Informática 36 (2012) 83-90 83 Multiple Attribute Decision Making Method Based on the Trapezoid Fuzzy Linguistic Hybrid Harmonic Averaging Operator Peide Liu and Yu Su School of Management Science and Engineering, Shandong University of Finance and Economics No.7366 Erhuandong Road, Lixia District, Ji'nan 250014, Shandong Province, P.R. China E-mail1:Peide.liu@gmail.com Keywords: the trapezoid fuzzy linguistic variables (TFLVs), the TFLHHA operator, multiple attribute decision making (MADM) Received: October 18, 2010 A new method is proposed to solve the multiple attribute decision making (MADM) problems with the trapezoid fuzzy linguistic variables (TFLVs) based on the trapezoid fuzzy linguistic hybrid harmonic averaging (TFLHHA ) operator. To begin with, this paper reviews the concept and operational rules of the TFLVs, the calculation method of the possibility degree with TFLVs, and the comparison method of TFLVs . Then, some operators are proposed, in order to aggregate the TFLVs , such as the trapezoid fuzzy linguistic weighted harmonic averaging (TFLWHA ) operator, the trapezoid fuzzy linguistic ordered weighted harmonic averaging (TFLOWHA ) operator, and the trapezoid fuzzy linguistic hybrid harmonic averaging (TFLHHA) operator. Furthermore, based on the TFLHHA operator, a new method solving the MADM problems with the TFLVs is proposed. Finally, an illustrative example is given to show the decision making steps, and it verifies the effectiveness of the developed method. Povzetek: Članek opisuje metodo za podporo odločanju, ki uporablja mehko logiko. 1 Introduction In the process of the multiple attribute decision making (MADM), the decision making information, given by the decision makers, often takes the form of the linguistic variables, because of the complexity and uncertainty of the objective things, and the ambiguity of human thinking. Therefore, the MADM under the linguistic context is an interesting research topic which has been receiving more and more attention in recent years [1-4]. Some operators were widely used to aggregate the decision making information in the process of the MADM. Bordogna et al. [5] developed a model within fuzzy set theory by the linguistic ordered weighted average ( LOWA ) operators for the group decision making in the linguistic context. Xu [6] proposed an approach to solve the multiple attribute group decision making problems with the uncertain linguistic information, based on the uncertain linguistic ordered weighted averaging ( ULOWA ) operator and the uncertain linguistic hybrid aggregation ( ULHA ) operator. Wu and Chen [7] introduced the linguistic weighted arithmetic averaging ( LWAA ) operator to aggregate the decision making information which took the form of the linguistic variables. Xu [8] developed some operators for aggregating the triangular fuzzy linguistic variables, such as the fuzzy linguistic averaging ( FLA ) operator, the fuzzy linguistic weighted averaging ( FLWA ) operator, the fuzzy linguistic ordered weighted averaging (FLOWA) operator, and the induced FLOWA (IFLOWA) operator. But in the real situation, the decision-makers sometimes can only provide the decision making information in the form of the trapezoid fuzzy linguistic variables ( TFLVs ). The trapezoid fuzzy linguistic variable ( TFLV ) generalizes the linguistic variable, the uncertain linguistic variable and the triangular fuzzy linguistic variable. So the research on the MADM problems with the TFLVs is very significant. But the related decision making methods based on the TFLVs are less. Xu [9] proposed the trapezoid fuzzy linguistic weighted averaging ( TFLWA ) operator to aggregate all the decision making information corresponding to each alternative, and he used the similarity measure to rank the decision alternatives and then the most desirable one is selected. Liang and Chen [10] proposed the trapezoid fuzzy linguistic weighted averaging (TFLWA) operator to aggregate the decision making information, and then all the alternatives were ranked by comparing the possibility degree of the TFLV. Based on these, this paper extends the OWHA operator [11] and the UCWHA operator [12] 84 Informatica 36 (2012) 75-82 X. Zhao et al. to deal with the MADM problems with the trapezoid fuzzy linguistic information, such as the trapezoid fuzzy linguistic weighted harmonic averaging ( TFLWHA) operator, and the trapezoid fuzzy linguistic ordered weighted harmonic averaging (TFLOWHA) operator. The TFLWHA operator only focuses on the attribute weight itself, but it ignores the position weight with respect to the attribute value; and the TFLOWHA operator focuses on the position weight with respect to the attribute value, but it ignores the weight of the attribute value itself. The two operators are one-sided. So in order to avoid the disadvantage of the two operators, the trapezoid fuzzy linguistic hybrid harmonic averaging ( TFLHHA ) operator is proposed to aggregate the attribute values which take the form of the TFLVs . According to the TFLHHA operator, the new method is proposed, which can solve MADM problems with the TFLVs directly. To do so, the remainder of this paper is structured as follows: In section 2, this paper reviews the concept and the operational rules of the TFLVs, and introduces the comparison method of the TFLVs , in which the calculation method of the possibility degree with the TFLVs is reviewed. In section 3, three operators are proposed in order to aggregate the TFLVs, such as the TFLWHA operator, the TFLOWHA operator, and the TFLHHA operator. In section 4, the decision making steps of the new method is proposed based on the TFLHHA operator. In section 5, an illustrative example is given to show the decision making steps, and it verifies the effectiveness of the developed method. The section 6 concludes this paper. 2 The Trapezoid Fuzzy Linguistic Variables 2.1 The definition of the trapezoid fuzzy linguistic variables Let S = ]s, \i = 1,2,—, t] be a linguistic term set with odd cardinality, any label st represents a possible value of the linguistic variable. Especially, ^ and st represent the lower and the upper values of the linguistic terms, respectively. For example, a linguistic term set S could be given as follows: S = { s =extremely poor, s2 =very poor, s3 =poor, s4 =slightly poor, s5 =fair, s6 =slightly good, s7 =good, s8 =very good, s9 =extremely good} Usually, in these cases, s and s . must satisfy the ' J following additional characteristics [13]: (1) The set S is ordered: st is worse than s ., if i < j ; (2) Maximum operator: max( s, sj ) = s , if s - ^ ; (3) Minimum operator: min(s, sy ) = sy , if s - s ■. Some calculation results, however, may not exactly match any linguistic labels in S in the calculation process. To preserve all the given information, the discrete term set S is extended to a continuous term set S = (s; \ s0 < s — sq, i e [0,q]} , where s. meets all the characteristics above and q(q > t) is a sufficient large positive integer. If s; e S , then we call st the original term, otherwise, we call s the virtual term. In general, the decision makers use the original linguistic terms to evaluate the alternatives, and the virtual linguistic terms can only appear in the process of the operation and ranking [13]. Definition 