Computer-Aided Design 39 (2007) 164–169www.elsevier.com/locate/cad

Improved method of multicriteria fuzzy decision-making based onvague sets

Jun Ye∗

Department of Mechatronic Engineering, Shaoxing College of Arts and Sciences, 508 Huancheng West Road, Shaoxing, Zhejiang Province 312000, PR China

Received 1 March 2006; accepted 23 November 2006

Abstract

An improved method is presented, which provides improved score functions to measure the degree of suitability of each of a set of alternatives,with respect to a set of criteria presented with vague values. The improved algorithm for score functions is introduced by taking into account theeffect of an unknown degree (hesitancy degree) of the vague values on the degree of suitability to which each alternative satisfies the decision-maker’s requirement. The meaning of the proposed function is more transparent than that of other existing functions, which are not reasonable insome cases. The proposed function illustrates that it has stronger discrimination in comparison with previous functions. The applicability of thisimproved multicriteria fuzzy decision-making approach is also demonstrated by means of examples. The improved method can be used to rankthe decision alternatives according to the decision criteria. The functions proposed in this paper can provide a more useful technique than previousfunctions, in order to efficiently help the decision-maker.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Fuzzy set; Vague set; Multicriteria fuzzy decision-making

1. Introduction

On many occasions, an engineering decision needs to bemade based on available data and information that are vague,imprecise, and uncertain, by nature. The decision-makingprocess of engineering schemes in the conceptual design phaseis one of these typical occasions, which frequently calls uponsome method of treating uncertain data and information.

In a conceptual design phase, designers usually presentmany alternatives. However, the subjective characteristics of thealternatives are generally uncertain and need to be evaluatedbased on the decision-maker’s insufficient knowledge andjudgments. The nature of this vagueness and uncertaintyis fuzzy, rather than random, especially when subjectiveassessments are involved in the decision-making process. Fuzzyset theory offers a possibility for handing these sorts of data andinformation involving the subjective characteristics of humannature in the decision-making process.

Since the theory of fuzzy sets [1] was proposed in 1965, ithas been used for handling fuzzy decision-making problems

∗ Tel.: + 86 575 8327323.E-mail address: yejun@mail.sxptt.zj.cn.

0010-4485/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2006.11.005

[2–8,10,11]. Kickert [3] has discussed the field of fuzzymulticriteria decision-making. Zimmermann [5] illustrated afuzzy set approach to multi-objective decision-making, andhe has compared some approaches to solve multi-attributedecision-making problems based on fuzzy set theory. Yager [6]presented a fuzzy multi-attribute decision-making method thatuses crisp weights, and he [7] introduced an ordered weightedaggregation operator and investigated its properties. Laarhovenet al. [8] presented a method for multi-attribute decision-making using fuzzy numbers as weights.

Vague sets, which Cau and Buehrer [9] presented, area generalized form of fuzzy sets. These sets were used byChen and Tan [10]. They presented some new techniques forhandling multicriteria fuzzy decision-making problems basedon vague set theory, where the characteristics of the alternativesare represented by vague sets. The proposed techniques useda score function, S, to evaluate the degree of suitability towhich an alternative satisfies the decision-maker’s requirement.Recently, Hong and Choi [11] proposed an accuracy function,H , to measure the degree of accuracy in the grades ofmembership of each alternative, with respect to a set of criteriarepresented by vague values. However, in some cases, thesefunctions do not give sufficient information about alternatives.

J. Ye / Computer-Aided Design 39 (2007) 164–169 165

In this paper, an improved score function is proposed toefficiently evaluate the degree of suitability of each alternative,with respect to a set of criteria represented by vague values.The proposed function can provide another useful way to helpthe decision-maker. The rest of this paper is organized asfollows. In Section 2, we briefly describe vague set theoryand the main results of Chen and Tan [10] and Hong andChoi [11]. In Section 3, we propose an improved score functionand some measures to handle multicriteria fuzzy decision-making problems. We conclude with some final remarks of theimproved method in Section 4.

