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Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets
Expert Systems with Applications 37 (2010) 8673–8678
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets Deng-Feng Li * School of Management, Fuzhou University, Fuzhou, Fujian 350108, China
a r t i c l e
i n f o
Keywords: Multiattribute decision making Uncertainty Generalized ordered weighted averaging operators (GOWA) Intuitionistic fuzzy set Fuzzy set
a b s t r a c t The intuitionistic fuzzy (IF) set characterized by two functions was a generalization of the fuzzy set. In this paper, we investigate multiattribute decision making (MADM) problems with ratings of alternatives being expressed using IF sets and attribute weights given as real numbers. Firstly, the generalized ordered weighted averaging (GOWA) operators introduced by Yager [Yager, R. (2004). Generalized OWA aggregation operators. Fuzzy Optimization and Decision Making, 3, 93–107] are extended to aggregate IF sets. Secondly, MADM problems with IF sets are formulated through transforming the ratings of alternatives on both qualitative and quantitative attributes into IF sets in a unified way. The method and procedure based on the extended GOWA operators are developed to solve the MADM problems with IF sets. Finally, the effectiveness and practicability of the proposed method are illustrated with a numerical example. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Atanassov (1986, 1999) introduced the concept of an intuitionistic fuzzy (IF) set characterized by a membership function and a non-membership function, which is a generalization of the fuzzy set introduced by Zadeh (1965). The IF set theory gives us the possibility to model hesitation and uncertainty by using an additional degree, i.e., the intuitionistic index. Over the last decades, the IF set theory has been applied to many different fields such as decision making (Atanassov, Pasi, and Yager, 2002; Herrera, Martinez, and Sanchez, 2005; Li, 2004, 2005; Szmidt and Kacprzyk, 2002, 2004), logic programming (Atanassov and Georgiev, 1993), topology (Abbas, 2005; Davvaz, Dudek, and Jun, 2006; Dudek, Davvaz, and Jun, 2005), medical diagnosis (De, Biswas, and Roy, 2001), pattern recognition (Li and Cheng, 2002; Wang and Xin, 2005) and machine learning as well as market prediction (Liang and Shi, 2003). In comparison with the fuzzy set, the IF set seems to be better suited for expressing a very important factor, i.e., the hesitation of the decision maker, which should be taken into account when we try to construct really adequate models and solutions of decision making problems. In this paper, ratings of alternatives on attributes are expressed with IF sets due to the fact that in some situations determining them precisely is usually depended on judgment and intuition of the decision maker. Therefore, investigation on mul-
* Present address: School of Management, Fuzhou University, No. 2, Xueyuan Road, Daxue New District, Fuzhou District, Fuzhou, Fujian 350108, China. Tel./fax: +86 0591 83768427. E-mail addresses: [email protected], [email protected]. 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.06.062
tiattribute decision making (MADM) problems with IF sets is of importance for both theory and application. MADM problems are an important type of multicriteria decision making (MCDM) problems and are wide spread in real-life decision situations (Erol and Ferrell, 2003; Herrera et al., 2005; Hwang and Yoon, 1981; Srinivasan and Shocker, 1973). MADM problems are solved using the scoring techniques such as the weighted aggregation operators based on multiple attribute value theory (Keeney and Raiffa, 1976). The classical weighted aggregation is usually known in the literature by the weighted average (WA) or simple additive weighting method (Yager, 1996). Another important aggregation operator within the class of weighted aggregation operators is the OWA operator (Yager, 1988, 1993). Since its appearance in 1988, the OWA operator has been used in a wide range of applications such as engineering, neural networks, data mining, decision making and image process as well as expert systems (Calvo, Mayor, and Mesiar, 2002; Torra, 2003; Xu, 2004). Recently, Yager (2004) introduced a generalization of the ordered weighted averaging (OWA) operators, which is called the generalized ordered weighted averaging (GOWA) operators. The GOWA operators are an extension of the OWA operators to the case in which adds to the OWA operators an additional parameter controlling the power to which the argument values are raised. They are also regarded as a generalization of combining the OWA operators with the generalized mean operator (Dyckhoff and Pedrycz, 1984). Yager studied many properties of these operators and several special cases, including generalized means, Hurwicz operator, min, max and the ordered weighted geometric (OWG) operator. The existing research of the GOWA operators mainly focused on MADM
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problems which values are crisp or fuzzy. In this paper, the GOWA operators are extended to aggregate IF sets. Then, a methodology based on the extended GOWA operators are developed to solve the MADM problems with IF sets. The rest of this paper is organized as follows. Section 2 reviews some basic concepts such as the IF set as well as the GOWA operators. In Section 3, the MADM problems with IF sets are formulated and the GOWA operators are extended to aggregate IF sets. The method and procedure for solving MADM problems with IF sets are developed in detailed. The proposed method is illustrated with a numerical example in Section 4. Short conclusion is given in Section 5. 2. Intuitionistic fuzzy sets and GOWA operators 2.1. The concept and operations of IF sets Definition 1 (Atanassov, 1986, 1999). Let X be a finite universal set. An IF set B in X is an object having the following form:
B ¼ fhx; lB ðxÞ; tB ðxÞijx 2 Xg where the functions
lB : X#½0; 1 x 2 X ! lB ðxÞ 2 ½0; 1
The score function D(A) represents the ‘‘net” membership degree. Definition 4 (Hong and Choi, 2000). Let A = hlA, tAi be an IF set. Then the accuracy function r of A is defined as follows
rðAÞ ¼ lA þ tA Obviously, r(A) 2 [0, 1]. It is easy to see that r(A) = lA + tA = 1 pA. Therefore, the accuracy function r(A) represents the non-hesitation degree. To rank two IF sets A = hlA, tAi and B = hlB, tBi, their score functions and accuracy functions need to be compared, respectively. In some literatures, the score function and the accuracy function are explained in terms of the mean and variance in Statistics. Thus, in general the score function is more important than the accuracy function. Then according to the lexicographic method, an order relation may be defined as follows. Definition 5. Let A = hlA, tAi and B = hlB, tBi be two IF sets. (1) (2) (2a) (2b) (2c)
If If If If If
D(A) > D(B) then A is bigger than B. D(A) = D(B) then r(A) = r(B) then A is equal to B; r(A) < r(B) then A is smaller than B; r(A) > r(B) then A is bigger than B.
and
tB : X#½0; 1 x 2 X ! tB ðxÞ 2 ½0; 1 define the degree of membership and degree of non-membership of the element x 2 X to the set B # X, respectively, and for every x 2 X, 0 6 lB(x) + tB(x) 6 1. Let
pB ðxÞ ¼ 1 lB ðxÞ tB ðxÞ which is called the intuitionistic index of the element x in the set B (Burillo and Bustince, 1995; Bustince and Burillo, 1996). It is the degree of indeterminacy membership of the element x to the set B. Obviously,
0 6 pB ðxÞ 6 1 If jBj = 1, i.e., there is only one element x in the set B, then B degenerates to B = {hx, lB(x), tB(x)i}, which usually is denoted by B = hlB, tBi for short. Definition 2 (Atanassov, 1986; De, Biswas, and Roy, 2000). Let B and Q be two IF sets in the set X and b P 0, then (1) (2) (3) (4)
B + Q = {hx, lB(x) + lQ(x) lB(x)lQ(x), tB(x)tQ(x)ijx 2 X}; BQ = {hx, lB(x)lQ(x), tB(x) + tQ(x) tB(x)tQ(x)ijx 2 X}; bB = {hx, 1 (1 lB(x))b, (tB(x))bijx 2 X}; Bb = {hx, (lB(x))b, 1 (1 tB(x))bijx 2 X}.
2.2. A ranking method for IF sets There are a few methods for ranking IF sets. Here the method based on the score function is chosen because of its concision and intuition.. Definition 3 (Chen and Tan, 1994). Let A = hlA, tAi be an IF set. Then the score function D of A is defined as follows
DðAÞ ¼ lA tA Obviously, D(A) 2 [1, 1].
