Multicriteria fuzzy decision-making problems based on vague set theory

Fuzzy Sets and Systems 114 (2000) 103–113

www.elsevier.com/locate/fss

Multicriteria fuzzy decision-making problems based on vague set theory Dug Hun Hong a; ∗ , Chang-Hwan Choi b a

School of Mechanical and Automotive Engineering, Catholic University of Taegu-Hyosung, Hayang-up, Kyongsan, 330, Kumrak 1-ri, Kyungbuk, 712-702, South Korea b Department of Mechanical Engineering, North Carolina State University, Raleigh, NC 27695, USA Received May 1997; received in revised form June 1998

Abstract Chen et al. (Fuzzy Sets and Systems 67 (1994) (163 –172)) present some techniques for handling multicriteria fuzzy decision-making problems based on vague set theory. They provide some functions to measure the degree of suitability of each alternative with respect to a set of criteria presented by vague values. However, in some cases, these functions do not give sucient information about alternatives. In this paper, we provide new functions to measure the degree of accuracy in the grades of membership of each alternative with respect to a set of criteria represented by vague values. The proposed functions give additional information about alternatives. The techniques proposed in this paper can provide more useful way than those of Chen to eciently help the decision-maker to make his decision. c 2000 Elsevier Science B.V. All rights reserved.

Keywords: Truth-membership function; Vague set; Fuzzy set; Multicriteria fuzzy decision making

1. Introduction Since the theory of fuzzy sets [12] was proposed in 1965, fuzzy set theory has been used for handling fuzzy decision-making problems [1–5, 7–10, 13, 14]. Kickert [7] has discussed the eld of fuzzy multicriteria decision making. Zimmermann [14] illustrated a fuzzy set approach to multiobjective decision making. Yager [10] presented a fuzzy multiattribute decision-making method, using crisp weights and he [11] introduced an ordered weighted aggregation operator and investigated the properties of the operator. Laarhoven et al. [8] presented a method for multiattribute decision making, using fuzzy numbers as weights. Zimmermann [14] has compared some approaches to solve multiattribute decision problems based on fuzzy set theory. Roughly speaking, a fuzzy set is a class with fuzzy boundaries. The fuzzy set A in the universe of discourse U , U = {u1 ; u2 ; : : : ; un }, is a set of ordered pairs {(u1 ; A (u1 )); (u2 ; A (u2 )); : : : ; (un ; A (un ))}, where A is the

This paper was supported by Non-directed Research Fund, Korea Research Foundation, 1998. Corresponding author. Tel.: +82 53 850-2712; fax: +82 53 850-2704. E-mail address: [email protected] (D.H. Hong).

∗

c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 2 7 1 - 1

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D.H. Hong, C.-H. Choi / Fuzzy Sets and Systems 114 (2000) 103–113

membership function of the fuzzy set A; A : U → [0; 1], and A (ui ) indicates the grade of membership of ui in A. When the universe of discourse U is a nite set, then the fuzzy set A can be represented by A = A (u1 )=u1 + A (u2 )=u2 + · · · + A (un )=un =

n X

A (ui )=ui :

i=1

When the universe of discourse U is an interval of real numbers between a and b, then a fuzzy set A is often written in the form Z b A (ui )=ui ; A= a