2.1[8]: Let sa, ^ e S, then we defined the distance between sa and sp as: d (s a, sp ) = \a~p (1) where sa, sp, sr,s^e S Definition 2.2[8]: Let s = [5^,5^,5^,5^] £ S , and the subscripts a , P, r, T] are non-decreasing numbers, and s^ and s indicate the interval in which the membership value is 1, with s and s indicating the lower and upper values of .v , respectively, then .v is called the trapezoid fuzzy linguistic variable ( TFLV ), which is characterized by the following membership function (see Figure 1): 0 d (se, sa ) d (sp, sa ) 1 d(se, sT) d(sr, sv) 0 s0 - se - sa sa- se- sp sp< se< sr sr- se< sv sT- se- s q (2) where S is the set of all the trapezoid fuzzy linguistic variables. Especially, if any two of a, /?, y, 7] are equal, then .v is reduced to a triangular fuzzy linguistic variable; if any three of a, P,r,T are equal, then .v is reduced to an uncertain linguistic variable [8]. MULTIPLE ATTRIBUTE DECISION MAKING. Informatica 36 (2012) 83-90 85 M®) Figure 1 A trapezoid fuzzy linguistic variable s 2.2 The operational rules and characteristics of the trapezoid fuzzy linguistic variables Let £ = , ¿^[s^s^s^s^] and s2 = , s^, s, st]_ ] e S be any three trapezoid fuzzy linguistic variables, and X e [0,1] and X e [0,1], then their operational rules are defined as follows: (1) s, © £2 = , s^, ^, ^ ] © , sPi, ^, sV2 ] = isa1+a2 > Spl+p2 > Sn+r2 > ] ; (2 )As = A[sa,s/},sy,sfJ]-, = [sXa , SXP , SXj , SXi ] (3) if 0 s2) = min{max( 7+vù - (a+p2) (7i - (a1 + P) + 7 -(^2 +^2) ,0},1} (3) Example 1: Let s^ =[s2,s3,s5,s6] and S2 = [,s4, .V-, sH, ,sy ] be two trapezoid fuzzy linguistic variables, then the possibility degree of s1 > s2 is: (5 + 6) - (4 + 5) p(s{ > s2) = min{max{- 0},1} 1 / / P / a = [S 1/, S y, s y, S y] /1 /j / P /a In addition, the trapezoid fuzzy linguistic variables have the following characteristics: (1) (2) (A©4)s = As©^; (3) A(s © s1) = As © Asx. 2.3 The comparison method of the trapezoid fuzzy linguistic variables Definition 2.3 [10]: Let .V, = [.s^, s^, .v.,, s ] and s2 = [.V(A, s^, s, St]_ ] be two trapezoid fuzzy linguistic variables, then the possibility degree of .v, > s2 is defined as follows: (5 + 6) - (2 + 3) + (8 + 9) - (4 + 5) = min{max{0.143,0},1} = 0.143 The characteristics of the possibility degree p(sl > s2) arc shown as follows [10]: Let s1 =[sai,sPi,s7i,Stli-\ , s2=[sai,sfii,sri,svj , s3 = [su , s^, s, s ] be any three trapezoid fuzzy linguistic variables, then (1)0s2) <1, 05j) <1; (2)p{s1>s2) + p{s2>s1) = \. Especially, if p(s1>s2) = p(s2>s1) , then P(*i ^ = Pi*2 ^ = |; (3) if p(s1 >s2)>— , and p(s2>s3)>- , then/^ (4) if p(s1 >s2)>— , and p(s2>s3)>- , then p(sx > s2) + p(s2 >§3)> p(s1 > s3) Let .V, and ,V; be two trapezoid fuzzy linguistic ' J variables, then the steps of the comparison method are shown as follows: (1) Utilize the formula (3) to compare the size of .v, and .v ,, and suppose that ptJ = p(sj > .v^). then we can contribute the possibility degree matrix P = (P,i) BXB ,where p„ > 0 , 1 Pj + Pj, =1 > Pn=~ > iJ = 1,2,—,n . We can 2 easily obtain the result that the matrix P = (pj )nxn is the complimentary judgment matrix [14]. (2) Sum all the elements of each rows of the possibility degree matrix, and rank the orders of the trapezoid fuzzy linguistic variables based on the values p , where 1 s s s s p a 7 n 86 Informatica 36 (2012) 83-90 P. Liu et al. pt = ^jPij (i = 1,2,-• -,/7 ). The larger the value of four trapezoid fuzzy linguistic variables, and w = (0.3,0.2,0.1,0.4) is the weight vector, then j=1 w. pi is, the larger the trapezoid fuzzy linguistic variable si TFL WHAw (v,, ,v2, ,v3, ,v4) = —- ) 1 is. Example 2: Let S1 = and S2 = [.S^.S^.S^.Sy] be