2.3. GOWA operators P Let a mapping f : Rn ? R such that f ða1 ; a2 ; . . . ; an Þ ¼ nj¼1 xj bj , T where x = (x1, x2, . . ., xn) is a weight vector which is correlative Pn with f, satisfying xj 2 [0, 1] (j = 1, 2, . . ., n) and j¼1 xj ¼ 1; bj is the jth largest one of all numerical values ak (k = 1, 2, . . ., n). The function f is called an OWA operator of n-dimension (Yager, 1988). Definition 6 (Yager, 2004). Let a mapping h : Rn ? R such that P 1=k k n , where x = (x1, x2, . . ., xn)T is a hða1 ; a2 ; . . . ; an Þ ¼ j¼1 xj bj weight vector which is correlative with h, satisfying xj 2 [0, 1] Pn (j = 1, 2, . . ., n) and j¼1 xj ¼ 1; bj is the jth largest one of all numerical values ak (k = 1, 2, . . ., n). k 2 (1, +1) is a parameter. The function h is called GOWA operators of n-dimension. Obviously, there are some properties for the GOWA operators h. (1) When k ? 1, if xj – 0 for all j, then h(a1, a2, . . ., an) = bn = min{aj}. The GOWA operator h reduces to the Min operator. However, if the weight vector x is such that x1 = 1 and xj = 0 for all j – 1 then even though k ? 1 we get h(a1, a2, . . ., a n) = b1 = max{aj}. Q x (2) When k ! 0; hða1 ; a2 ; . . . ; an Þ ¼ nj¼1 bj j . The GOWA operator h reduces to the OWG operator. P (3) When k ¼ 1; hða1 ; a2 ; . . . ; an Þ ¼ nj¼1 xj bj . The GOWA operator h reduces to the OWA operator. (4) When k ? +1, if xj – 0 for all j, then h(a1, a2, . . ., an) = b1 = max{aj}. The GOWA operator h reduces to the Max operator. It is interesting to note that if k ? +1 but the weight vector x is such that xn = 1 and xj = 0 for all j – n then h(a1, a2, . . ., an) = bn = min{aj}. 3. Extended GOWA operators with IF sets and MADM methodology 3.1. MADM problems with IF sets Suppose that there exist an alternative set A = {A1, A2, . . ., Am} which consists of m non-inferior alternatives from which the best
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alternative is to be selected. Each alternative is assessed on n attributes, both quantitatively and qualitatively. Denote the set of all attributes by X = {x1, x2, . . ., xn}. For quantitative attributes xj 2 X, alternatives Ai 2 A may be assessed using crisp values aij, which are obtained according to history data and Statistics. Since the physical dimensions and measurements of different quantitative attributes are different, so all values aij need to be normalized. Various normalization methods may be chosen according to the expert’s experience and the need in real situations. Here the formulae for relative degrees of membership and relative degrees of non-membership are chosen as follows
lij ¼
8 aij > < aj amax j
ðj 2 F 1 Þ
min > : d j aj
ðj 2 F 2 Þ
aij
ð1Þ
and
tij ¼
8 a ij > < bj amax j > :c
amin j
j aij
Definition 7. Let g be a mapping such that
ð5Þ
where x = (x1, x2, . . ., xn)T is a weight vector which is correlative Pn with g, satisfying xj 2 [0, 1] (j = 1, 2, . . ., n) and j¼1 xj ¼ 1; dj ¼ < lj ; tj > is the jth largest one of all IF sets ck (k = 1, 2, . . ., n) using the ranking methods of IF sets (for example, Definition 5); k 2 (0, +1) is a parameter which is chosen a prior according to characteristics and needs in real-life situations. Then g is called GOWA operators with IF sets, which are called as the extended GOWA operators for short. It is worthy to note that k should not take negative numbers since the negative power of the IF set dj has no meaning. According to Definition 2, it can be derived from Eq. (5) that n X
#1=k
xj ðhlkj ; 1 ð1 tj Þk iÞ
j¼1
(
ð2Þ
ðj 2 F 2 Þ
¼
n X h1 ð1 lkj Þxj ; ½1 ð1 tj Þk xj i
)1=k
j¼1
2
respectively, where F and F are the subscript sets of benefit quantitative attributes and cost quantitative attributes, and
amax ¼ max faij g; j
!1=k
xj dkj
j¼1
"
ðj 2 F Þ
n X
gðc1 ; c2 ; . . . ; cn Þ ¼
gðc1 ; c2 ; . . . ; cn Þ ¼
1
1
16i6m
3.2. GOWA operators with IF sets
amin ¼ min faij g j 16i6m
ð3Þ
aij 6 1 ðj 2 F 1 Þ amax j
and
0 6 lij þ tij ¼ ðdj þ cj Þ
amin j aij
¼ *" ¼
n n Y Y 1 ð1 lkj Þxj ; ½1 ð1 tj Þk xj j¼1
n Y 1 ð1 lkj Þxj j¼1
and aj 2 [0, 1], bj 2 [0, 1], dj 2 [0, 1] and cj 2 [0, 1] satisfying conditions 0 6 aj + bj 6 1 and 0 6 dj + cj 6 1. Values of parameters aj, bj, dj and cj are chosen a prior according to characteristics and needs in real situations. Obviously,
0 6 lij þ tij ¼ ðaj þ bj Þ
(*
6 1 ðj 2 F 2 Þ
Then, values aij of alternatives Ai 2 A on quantitative attributes xj 2 X can be transformed into IF sets rij = {hxj, lij, tiji}, respectively, which usually are denoted by rij = hlij, tiji for short. For qualitative attributes xj 2 X, alternatives Ai 2 A may be assessed using linguistic variables with designed terms a prior. Semantics of these terms are expressed with IF sets according to characteristics and needs in real situations. Thus, ratings of alternatives on qualitative and quantitative attributes may be expressed with IF sets in a unified way. Let the vector of IF sets of all n attributes for an alternative Ai 2 A be denoted by (ri1, ri2, . . ., rin) = (hli1, ti1i, hli2, ti2i, . . ., hlin, tini). Hence, a MADM problem with IF sets can be expressed concisely in the matrix format as follows
j¼1
#1=k
;1
( 1
+)1=k
n Y ½1 ð1 tj Þk xj
)1=k +
j¼1
ð6Þ Obviously, the following conclusions are derived. Q x (1) When k ! 0; gðc1 ; c2 ; . . . ; cn Þ ¼ nj¼1 dj j . The extended GOWA operator g reduces to the OWG operator using IF sets. P (2) When k ¼ 1; gðc1 ; c2 ; . . . ; cn Þ ¼ nj¼1 xj dj . The extended GOWA operator g reduces to the OWA operator using IF sets. (3) When k ? +1, if xj – 0 for all j, then g(c1, c2, . . ., cn) = d1. The extended GOWA operator g reduces to the Max operator using IF sets. d1 is the largest one of all IF sets cj (j = 1, 2, . . ., n) according to the ranking methods of IF sets. It is interesting to note that if k ? +1 but the weight vector x is such that xn = 1 and xj = 0 for all j – n then g(c1, c2, . . ., cn) = dn. 3.3. Decision making method and procedure based on the extended GOWA operators Assume that a MADM problem with IF sets be expressed as the IF decision matrix in Eq. (4). Then an algorithm and process of the MADM methodology based on the extended GOWA operators are summarized as follows. Step 1: Obtain the IF decision matrix F. Applying the method proposed in Section 3.1, the IF decision matrix F can be constructed. Step 2: Compute overall assessments of alternatives. Using Eq. (6) and the rows of the IF decision matrix F in Eq. (4), the overall assessments ri = hli, tii (i = 1, 2, . . ., m) of alternatives are computed, i.e.,
r i ¼ gðr i1 ; r i2 ; . . . ; rin Þ ð4Þ which is referred to as an IF decision matrix used to represent the MADM problem with IF sets.
respectively, where (ri1, ri2, . . ., rin) = (hli1, ti1i, hli2, ti2i, . . ., hlin, tini). Step 3: Rank the order of all alternatives. The ranking order of the alternative set X is generated according to the scores D(ri) and the accuracies r(ri) (i = 1, 2, . . ., m) using Definition 5.
