International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
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Ordering Alternatives under Fuzzy Multiple Criteria Decision Making via a Fuzzy Number Dominance Based Ranking Approach Ta-Chung Chu and Peerayuth Charnsethikul Abstract1 This work suggests ordering alternatives under fuzzy multiple criteria decision making (MCDM) via a fuzzy number dominance based ranking approach, where the ratings of alternatives versus qualitative criteria and the importance weights of all criteria are assessed in linguistic values represented by fuzzy numbers. The difference of one final fuzzy evaluation value over another is used to represent the dominance degree of one alternative over another. The membership function for the final fuzzy evaluation value of each alternative and the difference between each pair of final fuzzy evaluation values can be developed through α-cuts and interval arithmetic of fuzzy numbers. Formulas for the dominance degree can be clearly derived based on the developed membership function via integral development. A simple ranking procedure based on these dominance degrees is then proposed to order the alternatives. Finally a numerical example demonstrates the feasibility of the proposed model. Keywords: Fuzzy MCDM, fuzzy number dominance, ranking, integral.
1. Introduction A fuzzy multiple criteria decision making (MCDM)[1] model is to assess alternatives versus selected criteria through a committee of decision makers, where suitability of alternatives versus criteria and the importance weights of criteria can be evaluated in linguistic values represented by fuzzy numbers. Fuzzy set theory, initially proposed by Zadeh [2], has been extensively applied to objectively resolve the Corresponding Author: Ta-Chung Chu is with the Department of Management and Information Technology, Southern Taiwan University of Science and Technology, No.1, Nantai St., Yungkang District, Tainan City 710, Taiwan. E-mail:
[email protected] Peerayuth Charnsethikul is with the Department of Industrial Engineering, Faculty of Engineering, Kasetsart University, 50 Ngamwongwan Rd. Chatuchak, Bangkok, Thailand 10900. Email:
[email protected] Manuscript received 5 May 2010; revised 30 March 2012; accepted 28 May 2013.
uncertainties in human judgment and effectively reflect the ambiguities in the available information in an ill-defined multiple criteria decision making environment. A comparison and review of fuzzy MCDM methods can be found in Carlsson and Fuller [3], Ribeiro [4] and Triantaphyllou and Lin [5]. Some recent applications can be found in [6-20]. The final evaluation values of alternatives in most of the above fuzzy MCDM problems are usually still fuzzy numbers and these fuzzy numbers need a proper ranking approach to defuzzify them into crisp values for decision making. A comparison and review of many of the fuzzy number ranking methods can be seen in [21-31]. However, in spite of the merits, some of the applied ranking methods are computational complex and others are difficult to implement the connection between the ranking procedure and the final fuzzy evaluation values, limiting the applicability of the fuzzy MCDM model. To resolve the above limitations, this work suggests a fuzzy MCDM model with a fuzzy number dominance based ranking procedure, where ratings of alternatives versus qualitative criteria and the importance weights of all criteria are assessed in linguistic values [32] represented by triangular fuzzy numbers. Criteria are classified to qualitative and quantitative ones. The dominance degree of one alternative over another is a concept from Chen [9], which is developed based on measuring the difference of one final fuzzy evaluation value of the corresponding alternative over another. Through α-cuts and interval arithmetic of fuzzy numbers, membership functions for the final fuzzy evaluation value of each alternative and the difference between each pair of final fuzzy evaluation values can be developed. Formulas for the dominance degree of one alternative over another can be clearly derived via integral development based on the developed membership functions. A simple ranking procedure is then proposed to order the alternatives based on these dominance degrees. Finally a numerical example demonstrates the feasibility of the proposed model. The rest of this work is organized as follows. Section 2 briefly introduces fuzzy set theory. Section 3 introduces the suggested model. Meanwhile, an example is presented in Section 4 to demonstrate the computational process of the proposed model and conclusions are finally made in Section 5.