two trapezoid fuzzy linguistic variables, then we can compare the size of .V, with ,v2: (1) The possibility degree of sx > s2 is: >s2) = min{max( (5 + 6)-(4 + 5) (5 + 6) - (2 + 3) + (8 + 9) - (4 + 5) ,0},1} = min{max{0.143,0},1} = 0.143 and the possibility degree of s2 > .v, is: p(s2 > = min{max{- S + 9) - (2 + 3) ! + 9) - (4 + 5) + (5 + 6) - (2 + 3) = min{max{0.857,0},1} = 0.857 Then we can contribute the possibility degree matrix: " 0.5 0.143" 0.857 0.5 P = (Pj )2x2 = (2) p = £ pXj = 0.5 + 0.143 = 0.643, j=1 2 P 2 =Z P 2 j = 0 875 + 0.5 = 1.375, j=1 so px S, if " w. TFLWHAw(5j, s2, •-Jn) = CZ—) ' (4) where S is the set of all trapezoid fuzzy linguistic variables, and .v , ce S ( j = 1,2, •••,«) is the trapezoid fuzzy linguistic variable. w = (w,w2,—,wn) is the weight vector, and it is the weight of .v, , where n wi > 0, i = 1,2,—,n, £w, = 1, then TFLWHA is i=1 called the trapezoid fuzzy linguistic weighted harmonic averaging (TFLWHA )operator. Example 3: If^ = s2 = [s4,s5,s8,s9] s3 =[5s,56,57,59] and s4 = [s3,s4,s5,s7] e S are = ( 0.3 0.2 0.1 0.4 )- [S2,S3,S5,S6] [S4,S5,S8,S9] [S5,S6,S7,S9] [S3,S4, S5,S7] = ([s0.05 , S0.06, S0.1, S0.15 ) ® (s0.022, S0.025 , S0.04, S0.05 ] ® [S0.011, S0.014, S0.017, S0.02 ] ® [S0.057, S0.08, S0.1, S0.133 ]) = [S0.14, S0.179, S0.257, S0.353 ] = [S2.833, S3.891, S5.587, S7.143 ] Definition 3.2: Let TFLOWHA : S" -> S , if " CO 1' 2' ' n W i 1 TFLOWHA^ (A, s2,—,$„) = (£ -r-) (5) ,0},1} M rJ where S is the set of all trapezoid fuzzy linguistic variables , and s. . r, e S ( j = 1,2, •••,« ) are the ~ •th trapezoid fuzzy linguistic variables, r. is the j largest of si (z = 1,2,■■■,«), and 0) = (0)l,0)2,---,0)n) is the position weight vector with TFLOWHA , where (> 0 , j = 1,2,—,n , = 1 then j=1 TFLOWHA is called the trapezoid fuzzy linguistic ordered weighted harmonic averaging (TFL O WHA )operator. The characteristic of the TFLOWHA operator is: Firstly, The order of the trapezoid fuzzy linguistic variables is ranked, then the position weights are aggregated with them, but there is no relationship between C0j and .v ,, and COj is only associated with the jth position in the aggregation process, so (O = ,®2,—) is called the position weight vector. According to the real situation, the position weight vector (= ((,(2,—,(n) is determined. In this paper, the position weight is determined by the method which proposed in literature [15]. The formula is shown as follows: C —itr,z = 0, \---n-\ (6) Example 4: Let sx = and S2 = [,S4, .V-, .Ss, .Sy ] be two trapezoid fuzzy linguistic variables, and we already know that .V, < s2 (the calculation steps are shown in Example 1), then the position weight vector is ( = (C^,Ci1)=(0.5,0.5), 4 MULTIPLE ATTRIBUTE DECISION MAKING. Informatica 36 (2012) 83-90 87 TFLO WHAa (Sj, s2 ) = (■ 0.5 0.5 0.5 = ( -e- 0.5 r [s2, S3, s5, s6] [s4, s5, s8, s9] = ([S0.083 , S0.1, S0.167 , S0.25 ] ® [S0.056 , S0 0.0625, S0.1, S0.125 K139 , S0.1625 , S0.267 , S0.375 ] = [S2.667 , S3.745 , S6.154 , S7.217 ] The TFLWHA operator only focuses on the weight of the attribute value itself, but it ignores the position weight with respect to the attribute value; and the TFLOWHA operator focuses on the position weight with respect to the attribute value, but it ignores the weight of the attribute value itself. The two operators are one-sided. If the decision makers use these operators to aggregate the decision making information, some information may be lost. So in order to avoid the disadvantage of the two operators, the trapezoid fuzzy linguistic hybrid harmonic averaging ( TFLHHA ) operator is defined as follows: Definition 3.3: Let TFLHHA : S" -> S, if