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4. A numerical example
a11 2:0 ¼ 0:64 ¼ 0:8 2:5 amax 1 a21 2:5 l21 ¼ a1 max ¼ 0:8 ¼ 0:8 2:5 a1 a31 1:8 l31 ¼ a1 max ¼ 0:8 ¼ 0:58 2:5 a1 a41 2:2 ¼ 0:70 l41 ¼ a1 max ¼ 0:8 2:5 a1 a11 2:0 ¼ 0:08 t11 ¼ b1 max ¼ 0:1 2:5 a1 a21 2:5 ¼ 0:1 t21 ¼ b1 max ¼ 0:1 2:5 a1 a31 1:8 ¼ 0:07 t31 ¼ b1 max ¼ 0:1 2:5 a1
l11 ¼ a1
An extended air-fighter selection problem (Hwang and Yoon, 1981) is investigated in this section. Suppose that one country D plans to buy air-fighters from another country J. The Defense Department of the country J would provide the country D with characteristic data for four candidate air-fighters A1, A2, A3 and A4. The decision maker takes into consideration the following six attributes in evaluating the air-fighters, including maximum speed (x1), cruise radius (x2 ), maximum loading (x3), price (x4), reliability (x5) and maintenance (x6). x5 and x6 are qualitative attributes and their ratings are expressed using linguistic variables. The ratings or data of all air-fighters on attributes are given by the decision maker as in Table 1. The corresponding relations between linguistic variables and IF sets are given as in Table 2. According to Table 1, the decision matrix can be obtained as follows
and
t41 ¼ b1 ð7Þ
a41 2:2 ¼ 0:09 ¼ 0:1 2:5 amax 1
respectively. Thus, crisp values ai1 of Ai (i = 1, 2, 3, 4) on the attribute x1 can be transformed into the IF sets as follows
r11 ¼ h0:64; 0:08i; r31 ¼ h0:58; 0:07i; For the benefit attribute x1 (i.e., maximum speed), it is easily seen from Eq. (7) that
amax ¼ 2:5 1 Using Eqs. (1) and (2) with a1 = 0.8 and b1 = 0.1, the relative degrees of membership and relative degrees of non-membership for Ai (i = 1, 2, 3, 4) on the attribute x1 can be calculated as follows
r 21 ¼ h0:8; 0:1i; r 41 ¼ h0:70; 0:09i
respectively. Similarly, using Eqs. (1) and (2) with a2 = 0.9, b2 = 0.05, a3 = 0.85, b3 = 0.1, d4 = 0.75 and c4 = 0.2, crisp values aij of Ai(i = 1, 2, 3, 4) on the attributes xj (j = 2,3,4) can be transformed into IF sets rij, respectively. Thus, combining with Table 2, the decision matrix F (i.e., Eq. (7)) can be transformed into the IF decision matrix as follows:
ð8Þ
Assume that the weights which are correlative with the extended GOWA operators g be given as crisp numbers, namely,
Table 1 Decision information given by the decision maker. Air-fighters Attributes x1 (mach) x2 (mile 103) x3 (lb. 104) x4 ($ 106) x5 x6 A1 A2 A3 A4
2.0 2.5 1.8 2.2
1.5 2.7 2.0 1.8
2.0 1.8 2.1 2.0
5.5 6.5 4.5 5.0
Table 2 The relations between linguistic variables and IF sets. Linguistic variables
IF sets
Very high (VH) High (H) Medium (M) Low (L) Very low (VL)
h0.95, 0.05i h0.70, 0.25) h0.50, 0.40i h0.25, 0.70i h0.05, 0.95i
M L H M
VH M H M
x ¼ ð0:1; 0:15; 0:2; 0:15; 0:15; 0:25ÞT According to Eq. (8) and Definition 3, scores of r1j (j = 1, 2, . . ., 6) are obtained as follows
Dðr 11 Þ ¼ 0:56; Dðr 14 Þ ¼ 0:46;
Dðr 12 Þ ¼ 0:47; Dðr13 Þ ¼ 0:71; Dðr 15 Þ ¼ 0:1; Dðr 16 Þ ¼ 0:9
respectively. Obviously,
Dðr 16 Þ > Dðr 13 Þ > Dðr11 Þ > Dðr 12 Þ > Dðr 14 Þ > Dðr 15 Þ Hence, using Eq. (6), the overall assessment r1 = hl1, t1i of the alternative A1 can be computed as follows
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D.-F. Li / Expert Systems with Applications 37 (2010) 8673–8678 Table 3 Overall assessments of alternatives with different parameter values and scores and accuracies. A1
k
?0 1 2 ?+1
A2
A3
A4
r1
D(r1)
r(r1)
r2
D(r2)
r(r2)
r3
D(r3)
r(r3)
r4
D(r4)
r(r4)
(0.62, 0.18) (0.69, 0.11) (0.71, 0.11) (0.95, 0.05)
0.44 0.58 0.6 0.9
0.8 0.8 0.82 1.0
(0.52, 0.36) (0.64, 0.19) (0.66, 0.18) (0.90, 0.05)
0.16 0.45 0.48 0.85
0.88 0.83 0.84 0.95
(0.70, 0.17) (0.71, 0.14) (0.71, 0.13) (0.85, 0.10)
0.53 0.57 0.58 0.75
0.87 0.85 0.84 0.95
(0.60, 0.23) (0.62, 0.15) (0.63, 0.14) (0.81, 0.10)
0.37 0.47 0.49 0.71
0.83 0.77 0.77 0.91
r 1 ¼ ð0:1 r k16 þ 0:15 r k13 þ 0:2 r k11 þ 0:15 r k12 þ 0:15 r k14 þ 0:25 r k15 Þ
Table 4 The ranking orders of the alternatives for the different parameter values.