© 2013 TFSA
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
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A r
2. Fuzzy Set Theory A. Fuzzy sets A x, f A x | x U , where U is the universe of discourse, x is an element in U, A is a fuzzy set in U, f A x is the membership function of A at x [33]. The larger f A x , the stronger the grade of membership for x in A. B. Fuzzy numbers A real fuzzy number A is described as any fuzzy subset of the real line R with membership function f A which possesses the following properties [34]: (a) f A is a continuous mapping from R to [0,1]; (b) f A x 0, x , a ] ; (c) f A is strictly increasing on a , b ; (d) f A x 1, x b, c ; (e) f A is strictly decreasing on c , d ; (f) f A x 0, x [ d , ) ; where a b c d , A can be denoted as a , b , c , d . The membership function f A of the fuzzy number A can also be expressed as: f AL x , a x b bxc 1 , (1) f A x R f x c x d , A 0 , otherwise where f AL x and f AR x are left and right membership functions of A, respectively [33]. A fuzzy triangular number can be denoted as (a, b, c) [35]. C. α-cuts The α-cuts of fuzzy number A can be defined as A x | f A x , 0, 1 , where A is a non-empty bounded closed interval contained in R and can be denoted by A Al , Au , where Al and Au are its lower and upper bounds, respectively [33].
D. Arithmetic operations on fuzzy numbers Given fuzzy numbers A and B, A, B R , the α-cuts of A and B are A Al , Au and B Bl , Bu , respectively. By the interval arithmetic, some main operations of A and B can be expressed as follows [33]: (2) A B Al Bl , Au Bu
A B Al Bu , Au Bl A B Al Bl , Au Bu
(3) (4)
Al r , Au r , r R
(5)
E. Linguistic values A linguistic variable is a variable whose values are expressed in linguistic terms. Linguistic variable is a very helpful concept for dealing with situations which are too complex or not well-defined to be reasonably described by traditional quantitative expressions [32]. For example, the linguistic rating set {B, B.B&G, G, VG, E}, where B=Bad, B.B&G=Between Bad and Good, G=Good, VG=Very Good, E=Excellent, can be used to evaluate the suitability of alternatives versus qualitative criteria. These linguistic values can be further represented by triangular fuzzy numbers such as B=(0,0.1,0.3), B.B&G=(0.1,0.3,0.5), G=(0.3,0.5,0.7), VG=(0.5,0.7,0.9), E=(0.7,0.9,1.0) as shown in Fig. 1.
Figure 1. Linguistic values and their fuzzy numbers for alternatives versus qualitative criteria.
3. Model Development Assume that a committee of k decision-makers (i.e. Dt, t=1~k) is responsible for the selection of m alternative (i.e. Ai, i=1~m) under n criteria (Cj, j=1~n). Criteria are classified to qualitative, Cj, j = 1,…,g, and quantitative; quantitative criteria are further classified to benefit, Cj, j =g+1,…,h and cost,Cj, j = h+1,…,n, ones. Moreover, we assume that the importance weights of the criteria and the performance ratings under each of the qualitative criteria are assessed in linguistic values represented by positive triangular fuzzy numbers. A. Average importance weights Let w jt o jt , p jt , q jt , w jt R ,
j=1~n, t=1~k, be
the importance weight assigned by decision maker Dt to criterion Cj. w j o j , p j , q j is the averaged importance weight of criterion Cj assessed by the committee of k decision makers. wj
where
1 w j1 w j 2 w jk k
(6)
Ta-Chung Chu and Peerayuth Charnsethikul: Ordering Alternatives under Fuzzy Multiple Criteria Decision Making
oj
k
k
developed as follows: Through Eqs. (2)-(4), we obtain: xij wj (bij aij )( p j o j ) 2 aij ( p j o j )o j (bij aij ) aij o j ,
k
1 1 1 o jt , p j p jt , q j q jt . k t 1 k t 1 k t 1
B. Average ratings of alternatives versus qualitative criteria Let xijt (aijt , bijt , cijt ) , i=1,…,m, j=1,…,g, t=1,…,k, is the rating assigned to alternative Ai by decision maker Dt under criterion Cj. xij (aij , bij , cij ) is the averaged rating of alternative Ai versus criterion Cj assessed by the committee of decision makers.
1 xij xij1 xijk xijt k
(7)
k
k
1 1 1 aij aijt , bij bijt , cij cijt , j=1~g. k t 1 k t 1 k t 1
j=g+1~n.