where ui ∈ [a; b]. It is obvious that ∀ui ∈ U , the membership value A (ui ) is a single value between zero and one. Gau et al. [6] pointed out that this single value combines the evidence for ui ∈ U and the evidence against ui ∈ U , without indicating how much there is of each. They also pointed out that the single number tells us nothing about its accuracy. Thus Gau et al. presented the concepts of vague sets. They used a truthmembership function tA and false-membership function fA to characterize the lower bound on A . These lower bounds are used to create a subinterval on [0; 1], namely [tA (ui ); 1 − fA (ui )], to generalize the A (ui ) of fuzzy sets, where tA (ui ) 6 A (ui ) 6 1 − fA (ui ). For example, let A be a vague set with truth-membership function tA and false-membership function fA , respectively. If [tA (ui ); 1 − fA (ui )] = [0:5; 0:8], then we can see that tA (ui ) = 0:5; 1 − fA (ui ) = 0:8; fA (ui ) = 0:2. It can be interpreted as “the vote for resolution is 5 in favor, 2 against, and 3 abstentions”. Recently, Chen et al. [5] presented some new techniques for handling multicriteria fuzzy decision-making problems based on vague set theory, where the characteristics of the alternatives are presented by vague sets. The proposed techniques used a score function S to evaluate the degree of suitability to which an alternative satis es the decision-maker’s requirement. In this paper, we propose a new function to evaluate the degree of accuracy of vague sets. This proposed function is a new concept on vague set theory. The proposed function can provide another useful way to eciently help the decision-maker to make his decisions. The rest of this paper is organized as follows. In Section 2, we brie y review the theory of vague sets from [5, 6]. In Section 3, we summarize the main results of Chen et al. [5] and propose a new function and some measures to handle multicriteria fuzzy decision-making problems. The conclusions are discussed in Section 4. 2. Vague sets Let U be the universe of discourse, U = {u1 ; u2 ; : : : ; un }, with a generic element of U denoted by ui . A vague set A in U is characterized by a truth-membership function tA and a false-membership function fA , tA : U → [0; 1];

fA : U → [0; 1];

where tA (ui ) is a lower bound on the grade of membership of ui derived from the evidence for ui ; fA (ui ) is a lower bound on the negation of ui derived from the evidence against ui , and tA (ui ) + fA (ui ) 6 1. The grade of membership of ui in the vague set A is bounded to a subinterval [tA (ui ); 1 − fA (ui )] of [0; 1]. The vague value [tA (ui ); 1 − fA (ui )] indicates that the exact grade of membership A (ui ) of ui may be unknown. But it is bounded by tA (ui ) 6 A (ui ) 6 1 − fA (ui ), where tA (ui ) + fA (ui ) 6 1. For example, Fig. 1 shows a vague set in the universe of discourse U . When the universe of discourse U is continuous, a vague set A can be written as Z [tA (ui ); 1 − fA (ui )]=ui : A= U


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Fig. 1. A vague set.

When the universe of discourse U is discrete, a vague set A can be written as A=

n X

[tA (ui ); 1 − fA (ui )]=ui :

i=1

For example, let U be the universe of discourse, U = {6; 7; 8; 9; 10}. A vague set “LARGE” of U may be de ned by LARGE = [0:1; 0:2]=6 + [0:3; 0:5]=7 + [0:6; 0:8]=8 + [0:9; 1]=9 + [1; 1]=10: Deÿnition 2.1. Let x be a vague value, x = [tx ; 1−fx ], where tx ∈ [0; 1]; fx ∈ [0; 1]; tx +fx 6 1. The complement of the vague value x is denoted by x0 and is de ned by x0 = [fx ; 1 − tx ]: Deÿnition 2.2. Let x and y be two vague values, x = [tx ; 1−fx ] and y = [ty ; 1−fy ], where tx ∈ [0; 1]; ty ∈ [0; 1]; fx ∈ [0; 1]; fy ∈ [0; 1]; tx + fx 6 1; and ty + fy 6 1. The result of the minimum operation of the vague values x and y is a vague value z, written as z = x ∧ y = [tz ; 1 − fz ], where tz = Min(tx ; ty );

1 − fz = Min(1 − fx ; 1 − fy ):

Deÿnition 2.3. Let x and y be two vague values, x = [tx ; 1−fx ] and y = [ty ; 1−fy ], where tx ∈ [0; 1]; ty ∈ [0; 1]; fx ∈ [0; 1]; fy ∈ [0; 1]; and tx + fx 6 1; and ty + fy 6 1. The result of the maximum operation of the vague values x and y is a vague value c, written as c = a ∨ b = [tc ; 1 − fc ], where tc = Max(tx ; ty );

1 − fc = Max(1 − fx ; 1 − fy ):

Deÿnition 2.4. A vague set A is empty if and only if its truth-membership function and false-membership function are identically zero on the universe of discourse U . Let A be a vague set of the universe of discourse U with truth-membership function and false-membership function tA and fA , respectively, and let B be a vague set of U with truth-membership function and falsemembership function tB and fB , respectively. The notions of complement, union, and intersection of vague sets are de ned as follows.