2. Vague set theory and its decision-making methods

2.1. Vague sets

The vague set, which is a generalization of the concept of afuzzy set, has been introduced by Gau and Buehrer [9]. Vagueset theory is introduced as follows.

A vague set, A(x), in X , X = {x1, x2, . . . , xn}, ischaracterized by a truth-membership function, tA, and a false-membership function, f A, for the elements xi ∈ X to A(x) ∈

X , (i = 1, 2, . . . , n),

tA : X → [0, 1], f A : X → [0, 1],

where the functions tA(xi ) and f A(xi ) are constrained by thecondition 0 ≤ tA(xi ) + f A(xi ) ≤ 1. The truth-membershipfunction, tA(xi ), is a lower bound on the grade of membershipon the evidence for xi , and f A(xi ) is a lower bound on thenegation of xi derived from the evidence against xi . Thegrade of membership of xi in the vague set, A, is boundedto a subinterval [tA(xi ), 1 − f A(xi )] of [0, 1]. The vaguevalue [tA(xi ), 1 − f A(xi )] indicates that the exact grade ofmembership µA(xi ) of xi may be unknown, but it is boundedby tA(xi ) ≤ µA(xi ) ≤ 1 − f A(xi ). Fig. 1 shows a vague set inthe universe of discourse, X .

Let X be the universe of discourse, X = {x1, x2, . . . , xn},xi ∈ X . A vague set, A, of X can be represented by

A =

n∑i=1

[tA(xi ), 1 − f A(xi )]/xi , xi ∈ X.

For example, let X = {6, 7, 8, 9, 10}. A vague set “LARGE”of X may be defined by

LARGE = [0.1, 0.2]/6 + [0.3, 0.5]/7+ [0.6, 0.8]/8 + [0.9, 1]/9 + [1, 1]/10.

Note that we will omit the argument xi of tA(xi ) and f A(xi )

throughout this paper, unless they are needed for clarity.

Definition 1. The intersection of two vague sets, A(x) andB(x), is a vague set, C(x), written as C(x) = A(x) ∧ B(x).The truth-membership and false-membership functions are tCand fC , respectively, where tC = min(tA, tB), and 1 − fC =

min(1 − f A, 1 − fB). That is,

[tC , 1 − fC ] = [tA, 1 − f A] ∧ [tB, 1 − fB]

= [min(tA, tB), min(1 − f A, 1 − fB)].

Fig. 1. A vague set.

Definition 2. The union of two vague sets, A(x) and B(x), is avague set, C(x), written as C(x) = A(x) ∨ B(x). The truth-membership function and false-membership functions are tCand fC , respectively, where tC = max(tA, tB), and 1 − fC =

max(1 − f A, 1 − fB). That is,

[tC , 1 − fC ] = [tA, 1 − f A] ∨ [tB, 1 − fB]

= [max(tA, tB), max(1 − f A, 1 − fB)].

2.2. Existing decision-making methods

This section reviews the proposed functions and somemeasures of Chen and Tan [10], and Hong and Choi [11], tohandle multi-criteria 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 arepresented by the vague set:

Ai = {(C1,[ti1, 1 − fi1]),

(C2,[ti2, 1 − fi2]), . . . , (Cn,[tin, 1 − fin])},

where ti j indicates the degree to which the alternative Aisatisfies criteria C j , fi j indicates the degree to which thealternative Ai does not satisfy criteria C j , ti j ∈ [0, 1], fi j ∈

[0, 1], ti j + fi j ≤ 1, 1 ≤ j ≤ n, and 1 ≤ i ≤ m. Lett∗i j = 1 − fi j , where 1 ≤ j ≤ n and 1 ≤ i ≤ m. In thiscase, Ai can be rewritten as

Ai = {(C1,[ti1, t∗i1]), (C2,[ti2, t∗i2]), . . . , (Cn,[tin, t∗in])},

where 1 ≤ i ≤ m. In this case, the characteristics of thesealternatives can be represented as shown in Table 1.

