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Compromise ratio method for fuzzy multi-attribute group decision making
Applied Soft Computing 7 (2007) 807–817 www.elsevier.com/locate/asoc
Compromise ratio method for fuzzy multi-attribute group decision making Deng-Feng Li a,b,c,* a
Department of Sciences, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, Liaoning, China b Department Five, Dalian Naval Academy, No. 1, Xiaolong Street, Dalian 116018, Liaoning, China c The Missile Institute, Air Force Engineering University, Sanyuan, Shaanxi 713800, China Received 4 January 2005; received in revised form 7 February 2006; accepted 13 February 2006 Available online 2 May 2006
Abstract The aim of this paper is to develop a compromise ratio (CR) methodology for fuzzy multi-attribute group decision making (FMAGDM), which is an important part of decision support system. Owing to fuzziness being inherent in decision data and group decision making processes, the crisp values are inadequate to model real-life situations. In this paper, the weights of all attributes and the ratings of each alternative with respect to each attribute are described by linguistic terms which can be expressed in trapezoid fuzzy numbers. A fuzzy distance measure is developed to calculate difference between trapezoid fuzzy numbers. The compromise ratio method for FMAGDM is developed by introducing the ranking index based on the concept that the chosen alternative should be as close as possible to the ideal solution and as far away from the negative-ideal solution as possible simultaneously. The computation principle and procedure of the compromise ratio method are described in detail in this paper. Moreover the TOPSIS method which was developed for multi-attribute decision making (MADM) with crisp decision data is analyzed and extended to multiattribute group decision making (MAGDM) under fuzzy environments. A comparative analysis of the compromise ratio method and the extended fuzzy TOPSIS method is illustrated with a numerical example, showing their similarity and some differences. # 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy multi-attribute group decision making; Linguistic variable; Fuzzy number; Compromise ratio method; TOPSIS; Comparative analysis
1. Introduction Multi-attribute decision making (MADM) problems are wide spread in real life decision situations [6–10,12–21]. A MADM problem is to find a best compromise solution from all feasible alternatives assessed on multiple attributes, both quantitative and qualitative, and can be dealt with using several existing methods such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) developed by Huang and Yoon [12], which is one of the well-known MADM methods. The basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the positiveideal solution (PIS) and the farthest distance from the negativeideal solution (NIS). In the process of the TOPSIS method, the performance ratings and weights of the attributes are given as crisp values a priori. However, the highest ranked alternative by the TOPSIS method is the best in terms of the ranking index, * Tel.: +86 411 85856357; fax: +86 411 85856357. E-mail addresses: [email protected], [email protected]. 1568-4946/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2006.02.003
which does not mean that it is always the closest to the ideal solution (Section 2). Moreover, under many conditions, crisp data are inadequate or insufficient to model real-life decision problems [4,5,23,24]. Indeed, human judgments including preferences are vague or fuzzy in nature and as such it may not be appropriate to represent them by accurate numerical values. A more realistic approach could be to use linguistic variables to model human judgments [2,30], that is, to suppose that the ratings and weights of the attributes in the decision making problem are assessed by means of linguistic variables. In this paper, the TOPSIS method for solving MADM problems with crisp decision data is analyzed and further extended to multi-attribute group decision making (MAGDM) under fuzzy environments. A new compromise ratio methodology is developed to solve fuzzy multi-attribute group decision making (FMAGDM) problems. In this methodology, linguistic variables are used to capture fuzziness in decision information and decision making processes by means of a fuzzy decision matrix. The multiattribute ranking index is introduced based on the fuzzy
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distance measure, which is developed from the concept of Minkowski distance. The compromise ratio method is based on the concept that the chosen alternative should have the distance from the ideal solution as short as possible and the distance from the negative-ideal solution as far as possible simultaneously. The relative importance of these distances [8,9,25–27,29], which is a major concern in real-life decision situations, is considered in the developed method. A comparative analysis is illustrated with a numerical example, showing their similarity and some differences. The paper is organized as follows. A brief overview and analysis of the TOPSIS method is given in Section 2. A discussion on the TOPSIS method extended to the MAGDM problems in fuzzy environments is also presented in this section. The basic principle and procedure of the compromise ratio method are presented in Section 3. A comparative analysis of the compromise ratio method and the fuzzy extension of the TOPSIS method is given in Section 4. Section 5 gives a numerical example to show similarity and some differences of the compromise ratio method and the fuzzy extended TOPSIS method. A short conclusion is given in Section 6.
