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Handling multicriteria fuzzy decision-making problems based on vague set theory
FUZ '¥
sets and systems ELSEVIER
Fuzzy Sets and Systems 67 (1994) 163-172
Handling multicriteria fuzzy decision-making problems based on vague set theory Shyi-Ming Chen*, Jiann-Mean Tan Department of Computer and Information Science. National Chiao Tung University, Hsinchu, Taiwan, ROC Received December 1993; revised May 1994
Abstract
New techniques for handling multicriteria fuzzy decision-making problems based on vague set theory are presented. The proposed techniques allow the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria to be represented by vague values. Furthermore, the proposed techniques allow the decision-maker to assign a different degree of importance to each criteria. The techniques proposed in this paper can provide a useful way to efficiently help the decision-maker to make his decisions. Key words: False-membership function; Fuzzy set; Multicriteria fuzzy decision-making; Truth-membership function; Vague set
1. Introduction
In 1965, Zadeh presented the theory of fuzzy sets [18]. In recent years, fuzzy set theory has been used for handling fuzzy decision-making problems [1, 3, 6, 7, 12-14, 17, 19, 20]. Roughly speaking, a fuzzy set is a class with fuzzy boundaries. A fuzzy set A of the universe of discourse U, U = {u~, u2 . . . . . u,}, is a set of ordered pairs, {(Ul,#A(Ut)),(Uz,#A(U2)) . . . . . (U,,#A(U,))}, where /~A is the membership function of the fuzzy set A, /~A: U ~ [0, 1], and #A(Ul) indicates the grade of membership of ul in A; Vui ~ U, the membership value/~A (u~) is a single value between zero and one. G a u et al. [9] pointed out that this single value combines the evidence for u~ e 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, G a u et al. [9] presented the concepts of vague sets. They used a truth-membership function tA and false-membership functionfA to characterize the lower bounds on #a. These lower bounds are used to create a subinterval on [0, 1], namely IrA(ui), 1 - f A (Ul)], to generalize the #a(Ui) of fuzzy sets, where tA(ui) <~ #A(U~) <<. 1 --fA(U~). For example, let A be a vague set with truth-membership function ta and false-membership function fA, respectively. If [tA(u~), 1--fA(U~)] =
* Corresponding author. 0165-0114/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0165-01 14(94)001 57-3
S.-M. Chen, J.-M. Tan / Fuzzy Sets and Systems 67 (1994) 163-172
164
[0.5, 0.7], then we can see that tA(U~) = 0.5; 1 --fa(U~) = 0.7; fA(Ui) = 0.3. It can be interpreted as "the degree that object ui belongs to the vague set A is 0.5; the degree that object u~ does not belong to the vague set A is 0.3". As another example, in a voting model, the vague value [-0.5,0.7] can be interpreted as "the vote for resolution is 5 in favor, 3 against, and 2 abstentions". Kickert [12] pointed out that the multicriteria decision-making problem is a kind of problem that all the alternatives in the choice set can be evaluated according to a number of criteria; he also pointed out that the problem is to construct an evaluation procedure to rank the set of alternatives in order of preference. Kickert [12, pp. 61-84] has discussed the field of fuzzy multicriteria decision-making. Zimmermann [20] has pointed out that two major areas have evolved both of which concentrate on multicriteria decision-making: multiobjective decision-making and multiattribute decision-making. The multiobjective decision-making problem is often called the "vector-maximum" problem. Zimmermann [20] illustrated a fuzzy set approach to multiobjective decision-making. Yager [17] presented a fuzzy multiattribute decision-making method using crisp weights. Laarhoven et al. [13] presented a method for multiattribute decision-making using fuzzy numbers as weights. Zimmermann [20, pp. 125-192] has compared some approaches to solve multiattribute decision problems based on fuzzy set theory. For more details, please refer to [20]. In this paper, we present some new techniques for handling multicriteria fuzzy decision-making problems based on vague set theory, where the characteristics of the alternatives are represented by vague sets. Because the proposed techniques use the truth-membership function and the false-membership function to indicate the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria, respectively, and because the proposed techniques can allow each criteria to have a different degree of importance, the techniques proposed in this paper can provide a useful way to efficiently help the decision-maker to make his decisions.
2. Vague sets Let U be the universe of discourse, U = {Ul, u2 .... , u,}, with a generic element of U denoted by u~. A vague set A in U is characterized by a truth-membership function tA and a false-membership function fa,
tA: U
~
[0, 1],
(1)
fA : U ~
[0, 1],
(2)
where ta(Ui) is a lower bound on the grade of membership of u~ derived from the evidence for u~, fA(ui) is a lower bound on the negation of ui derived from the evidence against ug, and tA(U~) + fa(u~) ~< 1. The grade of membership of ui in the vague set A is bounded to a subinterval [tA(U~), 1 --fa(U~)] of [0, 1]. The vague value [tA(Ul), 1 --j~(U~)] indicates that the exact grade of membership I~a(U~)of ui may be unknown, but is bounded by tA(ui) <<,#n(Ui) ~< 1 --fa(ui), where tA(ui) +fA(Ui) ~< 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
A = fvEtA(ui), 1 -fA(ui)]/u~.
(3)
When the universe of discourse U is discrete, a vague set A can be written as
A = ~ EtA(Ul), 1 --fa(Ui)]/Ui. i=1
(4)
S.-M. Chen, J.-M. Tan / Fuzzy Sets and Systems 67 (1994) 163-172
~ l A(U~t) ~ . _ U~
165
tA(Ud
"~U
Fig. 1. A vague set.
F o r example, let U be the universe of discourse, U = {6, 7, 8, 9, 10}. A vague set " L A R G E " of U may be defined by L A R G E = [0.1,0.2]/6 + [0.3,0.5]/7 + [-0.6,0.8]/8 + [0.9, 1]/9 + [1, 1]/10. Definition 2.1. Let x be a vague value, x = [t~, 1 - f x ] , where tx e [ 0 , 1],f~ e [ 0 , 1], tx + f ~ ~ 1. The complement of the vague value x is denoted by x' and is defined by x'-
[f~, 1 -- tx].