e e f e g e j ij j ij j ij , , , j B, * * * d d d xij g j gij g j fij g j eij , , , j C, * d* d * d
The
(8)
D. Final fuzzy evaluation value of each alternative The final fuzzy evaluation value of each alternative Ai can be obtained by using the Simple Addictive Weighting [1] concept as follows:
j 1
xij w j , i=1, 2,…, m,
n n (b a )( p o ) 2 a ( p o )o (b a ) ij ij j j ij j j j ij ij j 1 j 1 n n aij o j , (bij cij )( p j q j ) 2 j 1 j 1
n n cij ( p j q j ) q j (bij cij ) cij q j j 1 j 1
(11)
Assume: Ei1
j 1
Fi 2
j 1
j 1
aij ( p j o j ) o j (bij aij ) , Ei 2
Vi
(bij aij )( p j o j ) ,
n
Fi1
n
n
n
(bij cij )( p j q j ) , j 1
n
cij ( p j q j ) q j (bij cij ) , j 1
aij o j , Yi
n
j 1
bij p j , Zi
n
cij q j . j 1
Applying the above assumption to Eq. (11), we have the following two equations to solve: (12) Ei1 2 Fi1 Vi x 0 2 (13) Ei 2 Fi 2 Z i x 0 Only root in [0, 1] are retained in Eq. (12) and (13). The left membership function, i.e. f SL x , and the right f SR i
x , of the final fuzzy
evaluation value Si can be produced as follows:
i
B g 1 ~ h, C h 1 ~ n.
n
j 1
membership function
* g max gij , ej min eij , d g j e j , j g 1 ~ n, j
Si
n
Si xij wj
i
where i
(10)
By applying Eq. (10) to Eq. (9), we obtain the -cut of Si as follows:
C. Normalization of ratings of alternatives versus quantitative criteria Quantitative criteria can be classified as benefit and cost ones. The benefit criterion has the characteristic: the larger the better. The cost criterion has the characteristic: the smaller the better. Values under both benefit and cost quantitative criteria can be either crisp or fuzzy. Values under quantitative criteria may have different units and then must be normalized into a comparable scale for calculation rationale. The following approach [36] is used to complete the normalization. This approach preserves by property where the ranges of normalized triangular fuzzy numbers belong to [0, 1]. Suppose rij (eij , f ij , g ij ) is the performance value of alternative Ai versus criteria Cj, normalization of the rij is as follows:
( bij cij )( p j q j ) 2 cij ( p j q j ) q j ( bij cij ) cij q j
where k
265
(9)
Here, Si is the final fuzzy evaluation values of each alternative Ai. The membership functions of the Si can be
1/2 Fi1 Fi12 4 Ei1 ( x Vi ) , V x Y , f SL x i i 2 Ei1 i
(14)
1/2 Fi 2 Fi22 4 Ei 2 ( x Zi ) ,Y xZ . f SR x i i 2 Ei 2 i
(15)
for convenience, Si can be displayed as:
Si Vi ,Yi ,Zi ; Ei1, Fi1;Ei 2 , Fi 2 , i 1 ~ m.
(16)
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E. Difference of two fuzzy numbers The difference of two fuzzy numbers, that is, the difference of one final fuzzy evaluation value over another from Chen [9], is applied to represent the dominance degree of one alternative over another. The membership function for the difference between each pair of final fuzzy evaluation values can be developed. Suppose Sh and Sk, h, k 1 ~ m, are two fuzzy numbers from the developed model. The difference of these two fuzzy numbers can be denoted as Thk Sh Sk . The membership function is developed through α-cuts and arithmetic operations as follows: (17) Thk , Thku Thkl
where , , Sh Shl , Shu S k S kl , S ku
, . Thkl Shl Sku Thku Shu Skl n n Ehk1 (bhj ahj )( p j o j ) (bkj ckj )( p j q j ), j 1 j 1
n n Ehk 2 (bhj chj )( p j q j ) (bkj akj )( p j o j ), j 1 j 1
n n Fhk 2 chj ( p j q j ) q j (bhj chj ) akj ( p j o j )o j (bkj akj ) , j 1 j 1
Vh Vk
n
ahj o j , j 1 n
akj o j , j 1
Yh Yk
n
bhj p j , j 1 n
bkj p j , j 1
Zh Zk
1/2
fTL hk
2 Fhk1 Fhk 4Ehk1 ( x Vh Z k ) 1 x 2Ehk1 1/2
2 Fhk1 Fhk 1 2Ehk1
0.
Proposition 2: Fhk1 Ehk1 Yh Yk Vh Zk . Proof: Ehk1 Fhk1 n n (bhj ahj )( p j o j ) (bkj ckj )( p j q j ) j 1 j 1 n ahj ( p j o j ) o j (bhj ahj ) j 1 n ckj ( p j q j ) q j (bkj ckj ) j 1 n n n n bhj p j ahj p j bhj o j ahj o j j 1 j 1 j 1 j 1 n n n n bkj p j ckj p j bkj q j ckj q j j 1 j 1 j 1 j 1
n
chj q j , j 1 n
ckj q j .