CO, TFLHHAW W(SJ, 52, • • •, 5n) = (E1 (7) M rJ where S is the set of all trapezoid fuzzy linguistic variables , and .v, . r, G S ( = 1,2) are the ~ •th trapezoid fuzzy linguistic variables, r, is the j largest of y ( 7 = 1,2, ■■ ), where w = (w1, w2, • • •, wn ) is the weight vector, and wi is the n weight of si. w, > 0 ( / = 1,2, •••,«), ^ w, = ' • and ¿=1 n is the balancing coefficient. O) = {(0^(02,- ■ -,(0n ) is the position weight vector with TFLHHA , where > 0 ( j = 1,2,—, n ), = 1, then TFLHHA j=1 is called the trapezoid fuzzy linguistic hybrid harmonic averaging (TFLHHA) operator. Example 5: Let ^ = [s2, , , ] and S2 = [.S^.S^.S^.Sy] be two trapezoid fuzzy linguistic variables. We already know that the position weight vector is o = (0.5,0.5) (the calculation steps are shown in Example 4), and the weight vector is w = (0.3,0.7) , given by the decision makers, then based on the method shown in section 2.2, we can calculate that: fY = = [s3 333 ,s5,s$ 333>^io] • and r2 y^AV [^2.857 5 ^3.571 ' ^5.714 ' ^6.429 ] ' Then, TFLHHA ,s2,-,sn) = (Z ^ M ri 0.5 0.5 = (■ -e- [S3.333 , S5 , S8.333 , S10 ] [S2.857 , S3.571, S5.714, S6.429 ] = ([S0.05 , S0.06 , S0.1, S0.15 ] e [S0.0778 , S0.0875 , S0.14 , S0.175 ]) = (S0.1278 , S0.1475 , S0.24 , S0.325 ) = (s 3.077 , S4.167 , S6.780 , S7.826 ) Especially, if w = i operator; if (=( 1 then • = fV 1/ .. n'/n' '/«' TFLHHA operator is reduced to TFLOWHA ' ,y,., V) , then TFLHHA n n n operator is reduced to the TFLWHA operator. Obviously, TFLOWHA operator and TFLWHA operator are extended from the TFLHHA operator. The TFLHHA operator focuses on not only the importance of the weight of the trapezoid fuzzy linguistic variables itself, but also the importance of the position weight of the trapezoid fuzzy linguistic variables. So this operator is better than the previous ones. 4 Multiple Attribute Decision Making Method Based on the Trapezoid Fuzzy Linguistic Variables A multiple attribute decision making problem under the fuzzy linguistic environment is represented as follows: Let X = (x,x2,•••,xn} be the set of the alternatives, and U = (u, u2, • • •,um } be the set of the attributes. Let w = (w,w2,•■■,wm)T be the weight vector of the attributes, and w . be the weight value of the jth attribute, where wj > 0 ( j = 1,2,., m ), m E wj = 1 , given by the decision makers directly. j=i Suppose that A = (ajj)nxm is the fuzzy linguistic decision matrix u1 u2 • um n n 88 Informatica 36 (2012) 83-90 P. Liu et al. A = ân a\: CI-, anl an2 ah â-, a„ where atj = [a^^a^',a\n e £ is the attribute value which takes the form of the trapezoid fuzzy linguistic variables, given by the decision makers, for the alternative x e X (i = 1,2, • • •,n) with respect to the attribute u. eU ( j = 1,2,—, m ). Let at = [¿7 ,, 0 of the attributes, n is the balancing ( j = \,2,-,m ), =1 j=i coefficient. Step 2: Utilize the formula (3) to construct the possibility degree matrixes Pi=(PjDmxm=(Pjk(aij with respect to the alternative x (i = 1,2,—, n ) , and sum all the elements of each rows of the possibility degree matrix P , then get the ranking vectors p« = (p«,pV>,---,p(I>), ( j = 1,2,• • •,m ), where p() p^l . Finally, rank the orders of attribute k=1 values af. ( / = 1,2,• • -,/7? ) with respect to the alternative X based on the values p(!) (j = 1,2, • • •,m). Step 3: Utilize the formula (6) to calculate the position weight vector 0) = (col,co2,---,G)m) of TFLHHA operator. Step 4: Utilize the formula (7) to calculate the combined attribute values zi = TFLHHA (s1 ,s2,- where i = 1,2,—, n . '>Sm m CO ;=1 rJ Step 5: Utilize the formula (3) to construct the possibility degree matrix P = (p^ )nxn , based on the combined attribute values z. of each alternative, then sum all the elements of each rows of the possibility n degree matrix, where p p^ ( i = 1,2,—, n ). j=1 Rank all the combined attribute values of each alternative and select the best alternative based on the values p . 