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Deÿnition 2.5. The complement of the vague set A is denoted by A0 , whose truth-membership function and false-membership function are tA0 and fA0 , respectively, where ∀ui ∈ U , tA0 (ui ) = fA (ui );

1 − fA0 (ui ) = 1 − tA (ui ):

Deÿnition 2.6. The union of the vague sets A and B is a vague set C, written as C = A ∨ B, whose truthmembership function and false-membership function are tC and fC , respectively, where ∀ui ∈ U , tC (ui ) = Max(tA (ui ); tB (ui ));

1 − fC (ui ) = Max(1 − fA (ui ); 1 − fB (ui )):

That is, ∀ui ∈ U , [tC (ui ); 1 − fC (ui )] = [tA (ui ); 1 − fA (ui )] ∨ [tB (ui ); 1 − fB (ui )] = [Max(tA (ui ); tB (ui )); Max(1 − fA (ui ); 1 − fB (ui ))]: Deÿnition 2.7. The intersection of the vague sets A and B is a vague set C, written as C = A ∧ B, whose truth-membership function and false-membership function are tC and fC , respectively, where ∀ui ∈ U , tC (ui ) = Min(tA (ui ); tB (ui ));

1 − fC (ui ) = Min(1 − fA (ui ); 1 − fB (ui )):

That is, ∀ui ∈ U , [tC (ui ); 1 − fC (ui )] = [tA (ui ); 1 − fA (ui )] ∧ [tB (ui ); 1 − fB (ui )] = [Min(tA (ui ); tB (ui )); Min(1 − fA (ui ); 1 − fB (ui ))]: 3. Multicriteria fuzzy decision-making problems This section reviews the research of Chen et al. [5] and proposes a new function and some measures to handle multicriteria fuzzy decision-making problems. Let A be a set of alternatives and let C be a set of criteria, where A = {A1 ; A2 ; : : : ; Am };

C = {C1 ; C2 ; : : : ; Cn }:

Assume that the characteristics of the alternative Ai are presented by the vague set shown as follows: Ai = {(C1 ; [ti1 ; (1 − fi1 )]); (C2 ; [tt2 ; (1 − fi2 )]); : : : ; (Cn ; (1 − fin )])}; where tij indicates the degree to which the alternative Ai satis es criteria Cj ; fij indicates the degree to which the alternative Ai does not satisfy criteria Cj ; tij ∈ [0; 1]; fij ∈ [0; 1]; tij + fij 6 1; 1 6 j 6 n, and 1 6 i 6 m. Let 1 − fij = tij∗ , where 1 6 j 6 n and 1 6 i 6 m. In this case, Ai can be rewritten as ∗ ∗ ]); (C2 ; [ti2 ; ti2 ]); : : : ; (Cn ; [tin ; tin∗ ])}; Ai = {(C1 ; [ti1 ; ti1

where 1 6 i 6 m. In this case, the characteristics of these alternatives can be represented by Table 1. Assume that there is a decision-maker who wants to choose an alternative which satis es the criteria Cj ; Ck ; : : : ; and Cp or which satis es the criteria Cs . This decision-maker’s requirement is represented by the following expression: Cj AND Ck AND · · · AND Cp OR Cs :


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Table 1 The characteristics of the alternatives