(19) The left and right membership functions of the difference Thk can be produced as follows: 1/2
,
(20)
Vh Z k x Yh Yk , 1/2
2 Fhk 2 Fhk 4 Ehk 2 ( x Z h Vk ) 2 x 2 Ehk 2
,
(21) Only two roots in [0, 1] are retained in Eqs. (20) and (21). For convenience, Thk can be denoted by: Yh Yk x Z h Vk .
Thk Vh Z k ,Yh Yk ,Z h Vk ;Ehk1, Fhk1; Ehk 2 , Fhk 2 ,
n
j 1
j 1
2 Fhk1 Fhk 1 4Ehk1( x Vh Zk ) L fT x hk 2Ehk1
n
j 1
Ehk 2 2 Fhk 2 Z h Vk x 0.
x Vh Z k .
Proof:
ahj p j ckj p j
n
j 1 n
j 1
ahj o j ckj q j
n
j 1 n
j 1
bhj o j
bkj q j
n
ahj o j j 1 n
ckj q j j 1
n
n
n
n
j 1
j 1
j 1
j 1
bhj p j bkj p j ahj o j ckj q j
By applying the above assumption and Eqs. (11)-(16) to Eq. (17), we have the following two simplified equations to solve: (18) Ehk1 2 Fhk1 Vh Zk x 0,
fTR hk
x 0 when
n n Fhk1 ahj ( p j o j ) o j (bhj ahj ) ckj ( p j q j ) q j (bkj ckj ) , j 1 j 1
Proposition 1:
Assume:
(22)
h, k 1~ m. fTL hk
Yh Yk Vh Z k .
Proposition 3: fTL x 1 when x Yh Yk . hk Proof: 1/2
2 Fhk1 Fhk 1 4Ehk1 ( x Vh Zk ) L fT x hk 2Ehk1
1/2
2 Fhk1 Fhk 1 4Ehk1(Yh Yk Vh Zk ) 2Ehk1 1/2
2 Fhk1 Fhk 4Ehk1( Fhk1 Ehk1) 1 2Ehk1
(by proposition 2)
1/2
Fhk1 (Fhk1 2Ehk1)2 2Ehk1
1.
Ta-Chung Chu and Peerayuth Charnsethikul: Ordering Alternatives under Fuzzy Multiple Criteria Decision Making
Proposition 4: fTR x 0 when x Z h Vk . hk Proof: Similar to proposition 1. Proposition 5: Ehk 2 Fhk 2 Yh Yk Zh Vk . Proof: Ehk 2 Fhk 2
n
j 1
(bhj chj )( p j q j )
n
(bkj akj )( p j o j ) j 1
n
chj ( p j q j ) q j (bhj chj ) j 1
n
j 1
j 1
chj p j
n
j 1
bhj q j
chj q j j 1 n
j 1
j 1
j 1
j 1
n
n
n
n
bkj p j akj p j bkj o j akj o j
n
chj p j akj p j
j 1 n
j 1
chj q j akj o j
j 1 n
j 1
bkj q j bkj o j
chj q j j 1
P1
x0 fT
hk
( x)dx, P2
x 0
akj o j j 1
n
n
n
j 1
j 1
j 1
j 1
bhj p j bkj p j chj q j akj o j
Situation 1: Yh Yk 0, dhk
3 2 F ( Z V ) Fhk 2 Fhk 2 4 Ehk 2 ( Z h Vk ) P1 hk 2 h k 2 2 Ehk 2 12 Ehk 2
P2
Proof: 1/2
2 Fhk 2 Fhk 2 4Ehk 2 ( x Zh Vk ) R fT x hk 2Ehk 2
1/2
2 Fhk 2 Fhk 2 4Ehk 2 (Yh Yk Zh Vk ) 2Ehk 2
P1 , P1 P2
where
Yh Yk Z h Vk .
Proposition 6: fTR x 1 when x Yh Yk . hk
fThk ( x)dx, P P1 P2 , P 0.
