such as society, economics, management, military and engineering technology. For example, the investment decisionmaking, project evaluation, the economic evaluation, the personnel evaluation etc. Since the object things are fuzzy, uncertainty and human thinking is ambiguous, the majority of the multi-attribute decision-making is uncertain and ambiguous, which is called the fuzzy multiple attribute decision-making (FMADM). Since Bellmanhe and Zadeh [1] initially proposed the basic model of fuzzy decision making based on the theory of fuzzy mathematics, FMADM has been receiving more and more attentions. Many achievements have been made on FMADM problems [2-5,7-21], TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is proposed by Hwang and Yoon [6], and it is a popular approach to MCDM problems. The basic principle is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. In the TOPSIS, the performance ratings and the weights of the criteria are given as crisp values. In many cases, crisp data are inadequate to model real life situations. Jahanshahloo et al [7] extends the TOPSIS method to the fuzzy decision making situations by considering interval numbers and defining crisp Euclidean distance between two interval numbers. Wang and Elhag[8] proposes a fuzzy TOPSIS method based on alpha level sets and presents a nonlinear programming (NLP) solution procedure by considering triangular fuzzy numbers. Liu and Zeng [9] proposes a new TOPSIS method to deal with the fuzzy multiple attribute group decision making problem based on the expected value operator of the trapezoidal fuzzy number when the fuzzy decision matrixes and the weights of the decision attributes and decision makers are all given by the trapezoidal fuzzy number. Tsaur et al. [10] convert the fuzzy MCDM problem into a crisp one via centroid defuzzification and then solve the non-fuzzy MCDM problem by the TOPSIS method. Chu and Lin [11] changed the fuzzy MCDM problem into a crisp one. Differing from the others, they first derive the membership functions of all the weighted ratings in a weighted normalized decision matrix and then convert them to crisp values by defuzzifying and then use TOPSIS method to solve this problem. The concept of the interval-valued fuzzy set is initially proposed by Gorzlczany[12]and Turksen[13], Some researchers focused on this research topic of interval-valued fuzzy numbers [12-18] in resent years, because the interval-valued fuzzy numbers are more general and better to express fuzzy information. Wang and Li [14-15] defined the expansion operation of the interval-valued fuzzy numbers, and proposed the concept and properties of the similarity coefficient based on the interval-valued fuzzy numbers. Hong and Lee [16] proposed the distance of the interval-valued fuzzy numbers. Ashtiani,et al[17] proposed definition of the interval-valued triangular fuzzy numbers and presented the extended TOPSIS group decision making method for 186 Informática 35 (2011 ) 185-196 P. Liu the interval-valued triangular fuzzy numbers. Wei and Chen[18]proposed similarity measures between the generalized interval-valued trapezoidal fuzzy numbers (GIVTFN) for risk analysis. This paper proposed an extended TOPSIS Method to solve the multiple attribute group decision making problems of which the attribute weights and values are given with the form of GIVTFN. In this paper we develop an extended TOPSIS method for multiple attribute group decision making based on the generalized interval-valued trapezoidal fuzzy numbers by defining the distance of GIVTFN. The remaining of this study is organized as follows. In the next section, we will briefly introduce the basic concept and the operation rules of the GIVTFN, and define the distance of GIVTFN. Section 3 describes the extended TOPSIS method to solve the multiple attribute group decision making problems by using GIVTFN. Section 4 gives a numerical example to explain validity of the decision-making steps and the method. The study is concluded in Section 5. 