A1 A2 .. . Ai .. . Am

C1

···

Cj

···

Ck

···

Cp

···

Cs

···

Cn

∗] [t11 ; t11 ∗] [t21 ; t21 .. . ∗] [ti1 ; ti1 .. . ∗ ] [tm1 ; tm1

··· ···

∗] [t1j ; t1j ∗] [t2j ; t2j .. . [tij ; tij∗ ] .. . ∗ ] [tmj ; tmj

··· ···

∗] [t1k ; t1k ∗] [t2k ; t2k .. . [tik ; tik∗ ] .. . ∗ ] [tmk ; tmk

··· ···

∗ ] [t1p ; t1p ∗ ] [t2p ; t2p .. . ∗] [tip ; tip .. . ∗ ] [tmp ; tmp

··· ···

∗] [t1s ; t1s ∗] [t2s ; t2s .. . [tis ; tis∗ ] .. . ∗ ] [tms ; tms

··· ···

∗] [t1n ; t1n ∗] [t2n ; t2n .. . ∗] [tin ; tin .. . ∗ ] [tmn ; tmn

··· ··· ··· ···

··· ··· ··· ···

··· ··· ··· ···

··· ··· ··· ···

··· ··· ··· ···

In this case, the degrees to which the alternative Ai satis es and does not satisfy the decision-maker’s requirement can be measured by the evaluation function E, ∗ ]) ∨ [tis ; tis∗ ] E(Ai ) = ([tij ; tij∗ ] ∧ [tik ; tik∗ ] ∧ · · · ∧ [tip ; tip ∗ )] ∨ [tis ; tis∗ ] = [Min(tij ; tik ; : : : ; tip ); Min(tij∗ ; tik∗ ; : : : ; tip ∗ ); tis∗ )] = [Max(Min(tij ; tik ; : : : ; tip ); tis ); Max(Min(tij∗ ; tik∗ ; : : : ; tip

= [tAi ; tA∗i ] = [tAi ; 1 − fAi ];

(1)

where ∧ and ∨ denote the minimum operator and the maximum operator of the vague values, respectively; E(Ai ) is a vague value, 1 6 i 6 m, and tAi = Max(Min(tij ; tik ; : : : ; tip ); tis );

∗ tA∗i = Max(Min(tij∗ ; tik∗ ; : : : ; tip ); tis∗ ):

Let x = [tx ; 1−fx ] be a vague value, where tx ∈ [0; 1]; fx ∈ [0; 1]; tx +fx 6 1. The score of x can be evaluated by the score function S shown as follows: S(x) = tx − fx ;

(2)

where S(x) ∈ [−1; +1]. Based on the score function S, the degree of suitability to which the alternative Ai satis es the decision-maker’s requirement can be measured as follows: S(E(Ai )) = tAi + tA∗i − 1;

(3)

where S(E(Ai )) ∈ [−1; +1]. The larger the value of S(E(AI )), the more the suitability to which the alternative Ai satis es the decision-maker’s requirement, where 1 6 i 6 m. Let S(E(A1 )) = p1 ; S(E(A2 )) = p2 ; .. . S(E(Am )) = pm : If S(E(Ai )) = pi and pi is the largest value among the values p1 ; p2 ; : : : ; and pm , then the alternative Ai is his best choice. Assume that the characteristics of the alternatives are shown in Table 1, and assume that there is a decisionmaker who wants to choose an alternative which satis es the criteria Cj ; Ck ; : : : ; and Cp or which satis es the

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criteria Cs , then the decision-maker’s requirement can be represented by the following expression: Cj AND Ck AND · · · AND Cp OR Cs : Assume that the degree of importance of the criteria Cj ; Ck ; : : : , and Cp presented by the decision-maker are wj ; wk ; : : : , and wp , respectively, where wj ∈ [0; 1]; wk ∈ [0; 1]; : : : ; wp ∈ [0; 1], and wj + wk + · · · + wp = 1. Then, the degree of suitability to which the alternative Ai satis es the decision-maker’s requirement can be measured by the weighting function W , ∗ ]) ∗ wp ; S([tis ; tis∗ ])): W (Ai ) = Max(S([tij ; tij∗ ]) ∗ wj + S([tik ; tik∗ ]) ∗ wk + · · · + S([tip ; tip

(4)

By applying (3), (4) can be rewritten into ∗ − 1) ∗ wp ; tis + tis∗ − 1); W (Ai ) = Max((tij + tij∗ − 1) ∗ wj + (tik + tik∗ − 1) ∗ wk + · · · + (tip + tip