1=k
¼ h½1 ð1 0:1 0:95k Þð1 0:15 0:81k Þð1 0:2 0:64k Þ ð1 0:15 0:5k Þð1 0:15 0:62k Þð1 0:25 0:5k Þ1=k ; 1 f1 ½1 ð1 0:9Þð1 0:05Þk ½1 ð1 0:85Þð1 0:1Þk
k
The ranking orders of the alternatives
The best alternatives
?0 1 2 ?+1
A3 A1 A4 A2 A1 A3 A4 A2 A1 A3 A4 A2 A1 A2 A3 A4
A3 A1 A1 A1
½1 ð1 0:8Þð1 0:08Þk ½1 ð1 0:85Þð1 0:03Þk ½1 ð1 0:85Þð1 0:16Þk ½1 ð1 0:75Þð1 0:4Þk g1=k i k
k
and
k
¼ h½1 ð1 0:1 0:95 Þð1 0:15 0:81 Þð1 0:2 0:64 Þ k
k
1=k
k
ð1 0:15 0:5 Þð1 0:15 0:62 Þð1 0:25 0:5 Þ k
k
r4 ¼ ð0:1 r k43 þ 0:15 r k41 þ 0:2 r k42 þ 0:15 r k44 þ 0:15 r k46 þ 0:25 r k45 Þ1=k
;
k
1 ½1 ð1 0:1 0:95 Þð1 0:15 0:9 Þð1 0:2 0:92 Þ k
k
k
1=k
ð1 0:15 0:97 Þð1 0:15 0:84 Þð1 0:25 0:6 Þ
i
Similarly, the overall assessments ri = hli,ti> of alternatives Ai(i = 2,3,4) can be obtained as follows
ð1 0:15 0:68k Þð1 0:15 0:5k Þð1 0:25 0:5k Þ1=k ; 1 f1 ½1 ð1 0:9Þð1 0:1Þk ½1 ð1 0:85Þð1 0:09Þk ½1 ð1 0:8Þð1 0:03Þk ½1 ð1 0:85Þð1 0:18Þk ½1 ð1 0:85Þð1 0:4Þk ½1 ð1 0:75Þð1 0:4Þk g1=k i
r 2 ¼ ð0:1 r k22 þ 0:15 r k21 þ 0:2 r k23 þ 0:15 r k24
¼ h½1 ð1 0:1 0:81k Þð1 0:15 0:7k Þð1 0:2 0:6k Þ
þ 0:15 r k26 þ 0:25 r k25 Þ1=k k
¼ h½1 ð1 0:1 0:81k Þð1 0:15 0:7k Þð1 0:2 0:6k Þ
ð1 0:15 0:68k Þð1 0:15 0:5k Þð1 0:25 0:5k Þ1=k ; k
k
¼ h½1 ð1 0:1 0:9 Þð1 0:15 0:8 Þð1 0:2 0:73 Þ
1 ½1 ð1 0:1 0:9k Þð1 0:15 0:91k Þð1 0:2 0:97k Þ 1=k
ð1 0:15 0:52k Þð1 0:15 0:5k Þð1 0:25 0:25k Þ
;
1 f1 ½1 ð1 0:9Þð1 0:05Þk ½1 ð1 0:85Þð1 0:1Þk k
k
½1 ð1 0:8Þð1 0:09Þ ½1 ð1 0:85Þð1 0:14Þ ½1 ð1 0:85Þð1 0:4Þk ½1 ð1 0:75Þð1 0:7Þk g1=k i ¼ h½1 ð1 0:1 0:9k Þð1 0:15 0:8k Þð1 0:2 0:73k Þ ð1 0:15 0:52k Þð1 0:15 0:5k Þð1 0:25 0:25k Þ1=k ; 1 ½1 ð1 0:1 0:95k Þð1 0:15 0:9k Þð1 0:2 0:91k Þ ð1 0:15 0:86k Þð1 0:15 0:6k Þð1 0:25 0:3k Þ1=k i r 3 ¼ ð0:1 r k33 þ 0:15 r k32 þ 0:2 r k34 þ 0:15 r k31 þ 0:15 r k35 þ 0:25 r k36 Þ1=k
ð1 0:15 0:82k Þð1 0:15 0:6k Þð1 0:25 0:6k Þ1=k i respectively. For some special values of the parameter k, ri = hli, tii (i = 1, 2, 3, 4) can be obtained as in Table 3. Corresponding scores D(ri) and accuracies r(ri) of ri = hli, tii (i = 1, 2, 3, 4) are also obtained as in Table 3. According to Table 3 and Definition 5, the ranking orders of alternatives Ai (i = 1, 2, 3, 4) can be generated for the different parameter values as in Table 4, respectively. It is easy to see from Table 4 that the best selection is the alternative (i.e., air-fighter) A1 for the special values of the parameter k except the situation k ? 0 in which the best selection is the airfighter A3. 5. Conclusions
¼ h½1 ð1 0:1 0:85k Þð1 0:15 0:67k Þð1 0:2 0:75k Þ ð1 0:58 0:5k Þð1 0:15 0:7k Þð1 0:25 0:7k Þ1=k ; 1 f1 ½1 ð1 0:9Þð1 0:1Þk ½1 ð1 0:85Þð1 0:04Þk ½1 ð1 0:8Þð1 0:2Þk ½1 ð1 0:85Þð1 0:07Þk ½1 ð1 0:85Þð1 0:25Þk ½1 ð1 0:75Þð1 0:25Þk g1=k i ¼ h½1 ð1 0:1 0:85k Þð1 0:15 0:67k Þð1 0:2 0:75k Þ ð1 0:58 0:5k Þð1 0:15 0:7k Þð1 0:25 0:7k Þ1=k ; 1 ½1 ð1 0:1 0:9k Þð1 0:15 0:96k Þð1 0:2 0:8k Þð 1 0:15 0:93k Þð1 0:15 0:75k Þð1 0:25 0:75k Þ1=k i
The GOWA operators are the extension of the OWA operators (Yager, 2004). As illustrated above, if the parameter k takes different values then the GOWA operators have different particular forms such as the OWG operator and the OWA operator as well as the Max operator. The GOWA operators are useful to deal with MADM problems, but the existing research on them mainly focused on crisp values or fuzzy sets. The fuzziness or uncertainty in the MADM problems with IF sets can be represented exactly since the IF set can model the hesitation of the decision maker. In this paper, the formulae for relative degrees of membership and relative degrees of non-membership are chosen to normalize the crisp values of alternatives on quantitative attributes; ratings of alternatives on qualitative attributes are expressed with
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linguistic variables corresponding to the designed IF sets a prior. The methodology for solving MADM problems with IF sets is developed based on the extended GOWA operators. The effectiveness and applicability of the proposed method is illustrated with an extended air-fighter selection problem. However, it may be expected that the proposed method can be applied to other similar MADM problems with IF sets. Acknowledgements This research was sponsored by the Natural Science Foundation of China (Nos. 70902041 and 70871117) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No. 2005-546). References Abbas, S. E. (2005). On intuitionistic fuzzy compactness. Information Sciences, 173, 75–91. Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96. Atanassov, K. T. (1999). Intuitionistic fuzzy sets. Heidelberg: Springer-Verlag. Atanassov, K. T., & Georgiev, C. (1993). Intuitionistic fuzzy prolog. Fuzzy Sets and Systems, 53, 121–128. Atanassov, K. T., Pasi, G., & Yager, R. R. (2002). Intuitionistic fuzzy interpretations of multi-person multi-criteria decision making. In Proceedings of 2002 1st international IEEE symposium intelligent systems, Varna (Vol. I, pp. 115–119). Burillo, P., & Bustince, H. (1995). Intuitionistic fuzzy relations (Part I). Mathware and Soft Computing, 2, 5–8. Bustince, H., & Burillo, P. (1996). Structures on intuitionistic fuzzy relations. Fuzzy Sets and Systems, 78, 293–303. Calvo, T., Mayor, G., & Mesiar, R. (2002). Aggregation operators: New trends and applications. New York: Physica-Verlag. Chen, S. M., & Tan, J. M. (1994). Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 67, 163–172. Davvaz, B., Dudek, W. A., & Jun, Y. B. (2006). Intuitionistic fuzzy Hv-submodules. Information Sciences, 176, 285–300. De, S. K., Biswas, R., & Roy, A. R. (2000). Some operations on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114, 477–484. De, S. K., Biswas, R., & Roy, A. R. (2001). An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets and Systems, 117, 209–213. Dudek, W., Davvaz, B., & Jun, Y. B. (2005). On intuitionistic fuzzy subhyperquasigroups of hyperquasigroups. Information Sciences, 170, 251–262.