2 The basic concept of the interval-valued trapezoidal fuzzy numbers 2.1 The generalized trapezoidal fuzzy numbers (1) The concept of the generalized trapezoidal fuzzy numbers Definition 1 [19]: The generalized trapezoidal fuzzy numbers can be defined as a vector A = (a], a2, cl , a4; H' -) (as shown in Figl), and the membership function li(x) : R —> [0,1] is defined as follows: ct(x) = x-a & ^ CX-y W-, X u a3 -a4 xw~, xe(ava2) x g (a2,ai) ■XWA, xG(a3,a4) XG (-00,^)^(04,00) If then A is reduced to a real number. Figure 1: The generalized trapezoidal fuzzy number A . (2) The operation of the generalized trapezoidal fuzzy number Suppose that a = (al,a2,a3,a4,wa) , b = (6,,62,63,64;w6-) are the generalized trapezoidal fuzzy numbers, then the operational rules of the generalized trapezoidal fuzzy number are shown as follows: [20] a e b = (q, a,, a3, a4; w.) 0 _, h,, h, b4; wi } (i) = +bl,a2 +b2,a3 +b3,a4 +b4,mm(ws,w£)) (2) (ii) = (c/, -b4,a2-b3,a3-b2,a4-b{,min(ii;.,w~)) (3) ct®b = (cil,a2, a3,a4,ws )«(/>./>../>, b4,w~) (iii) (4) = (a, b, c, c/\ mi n( ii^, ha )) (1) where ax < a2 < a3 < a4and>f - e [0,1]. The elements of the generalized trapezoidal fuzzy numbers x G R are real numbers, and its membership function Cl(x) is the regularly and continuous convex function, showing the membership degree to the fuzzy sets. If -1 < ax < an < a3 < a4 < 1, then A is called the normalized trapezoidal fuzzy number. Especially, if w- = 1 , then A is called the trapezoidal fuzzy number (a, ): if <:/, < = a, < a4 . then A is reduced to a triangular fuzzy number. where a = min(aj xbx,ax xb4,a4xbl,ct4 xb4), b = min{an xbn,a, xb3,a3 xb^,a3xbs) c = max(o, xbn,an xbs,as xbn,a3 xZ>3), d = max(aj * b.a. xb4,a4xbl,a4 xb4) If a1,a2,a3,a4,b1, b2, b3, b4 are the positive numbers, then ci®b = (a, xbya2 xb2,a3 xh,a4 ■ shown in Fig. 2, where 0wff)), JJ lIJ JJ lIJ JJ lIJ JJ lIJ (9) Figure 2: the interval-valued trapezoidal fuzzy numbers. (2) The operation of the interval-valued trapezoidal fuzzy numbers [18] (q h, .a. -h .a, -b2 .a, h ;trin(HV .n, )) (8) (iii) The product of two interval-valued trapezoidal fuzzy numbers A, B Mb^ 44,4^,4444^ = 4 xtf4 xb2,d A xiii^.^ft 4 x$ 4 x$4 xb"4 xff^mriw^w-)) ~ ~ ' ' A D (iv) The quotient of two interval-valued trapezoidal fuzzy numbers A, B : 1/b=[4444-W:¥),4444-W:¥) 41 % 4 Iff 4 ¡$4//> :nin(uru;v )) (v) The product between an interval-valued trapezoidal fuzzy number A and a constant number X (A > 0): (10) are the two interval-valued trapezoidal fuzzy numbers. 188 Informática 35 (2011 ) 185-196 P. Liu ÀA=Ax 2J/ Otof, (M, 2J/ (11) (2) The distance of two interval-valued trapezoidal fuzzy numbers is: d(A,B)=- 2.3 The distance of the interval-valued trapezoidal fuzzy numbers vJ'-Cv .v;)'-(v; V..M.V, xi) 4 (12) Suppose that A= B = » are any two generalized trapezoidal fuzzy numbers, then the distance of two interval-valued trapezoidal fuzzy numbcrs( A andB) is calculated as follows: (l)Utilize the formula (6) to calculate the coordinate of the COG point where d{ A, B) satisfies the following properties: (i) if A and B are the normalized interval-valued trapezoidal fuzzy numbers, then 0 < d{ A, B) < 1 (ii) A = Bc>d(A,B) = 0 (iii) d(A,B) = d(B,A) (iv) d(2, C) + d(&, J) > d(l, J) Obviously, the