(5)
Definition 2.2. Let x and y be two vague values, x = [t~, 1 - f x ] and y = [ty, 1 - f ~ ] , where tx ~[0, 1], tye [0, 1],fx ~ [0, 1],fy e [0, 1], t~ +fx ~< 1, and ty + f y ~< l. The result of the minimum operation of the vague values x and y is a vague value z, written as z = x @y = [t=, 1 - f z ] , where G -- Min(G, ty),
(6)
1 - f z = Min(1 - f ~ , 1 - f y ) .
(7)
Definition 2.3. Let x and y be two vague values, x = [t~, 1 - f x ] and y = [ty, 1 - ~ ] , where tx ~[0, 1], ty e l 0 , 1],fx e [ 0 , 1], a n d f y e l 0 , 1], tx + f , ~< 1, and tr + f y ~< 1. The result of the m a x i m u m operation of the vague values x and y is a vague value c, written as c = a Q b = [tc, 1 - f ~ ] , where tc = M a x ( t , , tr),
(8)
1 - f ~ = Max(1 - f x , 1 -fy).
(9)
Definition 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 false-membership function ts and fn, respectively. The notions of complement, union, and intersection of vague sets are defined as follows. Definition 2.5. The complement of the vague set A is denoted by A' whose truth-membership function and false-membership function are ta, and fA,, respectively, where Vui ~ U,
t~,(u,) = fA (u,)
(lO)
1 --fa'(ui) = 1 -- ta(ui).
(l 1)
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166
Definition 2.6. The union 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 ~'u~ ~ U, tc(Ui) = MaX(tA(ui), tB(Ui)),
(12)
1 - - f c ( U , ) = Max(1 --fa(u,), 1 --fB(U,)).
(13)
That is, Vu~ E U, Etc(Ui), 1 - fc(ui)] = [tA(Ui), 1 -- fA(Ui)] Q [tB(Ui), 1 --f,(ul)] = [MaX(tA(Uz), tB(U~)), Max(1 --fa(u~), 1 --fB(U~))].
(14)
Definition 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 Vu~ E U, tc(Ui) = Min(tA(ui), tB(ui)),
(15)
1 --fc(U~)= Min(1 --fA(u~), 1 --fB(u~)).
(16)
That is, Vui E U, [tc(Ui), 1 - fc(ui)] = [tA(Ui), 1 -- fA(U,)] ® [tB(Ui), 1 -- fB(ui)] = [Min(tA(U~), tB(U~)), Min(1 --fA(U~), 1 --fn(U~))].
(17)
3. Handling multicriteria fuzzy decision-making problems In this section, we present some new techniques for handling multicriteria fuzzy decision-making problems, where the characteristics of the alternatives are represented by vague sets. Let A be a set of alternatives and let C be a set of criteria, where A = {A,, A2 ..... A,,}, C = {C1, C2 . . . . . C,}. Assume that the characteristics of the alternative A~ is represented by the vague set shown as follows: Ai = {(C1, [t~l,(1 --f~)]), (C2, [t~z,(1 --f2)]) . . . . . (C,, [t~,,(1 -f~,)])},
(18)
where tij indicates the degree that the alternative A~ satisfies criteria C j, f~s indicates the degree that the alternative Ai does not satisfy criteria Cj, t~s ~[0, 1],f~j ~ [0, 1], t~j + f j ~< 1, 1 ~
{(C,, F t , , t*l]), (C2, [t,2, t*]), ..., (C., [t,,, t,.])},
(19)
where 1 ~< i ~< m. In this case, the characteristics of these alternatives can be represented by a table shown in Table 1. Assume that there is a decision-maker who wants to choose an alternative which satisfies the criteria Cj, Ck . . . . . and Cp or which satisfies the criteria Cs, then we can represent the decision-maker's requirement by the following expression: Cj AND
C k AND
... AND Cp OR Cs.
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167
Table 1 The characteristics of the alternatives C1
...
Cj
A, A2
[tl,,t*,] [t21,t~l]
... ---
[tlj, [tzj,
:
:
...
A,
[t,l, t*, ]
...
:
:
...
Am
[tml,
t*r.tl ]
.
...
Ck
...
Cp
t~j]
...
[tlk,t*k]
...
[ttp,
t~j]
... ...
[t2k,t~k] :
... ...
[tq, t*]
...
[t,k, t*]
...
...
:
...
[tmk, trek* ]
...
: : .
. [tmj, . . " tmj] .*
...
Cs
...
C,,
t*p]
...
[tl~,t*~]
...
[tl,,t*.]
[t2p, t*p] :
... ...
[tz~,t*,] :
... ...
[t2,,t*,] :
E%, tTp]
...
Et,~, r*,s]
...
Et,,,'t~',]
:
...
:
...
[t,,p, t*mp]
.
.
.
[tm~, . . t,,~] .*
:
[t,.,, " t,,.] *
In this case, the degrees that the alternative A/satisfies and not satisfies the decision-maker's requirement can be m e a s u r e d by the evaluation function E, E(Ai) = ([tij, t*] ® [tik, t~] ®...
® [t,p, t * ] ) © [t,s, t*]
= [Min(t~j, t~k. . . . . t,p), Min(t*, ti~ . . . . . t'p)] ©[t~,, t~*] [Max(Min(ti), tik .... , tip), tis), M a x ( M i n ( t * , t* . . . . . t'p), t ~* ) ]
=
(20)
= [_ta,, t*],
where ® and © denote the m i n i m u m o p e r a t o r and the m a x i m u m o p e r a t o r of the vague values, respectively, E(Ai) is a vague value, 1 ~< i ~< m, and tn, = Max(Min(ti), tik ..... tip), tis),
(21)
t* = M a x ( M i n ( t * , t~ ..... t*), tis).*
(22)
Let t* = 1 - f a , , where f~, = 1 - M a x ( M i n ( t * , ti* .... , tip), tis).
(23)
In this case, (20) can be rewritten as E(AI) = [tA,, 1 -- f a , ] .