Dyckhoff, H., & Pedrycz, W. (1984). Generalized means as model of compensative connectives. Fuzzy Sets and Systems, 14, 143–154. Erol, I., & Ferrell, W. G. (2003). A methodology for selection problems with multiple conflicting objectives and both qualitative and quantitative criteria. International Journal of Production Economics, 86, 187–199. Herrera, F., Martinez, L., & Sanchez, P. J. (2005). Managing non-homogeneous information in group decision making. European Journal of Operational Research, 166, 115–132. Hong, D. H., & Choi, C. H. (2000). Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 114, 103–113. Hwang, C. L., & Yoon, K. (1981). Multiple attributes decision making methods and applications. Berlin, Heidelberg: Springer. Keeney, R. L., & Raiffa, H. (1976). Multiattribute preferences under uncertainty: More than two attributes. Decisions with multiple objectives: Preferences and value tradeoffs. New York: Wiley. pp. 283–348. Li, D.-F. (2004). Some measures of dissimilarity in intuitionistic fuzzy structures. Journal of Computer and System Sciences, 68(1), 115–122. Li, D. F. (2005). Multiattribute decision making models and methods using intuitionistic fuzzy sets. Journal of Computer and System Science, 70, 73–85. Li, D.-F., & Cheng, C.-T. (2002). New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognition Letters, 23, 221–225. Liang, Z.-Z., & Shi, P.-F. (2003). Similarity measures on intuitionistic fuzzy sets. Pattern Recognition Letters, 24, 2687–2693. Srinivasan, V., & Shocker, A. D. (1973). Linear programming techniques for multidimensional analysis of preference. Psychometrica, 38, 337–342. Szmidt, E., & Kacprzyk, J. (2002). Using intuitionistic fuzzy sets in group decision making. Control and Cybernetics, 31, 1037–1053. Szmidt, E., & Kacprzyk, J. (2004). A concept of similarity for intuitionistic fuzzy sets and its use in group decision making. In Proceedings of international joint conference on neural networks and IEEE international conference on fuzzy systems, Budapest, Hungary (pp. 25–29). Torra, V. (2003). Information fusion in data mining. New York: Springer. Wang, W.-Q., & Xin, X.-L. (2005). Distance measures between intuitionistic fuzzy sets. Pattern Recognition Letters, 26, 2063–2069. Xu, Z. S. (2004). Uncertain multiple attribute decision making: Methods and applications. Beijing: Tsinghua University Press. Yager, R. (1988). On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions on Systems, Man and Cybernetics, 18, 183–190. Yager, R. (1993). Families of OWA operators. Fuzzy Sets and Systems, 59, 125–148. Yager, R. (1996). Quantifier guided aggregation using OWA operators. International Journal of Intelligent Systems, 11, 49–73. Yager, R. (2004). Generalized OWA aggregation operators. Fuzzy Optimization and Decision Making, 3, 93–107. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets Deng-Feng Li * School of Management, Fuzhou University, Fuzhou, Fujian 350108, China
a r t i c l e
i n f o
Keywords: Multiattribute decision making Uncertainty Generalized ordered weighted averaging operators (GOWA) Intuitionistic fuzzy set Fuzzy set
a b s t r a c t The intuitionistic fuzzy (IF) set characterized by two functions was a generalization of the fuzzy set. In this paper, we investigate multiattribute decision making (MADM) problems with ratings of alternatives being expressed using IF sets and attribute weights given as real numbers. Firstly, the generalized ordered weighted averaging (GOWA) operators introduced by Yager [Yager, R. (2004). Generalized OWA aggregation operators. Fuzzy Optimization and Decision Making, 3, 93–107] are extended to aggregate IF sets. Secondly, MADM problems with IF sets are formulated through transforming the ratings of alternatives on both qualitative and quantitative attributes into IF sets in a unified way. The method and procedure based on the extended GOWA operators are developed to solve the MADM problems with IF sets. Finally, the effectiveness and practicability of the proposed method are illustrated with a numerical example. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Atanassov (1986, 1999) introduced the concept of an intuitionistic fuzzy (IF) set characterized by a membership function and a non-membership function, which is a generalization of the fuzzy set introduced by Zadeh (1965). The IF set theory gives us the possibility to model hesitation and uncertainty by using an additional degree, i.e., the intuitionistic index. Over the last decades, the IF set theory has been applied to many different fields such as decision making (Atanassov, Pasi, and Yager, 2002; Herrera, Martinez, and Sanchez, 2005; Li, 2004, 2005; Szmidt and Kacprzyk, 2002, 2004), logic programming (Atanassov and Georgiev, 1993), topology (Abbas, 2005; Davvaz, Dudek, and Jun, 2006; Dudek, Davvaz, and Jun, 2005), medical diagnosis (De, Biswas, and Roy, 2001), pattern recognition (Li and Cheng, 2002; Wang and Xin, 2005) and machine learning as well as market prediction (Liang and Shi, 2003). In comparison with the fuzzy set, the IF set seems to be better suited for expressing a very important factor, i.e., the hesitation of the decision maker, which should be taken into account when we try to construct really adequate models and solutions of decision making problems. In this paper, ratings of alternatives on attributes are expressed with IF sets due to the fact that in some situations determining them precisely is usually depended on judgment and intuition of the decision maker. Therefore, investigation on mul-
* Present address: School of Management, Fuzhou University, No. 2, Xueyuan Road, Daxue New District, Fuzhou District, Fuzhou, Fujian 350108, China. Tel./fax: +86 0591 83768427. E-mail addresses: [email protected], [email protected]. 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.06.062
tiattribute decision making (MADM) problems with IF sets is of importance for both theory and application. MADM problems are an important type of multicriteria decision making (MCDM) problems and are wide spread in real-life decision situations (Erol and Ferrell, 2003; Herrera et al., 2005; Hwang and Yoon, 1981; Srinivasan and Shocker, 1973). MADM problems are solved using the scoring techniques such as the weighted aggregation operators based on multiple attribute value theory (Keeney and Raiffa, 1976). The classical weighted aggregation is usually known in the literature by the weighted average (WA) or simple additive weighting method (Yager, 1996). Another important aggregation operator within the class of weighted aggregation operators is the OWA operator (Yager, 1988, 1993). Since its appearance in 1988, the OWA operator has been used in a wide range of applications such as engineering, neural networks, data mining, decision making and image process as well as expert systems (Calvo, Mayor, and Mesiar, 2002; Torra, 2003; Xu, 2004). Recently, Yager (2004) introduced a generalization of the ordered weighted averaging (OWA) operators, which is called the generalized ordered weighted averaging (GOWA) operators. The GOWA operators are an extension of the OWA operators to the case in which adds to the OWA operators an additional parameter controlling the power to which the argument values are raised. They are also regarded as a generalization of combining the OWA operators with the generalized mean operator (Dyckhoff and Pedrycz, 1984). Yager studied many properties of these operators and several special cases, including generalized means, Hurwicz operator, min, max and the ordered weighted geometric (OWG) operator. The existing research of the GOWA operators mainly focused on MADM
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problems which values are crisp or fuzzy. In this paper, the GOWA operators are extended to aggregate IF sets. Then, a methodology based on the extended GOWA operators are developed to solve the MADM problems with IF sets. The rest of this paper is organized as follows. Section 2 reviews some basic concepts such as the IF set as well as the GOWA operators. In Section 3, the MADM problems with IF sets are formulated and the GOWA operators are extended to aggregate IF sets. The method and procedure for solving MADM problems with IF sets are developed in detailed. The proposed method is illustrated with a numerical example in Section 4. Short conclusion is given in Section 5. 2. Intuitionistic fuzzy sets and GOWA operators 2.1. The concept and operations of IF sets Definition 1 (Atanassov, 1986, 1999). Let X be a finite universal set. An IF set B in X is an object having the following form:
B ¼ fhx; lB ðxÞ; tB ðxÞijx 2 Xg where the functions
lB : X#½0; 1 x 2 X ! lB ðxÞ 2 ½0; 1
The score function D(A) represents the ‘‘net” membership degree. Definition 4 (Hong and Choi, 2000). Let A = hlA, tAi be an IF set. Then the accuracy function r of A is defined as follows
rðAÞ ¼ lA þ tA Obviously, r(A) 2 [0, 1]. It is easy to see that r(A) = lA + tA = 1 pA. Therefore, the accuracy function r(A) represents the non-hesitation degree. To rank two IF sets A = hlA, tAi and B = hlB, tBi, their score functions and accuracy functions need to be compared, respectively. In some literatures, the score function and the accuracy function are explained in terms of the mean and variance in Statistics. Thus, in general the score function is more important than the accuracy function. Then according to the lexicographic method, an order relation may be defined as follows. Definition 5. Let A = hlA, tAi and B = hlB, tBi be two IF sets. (1) (2) (2a) (2b) (2c)
If If If If If
D(A) > D(B) then A is bigger than B. D(A) = D(B) then r(A) = r(B) then A is equal to B; r(A) < r(B) then A is smaller than B; r(A) > r(B) then A is bigger than B.
and
tB : X#½0; 1 x 2 X ! tB ðxÞ 2 ½0; 1 define the degree of membership and degree of non-membership of the element x 2 X to the set B # X, respectively, and for every x 2 X, 0 6 lB(x) + tB(x) 6 1. Let
pB ðxÞ ¼ 1 lB ðxÞ tB ðxÞ which is called the intuitionistic index of the element x in the set B (Burillo and Bustince, 1995; Bustince and Burillo, 1996). It is the degree of indeterminacy membership of the element x to the set B. Obviously,
0 6 pB ðxÞ 6 1 If jBj = 1, i.e., there is only one element x in the set B, then B degenerates to B = {hx, lB(x), tB(x)i}, which usually is denoted by B = hlB, tBi for short. Definition 2 (Atanassov, 1986; De, Biswas, and Roy, 2000). Let B and Q be two IF sets in the set X and b P 0, then (1) (2) (3) (4)
B + Q = {hx, lB(x) + lQ(x) lB(x)lQ(x), tB(x)tQ(x)ijx 2 X}; BQ = {hx, lB(x)lQ(x), tB(x) + tQ(x) tB(x)tQ(x)ijx 2 X}; bB = {hx, 1 (1 lB(x))b, (tB(x))bijx 2 X}; Bb = {hx, (lB(x))b, 1 (1 tB(x))bijx 2 X}.