properties (i) and (iii) are satisfied. For the properties (ii), if A = B , then d( A,B) = 0 . If d( A,B) = 0 , then the COG of A is equal to B"s. so we approximately believe that A = B . which belongs to the generalized trapezoidal fuzzy numbers AL,AU ,BL,BU respectively. For properties (iv): In order to simplify the expression in distance formula, we suppose that d(A, É) + d(ë, C) > d(A, C) o (d{Â, Ê) + d(ë, C)J > d2 (A, C) o (d{Â, Ê) + d(ë, C)J - d2 (A, C) > 0 ofa -a,)2 +(/3,-J32)2 +(y -y2)2 +(tj ->]2f -oc,)2 +(J32-p3f +(r2-y3)2 +(>]2 ~rkf +2j[(q -a,)2 +(& -p2f +(7l ~r2f +('1 -a,)2 +(fi2 - A)2 +(r2-r3f +(n2 -[(q -a,)2 +(fil -A)2 +(7l-r3)2 -v3f]> o Suppose that £¿4 = (a, - a2)2 + (fl-p2)2 + {7l - y2)2 + fa - ^2)2 + (a2-a3)2 + (p2-p3)2 + (y2 - y3)2 + (rj2 -t]3)2 -[(a, -a3)2 + (A-p3)2 + (y -y3)2 + fa -%)2] dB=2^[(cxi -a,)2 -P2)2 My, ~r2)2 +(n + (£ -ft)2 +(r2 -y3f v £¿4 = (a, - a2)2 + (£ -P2f + {y -y2f + fa ~V2f + (a2 "1a3f + (P2 ~Ps? + fo " V3)2 + (V2 ~VÙ* - [(a, -a3)2+ (A -p3)2+ {y -y3)2+ fa - r,3)2 ] = 2(«! -a2)(a3-a2) + 2(fl -p2)(p3-p2) + 2{yx ~y2)(y3~y2) + 2fa - î]2)(î]3 -1]2)_ V dB=2^(0, -a2)2+(A -P2f+(n-r2f+h -?72)2]x[(a2 -a3f+(p2 -P3f+(r2 ~r3f Hn, ->hy = 2[(«i -a2)(a2 ~a3) + (fil ~P2)(fi2 -&) + (ri ~r2)(r2-h) + (rfi ->h)(>h _773)] AN EXTENDED TOPSIS METHOD FOR MULTIPLE. Informática 35 (2011 ) 185-196 189 where dC = 2[(a1 - a2)(a2 - a3)(y1 - y2)(y2 -y3) + (A ~ ß2)(ß2 ~ ß3){yl ~ y2)(y2 ~ y3) + (ril-ri2)(ri2-rii)(al-a2)(a2-ai)+ - rj2)(i]2 - rj3){ßx - ß2)(ß2 - ß3)] :.dA + dB> Oo d(A, B) + d(B, C) - d1 (A, C)> Oo d(A, B) + d(B, C) > d(A, C) 2.4 Utilize the interval-valued trapezoidal fuzzy numbers to represent the linguistic terms In the real decision making process, it is difficult to adopt the form of generalized interval-valued trapezoidal fuzzy numbers to give the attribute values and weights directly by the decision makers. So we usually adopt the form of linguistic terms. Wei and Chen [18] utilizes the interval-valued trapezoidal fuzzy numbers to represent the 9-member linguistic terms, (shown in Table 1) Table 1: A 9-member interval linguistic term set. linguistic terms (the attribute values) linguistic terms generalized interval-valued trapezoidal (weights)_fuzzy numbers_ Absolutely-poor(AP) Very-poor(VP) poor (P) Medium-poor(MP) Medium (F) Medium-good(MG) good(G) Very-good(VG) Absolutely-good(AG) Absolutely-low(AL) Very-low (VL) low (L) Medium-low(ML) Medium (M) Medium-high(MH) high(H) very-high (VH) Absolutely-high (AH) [(0.00,0.00 [(0.00,0.00 [(0.04,0.10, [(0.17,0.22, [(0.32,0.41 [(0.58,0.63. [(0.72,0.78 [(0.93,0.98, [(1.00,1.00, ,0.00,0.00;0.8). 0.02,0.07;0.8). ,0.18,0.23;0.8), ,0.36,0.42;0.8), 0.58,0.65;0.8), 0.80,0.86;0.8), 0.92,0.97;0.8), 1.00,1.00;0.8), 1.00,1.00;0.8), (0.00,0.00,0.00 (0.00,0.00,0.02 (0.04,0.10,0.18, (0.17,0.22,0.36 (0.32,0.41,0.58 (0.58,0.63,0.80. (0.72,0.78,0.92 (0.93,0.98,1.00, (1.00,1.00,1.00, ,0.00; 1.0)] 0.07;1.0)] ,0.23;1.0>] 0.42;1.0)] ,0.65;1.0)] 0.86;1.0)] 0.97; 1.0)] 1.00;1.0)] 1.00;1.0)] 3 Group decision making method 3.1 The description of the decision making problems Let E = jgj, e2, • • •, eq j be the set of decision makers in the group decision making, and A = {Al, A1, • • •, Am j be the set of alternatives, and C = \Cl, C2, • • •, Cn ] be the set of attributes. Suppose that is the attribute value given by the decision maker ek, where ai]k is an interval-valued trapezoidal fuzzy number for the alternative Ai with respect to the attribute C ■ and ~ ri L L L L L\ / U U U U t/\~| % = LHv %3>%-4> nig), Kl,^, % ) J is the attribute weight given by the decision maker ek, where CO kj is also an interval-valued trapezoidal fuzzy number. Let X - (\, A2, • • •, A ) be the weight vector of decision makers, where A, is a real number, and ^ Ak = 1. Then we use the attribute weights, the k=1 decision makers' weights, and the attribute values to rank the order of the alternatives. 