(5)

where W (Ai ) ∈ [−1; +1] and 1 6 i 6 m. Let W (A1 ) = p1 ; W (A2 ) = p2 ; .. . W (Am ) = pm : If W (Ai ) = pi and pi is the largest value among the values p1 ; p2 ; : : : ; and pm , then the alternative Ai is his best choice. We now consider the following two examples. Example 3.1. Let A1 and A2 be two alternatives, and let C1 ; C2 and C3 be three criteria. Assume that the characteristics of the alternatives are represented by the vague sets shown as follows: A1 = {(C1 ; [0; 1]); (C2 ; [0; 1]); (C3 ; [0; 1])}; A2 = {(C1 ; [0:5; 0:5]); (C2 ; [0:5; 0:5]); (C3 ; [0:5; 0:5])}; and assume that the decision-maker wants to choose an alternative which satis es the criteria C1 and C2 or which satis es the criteria C3 , then by applying (1), we can get E(A1 ) = ([0; 1] ∧ [0; 1]) ∨ [0; 1] = [0; 1]; E(A2 ) = ([0:5; 0:5] ∧ [0:5; 0:5]) ∨ [0:5; 0:5] = [0:5; 0:5]: By applying (2), we can get S(E(A1 )) = 0 + 1 − 1 = 0;

S(E(A2 )) = 0:5 + 0:5 − 1 = 0:

We still do not know which one is better. Example 3.2. Let A1 and A2 be two alternatives, and let C1 ; C2 and C3 be three criteria. Assume that the characteristics of the alternatives are represented by the vague sets shown as follows: A1 = {(C1 ; [0; 1]); (C2 ; [0; 1]); (C3 ; [0; 1])}; A2 = {(C1 ; [0:5; 0:5]); (C2 ; [0:5; 0:5]); (C3 ; [0:5; 0:5])};


109

and assume that the decision-maker wants to choose an alternative which satis es the criteria C1 and C2 or which satis es the criteria C3 , where the degree of importance of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, then by applying (5), we can get W (A1 ) = Max((0 + 1 − 1) ∗ 0:7 + (0 + 1 − 1) ∗ 0:3; 0 + 1 − 1) = Max(0; 0) = 0; W (A2 ) = Max((0:5 + 0:5 − 1) ∗ 0:7 + (0:5 + 0:5 − 1) ∗ 0:3; 0:5 + 0:5 − 1) = Max(0; 0) = 0: It is impossible to know which one is better. What is the dierence between alternative A1 and alternative A2 ? Here, some other measures are needed. In the following, we de ne an accuracy function H to evaluate the degree of accuracy of vague values. Let x = [tx ; 1 − fx ] be a vague value, where tx ∈ [0; 1]; fx ∈ [0; 1]; tx + fx 6 1. The degree of accuracy of x can be evaluated by the accuracy function H , shown as follows: H (x) = tx + fx :

(6)

where H (x) ∈ [0; 1]. The larger the value of H (x), the more the degree of accuracy of the grade of membership of vague value. Example 3.3. Let x be a vague value, where x = [0; 1]. That is, tx = 0; fx = 0. This means that we do not have any information about the grade of membership. Then, based on (6), the degree of accuracy at the grade of membership of vague value x is H (x) = 0 + 0 = 0: Example 3.4. Let x be a vague value, where x = [0:4; 0:7], that is tx = 0:4; fx = 0:3. Then, based on (6), the degree of accuracy at the grade of membership of vague value x is H (x) = 0:4 + 0:3 = 0:7: Example 3.5. Let x be a vague value, where x = [a; a]; 0 6 a 6 1, that is tx = a; fx = 1 − a. This means that we know that the exact grade of membership of vague value x. Then, based on (6), the degree of accuracy at the grade of membership of vague value x is H (x) = a + 1 − a = 1: The relation between the score function S and the accuracy function H is similar to the relation between mean and variance in statistics. Let A be a set of alternatives and let C be a set of criteria, where A = {A1 ; A2 ; : : : ; Am };

C = {C1 ; C2 ; : : : ; Cn }:

We assume that the characteristics of alternatives are shown in Table 1 and assume that there is a decisionmaker who wants to choose an alternative which the criteria Cj ; Ck ; : : : ; and Cp or which satis es the criteria Cs . Thus we can represent the decision-maker’s requirement by the following expression: Cj AND Ck AND · · · AND Cp OR Cs :

110


Then, based on (1) and (6), we can obtain H (E(Ai )) = tAi + 1 − tA∗i ;