Assume that there is a decision-maker who wants tochoose an alternative that satisfies the criteria C j , Ck, . . . , andC p, or which satisfies the criteria Cs . This decision-maker’srequirements are represented by the following expression:

C j AND Ck AND · · · AND C p OR Cs .

In this case, the degrees to which the Ai satisfies or does notsatisfy the decision-maker’s requirements can be measured by

166 J. Ye / Computer-Aided Design 39 (2007) 164–169

Table 1Characteristics of the alternatives

C1 . . . C j . . . Ck . . . C p . . . Cs . . . Cn

A1 [t11, t∗11] · · · [t1 j , t∗1 j ] · · · [t1k , t∗1k ] · · · [t1p, t∗1p] · · · [t1s , t∗1s ] · · · [t1n , t∗1n ]

A2 [t21, t∗21] · · · [t2 j , t∗2 j ] · · · [t2k , t∗2k ] · · · [t2p, t∗2p] · · · [t2s , t∗2s ] · · · [t2n , t∗2n ]

.

.

.... · · ·

.

.

. · · ·

.

.

. · · ·

.

.

. · · ·

.

.

. · · ·

.

.

.

Ai [ti1, t∗i1] · · · [ti j , t∗i j ] . . . [tik , t∗ik ] . . . [ti p, t∗i p] . . . [tis , t∗is ] . . . [tin , t∗in ]

.

.

.... · · ·

.

.

. · · ·

.

.

. · · ·

.

.

. · · ·

.

.

. · · ·

.

.

.

Am [tm1, t∗m1] · · · [tmj , t∗mj ] · · · [tmk , t∗mk ] · · · [tmp, t∗mp] · · · [tms , t∗ms ] · · · [tmn , t∗mn ]

the evaluation function E ,

E(Ai ) = ([ti j , t∗i j ] ∧ [tik, t∗ik] ∧ · · · ∧ [ti p, t∗i p]) ∨ [tis, t∗is]

= [min(ti j , tik, . . . , ti p), min(t∗i j , t∗ik, . . . , t∗i p)]

∨[tis, t∗is]

= [max(min(ti j , tik, . . . , ti p), tis),

max(min(t∗i j , t∗ik, . . . , t∗i p), t∗is)]

= [tAi , t∗Ai] = [tAi , 1 − f Ai ], (1)

where ∧ and ∨ denote the minimum operator and the maximumoperator of the vague values, respectively; E(Ai ) is a vaguevalue, 1 ≤ i ≤ m; tAi and t∗Ai

are as follows:

tAi = max(min(ti j , tik, . . . , ti p), tis),

t∗Ai= max(min(t∗i j , t∗ik, . . . , t∗i p), t∗is).

Chen and Tan [10] presented a score function S to measurethe degree of suitability of each alternative, with respect to a setof criteria presented by vague values.

Let x = [tx , 1 − fx ] be a vague value, where tx ∈ [0, 1],fx ∈ [0, 1], tx + fx ≤ 1. The score of x can be evaluated by thescore function, S, which is shown as follows:

S(x) = tx − fx , (2)

where S(x) ∈ [−1, +1]. Based on the score function, S, thedegree of suitability to which the alternative Ai satisfies thedecision-maker’s requirements can be measured as follows:

S(E(Ai )) = tAi + t∗Ai− 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 (1 ≤ i ≤ m)satisfies the decision-maker’s requirement. 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 valuesp1, p2, . . . , and pm , then the alternative Ai is the best choice.

Assuming that the characteristics of the alternatives areshown in Table 1, and assuming that there is a decision-maker who wants to choose an alternative which satisfies thecriteria C j , Ck, . . . , and C p, or which satisfies the criteria Cs ,

the decision-maker’s requirements can be represented by thefollowing expression:

C j AND Ck AND · · · AND C p OR Cs .