(24)
In the following, we present a score function S to evaluate the degree of suitability that an alternative satisfies the decision-maker's requirement. Let x = Its, 1 - f ~ ] be a vague value, where tx ~ [0, 1], J~, E [0, 1], tx + f , ~< 1. The score of x can be evaluated by the score function S shown as follows: S(x) = tx - f~,
(25)
where S(x) ~ [ - 1, + 1]. Based on the score function S, the degree of suitability that the alternative Ai satisfies the decision-maker's requirement can be measured. F r o m (20) and (24), we can see that E(Ai)= [tA,, t A* , ] = [-tA,, 1 - - f A , ] , where tA, = Max(Min(ti), rig ..... tip), t~s) t],
=
M a x ( M i n ( t * , tik* . . . . .
f,~,
=
1
--
* tip),
M a x ( M i n ( t * , tik* ....
* tis) * , tip),
* tis).
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S.-M. Chen, J.-M. Tan / Fuzzy Sets and Systems 67 (1994) 163-172
Thus, by applying (25), we can get S(~(A,)) = t A , - A , =
Max(Min(tij, tlk . . . . . tip), tis)
-
(1 - Max(Mm(tij, " * ti~ * . . . . . t*), tis)) *
= Max(Min(ti i, tik . . . . . tip), t,~) + Max(Min(t*, t* . . . . . t*), ti*) - 1 = ta, + t * , - 1,
(26)
where S(E(A~)) E [ - 1, + 1]. The larger the value of S(E(Ai)), the more the suitability that the alternative Ai satisfies the decision-maker's requirement, where 1 ~< i ~< m. Let S ( E ( A 1 ) ) = Pl
S(E(A2)) = P2
S(E(Am)) = p . .
If S(E(A~)) = p~ and Pi is the largest value among the values PI, P2 . . . . . and Pro, then the alternative Ai is his best choice. Example 3.1. Let A be a set of alternatives and C be a set of criteria, where
A = {A1,Az,A3,A4,A5}, C = { C I , C 2 , C3}.
Assume that the characteristics of the alternatives are represented by the vague sets shown as follows: A1 = {(C,, [0, 0]), (Cz, [0.8, 0.9]), (C3, [0.3, 0.4])}, A 2 = { ( C l , El.0, 1.0]), ( 6 2 , [0.6, 0.7]), (C3, [0.1, 0.2])},
A3 = {(C1, [0, 0]), (C2, [0.3, 0.4]), (C3, [0.8, 0.9])}, A , = {(C~, [0.7, 0.8]), (Cz, [0.1, 0.2]), (C3, [0.5, 0.6])}, A5 = {(C,, [0.8, 0.9]), (C2, [0.5, 0.6]), (C3, [0.3, 0.4])}, and assume that the decision-maker wants to choose an alternative which satisfies the criteria C1 and C2 or which satisfies the criteria C3, then the decision-maker's requirement can be expressed by an expression as follows: C1 A N D C2 O R C3. By applying (20), we can get
E ( A , ) = ([0,0] @[0.8,0.9])@[0.3,0.4] = [0.3,0.4], E(A2) = .([1.0, 1.0] @ [0.6, 0.7]) @[0.i, 0.2] = [0.6, 0.7], E(A3) = ([0,0] ®[0.3,0.4]) @[0.8,0.9] = [0.8,0.9],
E(A4) = ([0.7,0.8] @[0.1,0.2]) @[0.5,0.6] = [0.5,0.6], E(As) = ([0.8,0.9] @[0.5,0.6]) @[0.3,0.4] = [0.5,0.6].
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169
By a p p l y i n g (26), we c a n get
S(E(A1)) = 0.3 + 0.4 - 1 = --0.3, S(E(A2)) = 0.6 + 0.7 - 1 = 0.3, S(E(A3)) = 0.8 + 0.9 - 1 = 0.7, S(E(A,)) = 0.5 + 0.6 - 1 = 0.1, S(E(A5)) = 0.5 + 0.6 - 1 = 0.1. Therefore, we c a n see t h a t the a l t e r n a t i v e A3 is his best choice. T h e o r e m 3.1. Let a and b be two vague values, and S be the score function, where a = [ a l , a 2 ] , b = [ b l , b 2 ] . I f
S(a) >~ S(b), then al + a2 >~ bl + b2. P r o o f . By a p p l y i n g (25), we c a n get
S(a) = al + a2 - 1 S(b) = bl + b2 - -
1.
If S(a) >~ S(b), t h e n we c a n see t h a t al+a2-1~>
b l + b2 - 1.
T h i s yields
at + a2 ~ bl + b2.
[]
T h e o r e m 3.2. Let a, b, and c be three vague values, and S be the score function, where a = [ a l , a2], b = [ b l , b2], and c = [ c t , c 2 ] . I f S ( a Qc) >1 S(b Qc), then M i n ( a l , c l ) + M i n ( a 2 , c 2 ) / > M i n ( b t , c l ) + M i n ( b 2 , c 2 ) . P r o o f . By a p p l y i n g (6) a n d (7), we c a n get
a ® c = [al,a2] @ [ c l , c 2 ] = [ M i n ( a l , c l ) , M i n ( a 2 , c 2 ) ] b @ c = [bl,b2] ® [ c l , c 2 ] = [ M i n ( b l , C l ) , M i n ( b 2 , c 2 ) ] . By a p p l y i n g (26), we c a n get
S(a @ c) = M i n ( a l , c l ) + M i n ( a 2 , c 2 ) - 1 S(b ® c) = M i n ( b l , c l ) + M i n ( b 2 , c 2 ) - 1. If S(a Q c) >1 S(b ® c), t h e n we c a n see t h a t M i n ( a l , c l ) + M i n ( a 2 , c 2 ) -- 1 > / M i n ( b l , c l )
+ M i n ( b 2 , c 2 ) - 1.
T h i s yields Min(at,ct) + Min(a2,c2) >/Min(bl,Cl) + Min(b2,c2).
[]
T h e o r e m 3.3. Let a, b and c be three vague values, where a = [ a l , a 2 ] , b = [ b l , b 2 ] , and c -- [-cx,c2]. I f S(a 63 c) >~ S(b 63c), then M a x ( a t , c 1 ) + M a x ( a 2 , c 2 ) / > M a x ( b l , c i ) + M a x ( b 2 , c 2 ) .
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Proof. By applying (8) and (9), we can get a @ c = [aa, a2] Q [Ca, c2] = [Max(aa, c~), Max(a2, c2)] b @ c = [bl, b2] @ [Ca, c2] = [Max(bl, cl), Max(b2, c2)]. By applying (26), we can get S(a @c) = M a x ( a l , c l ) + M a x ( a z , c 2 ) - 1 S(b Qc) = Max(bl,c~) + Max(b2,c2) - 1.