5 Illustrative Examples In this section, a decision making problem of evaluating cars for buying (adapted from literature [1, 9]) is used to illustrate the new method. A decision maker intends to buy a car. Four types of cars x (i = 1,2,3,4 ) are available. He takes into account four attributes to decide which car he should buy: 1) Gj: economy, 2) G2: comfort, 3) G3 : design, and 4) G4: safety. The decision maker evaluates these four types of cars x (i = 1,2,3,4) under the attributes G- ( j = 1,2,3,4 ), where the weight vector is w = (0.3,0.2,0.1,0.4) given by the decision makers. He uses the linguistic term set: S ={ s =extremely poor, s2 =very poor, s3 =poor, s4 =slightly poor, s5 =fair, s6 =slightly good, s7 =good, s8 =very good, s9 =extremely good} and provides the linguistic decision making matrix A = (¿/;/ )4x4 : [S2, S3, S5, S6][S4, S5, S8, S9][S5, S6, S7 , S9][S3, S4, S5, S7 | V.. .V,. .V,. V 11 .V,. .V.,. .V.. \ 11 .V.. .V,. ,VS. V 11 .V.. .V,. .V.. ,VS [s4, S6, S8, S9 ][S4, S5, S6, S7 ][S6, S7 , S8, S9 ][S3, S4, S5, S6 [S5, S6, S7, S9 ] [S4, S7, S8, S9 ] [S3, S5, S6, S7 ] [S6 , S7, S8, S9 Step 1: Construct the weighted linguistic matrix A = (a..). . , where A = av = nw ^ =(^)4x4 ( j = 1,2,3,4 ). A = [S1. 67 , S2.5 , S4 . 17 , S5 ] [S5 , S6.25 , S10, S11.25 ] [^2.5 >S4.17>S5> S5.S3 ] [^6.25 ' S7.5 ' ^8.75 ' ^10 ] [S3 . 33 , S5 , S6. 67, S7. 5 ] [S3 . 33 , S4 . 17 , S5 , S5. §3 ] tS4.17 , S5 , S5.83 , S7.5 ] [S5 , S8.75 , S10 , S11.25 ] [S12.5, S15 , S17.5 , S22.5 ] [S1.875 , S2.5 , S3.125 , S4.375 ] [S10, S12.5 , S20, S22.5 ] [S2.5 , S3.125 , S4.375 , S5 ] [S15, S17.5 , S20, S22.5 ] [S1.875, S2.5 , S3.125 , S3.75 ] 7.^ 20 ' 22.5 [S7.5 , S12.5 , S15 , S17.5 ] [S3.75 , S4.375 , S5 , S5.625 ] 2.5 3.125 3.7 '3.75 , S4.375 , S5 , S5.625 J MULTIPLE ATTRIBUTE DECISION MAKING. Informatica 36 (2012) 83-90 89 Step 2: Utilize the formula (3) to construct the possibility degree matrixes with respect to each alternative x ( i = 1,2,3,4 ), and sum all the elements of each rows of the possibility degree matrix P , then get the ranking k=1 attribute values a{j (j = 1,2,3,4 ) with respect to the alternative xi based on the values p() (j = 1,2,3,4). " 0.5 0 0 0.59" 1 0.5 0 1 P1 = 1 1 0.5 1 0.41 0 0 0.5 p(1) = (P1(1), pf, pf, pD = (1.09,2 «13 >«12 >« 11 > «14 • " 0.5 0 0 0.66" 1 0.5 0 1 p2 = 1 1 0.5 1 0.34 0 0 0.5 p(2) = (1.16,2.5,3.5,0.84) «23 >«22 >«21 >«24 " 0.5 0.73 0 1 0.27 0.5 0 1 p3 = 1 1 0.5 1 0 0 0 0.5 p(3) = (2.23,1.77,3.5,0.5) Q^ 3 ^ j ^ Cl^ 2 ^ Cl^ 4 * P 4 = 0 0 0.5 1 0.5 0.0625 1 0.9375 0.5 0.22 0 0 0.78 1 1 0.5 p(4) = (1.28,2.5625,3.4375,0.72) «43 > «42 > «41 > «44 • Step 3: utilize the formula (6) to calculate the position vector of TFLHHA operator: c = (0.125,0.375,0.375,0.125) . Step 4: utilize the formula (7) to calculate the combined attribute values: ~ (S2.65 5 ^3.73 5 ^5.73 > S7.02 ) ' ~ (^3.67 5 S5.27 ' ^6.55 ' ^7.55 ) ' Z3 (^3.33 7 ^4.5 7 ^5.63'^6.53) — (^4 65 , S639 , S139 ,Sg s7 ) vectors p(l) = (p™, pf,• • •, pf) , ( 7 = 1,2,3,4 ), 4 where p() p. Finally, rank the orders of P = Step 5: utilize the formula (3) to construct the possibility degree matrix, based on z.: " 0.5 0.33 0.46 0.15" 0.67 0.5 0.66 0.29 0.54 0.34 0.5 0.12 0.85 0.71 0.88 0.5 then sum all the elements of each row of the possibility degree matrix, we can get px = 1.44 p2 = 2.12 p3 = 1.5 p4 = 2.94. Based on the values p , rank all