Assume that the weights of the criteria C j , Ck, . . . , and C p,presented by the decision-maker, are w j , wk, . . . , and wp,respectively, where w j ∈ [0, 1], wk ∈ [0, 1], . . . , wp ∈ [0, 1],and w j + wk + · · · + wp = 1. Then, the degree of suitabilityto which the alternative Ai satisfies the decision-maker’srequirements can be measured by the weighting function W ,

W (Ai ) = max{S([ti j , t∗i j ]) ∗ w j + S([tik, t∗ik]) ∗ wk

+ · · · + S([ti p, t∗i p]) ∗ wp, S([tis, t∗is])}. (4)

By applying Eq. (3), Eq. (4) can be rewritten as

W (Ai ) = max{(ti j + t∗i j − 1) ∗ w j + (tik + t∗ik − 1) ∗ wk

+ · · · + (ti p + t∗i p − 1) ∗ wp, (tis + t∗is − 1)}, (5)

where W (Ai ) ∈ [−1, +1] and 1 ≤ i ≤ m. Let

W (A1) = p1,

W (A2) = p2,

...

W (Am) = pm .

If W (Ai ) = pi , and pi is the largest value among the valuesp1, p2, . . . , and pm , then the alternative Ai is the best choice.

We now consider the following example.

Example 2.1. Let A1 and A2 be two alternatives, and let C1,C2, and C3 be three criteria. Assuming that the characteristicsof the alternatives are represented by the vague sets

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 assuming that the decision-maker wants to choose analternative that satisfies the criteria C1 and C2, or which satisfiesthe criteria C3, then by applying Eq. (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 Eq. (3), we can get

S(E(A1)) = 0 + 1 − 1 = 0,

S(E(A2)) = 0.5 + 0.5 − 1 = 0.

We do not know which is better.

J. Ye / Computer-Aided Design 39 (2007) 164–169 167

In Example 2.1, if the weights of the criteria C1 and C2entered by the decision-maker are 0.7 and 0.3, respectively, thenby applying Eq. (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 is better. What is thedifference between alternative A1 and alternative A2? Here,some other measures are needed.

In the above case, Hong and Choi [11] presented an accuracyfunction, H , to evaluate the degree of accuracy in the grades ofmembership of each alternative, with respect to a set of criteriarepresented by vague values.

Let x = [tx , 1 − fx ] be a vague value, where tx ∈ [0, 1],fx ∈ [0, 1], tx + fx ≤ 1. The degree of accuracy of x can beevaluated by the accuracy function, H , as follows:

H(x) = tx + fx , (6)

where H(x) ∈ [0, 1]. The larger the value of H(x), the greaterthe degree of accuracy of the grade of membership of the vaguevalue.

Based on the accuracy function, the degree of accuracy inthe grades of membership of the alternative Ai , which satisfiesthe decision-maker’s requirements, can be measured as follows:

H(E(Ai )) = tAi + 1 − t∗Ai, (7)

where H(E(Ai )) ∈ [0, 1]. The larger the value of H(E(Ai )),the greater the degree of accuracy in the grades of membershipof the alternative Ai .

In Example 2.1, we have obtained E(A1) = [0, 1] andE(A2) = [0.5, 0.5], with S(E(A1)) = S(E(A2)) = 0. Then,by applying Eq. (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 ofaccuracy than the alternative A1 in terms of grades ofmembership. Alternative A2 is thus the better choice.

Example 2.2. Let A1 and A2 be two alternatives, and let C1,C2, and C3 be three criteria. Assume that the characteristics ofthe alternatives are represented by the vague sets

A1 = {(C1, [0.2, 0.7]), (C2, [0.2, 0.7]), (C3, [0.2, 0.7])},

A2 = {(C1, [0.3, 0.8]), (C2, [0.3, 0.8]), (C3, [0.3, 0.8])},

and assume that the decision-maker wants to choose analternative that satisfies the criteria C1 and C2, or which satisfiesthe criterion C3; then by applying Eq. (1), we obtain

E(A1) = ([02, 0.7] ∧ [0.2, 0.7] ∨ [0.2, 0.7]) = [0.2, 0.7],

E(A2) = ([0.3, 0.8] ∧ [0.3, 0.8] ∨ [0.3, 0.8]) = [0.3, 0.8].