Obviously P1 indicates the area that alternative Ah dominates alternative Ak and P2 indicates the area that alternative Ah is dominated by alternative Ak, and clearly dhk dkh 1 . To develop the dominance degree, the following situations must be discussed.
n
n
(23)
where
n
n
j 1
n
n
j 1
bhj p j
n
some values, formulae for a dominance degree of Ah over Ak, denoted by dhk , can be produced through integral procedure [36, 37] as follows: P d hk 1 P
j1
membership functions in order to enhance applicability of the suggested model. Herein the concept for measuring the difference of two triangular fuzzy numbers from the Chen work [9] is used. If Thk 0 for [0,1] , alternative Ah absolutely dominates Ak. If Thk 0 for [0,1] , alternative Ah is absolutely dominated by Ak. If Thkl 0 and Thku 0 for
n
akj ( p j oj ) oj (bkj akj )
267
3/2
,
Fhk1 (Yh Yk Vh Z k ) 2 Ehk1
F 2 4E (Y Y V Z ) hk1 hk1 h k h k 2 12Ehk1 F 2 4E hk 2 ( Z h Vk ) hk 2
3/2
3/2
3 Fhk 1
F (Y Y ) hk 2 h k 2 Ehk 2
2 4E Fhk hk 2 (Yh Yk Z h Vk ) 2 2 12 Ehk 2
3/2
Proof:
1/2
2 Fhk 2 Fhk 4Ehk 2 ( Ehk 2 Fhk 2 ) 2 2Ehk 2
(by proposition 5) 1/2
Fhk 2 (Fhk 2 2Ehk 2 )2 2Ehk 2
1.
F. Develop dominance degree The dominance degree of one alternative over another is developed based on measuring the difference of one final fuzzy evaluation value of the corresponding alternative over another. Formulas for the dominance degree of one alternative over another can be clearly derived via integral procedure based on the developed
Figure 2. Yh Yk 0 .
Situation 1 can be shown as in Figure 2. Formulas for P1 and P2 in situation 1 can be developed as follows:
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
268
Z h Vk
0
P1
fTR ( x ) dx
F 2 4 E (Y Y V Z ) hk1 hk1 h k h k 2 12 Ehk1
hk
1/2
Z h Vk
0
2 Fhk 2 Fhk 4 Ehk 2 ( x Z h Vk ) 2 2 Ehk 2
0
dx
Y Y h
1/2
2 Z h Vk Fhk 2 Z h Vk Fhk 2 4 Ehk 2 ( x Z h Vk ) dx 0 0 2 Ehk 2 2 Ehk 2
Fhk 2 ( Z h Vk ) 0 2 Ehk 2
F 2 4E ( x Z V ) hk 2 h k hk 2 2 Ehk 2
0
Z h Vk
dx
F 2 4E ( x Z V ) hk 2 hk 2 h k Fhk 2 x 2 2 Ehk 2 Y Y 12 Ehk1 h k 0
F 2 hk 2
F 2 4E hk 2 ( Z h Vk ) hk 2
P2
dx
u
3/2
4 Ehk 2 ( x Z h Vk ) 2 12 Ehk 2
Z h Vk
4 Ehk 2 (Z h Vk ) 2 12 Ehk 2
P2
Yh Yk
V Z h
Fhk1
k
Fhk1 h k F 2 4 Ehk1 ( x Vh Z k ) hk1 2 2 Ehk1 V Z 12 Ehk 1 h
k
1/2
4 Ehk1 ( x Vh Z k ) 2 Ehk1
Y Y
Fhk1 (Yh Yk Vh Z k ) 2 Ehk1
3 Fhk 1
F (Y Y ) hk 2 h k 2 Ehk 2
2 4E Fhk hk 2 (Yh Yk Z h Vk ) 2 2 12 Ehk 2
3/2
P1 , P1 P2
dx
Yh Yk Vh Z k
Fhk1(Yh Yk ) 2 Ehk1
F 2 4 E (Y Y V Z ) hk1 h k h k hk1
3/2
2 4 E ( V Z ) Fhk hk1 h k 1
2 12 Ehk 1
f L ( x) Vh Z k Thk
P1
(24)
Yh Yk
F 2 hk1
3/2
3/2
where
3/2
F 2 4E hk 2 ( Z h Vk ) hk 2
Situation 2: Yh Yk 0, dhk
0 f L ( x) f R ( x) Vh Z k Thk Yh Yk Thk
Fhk1 (Yh Yk Vh Z k ) 2 Ehk1
for situation 1, can be easily obtained by using Eq. (24) and Eq. (25).