3.2 Normalize the decision-making information we need normalize the decision-making information, in order to eliminate the impact of different physical dimension to the decision-making result. Consider that there are generally benefit attributes ( /, ) and cost attributes (/2). The normalizing method is shown as follows: For benefit attributes, where mjk = max{a^k4) . y—'—'—'——,—,—,—,wpk) cL cE dL £ K mjk mjk mjk mjk ™jk ™jk y (13) For cost attributes, where njk = m\x\(a'jik] ) . *ijk\, 190 Informatica 35 (2011 ) 185-196 P. Liu "V '"ijk V {' Jk jk jk jk . L \ / jk jk jk jk . U n k Xyi4>^V )]} 1 ' k=1 1111 Ê )' Ê ), Z (4^3 ) >£ ); m,in<># ) i=l i=l i=l i=l , )> S ) ); mïn(wv-k ) k=1 ~ T/ii L L L\ s U U U U (7\~1 /7 /7 • 1 k=1 1111 N ¿=1 ¿=1 ¿=1 ¿=1 , Z ( )' )> i (H ) i (^<4 ); min(^ ) y A . k=\ k=1 3.4 Construct the weighted matrix Let V = be the weighted matrix, then: n % = [(v^v^.v^.v^iBTiXCv^v^.v^.v^;^)] = ~(x> n, . x> is. x> y4; min(w,J, 77L ))," <9 . L xL CO L ■I]I" ]1> xij2w j: (x> £, x> % x> ° x> m in« , 77 ? )) y ' 3.5 The extended TOPSIS decision making method (1) Determine the positive ideal solution and the ideal solution are i = negative ideal solution of the evaluation objects (15) (16) (17) Suppose that the positive ideal solution and the negative y~ = , then : v. = ,L+ ,,(7+ Î7 + . Î7 , Lv ' ;'2 ' j3 ' ;4 ' j ),(yn 5 y 2 , Vy3 ,V;4 ,cj . )] (m ax (vjj ), m ax (v£2 ), m ax (v£3 ), m ax ) ; m ax (m ^ )), y 2 y'3. y 4 > (m ax ( v^ ), m ax ( v£2 ), m ax ( ), m ax (v£4 ) ; m ax (et £ )) y'2 y'3 - y 4 > (18) AN EXTENDED TOPSIS METHOD FOR MULTIPLE. Informática 35 (2011 ) 185-196 191 V,. = L- L- ),(V fl u - u - u '",2 ,",1 >] (min(v^),min(v^),min(v^),min(v^4);min(s7j)), t 1 1 1 1 (min(v^),min(v^2),min(v^),min(v^4);min(s7^)) (19) (2) Calculate the weighted matrix and the COG of each attributes with respect to the positive ideal solution and the negative ideal solution Based on the formula (6), we can calculate the COG [(j', x)v ] of each element in the weighted matrix and the barycentric coordinates x)v+ ^J and ^(j-', of each element with respect to the positive ideal solution and the negative ideal solution. (3) Calculate the weighted distance between each project Ai and the ideal solution V and negative ideal solution V : (20) (21) where i = 1,2 • • • m, j = 1,2, • • • n . (4) Calculate the relative closeness coefficient C. dr C =■ d. +d. (22) , fill) vector of the decision makers is  = —,—,— . Each v3 3 3J decision maker utilizes the linguistic terms to assess the importance of each attribute, and the evaluation information of four volunteers is shown in Tables2, 3, 4 and 5, respectively [17], Table 2: The attribute weights given by three DMs. (5) Rank the alternatives We rank each alternative, based on the relative closeness coefficient. The bigger the relative closeness coefficient is, the better the alternative is, vice versa. 