By applying Eq. (7), we obtain

H(E(A1)) = 0.2 + 1 − 0.7 = 0.5,

H(E(A2)) = 0.3 + 1 − 0.8 = 0.5.

In this case, we still do not know which alternative is better.

3. Improved decision-making method

From the examples in Section 2, we can see that thesefunctions are not reasonable in some cases. This situation canbe problematic in a practical application. To overcome thisproblem, we propose an improved score function based on theunknown degrees of the vague values.

Definition 3. Let x be a vague value, x = [tx , 1 − fx ], wheretx ∈ [0, 1], fx ∈ [0, 1], tx + fx ≤ 1. The unknown degree,or hesitancy degree, of x is denoted by mx , and is defined bymx = 1 − tx − fx , and 0 ≤ mx ≤ 1.

For example, let A be a vague set with truth-membershipfunction, tA, and false-membership function, f A. If [tA, 1 −

f A] = [0.5, 0.7], then we can see that tA = 0.5, f A = 0.3,and m A = 0.2. This result can be interpreted as “the vote forresolution is 5 in favor, 3 against, and 2 abstentions”.

In the following, we propose an improved score function.Let x = [tx , 1 − fx ] be a vague value, where tx ∈ [0, 1],

fx ∈ [0, 1], mx ∈ [0, 1], and tx + fx ≤ 1. The score of x canbe evaluated by the modified score function, J , as follows:

J (x) = tx − fx + µmx = tx (1 − µ) + t∗x (1 + µ) − 1, (8)

where J (x) ∈ [−1, +1], and µ ∈ [−1, +1]. The parameter µ

is introduced by taking into account the effect of the unknowndegree mx on the score of x , and can be chosen according toactual cases. In Eq. (8), if µ > 0, the value of the third termµmx of J (x) is positive, and it tends to increase the score of xdue to its addition in the first term tx ; if µ < 0, the value of thethird term µmx is negative, then it tends to decrease the scoreof x due to its addition to the second item fx ; if µ = 0, thenthe improved score function J (x) is the same score functionas S(x) proposed by Chen et al. [10]. In the renewed vote forresolution, µmx can also be interpreted as: most abstentionsmay be in favor of µ > 0, most of them for µ < 0, or they stillkeep abstentions with µ = 0.

Based on Eqs. (1) and (8), the degree of suitability to whichthe alternative Ai satisfies the decision-maker’s requirementscan be measured as follows:

J (E(Ai )) = tAi (1 − µi ) + t∗Ai(1 + µi ) − 1, (9)

where J (E(Ai )) ∈ [−1, +1] and µi ∈ [−1, +1]. The larger thevalue of J (E(Ai )), the higher the degree of suitability to whichthe alternative Ai satisfies the decision-maker’s requirements.

In Example 2.1, we have obtained E(A1) = [0, 1] andE(A2) = [0.5, 0.5], with S(E(A1)) = S(E(A2)) = 0. If µ1and µ2 are 0.5 and 0.5, respectively, then by applying Eq. (9),we get

J (E(A1)) = 0(1 − 0.5) + 1(1 + 0.5) − 1 = 0.5,

J (E(A2)) = 0.5(1 − 0.5) + 0.5(1 + 0.5) − 1 = 0.

168 J. Ye / Computer-Aided Design 39 (2007) 164–169

Therefore, the alternative A1 is better than the alternative A2.In Example 2.2, we have obtained E(A1) = [0.2, 0.7] and

E(A2) = [0.3, 0.8], with H(E(A1)) = H(E(A2)) = 0.5.Similarly, choosing µ1 = µ2 = 0.5 and using Eq. (9), we canget

J (E(A1)) = 0.2(1 − 0.5) + 0.7(1 + 0.5) − 1 = 0.15,

J (E(A2)) = 0.3(1 − 0.5) + 0.8(1 + 0.5) − 1 = 0.35.