Similarly, P2 can be developed as follows: Yh Yk
3/2
(25)
3 2 Fhk 2 Fhk 2 4 Ehk 2 ( Z h Vk ) 2 12 Ehk 2 Fhk 2 ( Z h Vk ) Thus P1 2 Ehk 2 2 Fhk 2
Yh Yk
Therefore, the dominance degree, d hk P1 P1 , P P1 P2
0 3/2
3 Fhk 2
0
3/2
2 4 E Fhk hk 2 (Yh Yk Z h Vk ) 2 2 12 Ehk 2
F 2 4 E (Y Y V Z ) hk1 hk1 h k h k 2 12 Ehk1
3/2
dx
F (Y Y ) hk 2 h k 2 Ehk 2
1 du 2 Ehk 2 4 Ehk 2 1 2 3/2 u 2 3 8Ehk 2
0
hk
0
1/2
F 2 4E ( x Z V ) hk 2 h k hk 2 2 Ehk 2
fTR ( x)
k
1/2
2 Let u Fhk 2 4 Ehk 2 ( x Z h Vk ) du 4 Ehk 2 dx 1 dx du 4 Ehk 2
Z h Vk
3 Fhk 1
2 Fhk 2 Fhk 4 Ehk 2 ( x Z h Vk ) 2 Yh Yk 2 Ehk 2
dx
1/2
Z h Vk
3/2
Fhk 2 ( Z h Vk Yh Yk ) 2 Ehk 2
3 2 Fhk 2 Fhk 2 4 Ehk 2 (Yh Yk Z h Vk ) 2 12 Ehk 2 Fhk1 (Vh Z k ) . P2 2 Ehk1
F 2 4E (V Z ) hk1 hk1 h k 2 12 Ehk1
3/2
3 Fhk 1
3/2
,
3/2
Ta-Chung Chu and Peerayuth Charnsethikul: Ordering Alternatives under Fuzzy Multiple Criteria Decision Making
Proof:
3 2 Fhk 2 Fhk 2 4 Ehk 2 (Yh Yk Z h Vk ) 2 12 Ehk 2
P2
0
V Z h
k
269
3/2
(26)
fTL ( x)dx hk
1/2
2 Fhk1 Fhk 1 4 Ehk1 ( x Vh Z k ) Vh Z k 2 Ehk1
Figure 3. Yh Yk 0 .
P1
0
fTL ( x)dx hk
Yh Yk
0
Zh Yk
Y Y h
k
fTR ( x)dx
fTL (x)dx hk
3/2
dx Yh Yk
F (Y Y ) hk1 h k 2 Ehk1
3/2
2 4 E ( V Z ) Fhk 1 hk1 h k
3/2
2 12 Ehk 1 Z h Yk R fT ( x)dx hk Y Y
h
Vh Z k
(27)
P1 , P1 P2
for situation 2, can be easily obtained by using Eq. (26) and Eq. (27). Situation 3: Yh Yk 0, dhk
P1 , P1 P2
where
0
F 2 4 E (Y Y V Z ) hk1 h k h k hk1
0
3 Fhk 1
P
F 2 4E ( x V Z ) hk1 hk1 h k 2 12 Ehk1
3/2
3/2
Therefore, the dominance degree, d hk P1
1/2
F hk1 x 2 Ehk1 0
F 2 4E ( x V Z ) hk1 hk1 h k Fhk1 x 2 2 Ehk1 V Z 12 Ehk1 h k 2 Fhk1 (Vh Z k ) Fhk1 4Ehk1 (Vh Zk ) 2 2Ehk1 12Ehk 1
hk
2 Yh Yk Fhk1 Fhk1 4 Ehk1 ( x Vh Z k ) 0 2 Ehk1 Yh Yk
dx
0
Situation 2 can be shown as in Figure 3. Formulas for P1 and P2 in situation 2 can be developed as follows: Yh Yk
Yh Yk
3 2 Fhk 2 ( Z h Vk ) Fhk 2 Fhk 2 4 Ehk 2 (Z h Vk ) P1 2 2 Ehk 2 12 Ehk 2
2 Fhk1 (Vh Z k ) Fhk1 4Ehk1 (Vh Z k ) P2 2 2 Ehk1 12 Ehk 1
3/2
3/2
,
3 Fhk 1
Proof:
k
1/2
2 Fhk 2 Fhk 2 4 Ehk 2 ( x Z h Vk ) Yh Yk 2 Ehk 2
Z h Vk
F 2 4E ( x Z V ) hk 2 hk 2 h k Fhk 2 x 2 2 Ehk 2 Y Y 12 Ehk1 h k Z h Vk
3/2
dx Z h Vk
Yh Yk
F ( Z Vk Yh Yk ) hk 2 h 2 Ehk 2 3 2 Fhk 2 Fhk 2 4 Ehk 2 (Yh Yk Z h Vk ) 2 12 Ehk 2 Fhk1 (Yh Yk ) P1 2 Ehk1
Figure 4. Yh Yk 0 .