(7)

where H (E(Ai )) ∈ [0; 1]. The larger the value of H (E(Ai )), the more the degree of accuracy in the grades of membership of the alternative Ai , where 1 6 i 6 m. In Example 3.1, E(A1 ) = [0; 1] and E(A2 ) = [0:5; 0:5] with S(E(A1 )) = S(E(A2 )) = 0. Then, by applying (7), we get H (E(A1 )) = 0 + 1 − 1 = 0;

H (E(A2 )) = 0:5 + 1 − 0:5 = 1:

Therefore, the alternative A2 has a greater degree of accuracy than the alternative A2 in the grades of memberships. In this case, which one is better? It may depend on the decision-maker’s mind or policy. A conservative person might choose the alternative A2 , but an aggressive person may choose the alternative A1 . This measure provides an additional useful information to eciently help the decision-maker to make his decisions. The following two theorems are easy to check. Theorem 3.1. Let a be a vague value; and H be the accuracy function; where a = [a1 ; a2 ]. Then H (a) = 0 if and only if a1 = 0; a2 = 1. Theorem 3.2. Let a be a vague value; and H be the accuracy function; where a = [a1 ; a2 ]. Then H (a) = 1 if and only if a1 = a2 . The following three theorems can be proved easily in a similar manner as in [5]. We do not give the proof here. Theorem 3.3. Let a and b be two vague values; and H be the accuracy function; where a = [a1 ; a2 ]; b = [b1 ; b2 ]. Then H (a) ¿ H (b) if and only if a1 + b2 ¿ b1 + a2 . Theorem 3.4. Let a; b and c be three vague values; and H be the accuracy function; where a = [a1 ; a2 ]; b = [b1 ; b2 ]; and c = [c1 ; c2 ]. Then H (a ∧ c) ¿ H (b ∧ c) if and only if Min(a1 ; c1 ) + Min(b2 ; c2 ) ¿ Min(b1 ; c1 ) + Min(a2 ; c2 ). Theorem 3.5. Let a; b and c be three vague values; where a = [a1 ; a2 ]; b = [b1 ; b2 ]; and c = [c1 ; c2 ]. Then H (s ∨ c) ¿ H (b ∨ c) if and only if Max(a1 ; c1 ) + Max(b2 ; c2 ) ¿ Max(b1 ; c1 ) + Max(a2 ; c2 ). In the following, we present a weighted technique for handling multicriteria fuzzy decision-making problems. Assume that the characteristics of the alternatives are shown in Table 1, and assume that there is a decision maker who wants to choose an alternative which satis es the criteria Cj ; Ck ; : : : ; and Cp or which satis es the criteria Cs , then the decision-maker’s requirement can be represented by the following expression: Cj AND Ck AND · · · AND Cp OR Cs : Assume that the degree of importance of the criteria Cj ; Ck ; : : : ; and Cp entered by the decision-maker are wj ; wk ; : : : ; and wp , respectively, where wj ∈ [0; 1]; wk ∈ [0; 1]; : : : ; wp ∈ [0; 1]; and wj + wk + · · · + wp = 1. Let ∗ ]) ∗ wp ; T∩l=j; k;:::;p Cl (Ai ) = H ([tij ; tij∗ ]) ∗ wj + H ([tik ; tik∗ ]) ∗ wk + · · · + H ([tip ; tip

TCs (Ai ) = H ([tis ; tis∗ ]):

(8)


111

By applying (7), we can get ∗ + 1) ∗ wp ; T∩l=j; k;:::;p Cl (Ai ) = (tij − tij∗ + 1) ∗ wj + (tik − tik∗ + 1) ∗ wk + · · · + (tip − tip

TCs (Ai ) = tis − tis∗ + 1: Similarly, we de ne ∗ ]); W∩l=j; k;:::;p Cl (Ai ) = S[tij ; tij∗ ] ∗ wj + S([tik ; tik∗ ] ∗ wk + · · · + S([tip ; tip

WCs (Ai ) = S([tis ; tis∗ ]):

(9)