4 Illustrative example Suppose that a Telecommunication Company intends to choose a manager for R&D department from four volunteers named Al, A2, A3 and A4. The decision making committee assesses the four concerned volunteers based on five attributes: (1) the proficiency in identifying research areas (CI), (2) the proficiency in administration (C2), (3) the personality (C3), (4) the past experience (C4) and (5) the self-confidence (C5). The number of the committee members is three, labeled as DM1, DM2 and DM3, respectively, and the weight ci c2 c. 3 C4 c5 DM1 DM2 DM3 VH VH VH H H VH M H MH H MH MH MH VH M Table 3: The evaluation information given by DM1. ci C2 C3 C4 C5 VG VG VG VG VG a2 G VG VG VG MG a3 VG MG G G G a4 G F F G MG Table 4: The evaluation information given by DM2. ci C2 C3 C4 C5 G MG G G VG a2 G VG VG VG MG a3 G G MG VG G a4 VG F MG F G Table 5: The evaluation information given by DM3. ci C2 C3 C4 C5 MG F G VG VG a2 MG MG G MG G a3 VG VG VG VG MG a4 MG VG MG VG F The decision making steps are shown as follows: (1) Convert the linguistic terms into the interval-valued trapezoidal fuzzy numbers, and then get: 192 Informática 35 (2011 ) 185-196 P. Liu CO kj 3x5 M [(0.9 3 [(0.9 3 [(0.9 3 [(0 .72 , [(0 .72 , [(0.58, [(0 .72 , [(0.58, [(0.58, [(0.9 3, [(0 .72 , [(0.9 3, [(0.3 2, [(0.58, [(0.3 2, ,0.98,1 ,0.98,1 ,0.98,1 ,0.7 8,0 ,0.7 8,0 ,0.6 3,0 ,0.7 8,0 ,0.6 3,0 ,0.6 3,0 ,0.9 8,1 ,0.7 8,0 ,0.9 8,1 0.41,0. 0.63,0. 0.41,0. .00,1 .00,1 .00,1 9 2,0 9 2,0 8 0,0 9 2,0 8 0,0 8 0,0 00,1 9 2,0 00,1 5 8,0. 8 0,0. 5 8,0. .0 0 ;0 .0 0 ;0 .0 0 ;0 9 7 ; 0 . 9 7 ; 0 . 8 6 ; 0 . 9 7 ; 0 . 8 6 ; 0 . 8 6 ; 0 . 0 0 ;0 . 9 7 ; 0 . 0 0 ;0 . 6 5 ; 0 . 8 6 ; 0 . 6 5:0. .8),(0.93 .8),(0.93 .8),(0.93 .8 ),(0 .7 2 ,8),(0 ,8),(0 ,8),(0 ,8),(0 ,8),(0 .8 ),(0 .9 3 .8 ),(0 .7 2 .8 ),(0 .9 3 8 ),(0 .3 2 8 ),(0 .5 8 8 ),(0 .3 2 .7 2 .5 8 .7 2 .5 8 .5 8 ,0.98,1 ,0.98,1 ,0.98,1 0.7 8,0 0.7 8,0 0.6 3,0 0.7 8,0 0.6 3,0 0.6 3,0 0.98,1 0.7 8,0 0.98,1 0.4 1,0 0.6 3,0 0.4 1,0 .00,1 .00 .00,1 .00 .00,1 .00 .9 2,0.9 7 .9 2,0.9 7 .8 0,0.8 6 .9 2,0.9 7 .8 0,0.8 6 .8 0,0.8 6 .00,1 .00 .9 2,0.9 7 .00,1 .00 5 8,0.6 5; 8 0,0.8 6; .5 8,0.6 5; ,1.0)] ,1.0)] ,1.0)] 1 .0)] 1 .0)] 1 .0)] 1 .0)] 1 .0)] 1 .0)] 1 .0)] 1 .0)] 1 .0)] 1 .0)]-1 .0)] 1 .0)] [(0.9 3,0 [(0.72,0 [(0.9 3,0 [(0.72,0 [(0.9 3 ,0 [(0.9 3 ,0 [(0.5 8,0 [(0.3 2,0 [(0.9 3 ,0 [(0.9 3 ,0 [(0.72,0 [(0.3 2,0 [(0.9 3 ,0 [(0.9 3 ,0 [(0.72,0 [(0.72,0 [(0.9 3,0. [(0.5 8,0. [(0.72,0. [(0.5 8,0. .98,1 .7 8,0 .98,1 .7 8,0 98,1 98,1 6 3,0 4 1 ,0 98,1 98,1 7 8,0 4 1 ,0 98,1 98,1 7 8,0 7 8,0 98,1 . 6 3,0. 7 8,0. 6 3,0. .00,1.00 .9 2,0.9 7 .00,1.00 .9 2,0.9 7 00,1.00 00,1.00 80,0.86 5 8,0.6 5 00,1.00 00,1.00 92,0.97 5 8,0.65 00,1.00 00,1.00 92,0.97 92,0.97 00,1.00 8 0,0.86 9 2,0.97 8 0,0.86 0.8),(0.9 3,0 0.8),(0.72,0 0.8),(0.9 3,0 0.8),(0.72,0 0.8),(0.93,0 0.8),(0.93,0 0 ,8),(0 .5 8,0 0.8),(0.32,0 0.8),(0.93,0 0.8),(0.93,0 0.8),(0.72,0 0.8),(0.32,0 0.8),(0.93,0 0.8),(0.93,0 0.8),(0.72,0 0.8),(0.72,0 0 .8 ),(0 .9 3,0. 0 .8 ),(0 .5 8,0 . 0.8),(0.72,0. 0 .8 ),(0 .5 8,0 . (0.7 2,0 (0.7 2,0 (0.7 2,0 (0.9 3,0 (0.5 8,0. (0.9 3,0. (0.7 2,0. (0.3 2,0. (0.7 2,0. (0.9 3,0. (0.5 8,0. (0.5 8,0. (0.7 2,0. (0.9 3,0. (0.9 3,0. (0.3 2,0. (0.9 3,0. (0.5 8,0. (0.7 2,0. (0.7 2,0. .7 8,0 .7 8,0 .7 8,0 .98,1 .6 3 ,0 .9 8,1 .7 8 ,0 .4 1 ,0 .7 8 ,0 .9 8,1 .6 3 ,0 .6 3 ,0 .7 8 ,0 .9 8,1 .9 8,1 .4 1 ,0 9 8,1 6 3,0 7 8,0 7 8,0 .9 2,0.9 7 .9 2,0.9 7 .9 2,0.9 7 .00,1.00 .8 0,0.8 6 .00,1.00 .9 2,0.9 7 .5 8,0.6 5 .9 2,0.9 7 .00,1.00 .8 0,0.8 6 .8 0,0.8 6 .9 2,0.9 7 .00,1.00 .00,1.00 .5 8,0.6 5 .0 0,1.0 0; .8 0,0.8 6; .9 2,0.9 7; .9 2,0.9 7; ■ 8),(0 ■ 8),(0 ■ 8),(0 ■ 8),(0 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 8),(0. 0 .8 ),(0 . 0 .8 ),(0 . .7 2,0 .7 2,0 .7 2,0 .9 3 ,0 5 8,0 9 3,0 7 2,0 3 2,0 7 2,0 9 3,0 5 8,0 5 8,0 7 2,0 9 3,0 9 3,0 3 2,0 9 3,0. 5 8,0. 7 2,0. 7 2,0. 9 8,1 7 8 ,0 9 8,1 78,0 .9 8,1. .9 8,1. .6 3,0. .4 1,0. .9 8,1. .9 8,1. .7 8,0. .4 1,0. .9 8,1. .9 8,1. .7 8,0. .78,0 . .9 8,1. .6 3,0. .7 8,0. .6 3,0. .7 8,0 .7 8,0 .7 8,0 .9 8,1 6 3,0 9 8,1 7 8,0 4 1,0 7 8,0 9 8,1 6 3 ,0 6 3,0 7 8,0 9 8,1 9 8,1. 4 1,0. 9 8,1. 6 3,0. 7 8,0. 