If S(a Q c) >>-S(b © c), then we can get Max(al,cl) + Max(a2,c2)-
1 ~> M a x ( b l , C l ) + M a x ( b 2 , c 2 ) - 1.
This yields M a x ( a l , c l ) + Max(a2,c2) ~> Max(bl,Cl) + Max(b2,c2).
[]
Previously, we assumed that all criteria have the same degree of importance. However, if we can allow each criteria to have a different degree of importance, then there is room for more flexibility. 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 satisfies the criteria Ci, Ck . . . . . and Cp or which satisfies the criteria Cs, then the decision-maker's requirement can be represented by the following expression: Cj AND CR AND ... AND CpOR 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 E [-0, 1], Wk E [0, 1] .... , Wp e [0, 1], and wj + Wk + "" + Wp = 1. Then, the degree of suitability that the alternative Ai satisfies the decision-maker's requirement can be measured by the weighting function W, W ( A I ) . M a x ( S.( [ t i j , t*]) . * wj +. S([tik, t~k]) * Wk +
+ S([tlp, t*']) * Wp, S([tls , tis]) ).*
(27)
By applying (26), we can see that (27) can be rewritten into W ( A i ) = Max((t~j + t* - 1) • wj + (t~k + ti* -- 1) * Wk + "'" + (t~p + t * -- 1) * Wp, t~ + t~* -- I),
(28)
where W ( A i ) e [-- 1, + 1] and 1 ~< i ~< m. Let W(A1) = Pl, W ( A 2 ) = P2,
W ( A , , ) = p,..
If W ( A i ) = p~ and Pi is the largest value among the values Pl ,Pz . . . . . and p,,, then the alternative A~ is his best choice.
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171
E x a m p l e 3.2. Let AI, A2, A3, A4, and A5 be five 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: A~ = {(C1, [0.5, 0.7]), (Ca, [0.8, 0.9]), (C3, [0.3, 0.4])}, A2 -- {(C~, El, 1]), (C2, [0.7, 0.83), (C3, I-0.1, 0.2])}, A 3 = {(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 satisfies the criteria C~ and C2 or which satisfies the criteria C3, where the degrees of i m p o r t a n c e of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, then by applying (28), we can get
W(A1) = Max((0.5 + 0.7 - 1),0.7 + (0.8 + 0.9 - 1), 0.3,0.3 + 0.4 - 1) = Max(0.35, - 0 . 3 ) = 0.35, W(A2) = Max((1 + 1 -- 1 ) , 0 . 7 + (0.7 + 0.8 -- 1), 0.3),0.1 + 0.2 - 1) = Max(0.85,--0.7) = 0.85, W(A3) = Max((0 + 0 - 1)* 0.7 + (0.4 + 0.5 - 1 ) , 0.3, 0.8 + 0.9 - 1) = M a x ( - 0 . 7 3 , 0.7) = 0.7, W(A4) = Max((0.8 + 0.9 - 1 ) , 0 . 7 + (0.1 + 0.2 - 1),0.3, 0.5 + 0.6 - 1) = Max(0.28, 0.1) = 0.28, W(As) = Max((0.7 + 0.8 - 1).0.7 + (0.5 + 0.6 - 1).0.3, 0.1 + 0.2 - 1) = Max(0.38,-0.7) = 0.38. Therefore, we can see that the alternative A2 is his best choice.
4. Conclusions In this paper, we have presented some new techniques for handling multicriteria fuzzy decision-making problems, where the characteristics of the alternatives are represented by vague sets. The p r o p o s e d techniques allow the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria to be represented by vague values, respectively. F u r t h e r m o r e , the p r o p o s e d techniques allow the decision-maker to assign a different degree of i m p o r t a n c e to each criteria. Some examples are presented in Section 3 to illustrate the fuzzy decision-making process. F r o m these examples, we can see that the p r o p o s e d techniques can provide a useful way to efficiently help the decision-maker to m a k e his decisions. The
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p r o p o s e d techniques differ from p r e v i o u s a p p r o a c h e s for m u l t i c r i t e r i a fuzzy d e c i s i o n - m a k i n g due to the fact that the p r o p o s e d techniques use vague set t h e o r y r a t h e r t h a n fuzzy set theory.
Acknowledgement T h e a u t h o r s w o u l d like to t h a n k the referees for p r o v i d i n g very helpful c o m m e n t s a n d suggestions. Their insight a n d c o m m e n t s led to a b e t t e r p r e s e n t a t i o n of the ideas expressed in this paper. This w o r k was s u p p o r t e d in p a r t by the N a t i o n a l Science Council, R e p u b l i c of China, u n d e r G r a n t N S C 83-0408-E-009-041.
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R. Bellman and L.A. Zadeh, Decision-making in a fuzzy environment, Management Sci. 17 (1990) 141 164. C.L. Chang, Introduction to Artificial Techniques (JMA Press, Texas, 1985). S.M. Chen, A new approach to handling fuzzy decisionmaking problems, IEEE Trans. Systems, Man, Cybern. 18 (1988) 1012 1016. S.M. Chen, A new approach to inexact reasoning for rule-based systems, Cybern. Systems Internat. J. 23 (1992) 561 582. S.M. Chen, A weighted fuzzy reasoning algorithm for medical diagnosis, Dec. Support Systems 11 (1993) 37-43. S.M. Chen, J.S. Ke and J.F. Chang, Techniques for handling multicriteria fuzzy decision-making problems, Proc. 4th lnternat. Syrup. on Computer and Information Sciences (Ceseme, Turkey, 1989) 919-925. S.M. Chen, J.S. Ke and J.F. Chang, An efficient algorithm to handle medical diagnostic problems, Cybern. Systems lnternat. J. 21 (1990) 377-387. S.M. Chen, J.S. Ke and J.F. Chang, An inexact reasoning technique based on extended fuzzy production rules, Cybern. Systems Internat. J. 22 (1991) 151-171. W.L. Gau and D.J. Buehrer, Vague sets, IEEE Trans. Systems Man, Cybern. 23 (1993) 610-614. M.B. Gorzalczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems 21 (1987) 1-17. M.B. Gorzalczany, An interval-valued fuzzy inference method some basic properties, Fuzzy Sets and Systems 31 (1989) 243-251. W.J.M. Kickert, Fuzzy Theories on Decision-Making: A Critical Review (Kluwer Boston Inc., Boston, 1978). P.J.M. Laarhoven and W. Pedrycz, A fuzzy extension of Satty's priority theory, Fuzzy Sets and Systems 11 (1983) 229 241. H. Maeda and S. Murakami, A fuzzy decision-making method and its application to a company choice problem, Inform. Sci. 45 (1988) 331-346. C.V. Negoita, Expert Systems and Fuzzy Systems (Benjamin/Cummings Publishing Company, California, 1985). H. Xingui, Weighted fuzzy logic and its applications, Proc. 12th Ann. lnternat. Computer Software and Applications Conf. (Chicago, Illinois, 1988) 485-489. R.R. Yager, Fuzzy decision making including unequal objectives, Fuzzy Sets and Systems 1 (1978) 87 95. L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338 356. L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision process, IEEE Trans. Systems Man, Cybern. 3 (1973) 28-44. H.J. Zimmermann, Fuzzy Sets, Decision Making, and Expert Systems (Kluwer Academic Publishers, Boston, 1987).