2.2. A ranking method for IF sets There are a few methods for ranking IF sets. Here the method based on the score function is chosen because of its concision and intuition.. Definition 3 (Chen and Tan, 1994). Let A = hlA, tAi be an IF set. Then the score function D of A is defined as follows
DðAÞ ¼ lA tA Obviously, D(A) 2 [1, 1].
2.3. GOWA operators P Let a mapping f : Rn ? R such that f ða1 ; a2 ; . . . ; an Þ ¼ nj¼1 xj bj , T where x = (x1, x2, . . ., xn) is a weight vector which is correlative Pn with f, satisfying xj 2 [0, 1] (j = 1, 2, . . ., n) and j¼1 xj ¼ 1; bj is the jth largest one of all numerical values ak (k = 1, 2, . . ., n). The function f is called an OWA operator of n-dimension (Yager, 1988). Definition 6 (Yager, 2004). Let a mapping h : Rn ? R such that P 1=k k n , where x = (x1, x2, . . ., xn)T is a hða1 ; a2 ; . . . ; an Þ ¼ j¼1 xj bj weight vector which is correlative with h, satisfying xj 2 [0, 1] Pn (j = 1, 2, . . ., n) and j¼1 xj ¼ 1; bj is the jth largest one of all numerical values ak (k = 1, 2, . . ., n). k 2 (1, +1) is a parameter. The function h is called GOWA operators of n-dimension. Obviously, there are some properties for the GOWA operators h. (1) When k ? 1, if xj – 0 for all j, then h(a1, a2, . . ., an) = bn = min{aj}. The GOWA operator h reduces to the Min operator. However, if the weight vector x is such that x1 = 1 and xj = 0 for all j – 1 then even though k ? 1 we get h(a1, a2, . . ., a n) = b1 = max{aj}. Q x (2) When k ! 0; hða1 ; a2 ; . . . ; an Þ ¼ nj¼1 bj j . The GOWA operator h reduces to the OWG operator. P (3) When k ¼ 1; hða1 ; a2 ; . . . ; an Þ ¼ nj¼1 xj bj . The GOWA operator h reduces to the OWA operator. (4) When k ? +1, if xj – 0 for all j, then h(a1, a2, . . ., an) = b1 = max{aj}. The GOWA operator h reduces to the Max operator. It is interesting to note that if k ? +1 but the weight vector x is such that xn = 1 and xj = 0 for all j – n then h(a1, a2, . . ., an) = bn = min{aj}. 3. Extended GOWA operators with IF sets and MADM methodology 3.1. MADM problems with IF sets Suppose that there exist an alternative set A = {A1, A2, . . ., Am} which consists of m non-inferior alternatives from which the best
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alternative is to be selected. Each alternative is assessed on n attributes, both quantitatively and qualitatively. Denote the set of all attributes by X = {x1, x2, . . ., xn}. For quantitative attributes xj 2 X, alternatives Ai 2 A may be assessed using crisp values aij, which are obtained according to history data and Statistics. Since the physical dimensions and measurements of different quantitative attributes are different, so all values aij need to be normalized. Various normalization methods may be chosen according to the expert’s experience and the need in real situations. Here the formulae for relative degrees of membership and relative degrees of non-membership are chosen as follows
lij ¼
8 aij > < aj amax j
ðj 2 F 1 Þ
min > : d j aj
ðj 2 F 2 Þ
aij
ð1Þ
and
tij ¼
8 a ij > < bj amax j > :c
amin j
j aij
Definition 7. Let g be a mapping such that
ð5Þ
where x = (x1, x2, . . ., xn)T is a weight vector which is correlative Pn with g, satisfying xj 2 [0, 1] (j = 1, 2, . . ., n) and j¼1 xj ¼ 1; dj ¼ < lj ; tj > is the jth largest one of all IF sets ck (k = 1, 2, . . ., n) using the ranking methods of IF sets (for example, Definition 5); k 2 (0, +1) is a parameter which is chosen a prior according to characteristics and needs in real-life situations. Then g is called GOWA operators with IF sets, which are called as the extended GOWA operators for short. It is worthy to note that k should not take negative numbers since the negative power of the IF set dj has no meaning. According to Definition 2, it can be derived from Eq. (5) that n X
#1=k
xj ðhlkj ; 1 ð1 tj Þk iÞ
j¼1
(
ð2Þ
ðj 2 F 2 Þ
¼
n X h1 ð1 lkj Þxj ; ½1 ð1 tj Þk xj i
)1=k
j¼1
2
respectively, where F and F are the subscript sets of benefit quantitative attributes and cost quantitative attributes, and
amax ¼ max faij g; j
!1=k
xj dkj
j¼1
"
ðj 2 F Þ
n X
gðc1 ; c2 ; . . . ; cn Þ ¼
gðc1 ; c2 ; . . . ; cn Þ ¼
1
1
16i6m
3.2. GOWA operators with IF sets
amin ¼ min faij g j 16i6m
ð3Þ
aij 6 1 ðj 2 F 1 Þ amax j
and
0 6 lij þ tij ¼ ðdj þ cj Þ
amin j aij
¼ *" ¼
n n Y Y 1 ð1 lkj Þxj ; ½1 ð1 tj Þk xj j¼1
n Y 1 ð1 lkj Þxj j¼1
and aj 2 [0, 1], bj 2 [0, 1], dj 2 [0, 1] and cj 2 [0, 1] satisfying conditions 0 6 aj + bj 6 1 and 0 6 dj + cj 6 1. Values of parameters aj, bj, dj and cj are chosen a prior according to characteristics and needs in real situations. Obviously,
0 6 lij þ tij ¼ ðaj þ bj Þ
(*
6 1 ðj 2 F 2 Þ
Then, values aij of alternatives Ai 2 A on quantitative attributes xj 2 X can be transformed into IF sets rij = {hxj, lij, tiji}, respectively, which usually are denoted by rij = hlij, tiji for short. For qualitative attributes xj 2 X, alternatives Ai 2 A may be assessed using linguistic variables with designed terms a prior. Semantics of these terms are expressed with IF sets according to characteristics and needs in real situations. Thus, ratings of alternatives on qualitative and quantitative attributes may be expressed with IF sets in a unified way. Let the vector of IF sets of all n attributes for an alternative Ai 2 A be denoted by (ri1, ri2, . . ., rin) = (hli1, ti1i, hli2, ti2i, . . ., hlin, tini). Hence, a MADM problem with IF sets can be expressed concisely in the matrix format as follows
j¼1
#1=k
;1
( 1
+)1=k
n Y ½1 ð1 tj Þk xj
)1=k +
j¼1
ð6Þ Obviously, the following conclusions are derived. Q x (1) When k ! 0; gðc1 ; c2 ; . . . ; cn Þ ¼ nj¼1 dj j . The extended GOWA operator g reduces to the OWG operator using IF sets. P (2) When k ¼ 1; gðc1 ; c2 ; . . . ; cn Þ ¼ nj¼1 xj dj . The extended GOWA operator g reduces to the OWA operator using IF sets. (3) When k ? +1, if xj – 0 for all j, then g(c1, c2, . . ., cn) = d1. The extended GOWA operator g reduces to the Max operator using IF sets. d1 is the largest one of all IF sets cj (j = 1, 2, . . ., n) according to the ranking methods of IF sets. It is interesting to note that if k ? +1 but the weight vector x is such that xn = 1 and xj = 0 for all j – n then g(c1, c2, . . ., cn) = dn. 3.3. Decision making method and procedure based on the extended GOWA operators Assume that a MADM problem with IF sets be expressed as the IF decision matrix in Eq. (4). Then an algorithm and process of the MADM methodology based on the extended GOWA operators are summarized as follows. Step 1: Obtain the IF decision matrix F. Applying the method proposed in Section 3.1, the IF decision matrix F can be constructed. Step 2: Compute overall assessments of alternatives. Using Eq. (6) and the rows of the IF decision matrix F in Eq. (4), the overall assessments ri = hli, tii (i = 1, 2, . . ., m) of alternatives are computed, i.e.,
r i ¼ gðr i1 ; r i2 ; . . . ; rin Þ ð4Þ which is referred to as an IF decision matrix used to represent the MADM problem with IF sets.