We then de ne the range of the alternative Ai which satis es the criteria Cj ; Ck ; : : : ; and Cp or which satis es the criteria Cs as follows: 1 − T∩l=j; k;:::;p Cl (Ai ) 1 − T∩l=j; k;:::;p Cl (Ai ) ; ; W∩l=j; k;:::;p Cl (Ai ) + R∩l=j; k;:::;p Cl (Ai ) = W∩l=j; k;:::;p Cl (Ai ) − 2 2 1 − TCs (Ai ) 1 − TCs (Ai ) ; WCs (Ai ) + ; RCs (Ai ) = WCs (Ai ) − 2 2 where R∩l=j; k;:::;p Cl (Ai ) ⊂ [−1; 1], RCs (Ai ) ⊂ [−1; 1] and 1 6 i 6 m. Let R(Ai ) = (Rmin (Ai ); Rcenter (Ai ); Rmax (Ai )), where 1 − T∩l=j; k;:::;p Cl (Ai ) 1 − TCs (Ai ) ; WCs (Ai ) + ; Rmax (Ai ) = Max W∩l=j; k;:::;p Cl (Ai ) + 2 2 1 − T∩l=j; k;:::;p Cl (Ai ) 1 − TCs (Ai ) ; ; WCs (Ai ) − Rmin (Ai ) = Max W∩l=j; k;:::;p Cl (Ai ) − 2 2

(10)

Rcenter (Ai ) = Max(W∩l=j; k;:::;p Cl (Ai ); WCs (Ai )): There are many measures of rank for interval date. For example, max–min method, max–max method and max–center method. Let R(A1 ) = (p1l ; p1c ; p1r ); R(A2 ) = (p2l ; p1c ; p2r ); .. . l r ; p1c ; pm ): R(Am ) = (pm l , Max–min method: If R(Ai ) = (pil ; pic ; pir ) and pil is the largest value among the values p1l ; p2l ; : : : ; and pm then the alternative Ai is the decision-maker’s best choice. Max–max method: If R(Ai ) = (pil ; pic ; pir ) and pir is the largest value among the values p1r ; p2r ; : : : ; and r , then the alternative Ai is the decision-maker’s best choice. pm Max–center method: If R(Ai ) = (pil ; pic ; pir ) and pic is the largest value among the values p1c ; p2c ; : : : ; and c , then the alternative Ai is the decision-maker’s best choice. pm It is noted that max–center method is the same as that proposed by Chen et al. [5]. We rst consider the following critical example.

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Example 3.6. Let A1 , A2 and A3 be three alternatives, and let C1 ; C2 and C3 be three criteria. Assume that the characteristics of the alternatives are represented by the vague sets shown as follows: A1 = {(C1 ; [0; 1]); (C2 ; [0; 1]); (C3 ; [0; 1])}; A2 = {(C1 ; [0:5; 0:5]); (C2 ; [0:5; 0:5]); (C3 ; [0:5; 0:5])}; A3 = {(C1 ; [0:4; 0:7]); (C2 ; [0:4; 0:7]); (C3 ; [0:4; 0:7])}; and assume that the decision-maker wants to choose an alternative which satis es the criteria C1 and C2 or which satis es the criteria C3 , where the degree of importance of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, then by applying (8) – (10), we can easily get R(A1 ) = (−0:5; 0; 0:5);

R(A2 ) = (0; 0; 0);

R(A3 ) = (−0:05; 0:1; 0:25):

Therefore, we can see that applying the max–max method, alternative A1 is his best choice, applying the max–min method, alternative A2 is his best choice, and applying the max– center method, alternative A3 is his best choice. The following example is shown in [5]. Example 3.7. Let A1 ; A2 ; A3 ; A4 , and A5 be ve alternatives, and let C1 ; C2 ; and C3 be three criteria. Assume that the characteristics of the alternatives are represented by the vague sets shown as follows: A1 = {(C1 ; [0:5; 0:7]); (C2 ; [0:8; 0:9]); (C3 ; [0:3; 0:4])}; A2 = {(C1 ; [1; 1]); (C2 ; [0:7; 0:8]); (C3 ; [0:1; 0:2])}; A3 = {(C1 ; [0; 0]); (C2 ; [0:4; 0:5]); (C3 ; [0:8; 0:9])}; A4 = {(C1 ; [0:8; 0:9]); (C2 ; [0:1; 0:2]); (C3 ; [0:5; 0:6])}; A5 = {(C1 ; [0:7; 0:8]); (C2 ; [0:5; 0:6]); (C3 ; [0:1; 0:2])}; and assume that the decision-maker wants to choose an alternative which satis es the criteria C1 and C2 or which satis es the criteria C3 , where the degrees of importance of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, then by applying (8) – (10) and the result of [5], we can get WC1 ∩C2 (A1 ) = 0:35;