sets and systems ELSEVIER
Fuzzy Sets and Systems 67 (1994) 163-172
Handling multicriteria fuzzy decision-making problems based on vague set theory Shyi-Ming Chen*, Jiann-Mean Tan Department of Computer and Information Science. National Chiao Tung University, Hsinchu, Taiwan, ROC Received December 1993; revised May 1994
Abstract
New techniques for handling multicriteria fuzzy decision-making problems based on vague set theory are presented. The proposed techniques allow the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria to be represented by vague values. Furthermore, the proposed techniques allow the decision-maker to assign a different degree of importance to each criteria. The techniques proposed in this paper can provide a useful way to efficiently help the decision-maker to make his decisions. Key words: False-membership function; Fuzzy set; Multicriteria fuzzy decision-making; Truth-membership function; Vague set
1. Introduction
In 1965, Zadeh presented the theory of fuzzy sets [18]. In recent years, fuzzy set theory has been used for handling fuzzy decision-making problems [1, 3, 6, 7, 12-14, 17, 19, 20]. Roughly speaking, a fuzzy set is a class with fuzzy boundaries. A fuzzy set A of the universe of discourse U, U = {u~, u2 . . . . . u,}, is a set of ordered pairs, {(Ul,#A(Ut)),(Uz,#A(U2)) . . . . . (U,,#A(U,))}, where /~A is the membership function of the fuzzy set A, /~A: U ~ [0, 1], and #A(Ul) indicates the grade of membership of ul in A; Vui ~ U, the membership value/~A (u~) is a single value between zero and one. G a u et al. [9] pointed out that this single value combines the evidence for u~ e 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, G a u et al. [9] presented the concepts of vague sets. They used a truth-membership function tA and false-membership functionfA to characterize the lower bounds on #a. These lower bounds are used to create a subinterval on [0, 1], namely IrA(ui), 1 - f A (Ul)], to generalize the #a(Ui) of fuzzy sets, where tA(ui) <~ #A(U~) <<. 1 --fA(U~). For example, let A be a vague set with truth-membership function ta and false-membership function fA, respectively. If [tA(u~), 1--fA(U~)] =
* Corresponding author. 0165-0114/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0165-01 14(94)001 57-3
S.-M. Chen, J.-M. Tan / Fuzzy Sets and Systems 67 (1994) 163-172
164
[0.5, 0.7], then we can see that tA(U~) = 0.5; 1 --fa(U~) = 0.7; fA(Ui) = 0.3. It can be interpreted as "the degree that object ui belongs to the vague set A is 0.5; the degree that object u~ does not belong to the vague set A is 0.3". As another example, in a voting model, the vague value [-0.5,0.7] can be interpreted as "the vote for resolution is 5 in favor, 3 against, and 2 abstentions". Kickert [12] pointed out that the multicriteria decision-making problem is a kind of problem that all the alternatives in the choice set can be evaluated according to a number of criteria; he also pointed out that the problem is to construct an evaluation procedure to rank the set of alternatives in order of preference. Kickert [12, pp. 61-84] has discussed the field of fuzzy multicriteria decision-making. Zimmermann [20] has pointed out that two major areas have evolved both of which concentrate on multicriteria decision-making: multiobjective decision-making and multiattribute decision-making. The multiobjective decision-making problem is often called the "vector-maximum" problem. Zimmermann [20] illustrated a fuzzy set approach to multiobjective decision-making. Yager [17] presented a fuzzy multiattribute decision-making method using crisp weights. Laarhoven et al. [13] presented a method for multiattribute decision-making using fuzzy numbers as weights. Zimmermann [20, pp. 125-192] has compared some approaches to solve multiattribute decision problems based on fuzzy set theory. For more details, please refer to [20]. In this paper, we present some new techniques for handling multicriteria fuzzy decision-making problems based on vague set theory, where the characteristics of the alternatives are represented by vague sets. Because the proposed techniques use the truth-membership function and the false-membership function to indicate the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria, respectively, and because the proposed techniques can allow each criteria to have a different degree of importance, the techniques proposed in this paper can provide a useful way to efficiently help the decision-maker to make his decisions.
2. Vague sets Let U be the universe of discourse, U = {Ul, u2 .... , u,}, with a generic element of U denoted by u~. A vague set A in U is characterized by a truth-membership function tA and a false-membership function fa,
tA: U
~
[0, 1],
(1)
fA : U ~
[0, 1],
(2)
where ta(Ui) is a lower bound on the grade of membership of u~ derived from the evidence for u~, fA(ui) is a lower bound on the negation of ui derived from the evidence against ug, and tA(U~) + fa(u~) ~< 1. The grade of membership of ui in the vague set A is bounded to a subinterval [tA(U~), 1 --fa(U~)] of [0, 1]. The vague value [tA(Ul), 1 --j~(U~)] indicates that the exact grade of membership I~a(U~)of ui may be unknown, but is bounded by tA(ui) <<,#n(Ui) ~< 1 --fa(ui), where tA(ui) +fA(Ui) ~< 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
A = fvEtA(ui), 1 -fA(ui)]/u~.