respectively, where (ri1, ri2, . . ., rin) = (hli1, ti1i, hli2, ti2i, . . ., hlin, tini). Step 3: Rank the order of all alternatives. The ranking order of the alternative set X is generated according to the scores D(ri) and the accuracies r(ri) (i = 1, 2, . . ., m) using Definition 5.
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4. A numerical example
a11 2:0 ¼ 0:64 ¼ 0:8 2:5 amax 1 a21 2:5 l21 ¼ a1 max ¼ 0:8 ¼ 0:8 2:5 a1 a31 1:8 l31 ¼ a1 max ¼ 0:8 ¼ 0:58 2:5 a1 a41 2:2 ¼ 0:70 l41 ¼ a1 max ¼ 0:8 2:5 a1 a11 2:0 ¼ 0:08 t11 ¼ b1 max ¼ 0:1 2:5 a1 a21 2:5 ¼ 0:1 t21 ¼ b1 max ¼ 0:1 2:5 a1 a31 1:8 ¼ 0:07 t31 ¼ b1 max ¼ 0:1 2:5 a1
l11 ¼ a1
An extended air-fighter selection problem (Hwang and Yoon, 1981) is investigated in this section. Suppose that one country D plans to buy air-fighters from another country J. The Defense Department of the country J would provide the country D with characteristic data for four candidate air-fighters A1, A2, A3 and A4. The decision maker takes into consideration the following six attributes in evaluating the air-fighters, including maximum speed (x1), cruise radius (x2 ), maximum loading (x3), price (x4), reliability (x5) and maintenance (x6). x5 and x6 are qualitative attributes and their ratings are expressed using linguistic variables. The ratings or data of all air-fighters on attributes are given by the decision maker as in Table 1. The corresponding relations between linguistic variables and IF sets are given as in Table 2. According to Table 1, the decision matrix can be obtained as follows
and
t41 ¼ b1 ð7Þ
a41 2:2 ¼ 0:09 ¼ 0:1 2:5 amax 1
respectively. Thus, crisp values ai1 of Ai (i = 1, 2, 3, 4) on the attribute x1 can be transformed into the IF sets as follows
r11 ¼ h0:64; 0:08i; r31 ¼ h0:58; 0:07i; For the benefit attribute x1 (i.e., maximum speed), it is easily seen from Eq. (7) that
amax ¼ 2:5 1 Using Eqs. (1) and (2) with a1 = 0.8 and b1 = 0.1, the relative degrees of membership and relative degrees of non-membership for Ai (i = 1, 2, 3, 4) on the attribute x1 can be calculated as follows
r 21 ¼ h0:8; 0:1i; r 41 ¼ h0:70; 0:09i
respectively. Similarly, using Eqs. (1) and (2) with a2 = 0.9, b2 = 0.05, a3 = 0.85, b3 = 0.1, d4 = 0.75 and c4 = 0.2, crisp values aij of Ai(i = 1, 2, 3, 4) on the attributes xj (j = 2,3,4) can be transformed into IF sets rij, respectively. Thus, combining with Table 2, the decision matrix F (i.e., Eq. (7)) can be transformed into the IF decision matrix as follows:
ð8Þ
Assume that the weights which are correlative with the extended GOWA operators g be given as crisp numbers, namely,
Table 1 Decision information given by the decision maker. Air-fighters Attributes x1 (mach) x2 (mile 103) x3 (lb. 104) x4 ($ 106) x5 x6 A1 A2 A3 A4
2.0 2.5 1.8 2.2
1.5 2.7 2.0 1.8
2.0 1.8 2.1 2.0
5.5 6.5 4.5 5.0
Table 2 The relations between linguistic variables and IF sets. Linguistic variables
IF sets
Very high (VH) High (H) Medium (M) Low (L) Very low (VL)
h0.95, 0.05i h0.70, 0.25) h0.50, 0.40i h0.25, 0.70i h0.05, 0.95i
M L H M
VH M H M
x ¼ ð0:1; 0:15; 0:2; 0:15; 0:15; 0:25ÞT According to Eq. (8) and Definition 3, scores of r1j (j = 1, 2, . . ., 6) are obtained as follows
Dðr 11 Þ ¼ 0:56; Dðr 14 Þ ¼ 0:46;
Dðr 12 Þ ¼ 0:47; Dðr13 Þ ¼ 0:71; Dðr 15 Þ ¼ 0:1; Dðr 16 Þ ¼ 0:9
respectively. Obviously,
Dðr 16 Þ > Dðr 13 Þ > Dðr11 Þ > Dðr 12 Þ > Dðr 14 Þ > Dðr 15 Þ Hence, using Eq. (6), the overall assessment r1 = hl1, t1i of the alternative A1 can be computed as follows
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D.-F. Li / Expert Systems with Applications 37 (2010) 8673–8678 Table 3 Overall assessments of alternatives with different parameter values and scores and accuracies. A1
k
?0 1 2 ?+1
A2
A3
A4
r1
D(r1)
r(r1)
r2
D(r2)
r(r2)
r3
D(r3)
r(r3)
r4
D(r4)
r(r4)
(0.62, 0.18) (0.69, 0.11) (0.71, 0.11) (0.95, 0.05)
0.44 0.58 0.6 0.9
0.8 0.8 0.82 1.0
(0.52, 0.36) (0.64, 0.19) (0.66, 0.18) (0.90, 0.05)
0.16 0.45 0.48 0.85
0.88 0.83 0.84 0.95
(0.70, 0.17) (0.71, 0.14) (0.71, 0.13) (0.85, 0.10)
0.53 0.57 0.58 0.75
0.87 0.85 0.84 0.95
(0.60, 0.23) (0.62, 0.15) (0.63, 0.14) (0.81, 0.10)
0.37 0.47 0.49 0.71
0.83 0.77 0.77 0.91
r 1 ¼ ð0:1 r k16 þ 0:15 r k13 þ 0:2 r k11 þ 0:15 r k12 þ 0:15 r k14 þ 0:25 r k15 Þ
Table 4 The ranking orders of the alternatives for the different parameter values.