combined attribute values of each alternative and select the best alternative, then we can get x4 > x2 > x3 > x, so the best alternative is x4 . In order to verify the effective of this method, we utilized the method shown in literature [9] to solve this illustrate example. Stepl: From the linguistic decision making matrix^ = (cijj)n m. we can get the vector of the ideal point of the attribute values corresponding to the alternative X( (/ = 1,2,3,4): i=(I1,I2,i3,I4), and I3 — iS6 ?S7?S8?S9 — (^6 5 ^7 ' ' ^9 ) Step2: Utilize the TFLWA operator to derive the overall values z, ( z = 1,2,3,4 )of the alternative (i = 1,2,3,4) and z of the ideal point I — 0*3.1 5^4.1 '^5.8 5^7.3)' — (^3.9 '^5.2 '^6.8 >S7.s\ Z3 = (53 8,55 1,564,574),Z4 = (S5,S65,S75,Sgg) ^ ~ (S5.5 n S6.7 n ^8 ?S9 ) Step3: We get the similarity degree s(z,z;) between z and z, (z = 1,2,3,4 ) based on the similarity degree formula s(z,zj) = 0.876, s(z,z2) = 0.924, s(z,z3) = 0.910, s(z,z4) = 0.981 Step 4: Rank the order of s(z,z;) (z =1,2,3,4 ), then we can get: x > x > x > x . Analysis: The order calculated by this method is the same as the order calculated by the method proposed in literature [9], so it is demonstrated that the method proposed in this paper is feasible and effective, and it is also verified that the TFLHHAoperator is effective. It provided the new idea to solve the MADM problems under the linguistic context, and it provided the new idea of aggregating the trapezoid fuzzy linguistic variables in the MADM problems. 90 Informatica 36 (2012) 83-90 P. Liu et al. 6 Conclusions This paper proposed a new method of the MADM problems based on the TFLHHA operator. The new method can deal with the MADM problems where the decision making information takes the form of the TFLVs directly, and makes the computation process of the TFLVs easily without the loss of the information. This method is easy to use and understand, and it enriched and developed the theory and method of the MADM, This method can solve these MADM problems where the attribute values take the form of the fuzzy linguistic variables, such as fuzzy linguistic variables, the uncertain fuzzy linguistic variables, the triangular fuzzy linguistic variables, the trapezoid fuzzy linguistic variables, and the mixed fuzzy linguistic variables, if we can transform these fuzzy linguistic variables into the trapezoid fuzzy linguistic variables. But this method can only solve the MADM problem under the linguistic context. So it is the limitation of this paper. In the future, we will apply this method to solve the real-life MADM problems in the linguistic context, and we will continued working in the decision making method of the MADM problems with the TFLV . Acknowledgment This paper is supported by the Humanities and Social Sciences Research Project of Ministry of Education of China (No.10YJA630073 and No.09YJA630088), the Natural Science Foundation of Shandong Province (No. ZR2011FM036), the Social Science Planning Project Fund of Shandong Province (09BSHJ03), the Soft Science Project Fund of Shandong Province (2009RKA376), and the Doctor Foundation of Shandong Economic University. The authors also would like to express appreciation to the anonymous reviewers for their very helpful comments on improving the paper. References [1] Herrera, F., Martínez, L.(2000). An Approach for Combining Numerical and Linguistic Information Based on the 2-Tuple Fuzzy Linguistic Representation Model in Decision Making, International Journal of Uncertainty, Fuzziness and Knowledge -Based Systems, 8, pp. 539-562. [2] Li, D.F., Yang, J.B.(2004). 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