(3)
When the universe of discourse U is discrete, a vague set A can be written as
A = ~ EtA(Ul), 1 --fa(Ui)]/Ui. i=1
(4)
S.-M. Chen, J.-M. Tan / Fuzzy Sets and Systems 67 (1994) 163-172
~ l A(U~t) ~ . _ U~
165
tA(Ud
"~U
Fig. 1. A vague set.
F o r example, let U be the universe of discourse, U = {6, 7, 8, 9, 10}. A vague set " L A R G E " of U may be defined by L A R G E = [0.1,0.2]/6 + [0.3,0.5]/7 + [-0.6,0.8]/8 + [0.9, 1]/9 + [1, 1]/10. Definition 2.1. Let x be a vague value, x = [t~, 1 - f x ] , where tx e [ 0 , 1],f~ e [ 0 , 1], tx + f ~ ~ 1. The complement of the vague value x is denoted by x' and is defined by x'-
[f~, 1 -- tx].
(5)
Definition 2.2. Let x and y be two vague values, x = [t~, 1 - f x ] and y = [ty, 1 - f ~ ] , where tx ~[0, 1], tye [0, 1],fx ~ [0, 1],fy e [0, 1], t~ +fx ~< 1, and ty + f y ~< l. The result of the minimum operation of the vague values x and y is a vague value z, written as z = x @y = [t=, 1 - f z ] , where G -- Min(G, ty),
(6)
1 - f z = Min(1 - f ~ , 1 - f y ) .
(7)
Definition 2.3. Let x and y be two vague values, x = [t~, 1 - f x ] and y = [ty, 1 - ~ ] , where tx ~[0, 1], ty e l 0 , 1],fx e [ 0 , 1], a n d f y e l 0 , 1], tx + f , ~< 1, and tr + f y ~< 1. The result of the m a x i m u m operation of the vague values x and y is a vague value c, written as c = a Q b = [tc, 1 - f ~ ] , where tc = M a x ( t , , tr),
(8)
1 - f ~ = Max(1 - f x , 1 -fy).
(9)
Definition 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 false-membership function ts and fn, respectively. The notions of complement, union, and intersection of vague sets are defined as follows. Definition 2.5. The complement of the vague set A is denoted by A' whose truth-membership function and false-membership function are ta, and fA,, respectively, where Vui ~ U,
t~,(u,) = fA (u,)
(lO)
1 --fa'(ui) = 1 -- ta(ui).
(l 1)
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166
Definition 2.6. The union 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 ~'u~ ~ U, tc(Ui) = MaX(tA(ui), tB(Ui)),
(12)
1 - - f c ( U , ) = Max(1 --fa(u,), 1 --fB(U,)).
(13)
That is, Vu~ E U, Etc(Ui), 1 - fc(ui)] = [tA(Ui), 1 -- fA(Ui)] Q [tB(Ui), 1 --f,(ul)] = [MaX(tA(Uz), tB(U~)), Max(1 --fa(u~), 1 --fB(U~))].
(14)
Definition 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 Vu~ E U, tc(Ui) = Min(tA(ui), tB(ui)),
(15)
1 --fc(U~)= Min(1 --fA(u~), 1 --fB(u~)).
(16)
That is, Vui E U, [tc(Ui), 1 - fc(ui)] = [tA(Ui), 1 -- fA(U,)] ® [tB(Ui), 1 -- fB(ui)] = [Min(tA(U~), tB(U~)), Min(1 --fA(U~), 1 --fn(U~))].
(17)
3. Handling multicriteria fuzzy decision-making problems In this section, we present some new techniques for handling multicriteria fuzzy decision-making problems, where the characteristics of the alternatives are represented by vague sets. Let A be a set of alternatives and let C be a set of criteria, where A = {A,, A2 ..... A,,}, C = {C1, C2 . . . . . C,}. Assume that the characteristics of the alternative A~ is represented by the vague set shown as follows: Ai = {(C1, [t~l,(1 --f~)]), (C2, [t~z,(1 --f2)]) . . . . . (C,, [t~,,(1 -f~,)])},
(18)
where tij indicates the degree that the alternative A~ satisfies criteria C j, f~s indicates the degree that the alternative Ai does not satisfy criteria Cj, t~s ~[0, 1],f~j ~ [0, 1], t~j + f j ~< 1, 1 ~
{(C,, F t , , t*l]), (C2, [t,2, t*]), ..., (C., [t,,, t,.])},
(19)
where 1 ~< i ~< m. In this case, the characteristics of these alternatives can be represented by a table shown in Table 1. Assume that there is a decision-maker who wants to choose an alternative which satisfies the criteria Cj, Ck . . . . . and Cp or which satisfies the criteria Cs, then we can represent the decision-maker's requirement by the following expression: Cj AND
C k AND
... AND Cp OR Cs.
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167
Table 1 The characteristics of the alternatives C1
...
Cj
A, A2
[tl,,t*,] [t21,t~l]
... ---
[tlj, [tzj,
:
:
...
A,
[t,l, t*, ]
...
:
:
...
Am
[tml,
t*r.tl ]
.
...
Ck
...
Cp
t~j]
...
[tlk,t*k]
...
[ttp,
t~j]
... ...
[t2k,t~k] :
... ...
[tq, t*]
...
[t,k, t*]
...
...
:
...
[tmk, trek* ]
...
: : .
. [tmj, . . " tmj] .*
...
Cs
...
C,,
t*p]
...
[tl~,t*~]
...
[tl,,t*.]
[t2p, t*p] :
... ...
[tz~,t*,] :
... ...
[t2,,t*,] :
E%, tTp]
...
Et,~, r*,s]
...
Et,,,'t~',]
:
...
:
...
[t,,p, t*mp]
.
.
.
[tm~, . . t,,~] .*
:
[t,.,, " t,,.] *
In this case, the degrees that the alternative A/satisfies and not satisfies the decision-maker's requirement can be m e a s u r e d by the evaluation function E, E(Ai) = ([tij, t*] ® [tik, t~] ®...