1=k
¼ h½1 ð1 0:1 0:95k Þð1 0:15 0:81k Þð1 0:2 0:64k Þ ð1 0:15 0:5k Þð1 0:15 0:62k Þð1 0:25 0:5k Þ1=k ; 1 f1 ½1 ð1 0:9Þð1 0:05Þk ½1 ð1 0:85Þð1 0:1Þk
k
The ranking orders of the alternatives
The best alternatives
?0 1 2 ?+1
A3 A1 A4 A2 A1 A3 A4 A2 A1 A3 A4 A2 A1 A2 A3 A4
A3 A1 A1 A1
½1 ð1 0:8Þð1 0:08Þk ½1 ð1 0:85Þð1 0:03Þk ½1 ð1 0:85Þð1 0:16Þk ½1 ð1 0:75Þð1 0:4Þk g1=k i k
k
and
k
¼ h½1 ð1 0:1 0:95 Þð1 0:15 0:81 Þð1 0:2 0:64 Þ k
k
1=k
k
ð1 0:15 0:5 Þð1 0:15 0:62 Þð1 0:25 0:5 Þ k
k
r4 ¼ ð0:1 r k43 þ 0:15 r k41 þ 0:2 r k42 þ 0:15 r k44 þ 0:15 r k46 þ 0:25 r k45 Þ1=k
;
k
1 ½1 ð1 0:1 0:95 Þð1 0:15 0:9 Þð1 0:2 0:92 Þ k
k
k
1=k
ð1 0:15 0:97 Þð1 0:15 0:84 Þð1 0:25 0:6 Þ
i
Similarly, the overall assessments ri = hli,ti> of alternatives Ai(i = 2,3,4) can be obtained as follows
ð1 0:15 0:68k Þð1 0:15 0:5k Þð1 0:25 0:5k Þ1=k ; 1 f1 ½1 ð1 0:9Þð1 0:1Þk ½1 ð1 0:85Þð1 0:09Þk ½1 ð1 0:8Þð1 0:03Þk ½1 ð1 0:85Þð1 0:18Þk ½1 ð1 0:85Þð1 0:4Þk ½1 ð1 0:75Þð1 0:4Þk g1=k i
r 2 ¼ ð0:1 r k22 þ 0:15 r k21 þ 0:2 r k23 þ 0:15 r k24
¼ h½1 ð1 0:1 0:81k Þð1 0:15 0:7k Þð1 0:2 0:6k Þ
þ 0:15 r k26 þ 0:25 r k25 Þ1=k k
¼ h½1 ð1 0:1 0:81k Þð1 0:15 0:7k Þð1 0:2 0:6k Þ
ð1 0:15 0:68k Þð1 0:15 0:5k Þð1 0:25 0:5k Þ1=k ; k
k
¼ h½1 ð1 0:1 0:9 Þð1 0:15 0:8 Þð1 0:2 0:73 Þ
1 ½1 ð1 0:1 0:9k Þð1 0:15 0:91k Þð1 0:2 0:97k Þ 1=k
ð1 0:15 0:52k Þð1 0:15 0:5k Þð1 0:25 0:25k Þ
;
1 f1 ½1 ð1 0:9Þð1 0:05Þk ½1 ð1 0:85Þð1 0:1Þk k
k
½1 ð1 0:8Þð1 0:09Þ ½1 ð1 0:85Þð1 0:14Þ ½1 ð1 0:85Þð1 0:4Þk ½1 ð1 0:75Þð1 0:7Þk g1=k i ¼ h½1 ð1 0:1 0:9k Þð1 0:15 0:8k Þð1 0:2 0:73k Þ ð1 0:15 0:52k Þð1 0:15 0:5k Þð1 0:25 0:25k Þ1=k ; 1 ½1 ð1 0:1 0:95k Þð1 0:15 0:9k Þð1 0:2 0:91k Þ ð1 0:15 0:86k Þð1 0:15 0:6k Þð1 0:25 0:3k Þ1=k i r 3 ¼ ð0:1 r k33 þ 0:15 r k32 þ 0:2 r k34 þ 0:15 r k31 þ 0:15 r k35 þ 0:25 r k36 Þ1=k
ð1 0:15 0:82k Þð1 0:15 0:6k Þð1 0:25 0:6k Þ1=k i respectively. For some special values of the parameter k, ri = hli, tii (i = 1, 2, 3, 4) can be obtained as in Table 3. Corresponding scores D(ri) and accuracies r(ri) of ri = hli, tii (i = 1, 2, 3, 4) are also obtained as in Table 3. According to Table 3 and Definition 5, the ranking orders of alternatives Ai (i = 1, 2, 3, 4) can be generated for the different parameter values as in Table 4, respectively. It is easy to see from Table 4 that the best selection is the alternative (i.e., air-fighter) A1 for the special values of the parameter k except the situation k ? 0 in which the best selection is the airfighter A3. 5. Conclusions
¼ h½1 ð1 0:1 0:85k Þð1 0:15 0:67k Þð1 0:2 0:75k Þ ð1 0:58 0:5k Þð1 0:15 0:7k Þð1 0:25 0:7k Þ1=k ; 1 f1 ½1 ð1 0:9Þð1 0:1Þk ½1 ð1 0:85Þð1 0:04Þk ½1 ð1 0:8Þð1 0:2Þk ½1 ð1 0:85Þð1 0:07Þk ½1 ð1 0:85Þð1 0:25Þk ½1 ð1 0:75Þð1 0:25Þk g1=k i ¼ h½1 ð1 0:1 0:85k Þð1 0:15 0:67k Þð1 0:2 0:75k Þ ð1 0:58 0:5k Þð1 0:15 0:7k Þð1 0:25 0:7k Þ1=k ; 1 ½1 ð1 0:1 0:9k Þð1 0:15 0:96k Þð1 0:2 0:8k Þð 1 0:15 0:93k Þð1 0:15 0:75k Þð1 0:25 0:75k Þ1=k i
The GOWA operators are the extension of the OWA operators (Yager, 2004). As illustrated above, if the parameter k takes different values then the GOWA operators have different particular forms such as the OWG operator and the OWA operator as well as the Max operator. The GOWA operators are useful to deal with MADM problems, but the existing research on them mainly focused on crisp values or fuzzy sets. The fuzziness or uncertainty in the MADM problems with IF sets can be represented exactly since the IF set can model the hesitation of the decision maker. In this paper, the formulae for relative degrees of membership and relative degrees of non-membership are chosen to normalize the crisp values of alternatives on quantitative attributes; ratings of alternatives on qualitative attributes are expressed with
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linguistic variables corresponding to the designed IF sets a prior. The methodology for solving MADM problems with IF sets is developed based on the extended GOWA operators. The effectiveness and applicability of the proposed method is illustrated with an extended air-fighter selection problem. However, it may be expected that the proposed method can be applied to other similar MADM problems with IF sets. Acknowledgements This research was sponsored by the Natural Science Foundation of China (Nos. 70902041 and 70871117) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No. 2005-546). References Abbas, S. E. (2005). On intuitionistic fuzzy compactness. Information Sciences, 173, 75–91. Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96. Atanassov, K. T. (1999). Intuitionistic fuzzy sets. Heidelberg: Springer-Verlag. Atanassov, K. T., & Georgiev, C. (1993). Intuitionistic fuzzy prolog. Fuzzy Sets and Systems, 53, 121–128. Atanassov, K. T., Pasi, G., & Yager, R. R. (2002). Intuitionistic fuzzy interpretations of multi-person multi-criteria decision making. In Proceedings of 2002 1st international IEEE symposium intelligent systems, Varna (Vol. I, pp. 115–119). Burillo, P., & Bustince, H. (1995). Intuitionistic fuzzy relations (Part I). Mathware and Soft Computing, 2, 5–8. Bustince, H., & Burillo, P. (1996). Structures on intuitionistic fuzzy relations. Fuzzy Sets and Systems, 78, 293–303. Calvo, T., Mayor, G., & Mesiar, R. (2002). Aggregation operators: New trends and applications. New York: Physica-Verlag. Chen, S. M., & Tan, J. M. (1994). Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 67, 163–172. Davvaz, B., Dudek, W. A., & Jun, Y. B. (2006). Intuitionistic fuzzy Hv-submodules. Information Sciences, 176, 285–300. De, S. K., Biswas, R., & Roy, A. R. (2000). Some operations on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114, 477–484. De, S. K., Biswas, R., & Roy, A. R. (2001). An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets and Systems, 117, 209–213. Dudek, W., Davvaz, B., & Jun, Y. B. (2005). On intuitionistic fuzzy subhyperquasigroups of hyperquasigroups. Information Sciences, 170, 251–262.
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