7 8,0. .00,1 .00 .9 2,0.9 7 .00,1 .00 .9 2,0.9 7 00,1.00 00,1.00 80,0.86 5 8,0.6 5 00,1.00 00,1.00 92,0.97 5 8,0.6 5 00,1.00 00,1.00 92,0.97 92,0.97 0 0,1 .0 0, 8 0,0.8 6, 9 2,0.9 7, 8 0,0.8 6, .9 2,0.9 7 .9 2,0.9 7 .9 2,0.9 7 .00,1.00 .8 0,0.8 6 .0 0,1.0 0 .9 2,0.9 7 .5 8,0.6 5 .9 2,0.9 7 .0 0,1.0 0 .8 0,0.8 6 .8 0,0.8 6 .9 2,0.9 7 .0 0,1.0 0 .00,1.00 .5 8,0.6 5 0 0,1.0 0, 8 0,0.8 6. 9 2,0.97, 9 2,0.97, 1 -00)], 1 -00)], 1 -00)], 1 -00)], 00)] 00)] 00)] 00)] 00)] 00)] 00)] 00)] 00)] 00)] 1 .00)] 1 .00)] 1 .0)] 1.0)] 1 .0)] 1 .0)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ;i .00)] ,1 .0)] ,1 .0)] ,1 .0)] ,1 .0)] AN EXTENDED TOPSIS METHOD FOR MULTIPLE. Informática 35 (2011 ) 185-196 193 "[(0.58 [(0.58 [(0.9 3 [(0.58 [(0.3 2 [(0.58 [(0.9 3 [(0.9 3 [(0.7 2 [(0.7 2 [(0.9 3 [(0.58 [(0.9 3 [(0.58 [(0.9 3 [(0.9 3 [(0.9 3, [(0.7 2, [(0.58, [(0.3 2, ,0.63 ,0.63 ,0.98 ,0.63 0.4 1 0.6 3 0.9 8 0.9 8 0.7 8 0.7 8 0.9 8 0.6 3 0.9 8 0.6 3 0.9 8 0.9 8 0.98, 0.78, 0.63, 0.41, ,0.8 0 ,0.8 0 ,1.00 ,0.8 0 0.5 8 0.8 0 1.0 0 1.0 0 0.9 2 0.9 2 1.0 0 0.8 0 1.0 0 0.8 0 1.0 0 1.0 0 1.0 0 0.9 2 0.8 0 0.5 8 0.8 6 0.8 6 1 .0 0 0.8 6 0.6 5 8 6 0 0 0 0 97 97 0 0 8 6 0 0 8 6 0 0 0 0 1.0 0 0.9 7 0.8 6 0.6 5 8),(0 8),(0 8),(0 8),(0 8),(0 8),(0 8),(0 0 .8 ),(0 0 .8 ),(0 8),(0 8),(0 8),(0 8),(0 8),(0 8),(0 8),(0 8),(0. 0 .8 ),(0 . 0 .8 ),(0 . 0 .8 ),(0 . .5 8,0 .5 8,0 .9 3,0 .5 8,0 3 2,0 5 8,0 9 3,0 9 3,0 7 2,0 7 2,0 9 3,0 5 8,0 9 3,0 5 8,0 9 3,0 9 3,0 9 3,0. 7 2,0. 5 8,0. 3 2,0. .6 3,0.8 0 .6 3,0.8 0 .98,1 .00 .6 3,0.8 0 4 1,0.58 6 3,0.80 9 8,1.00 9 8,1.00 7 8,0.92 7 8,0.92 9 8,1.00 6 3,0.80 9 8,1.00 6 3,0.80 9 8,1.0 0, 9 8,1.0 0, 9 8,1.0 0, 7 8,0.9 2, 6 3,0.8 0, 4 1,0.58, ,0.8 6 ,0.8 6 ,1.00 ,0.8 6 ,0.6 5 ,0.8 6 ,1.00 ,1.00 ,0.9 7 ,0.9 7 ,1.00 ,0.8 6 ,1.00 ,0.8 6 ,1.00: ,1.00 ,1.00, 0.97, 0.86, 0.65, ;i .00)] ;i .00)] ;i .00)] ;i .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .00)] 1 .0)] 1 .0)] 1 .0)] 1 .0)] (2) Combine the individual preferences in order to obtain a collective preference value of each alternative: ~ [(0.743,0.797,0.907,0.943;0.800), (0.743,0.797,0.907,0.943;1.000)] [(0.673,0.730,0.880,0.933;0.800),(0.673,0.730,0.880,0.933;1.000)] [(0.860,0.913,0.973,0.990;0.800),(0.860,0.913,0.973,0.990;1.000)] [(0.743,0.797,0.907,0.943;0.800),(0.743,0.797,0.907,0.943;1.000)] [(0.610,0.673,0.793,0.837;0.800),(0.610,0.673,0.793,0.837;1.000)] [(0.813,0.863,0.933,0.953;0.800),(0.813,0.863,0.933,0.953;1.000)] [(0.743,0.797,0.907,0.943;0.800),(0.743,0.797,0.907,0.943;1.000)] [(0.523,0.600,0.720,0.767;0.800),(0.523,0.600,0.720,0.767;1.000)] [(0.790,0.847,0.947,0.980;0.800),(0.790,0.847,0.947,0.980;1.000)] [(0.860,0.913,0.973,0.990;0.800),(0.860,0.913,0.973,0.990;1.000)] [(0.743,0.797,0.907,0.943;0.800),(0.743,0.797,0.907,0.943;1.000)] [(0.493,0.557,0.727,0.790;0.800),(0.493,0.557,0.727,0.790;1.000)] [(0.860,0.913,0.973,0.990;0.800),(0.860,0.913,0.973,0.990;1.000)] [(0.813,0.863,0.933,0.953;0.800),(0.813,0.863,0.933,0.953;1.000)] [(0.860,0.913,0.973,0.990;0.800),(0.860,0.913,0.973,0.990;1.000)] [(0.657,0.723,0.833,0.873;0.800),(0.657,0.723,0.833,0.873;1.000)] [(0.930,0.980,1.000,1.000;0.800), (0.930,0.980,1.000,1.000;1.000)]_ [(0.627,0.680,0.840,0.897;0.800),(0.627,0.680,0.840,0.897;1.000)] [(0.673,0.730,0.880,0.933;0.800),(0.673,0.730,0.880,0.933;1.000)] [(0.540,0.607,0.767,0.827;0.800),(0.540,0.607,0.767,0.827;1.000)] [[(0.9 3 0,0.9 80,1.000,1.000 ;0.80 0), (0.9 3 0,0.9 8 0,1.000,1.000; 1.000)], [(0.62 7,0.6 80,0.840,0.8 97 ;0.8 00),(0.627,0.6 80,0.840,0.8 97; 1.000)], [(0.62 7,0.6 80,0.840,0.8 97 ;0.8 00),(0.627,0.6 80,0.840,0.8 97; 1.000)], [(0.9 3 0,0.9 80,1.000,1.000 ;0.8 00),(0.9 3 0,0.9 80,1.0 00,1.0 00; 1.000)], [(0.320,0.4 10,0.5 80,0.650 ;0.800),(0.320,0.4 10,0.5 80,0.6 50; 1.000)]] M. CO : (3)Calculate the weighted decision making matrix: 194 Informática 35 (2011 ) 185-196 P. Liu [U [(0.691,0.781,0.907,0.943 ;0.800),(0.691,0.781,0.907,0.943; 1.000)] [(0.626,0.7 15,0.880,0.93 3 ;0.800),(0.626,0.7 15,0.880,0.93 3; 1.000)] [(0.800,0.8 95,0.973,0.990 ;0.800),(0.800,0.8 95,0.97 3,0.990; 1.000)] [(0.691,0.781,0.907,0.943 ;0.800),(0.691,0.781,0.907,0.943; 1.000)] [(0.382,0.458,0.666,0.750;0.800),(0.3 82,0.458,0.666,0.750;1.000)] [(0.5 1 0,0.587,0.784,0.855 ;0.800),(0.5 10,0.587,0.784,0.855; 1.000)] [(0.466,0.5 42,0.7 62,0.846 ;0.8 00),(0.4 66,0.5 42,0.7 62,0.846 ;1.000)] [(0.3 28,0.408,0.605,0.687;0.800),(0.3 28,0.408,0.605,0.687;1.000)] [(0.495,0.576,0.7 95,0.879 ;0.800),(0.4 95,0.576,0.795,0.879; 1.000)] [(0.5 3 9,0.621,0.8 18,0.888 ;0.800), (0.5 3 9,0.621,0.8 18,0.888 ;1.000)] [(0.466,0.5 42,0.7 62,0.846 ;0.8 00),(0.4 66,0.5 42,0.7 62,0.846 ;1.000)] [(0.3 09,0.3 79,0.6 1 0,0.708 ;0.800),(0.3 09,0.3 79,0.610,0.708; 1.000)] [(0.800,0.895,0.973,0.990;0.800),(0.800,0.895,0.973,0.990;1.000)] [(0.756,0.846,0.933,0.953 ;0.800),(0.756,0.846,0.933,0.953 ;1.000)] [(0.800,0.895,0.973,0.990;0.800),(0.800,0.895,0.973,0.990;1.000)] [(0.61 1,0.709,0.833,0.873 ;0.800),(0.61 1,0.709,0.833,0.873 ;1.000)] [(0.29 8,0.402,0.5 80,0.650 ;0.800),(0.298,0.402,0.5 80,0.650; 1.000)]" [(0.20 1,0.279,0.487,0.583 ;0.800),(0.201,0.279,0.487,0.583 ;1.000)] [(0.21 5,0.299,0.5 1 0,0.607 ;0.800),(0.215,0.299,0.5 10,0.607; 1.000)] [(0.173,0.249,0.445,0.537;0.800),(0.1 73,0.249,0.445,0.537;1.000)] (4) Determine the positive ideal solution and the negative ideal solution : [[(0.800,0.895,0.973,0.990;0.800), (0.800,0.895,0.973,0.990;1.000)], [(0.510,0.587,0.784,0.855 ;0.800),(0.510,0.587,0.784,0.855 ;1.000)], [(0.539,0.621,0.818,0.888 ;0.800),(0.5 3 9,0.621,0.818,0.888; 1.000)], [(0.800,0.895,0.973,0.990;0.800),(0.800,0.895,0.973,0.990;1.000)], [(0.298,0.402,0.580,0.650;0.800),(0.298,0.402,0.580,0.650;1.000)]] [[(0.626,0.715,0.880,0.933;0.800),(0.626,0.715,0.880,0.933; 1.000)], [(0.328,0.408,0.605,0.687;0.800),(0.328,0.408,0.605,0.687;1.000)], [(0.309,0.379,0.610,0.708;0.800),(0.309,0.379,0.610,0.708;1.000)], [(0.611,0.709,0.8 3 3,0.87 3 ;0.800), (0.611,0.709,0.83 3,0.873 ;1.000)], [(0.173,0.249,0.445,0.537;0.800),(0.173,0.249,0.445,0.537;1.000)]] (5) Calculate the weighted matrix and the COG of each attributes with respect to the positive ideal solution and the negative ideal solution (>', x) : [(0.3333,0.8283),(0.4166,0.8283)],[(0.3422,0.5645),(0.4278,0.5645)],[(0.3429,0.6 863), [(0.3381,0.7873),(0.4227,0.7873)],[(0.3427,0.6837),(0.4284,0.6837)],[(0.3418,0.7 159), [(0.3215,0.9107),(0.4019,0.9107)],[(0.3438,0.6540),(0.4298,0.6540)],[(0.3438,0.6540), [(0.3333,0.8283),(0.4166,0.8283)],[(0.3397,0.5071),(0.4246,0.5071)],[(0.3441,0.5 026), (0.4287,0.6863)],[(0.321 5,0.9107),(0.4019,0.9107)],[(0.3341,0.4809),(0.4176,0.4809)] (0.4273,0.71 59)],[(0.3258,0.8691),(0.4072,0.8691)],[(0.3393,0.3880),(0.4242,0.38 80)] (0.4298,0.6540)],[(0.321 5,0.9107),(0.4019,0.9107)],[(0.3386,0.4084),(0.4233,0.40 84)] (0.4301,0.5026)],[(0.3299,0.7540),(0.4123,0.7540)],[(0.3383,0.3515),(0.4229,0.35 15)] _ [[(0.3215,0.9107),(0.4019,0.9107),(0.3427,0.6837)],[(0.4284,0.6837),(0.3418,0.7159), (0.4273,0.7159),(0.3215,0.9107)],[(0.4019,0.9107),(0.3341,0.4809),(0.4176,0.4809)]] _ [[(0.3381,0.7873),(0.4227,0.7873)],[(0.3397,0.5071),(0.4246,0.5071)],[(0.3441,0.5 026), 5 _ (0.4301,0.5026)],[(0.3299,0.7540),(0.4123,0.7540)],[(0.3383,0.3515),(0.4229,0.35 15)]] V V = [O'A L = (y>x\ (yA V AN EXTENDED TOPSIS METHOD FOR MULTIPLE. Informática 35 (2011 ) 185-196 195 (6) Calculate the weighted distance