® [t,p, t * ] ) © [t,s, t*]
= [Min(t~j, t~k. . . . . t,p), Min(t*, ti~ . . . . . t'p)] ©[t~,, t~*] [Max(Min(ti), tik .... , tip), tis), M a x ( M i n ( t * , t* . . . . . t'p), t ~* ) ]
=
(20)
= [_ta,, t*],
where ® and © denote the m i n i m u m o p e r a t o r and the m a x i m u m o p e r a t o r of the vague values, respectively, E(Ai) is a vague value, 1 ~< i ~< m, and tn, = Max(Min(ti), tik ..... tip), tis),
(21)
t* = M a x ( M i n ( t * , t~ ..... t*), tis).*
(22)
Let t* = 1 - f a , , where f~, = 1 - M a x ( M i n ( t * , ti* .... , tip), tis).
(23)
In this case, (20) can be rewritten as E(AI) = [tA,, 1 -- f a , ] .
(24)
In the following, we present a score function S to evaluate the degree of suitability that an alternative satisfies the decision-maker's requirement. Let x = Its, 1 - f ~ ] be a vague value, where tx ~ [0, 1], J~, E [0, 1], tx + f , ~< 1. The score of x can be evaluated by the score function S shown as follows: S(x) = tx - f~,
(25)
where S(x) ~ [ - 1, + 1]. Based on the score function S, the degree of suitability that the alternative Ai satisfies the decision-maker's requirement can be measured. F r o m (20) and (24), we can see that E(Ai)= [tA,, t A* , ] = [-tA,, 1 - - f A , ] , where tA, = Max(Min(ti), rig ..... tip), t~s) t],
=
M a x ( M i n ( t * , tik* . . . . .
f,~,
=
1
--
* tip),
M a x ( M i n ( t * , tik* ....
* tis) * , tip),
* tis).
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S.-M. Chen, J.-M. Tan / Fuzzy Sets and Systems 67 (1994) 163-172
Thus, by applying (25), we can get S(~(A,)) = t A , - A , =
Max(Min(tij, tlk . . . . . tip), tis)
-
(1 - Max(Mm(tij, " * ti~ * . . . . . t*), tis)) *
= Max(Min(ti i, tik . . . . . tip), t,~) + Max(Min(t*, t* . . . . . t*), ti*) - 1 = ta, + t * , - 1,
(26)
where S(E(A~)) E [ - 1, + 1]. The larger the value of S(E(Ai)), the more the suitability that the alternative Ai satisfies the decision-maker's requirement, where 1 ~< i ~< m. Let S ( E ( A 1 ) ) = Pl
S(E(A2)) = P2
S(E(Am)) = p . .
If S(E(A~)) = p~ and Pi is the largest value among the values PI, P2 . . . . . and Pro, then the alternative Ai is his best choice. Example 3.1. Let A be a set of alternatives and C be a set of criteria, where
A = {A1,Az,A3,A4,A5}, C = { C I , C 2 , C3}.
Assume that the characteristics of the alternatives are represented by the vague sets shown as follows: A1 = {(C,, [0, 0]), (Cz, [0.8, 0.9]), (C3, [0.3, 0.4])}, A 2 = { ( C l , El.0, 1.0]), ( 6 2 , [0.6, 0.7]), (C3, [0.1, 0.2])},
A3 = {(C1, [0, 0]), (C2, [0.3, 0.4]), (C3, [0.8, 0.9])}, A , = {(C~, [0.7, 0.8]), (Cz, [0.1, 0.2]), (C3, [0.5, 0.6])}, A5 = {(C,, [0.8, 0.9]), (C2, [0.5, 0.6]), (C3, [0.3, 0.4])}, and assume that the decision-maker wants to choose an alternative which satisfies the criteria C1 and C2 or which satisfies the criteria C3, then the decision-maker's requirement can be expressed by an expression as follows: C1 A N D C2 O R C3. By applying (20), we can get
E ( A , ) = ([0,0] @[0.8,0.9])@[0.3,0.4] = [0.3,0.4], E(A2) = .([1.0, 1.0] @ [0.6, 0.7]) @[0.i, 0.2] = [0.6, 0.7], E(A3) = ([0,0] ®[0.3,0.4]) @[0.8,0.9] = [0.8,0.9],
E(A4) = ([0.7,0.8] @[0.1,0.2]) @[0.5,0.6] = [0.5,0.6], E(As) = ([0.8,0.9] @[0.5,0.6]) @[0.3,0.4] = [0.5,0.6].
S.-M. Chen, J.-M. Tan / Fuzzy Sets and Systems 67 (1994) 163-172
169
By a p p l y i n g (26), we c a n get
S(E(A1)) = 0.3 + 0.4 - 1 = --0.3, S(E(A2)) = 0.6 + 0.7 - 1 = 0.3, S(E(A3)) = 0.8 + 0.9 - 1 = 0.7, S(E(A,)) = 0.5 + 0.6 - 1 = 0.1, S(E(A5)) = 0.5 + 0.6 - 1 = 0.1. Therefore, we c a n see t h a t the a l t e r n a t i v e A3 is his best choice. T h e o r e m 3.1. Let a and b be two vague values, and S be the score function, where a = [ a l , a 2 ] , b = [ b l , b 2 ] . I f
S(a) >~ S(b), then al + a2 >~ bl + b2. P r o o f . By a p p l y i n g (25), we c a n get
S(a) = al + a2 - 1 S(b) = bl + b2 - -
1.
If S(a) >~ S(b), t h e n we c a n see t h a t al+a2-1~>
b l + b2 - 1.
T h i s yields
at + a2 ~ bl + b2.
[]
T h e o r e m 3.2. Let a, b, and c be three vague values, and S be the score function, where a = [ a l , a2], b = [ b l , b2], and c = [ c t , c 2 ] . I f S ( a Qc) >1 S(b Qc), then M i n ( a l , c l ) + M i n ( a 2 , c 2 ) / > M i n ( b t , c l ) + M i n ( b 2 , c 2 ) . P r o o f . By a p p l y i n g (6) a n d (7), we c a n get
a ® c = [al,a2] @ [ c l , c 2 ] = [ M i n ( a l , c l ) , M i n ( a 2 , c 2 ) ] b @ c = [bl,b2] ® [ c l , c 2 ] = [ M i n ( b l , C l ) , M i n ( b 2 , c 2 ) ] . By a p p l y i n g (26), we c a n get
S(a @ c) = M i n ( a l , c l ) + M i n ( a 2 , c 2 ) - 1 S(b ® c) = M i n ( b l , c l ) + M i n ( b 2 , c 2 ) - 1. If S(a Q c) >1 S(b ® c), t h e n we c a n see t h a t M i n ( a l , c l ) + M i n ( a 2 , c 2 ) -- 1 > / M i n ( b l , c l )
+ M i n ( b 2 , c 2 ) - 1.