Obviously, the alternative A2 has a greater degree ofsuitability than the alternative A1. Therefore, the alternative A2is the better choice.

This measurement method provides additional usefulinformation to efficiently help the decision-maker. Thefollowing five theorems can be introduced according to theproposed measure function.

Theorem 1. Let a = [a1, a2] be a vague value, and J be thescore function. Then J (a) = 0 if and only if a1(1−µa)+a2(1+

µa) = 1, where µa ∈ [−1, +1].

Theorem 2. Let a = [a1, a2] be a vague value, and J be thescore function. Then J (a) = 1 if and only if a1(1−µa)+a2(1+

µa) = 2, where µa ∈ [−1, +1].

Theorem 3. Let a = [a1, a2] and b = [b1, b2] be two vaguevalues, and J be the score function. Then J (a) ≥ J (b) if andonly if a1(1 − µa) + b2(1 + µb) ≥ b1(1 − µb) + a2(1 + µa),where µa ∈ [−1, +1] and µb ∈ [−1, +1].

Theorem 4. Let a = [a1, a2], b = [b1, b2], and c = [c1, c2]

be three vague values, and J be the score function. ThenJ (a ∧ c) ≥ J (b ∧ c) if and only if min(a1(1 − µa), c1(1 −

µc))+ min(b2(1 +µb), c2(1 +µc)) ≥ min(b1(1 −µb), c1(1 −

µc)) + min(a2(1 + µa), c2(1 + µc)), where µa ∈ [−1, +1],µb ∈ [−1, +1], and µc ∈ [−1, +1].

Theorem 5. Let a = [a1, a2], b = [b1, b2] and c = [c1, c2]

be three vague values, and J be the score function. ThenJ (a ∨ c) ≥ J (b ∨ c) if and only if max(a1(1 − µa), c1(1 −

µc))+max(b2(1+µb), c2(1+µc)) ≥ max(b1(1−µb), c1(1−

µc)) + max(a2(1 + µa), c2(1 + µc)), where µa ∈ [−1, +1],µb ∈ [−1, +1], and µc ∈ [−1, +1].

The above theorems are easy to verify. The proofs are similarto those of the theorems in [10] (omitted).

In the following, we present a weighted techniquefor handling multi-criteria fuzzy decision-making problems.Assuming that there is a decision-maker who wants to choosean alternative that satisfies the criteria C = {C1, C2, . . . , Ck},or which satisfies the criteria D = {D1, D2, . . . , Dt }, thedecision-maker’s requirements can be represented by thefollowing expression:

{C1 AND C2 AND · · · AND Ck}

OR {D1 AND D2 AND · · · AND Dt }

Assume that the weights of the criteria C = {C p|p =

1, 2, . . . , k} and D = {Dq |q = 1, 2, . . . , t}, entered bythe decision-maker, are W = {wp|p = 1, 2, . . . , k} and

V = {vq |q = 1, 2, . . . , t}, respectively, where wp ∈ [0, 1],∑kp=1 wp = 1, vq ∈ [0, 1], and

∑tq=1 vq = 1; and

assume that the decision-maker enters the parameters µCp(p = 1, 2, . . . , k) and µDq (q = 1, 2, . . . , t), where µCp ∈

[−1, +1] and µDq ∈ [−1, +1]; then the degree to which thealternativeAi satisfies the decision-maker’s requirements can bemeasured by the weighting function R,

R(Ai ) = max{J ([tiC1 , t∗iC1]) ∗ w1 + J ([tiC2 , t∗iC2

]) ∗ w2

+ · · · + J ([tiCk , t∗iCk]) ∗ wk,

J ([ti D1 , t∗i D1]) ∗ v1 + J ([ti D2 , t∗i D2

]) ∗ v2

+ · · · + J ([ti Dt , t∗i Dt]) ∗ vt }. (10)