3/2
Situation 3 can be shown as in Figure 4. Formulas for P1 and P2 in situation 3 can be developed as follows:
3/2 3/2 F 2 4 E (Y Y V Z ) F 2 4 E ( V Z ) hk1 h k h k hk1 h k hk1 hk1 2 12 Ehk 1
F ( Z V Yh Yk ) hk 2 h k 2 Ehk 2
P1
Z h Vk
0
2 Z h Vk Fhk 2 Fhk 2
0
fTR ( x)dx hk
1/2
4 E hk 2 ( x Z h Vk ) 2 Ehk 2
dx
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Fhk 2 ( Z h Vk ) 2 Ehk 2
3 2 Fhk 2 Fhk 2 4 Ehk 2 ( Z h Vk ) 2 12 Ehk 2
P2
0
V Z h
k
3/2
(28)
fTL ( X )dx hk
1/2
2 Fhk1 Fhk 4 Ehk1 ( x Vh Z k ) 1 Vh Z k 2 Ehk1 F (V Z k ) hk1 h 2Ehk1
0
F 2 4E (V Z ) hk1 hk1 h k 2 12 Ehk 1
3/2
3 Fhk 1
dx
By Eq. (6), the average weights of criteria can be obtained and also shown in Table 2. Meanwhile, the decision-makers employ the linguistic values and corresponding fuzzy numbers shown in Fig. 1 in Section 2.5 to evaluate the suitability of alternative locations versus different qualitative criteria as shown in Tables 3-5. By Eq. (7), the average ratings of criteria can be produced and are also displayed in Tables 3-5. The evaluation data of alternative locations versus quantitative criteria are shown in Table 6. And by Eq. (8), the normalized data is also shown in Table 6.
(29)
Therefore, the dominance degree, d hk P1 P
P1 , P1 P2
for situation 3, can be easily obtained by using Eq. (28) and Eq. (29). G. A ranking approach based on dominance degree Formulae for the dominance degree, d hk , of the alternative Ah over Ak are developed through Eqs. (23)-(29). When d hk 0.5 , Ah dominates Ak; when d hk 0.5 , Ak dominates Ah; when d hk 0.5 , Ah and Ak are equal; and these three situations become the same situations for the preference degree mentioned in [9] when Thk is a linear fuzzy number. Finally, a ranking approach based on the dominance degree is suggested to order the alternatives. Let d ( Ah ) Max d hk , h k , k h
h, k 1 ~ m. Obviously, if d ( Ah ) d ( Ak ) , then Ah Ak ;
if d ( Ah ) d ( Ak ) , then Ah Ak ; if d ( Ah ) d ( Ak ) , then Ah Ak .
4. Numerical Example A hypothetical facility location selection problem is applied to demonstrate the computational process of the proposed model. Assume that a high-tech manufacturing company needs to select a location to construct a plant. After an initial screening, four alternative locations A1, A2, A3 and A4 are chosen for further evaluation. A committee of three decision-makers, D1, D2 and D3 is formed to determine the most suitable alternative location. Further assume that five criteria are selected and categorized by the decision makers as shown in Table 1. Furthermore, assume that the decision-makers employ the linguistic values and their corresponding fuzzy numbers shown in Fig. 5 to evaluate the importance weights of the criteria as shown in Table 2.