T h i s yields Min(at,ct) + Min(a2,c2) >/Min(bl,Cl) + Min(b2,c2).
[]
T h e o r e m 3.3. Let a, b and c be three vague values, where a = [ a l , a 2 ] , b = [ b l , b 2 ] , and c -- [-cx,c2]. I f S(a 63 c) >~ S(b 63c), then M a x ( a t , c 1 ) + M a x ( a 2 , c 2 ) / > M a x ( b l , c i ) + M a x ( b 2 , c 2 ) .
170
S.-M. Chen, J.-M. Tan / Fuzz)' Sets and Systems 67 (1994) 163-172
Proof. By applying (8) and (9), we can get a @ c = [aa, a2] Q [Ca, c2] = [Max(aa, c~), Max(a2, c2)] b @ c = [bl, b2] @ [Ca, c2] = [Max(bl, cl), Max(b2, c2)]. By applying (26), we can get S(a @c) = M a x ( a l , c l ) + M a x ( a z , c 2 ) - 1 S(b Qc) = Max(bl,c~) + Max(b2,c2) - 1.
If S(a Q c) >>-S(b © c), then we can get Max(al,cl) + Max(a2,c2)-
1 ~> M a x ( b l , C l ) + M a x ( b 2 , c 2 ) - 1.
This yields M a x ( a l , c l ) + Max(a2,c2) ~> Max(bl,Cl) + Max(b2,c2).
[]
Previously, we assumed that all criteria have the same degree of importance. However, if we can allow each criteria to have a different degree of importance, then there is room for more flexibility. 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 satisfies the criteria Ci, Ck . . . . . and Cp or which satisfies the criteria Cs, then the decision-maker's requirement can be represented by the following expression: Cj AND CR AND ... AND CpOR 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 E [-0, 1], Wk E [0, 1] .... , Wp e [0, 1], and wj + Wk + "" + Wp = 1. Then, the degree of suitability that the alternative Ai satisfies the decision-maker's requirement can be measured by the weighting function W, W ( A I ) . M a x ( S.( [ t i j , t*]) . * wj +. S([tik, t~k]) * Wk +
+ S([tlp, t*']) * Wp, S([tls , tis]) ).*
(27)
By applying (26), we can see that (27) can be rewritten into W ( A i ) = Max((t~j + t* - 1) • wj + (t~k + ti* -- 1) * Wk + "'" + (t~p + t * -- 1) * Wp, t~ + t~* -- I),
(28)
where W ( A i ) e [-- 1, + 1] and 1 ~< i ~< m. Let W(A1) = Pl, W ( A 2 ) = P2,
W ( A , , ) = p,..
If W ( A i ) = p~ and Pi is the largest value among the values Pl ,Pz . . . . . and p,,, then the alternative A~ is his best choice.
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171
E x a m p l e 3.2. Let AI, A2, A3, A4, and A5 be five 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: A~ = {(C1, [0.5, 0.7]), (Ca, [0.8, 0.9]), (C3, [0.3, 0.4])}, A2 -- {(C~, El, 1]), (C2, [0.7, 0.83), (C3, I-0.1, 0.2])}, A 3 = {(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 satisfies the criteria C~ and C2 or which satisfies the criteria C3, where the degrees of i m p o r t a n c e of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, then by applying (28), we can get
W(A1) = Max((0.5 + 0.7 - 1),0.7 + (0.8 + 0.9 - 1), 0.3,0.3 + 0.4 - 1) = Max(0.35, - 0 . 3 ) = 0.35, W(A2) = Max((1 + 1 -- 1 ) , 0 . 7 + (0.7 + 0.8 -- 1), 0.3),0.1 + 0.2 - 1) = Max(0.85,--0.7) = 0.85, W(A3) = Max((0 + 0 - 1)* 0.7 + (0.4 + 0.5 - 1 ) , 0.3, 0.8 + 0.9 - 1) = M a x ( - 0 . 7 3 , 0.7) = 0.7, W(A4) = Max((0.8 + 0.9 - 1 ) , 0 . 7 + (0.1 + 0.2 - 1),0.3, 0.5 + 0.6 - 1) = Max(0.28, 0.1) = 0.28, W(As) = Max((0.7 + 0.8 - 1).0.7 + (0.5 + 0.6 - 1).0.3, 0.1 + 0.2 - 1) = Max(0.38,-0.7) = 0.38. Therefore, we can see that the alternative A2 is his best choice.
4. Conclusions In this paper, we have presented some new techniques for handling multicriteria fuzzy decision-making problems, where the characteristics of the alternatives are represented by vague sets. The p r o p o s e d techniques allow the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria to be represented by vague values, respectively. F u r t h e r m o r e , the p r o p o s e d techniques allow the decision-maker to assign a different degree of i m p o r t a n c e to each criteria. Some examples are presented in Section 3 to illustrate the fuzzy decision-making process. F r o m these examples, we can see that the p r o p o s e d techniques can provide a useful way to efficiently help the decision-maker to m a k e his decisions. The
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p r o p o s e d techniques differ from p r e v i o u s a p p r o a c h e s for m u l t i c r i t e r i a fuzzy d e c i s i o n - m a k i n g due to the fact that the p r o p o s e d techniques use vague set t h e o r y r a t h e r t h a n fuzzy set theory.
Acknowledgement T h e a u t h o r s w o u l d like to t h a n k the referees for p r o v i d i n g very helpful c o m m e n t s a n d suggestions. Their insight a n d c o m m e n t s led to a b e t t e r p r e s e n t a t i o n of the ideas expressed in this paper. This w o r k was s u p p o r t e d in p a r t by the N a t i o n a l Science Council, R e p u b l i c of China, u n d e r G r a n t N S C 83-0408-E-009-041.
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