between each project Ai and the ideal solution and negative ideal solution: d+ =(0.1050 0.1140 0.0707 0.2500) d ~ = (0.2002 0.2136 0.2097 0.0292) (7) Calculate the relative closeness coefficient: C = (0.6560 0.6520 0.7479 0.1046 ) (8) Rank the alternatives: Based on relative closeness coefficient, we can rank the order of each alternatives: >- at >- a2 >- a4. (9) Analysis: In this example, our approach produces the same ranking as the literature [17], which proves the approach presented in this paper is effective. Comparing with the literatures [7-11], all of them utilize the TOPSIS method to deal with the decision making problems under the fuzzy information environment. The method in this paper, however, can deal with the more complex decision making problems under the generalized interval-valued trapezoidal fuzzy information environment. Comparing with the literature [17], in the fuzzy information, this method solves the FMADA problem based on the generalized interval-valued trapezoidal fuzzy information, and the literature [17] solves the FMADA problem based on the interval-valued triangular fuzzy numbers. In decision making method, literature [17] firstly uses the lower limits and the upper limits of the interval-valued triangular fuzzy numbers to calculate the relative closeness coefficient based on the TOPSIS method, then it uses the mean closeness coefficient to rank the order of the alternatives, so this method is not considering the interval-valued triangular fuzzy numbers as a whole; in this study, we proposed the extended TOPSIS based on the definition of the distance and the comparison method between the generalized interval-valued trapezoidal fuzzy numbers. Comparing with the literature [18], both of them are the decision making problems based on the generalized interval-valued trapezoidal fuzzy numbers. The literature [18] ranks the order of the alternatives based on the similarity which is hard to calculate. The method proposed in this paper, however, is easy to calculate the similarity. 5 Conclusion Fuzzy multiple attribute decision making (FMADM) is wildly used in various areas. The interval-valued trapezoidal fuzzy numbers can be precisely express the attribute values and weights of the decision making process. This study proposes an extended TOPSIS method for solving the MADM problems which the attribute weights and values are given with the form of GIVTFN and the decision making steps. This method is simple and easy to understand. This method constantly enriches and develops the theory and method of FMADM, and it proposes a new idea for solving the FMADM problems. Acknowledgement This paper is supported by the Humanities and Social Sciences Research Project of Ministry of Education of China(No.09YJA630088), and the Natural Science Foundation of Shandong Province (No. ZR2009HL022). The authors also would like to express appreciation to the anonymous reviewers for their very helpful comments on improving the paper. References [1] Bellman R.E., Zadeh L.A. (1970). Decision-making in a fuzzy environment, management science, 171, pp.41-164. [2] Hwang C.L., Yoon K. (1981). Multiple Attributes Decision Making Methods and Applications, Springer, Berlin Heidelberg. [3] Zavadskas E.K, Kaklauskas A., Turskis Z. (2009) .Multi-Attribute Decision-Making Model by Applying Grey Numbers. INFORMATICA, 20(2), pp.305-320. [4] Liu P.D. (2009a). Multi-Attribute Decision-Making Method Research Based on Interval Vague Set and TOPSIS Method. Technological and Economic Development of Economy, 15(3), pp. 453-463. [5] Liu P.D.(2009b). 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