WC3 (A1 ) = − 0:3;

TC1 ∩C2 (A1 ) = 0:83;

TC3 (A1 ) = 0:9;

and hence R(A1 ) = (0:295; 0:35; 0:465): WC1 ∩C2 (A2 ) = 0:85;

WC3 (A2 ) = − 0:7;

TC1 ∩C2 (A2 ) = 0:97;

TC3 (A2 ) = 0:9;

TC1 ∩C2 (A3 ) = 0:97;

TC3 (A3 ) = 0:9;

and hence R(A2 ) = (0:835; 0:85; 0:865); WC1 ∩C2 (A3 ) = − 0:73;

WC3 (A3 ) = 0:7;

and hence R(A3 ) = (0:65; 0:7; 0:75); WC1 ∩C2 (A4 ) = 0:28;

WC3 (A4 ) = 0:1;

TC1 ∩C2 (A4 ) = 0:54;

TC3 (A4 ) = 0:9;


113

Table 2

Rmax Rmin Rcenter

A1

A2

A3

A4

A5

Best choice

0.465 0.295 0.35

0.865 0.835 0.85

0.75 0.65 0.7

0.51 0.05 0.28

0.43 0.33 0.38

A2 A2 A2

and hence R(A4 ) = (0:05; 0:28; 0:51); WC1 ∩C2 (A5 ) = 0:38;

WC3 (A5 ) = − 0:7;

TC1 ∩C2 (A5 ) = 0:9;

TC3 (A5 ) = 0:9;

and hence R(A5 ) = (0:33; 0:38; 0:43): From Table 2, applying all of these three methods, we can see that the alternative A2 is his best choice. 4. Conclusion In this paper, we have provided new functions to measure the degree of accuracy in the grades of membership of each alternative with respect to a set criteria to be represented by vague values for handling multicriteria fuzzy decision-making problems. We also used examples to illustrate the application of the proposed function. The proposed techniques can provide more useful ways than those of Chen et al. [5] to eciently help the decision-maker to make his decision. References [1] R. Bellman, L.A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17 (1990) 141–164. [2] S.M. Chen, A new approach to handling fuzzy decisionmaking problems, IEEE Trans. Systems Man Cybernet. 18 (1988) 1012–1016. [3] S.M. Chen, J.S. Ke, J.F. Chang, Techniques for handling multicriteria fuzzy decision-making problems, Proc. 4th Internat. Symp. on Computer and Information Sciences, Ceseme, Turkey, 1989, pp. 919–925. [4] S.M. Chen, J.S. Ke, J.F. Chang, An ecient algorithm to handle diagnostic problems, Cylbern, Systems Internat. J. 21 (1990) 377–387. [5] S.M. Chen, J.M. Tan, Handling multicriteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems 67 (1994) 163–172. [6] W.L. Gau, D.J. Buehrer, Vague sets, IEEE Trans. Systems Man Cybernet. 23 (1993) 610 – 614. [7] W.J.M. Kickert, Fuzzy Theories on Decision Making: A Critical Review, Kluwer, Boston, 1978. [8] P.J.M. Laarhoven, W. Pedrycz, A fuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems 11 (1983) 229–241. [9] H. Maeda, S. Murakami, A fuzzy decision-making method and its application to a company choice problem, Inform. Sci. 45 (1988) 331–346. [10] R.R. Yager, Fuzzy decision making including unequal objectives, Fuzzy Sets and Systems 1 (1978) 87–95. [11] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Systems Man Cybernet. 18 (1988) 183–190 [12] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338 –356. [13] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision process, IEEE Trans. Systems Man Cybernet. 3 (1973) 28 – 44. [14] H.J. Zimmermann, Fuzzy Sets, Decision Making, and Expert Systems, Kluwer Academic Publishers, Boston, 1987.