Figure 5. Membership functions for linguistic values (UI=Unimportant=(0,0,0.2), B. UI & I=Between Unimportant and Important=(0.1,0.3,0.5), I= Important=(0.3,0.5,0.7), VI=Very Important=(0.5,0.7,0.9), AI=Absolutely Important=(0.8,1,1)). Table 1. Categorized facility location selection criteria. Qualitative Criteria Availability of skilled workers, C1 Power availability, C2 Transportation availability, C3
Quantitative Criteria Community attitude, C4 (pt., larger, better) Investment cost, C5 ($, smaller, better)
Table 2. The importance weights of the criteria and the average weights. Attributes C1 C2 C3 C4 C5
Decision makers D1 D2 I AI B. UI & I VI I VI AI AI B. UI & I I
D3 AI VI I VI I
Average weights (wj) (0.6333, 0.8333, 0.9000) (0.3667, 0.5667, 0.7667) (0.3667, 0.5667, 0.7667) (0.7000, 0.9000, 0.9667) (0.2333, 0.4333, 0.6333)
Table 3. Ratings of locations under C1 and average ratings. Alternatives A1 A2 A3 A4
D1 E E G G
Decision makers D2 D3 VG VG E VG B. B&G G VG E
Average ratings (xi1) (0.5667, 0.7667, 0.9333) (0.6333, 0.8333, 0.9667) (0.2333, 0.4333, 0.6333) (0.5000, 0.7000, 0.8667)
Table 4. Ratings of locations under C2 and average ratings. Alternatives A1 A2 A3 A4
D1 VG G VG E
Decision makers D2 D3 G VG G VG G G B. B&G VG
Average ratings (xi2) (0.4333, 0.6333, 0.8333) (0.3667, 0.5667, 0.7667) (0.3667, 0.5667, 0.7667) (0.4333, 0.6333, 0.8000)
Ta-Chung Chu and Peerayuth Charnsethikul: Ordering Alternatives under Fuzzy Multiple Criteria Decision Making
d ( A1 ) Max d12 , d13 , d14
Table 5. Ratings of locations under C3 and average ratings. Alternatives
D1 VG VG G G
A1 A2 A3 A4
Decision makers D2 D3 E VG E G VG B. B&G E G
271
Max 0.679, 0.962, 0.665 0.962;
Average ratings (xi3)
d ( A2 ) Max d 21 , d 23 , d 24
(0.5667, 0.7667, 0.9333) (0.5000, 0.7000, 0.8667) (0.3000, 0.5000, 0.7000) (0.4333, 0.6333, 0.8000)
Max 0.321, 0.893, 0.456 0.893;
d ( A3 ) Max d 31 , d 32 , d 34
Max 0.038, 0.107, 0.090 0.107;
By Eqs. (9)-(16), Pi, i=1~4, can be obtained and displayed as in Table 7. By Eqs. (17)-(22), the difference of two final fuzzy evaluation values Thk , h, k 1~ 4 , can be obtained and displayed as in Table 8. By Eqs. (23)-(29), the dominance degree, d hk , h, k 1~ 4 , can be obtained and displayed as in Table 9.
because Obviously, d ( A1 ) d ( A4 ) d ( A2 ) d ( A3 ) 0.962>0.910>0.893>0.107. Therefore the ranking ordering of the four alternatives is A1 A4 A2 A3 .
Table 6. Quantitative values of locations and normalized values.
5. Conclusions
Alternatives A1 A2 A3 A4
Community attitude (0~100 points), C4 93 88 83 89
Investment cost (million $), C5 (48,53,58) (46,50,54) (52,56,62) (41,45,48)
normalized points xi4
normalized costs xi5
1 0.5 0 0.6
(0.190,0.429,0.667) (0.381,0.571,0.762) (0.000,0.286,0.476) (0.667,0.810,1.000)
Table 7. Final fuzzy evaluation values of alternatives. Si S1 S2 S3 S4
Vi 1.470 1.158 0.392 1.210
Yi 2.518 2.110 1.089 2.192
Zi 3.583 3.088 1.996 3.220
Ei1 0.168 0.158 0.177 0.149
Fi1 0.880 0.794 0.520 0.833
Ei2 0.132 0.120 0.131 0.116
Fi2 -1.197 -1.099 -1.038 -1.144
Table 8. The difference of two final fuzzy evaluation values. Thk
Vh Z k
Yh Yk
Z h Vk
Ehk1
Fhk1
Ehk 2
Fhk 2
T12 T13 T14 T23 T24 T34
-1.618 -0.526 -1.750 -0.838 -2.062 -2.828
0.408 1.429 0.326 1.020 -0.082 -1.103
2.426 3.191 2.373 2.696 1.878 0.786
0.047 0.036 0.052 0.027 0.042 0.061
1.979 1.918 2.024 1.832 1.938 1.664
-0.026 -0.045 -0.017 -0.057 -0.028 -0.017
-1.991 -1.717 -2.031 -1.619 -1.932 -1.871
Table 9. Dominance degree of one alternative over another. dhk
Yh Yk
d12 d13
P1
P2
P1
P2
P
>0
1.381
0.653
2.034
0.679
0.321 ( d 21 )
>0
1.800
0.072
1.872
0.962
0.038 ( d31 )
d 23
>0
1.590
0.191
1.781
0.893
0.107 ( d32 )
d14