By applying Eq. (9), Eq. (10) can be rewritten into

R(Ai ) = max{(tiC1(1 − µiC1) + t∗iC1(1 + µiC1) − 1) ∗ w1

+ (tiC2(1 − µiC2) + t∗iC2(1 + µiC2) − 1) ∗ w2

+ · · · + (tiCk (1 − µiCk ) + t∗iCk(1 + µiCk ) − 1) ∗ wk,

(ti D1(1 − µi D1) + t∗i D1(1 + µi D1) − 1) ∗ v1

+ (ti D2(1 − µi D2) + t∗i D2(1 + µi D2) − 1) ∗ v2

+ · · · + (ti Dt (1 − µi Dt ) + t∗i Dt(1 + µi Dt ) − 1) ∗ vt }, (11)

where R(Ai ) ∈ [−1, +1] and 1 ≤ i ≤ m. Let

R(A1) = p1,

R(A2) = p2,

...

R(Am) = pm .

If R(Ai ) = pi and pi is the largest value among the values p1,p2, . . . , and pm , then the alternative Ai is the best choice.

The following example is shown in [10,11].

Example 3.1. Let A1, A2, A3, A4, and A5 be alternatives,and let C1, C2, and D1 be three criteria. Assuming that thecharacteristics of the alternatives are represented by the vaguesets:

A1 = {(C1, [0.5, 0.7]), (C2, [0.8, 0.9]), (D1, [0.3, 0.4])},

A2 = {(C1, [1, 1]), (C2, [0.7, 0.8]), (D1, [0.1, 0.2])},

A3 = {(C1, [0, 0]), (C2, [0.4, 0.5]), (D1, [0.8, 0.9])},

A4 = {(C1, [0.8, 0.9]), (C2, [0.1, 0.2]), (D1, [0.5, 0.6])},

A5 = {(C1, [0.7, 0.8]), (C2, [0.5, 0.6]), (D1, [0.1, 0.2])}.

Assume that the decision-maker wants to choose an alternativethat satisfies the criteria C1 AND C2, or which satisfies thecriterion D1, where the weights of the criteria C1 and C2entered by the decision-maker are 0.7 and 0.3, respectively, andµiC1, µiC2, and µi D1 are 0.5, 0.5 and 0.5 (i = 1, 2, 3, 4, 5),respectively.

By applying Eq. (8), we get

J [0.5, 0.7] = 0.3000, J [0.8, 0.9] = 0.7500,

J [0.3, 0.4] = −0.2500.

J. Ye / Computer-Aided Design 39 (2007) 164–169 169

By applying Eqs. (10) and (11), we obtain

R(A1) = max(J [0.5, 0.7] ∗ 0.7 + J [0.8, 0.9] ∗ 0.3,

J [0.3, 0.4]) = 0.4350.

By the same calculation, we get

R(A2) = 0.8650, R(A3) = 0.7500,

R(A4) = 0.3300, R(A5) = 0.4300.

Obviously, the order of quality is R(A2), R(A3), R(A1),R(A5), and R(A4).

Therefore, we can see that the alternative A2 is the bestchoice. This result is the same as in [10,11].

4. Conclusions and further research

In this paper, we have provided improved score functionsto measure the degree of suitability of each alternative, withrespect to a set of criteria to be represented by vague values forhandling multi-criteria fuzzy decision-making problems. Wealso used examples to illustrate the application of the proposedfunction. The improved method can be used to rank the decisionalternatives according to the decision criteria, and then thetechniques proposed in this paper can provide more usefulapproaches than those of Cheng and Tan [10], and Hong andChoi [11], to efficiently help the decision-maker. But in theimproved method, a new problem is how to choose a reasonableparameter µ for the improved score function. This issue will beinvestigated in future work.

Acknowledgment

This work was supported by Natural Science Foundation ofZhejiang Province, PR China under Grant No. M603070.

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