P attribute f i, where vi 0 (i = 1, 2, . . ., m) and m i¼1 vi ¼ 1. Denote a weight vector by v = (v1, v2, . . ., vm)T. The above MADM problem can be dealt with using the TOPSIS method developed by Huang and Yoon [12]. The basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. The procedure of the TOPSIS method consists of the following steps: (1) Calculate the normalized decision matrix. The normalized value rij (i = 1, 2, . . ., m; j = 1, 2, . . ., n) of f ij is calculated as fi j ffi ri j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 f j¼1 i j
(1)
(2) Calculate the weighted normalized decision matrix. The weighted normalized value vi j (i = 1, 2, . . ., m; j = 1, 2, . . ., n) of rij is calculated as vi j ¼ v i r i j
(2) *+
2. The TOPSIS method and its extension to FMAGDM 2.1. The TOPSIS method A MADM problem is to find a best compromise solution from all feasible alternatives assessed on multiple attributes [5,12,14,16]. Suppose the decision maker has to choose one of or rank n efficient/non-inferior alternatives xj (j = 1, 2, . . ., n) based on m attributes f i (i = 1, 2, . . ., m). Denote an alternative set by X = {x1, x2, . . ., xn} and an attribute set by F = {f 1, f 2, . . ., f m). In general, attributes can be classified into two types: benefit attributes and cost attributes. In other words, the attribute set F can be divided into two subsets: F 1 and F 2, where F k (k = 1, 2) is the subset of benefit attributes and cost attributes, respectively. Furthermore, F = F 1 [ F 2 and F 1 \ F 2 = 1, where 1 is an empty set. Then the MADM model can be built as follows: maxf fi ðx j Þj fi 2 F 1 g minf fi ðx j Þj fi 2 F 2 g s:t: x j 2 X The alternative set X = {x1, x2, . . ., xn} and the attribute set F = {f 1, f 2, . . ., f m} are finite, so it is very convenient to denote the rating of alternative xj (j = 1, 2, . . ., n) on attribute f i (i = 1, 2, . . ., m) by f ij, i.e., f ij = f i(xj). Then a MADM problem can be concisely expressed as the following decision matrix:
(3) Determine the ideal solution x and the negative-ideal solution x*, whose weighted normalized value vectors are þ þ denoted by vþ ¼ ðvþ ¼ ðv 1 ; v2 ; 1 ; v2 ; . . . ; vm Þ and v . . . ; vm Þ, respectively. For every ith attribute function (i = 1, 2, . . ., m), 8 < max fvi j g ð fi 2 F 1 g 1 jn þ (3) vi ¼ : min fvi j g ð fi 2 F 2 g 1 jn
and v i
¼
8 < min fvi j g
ð fi 2 F 1 g
: max fvi j g
ð fi 2 F 2 g
1 jn 1 jn
(4)
(4) Calculate the separation measures, using the m-dimensional Euclidean distance. The separation of each alternative xj (j = 1, 2, . . ., n) from the ideal solution x*+ is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 þ D ðx j Þ ¼ (5) ðvi j vþ i Þ i¼1
Similarly, the separation of each alternative xj (j = 1, 2, . . ., n) from the negative-ideal solution x* is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 D ðx j Þ ¼ (6) ðvi j v i Þ i¼1
(5) Calculate the relative closeness to the ideal solution. The relative closeness of each alternative xj (j = 1, 2, . . ., n) with respect to x*+ is defined as C ðx j Þ ¼ In decision making process, different attributes have different importance. Suppose vi (i = 1, 2, . . ., m) is the relative weight of
D ðx j Þ Dþ ðx j Þ þ D ðx j Þ
(7)
(6) Rank the preference order of all alternatives xj (j = 1, 2, . . ., n) by the C*(xj) in decreasing order.
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
According to the formulation of the ranking index C*(xj) (j = 1, 2, . . ., n) in Eq. (7), alternative xj is better than xk if C*(xj) > C*(xk), i.e., D ðx j Þ D ðxk Þ > þ D ðxk Þ þ D ðxk Þ j Þ þ D ðx j Þ
Dþ ðx j Þ < Dþ ðxk Þ and
ðBÞ
Alternatives
Ranking order
x1
x2
x3
0.114 0.365 0.762
0.108 0.280 0.722
0.365 0.114 0.238
x2 x1 x3 x1 x2 x3 x1 x2 x3
D ðx j Þ > D ðxk Þ;
or þ
Decision matrix
D (xj) D*(xj) C*(xj)
which will hold if ðAÞ
Table 1 Decision results obtained by the TOPSIS method
*+
Dþ ðx
809
However, x1 is not the closest to the ideal solution x*+, i.e., D*+(x1) = 0.114, D*+(x2) = 0.108. According to condition (B) as stated above, in this case, x1 is ranked best by the TOPSIS method, although it is not the closest to the ideal solution, because
þ
D ðx j Þ > D ðxk Þ and
D ðx j Þ > D ðxk Þ; but Dþ ðx j Þ <
Dþ ðxk ÞD ðx j Þ : D ðxk Þ
Condition (A) shows the ‘‘regular’’ situation, when alternative xj is better than xk because it is closer to the ideal solution x*+ and farther from the negative-ideal solution x*. On the contrary, condition (B) shows that an alternative xj could be better than xk even though xj is farther from ideal solution x*+ than xk. Let xk be the alternative with D*+(xk) = D*(xk) and C*(xk) = 0.5. In this case, all alternatives xj with D*+(xj) > D*+(xk) and D*+(xj) < D*(xj) are better ranked than xk, although xk is closer to the ideal solution x*+. For example, a mountain climber (beginner) must choose an alternative (destination) from a set of three alternatives, i.e., the alternative set X = {x1, x2, x3}. The alternatives are evaluated according to two attributes: risk and altitude [26]. Risk and altitude are denoted as f 1 and f 2, respectively. Risk f 1 is a cost attribute and evaluated using subjective scales: 1–5. Altitude f 2 is a benefit attribute and evaluated in meters above the sea (m.a.s). Denote the attribute set as F = {f 1, f 2}. Attribute values f ij of alternatives are given as the following decision matrix:
(8)
Let us suppose that both attributes f 1 and f 2 are equally important, i.e., the weights of attributes are vi = 1/2 (i = 1, 2). Using Eq. (1), the normalized decision matrix of the decision matrix Yf is computed as follows:
The decision results obtained by the TOPSIS method are presented in Table 1 using Eqs. (5)–(7). Analyzing the decision results of the TOPSIS method, we find that the decision results from the TOPSIS method are interesting due to the following. According to C*(xj) (j = 1, 2, 3) as in Table 1, the best solution is alternative x1, i.e., C*(x1) = 0.762, C*(x1) = 0.722; and it is the best according to D*(xj) (j = 1, 2, 3), i.e., D*(x1) = 0.365, D*(x2) = 0.280.
Dþ ðx1 Þ ¼ 0:114 < 0:141 ¼ ¼
0:108 0:365 0:280
Dþ ðx2 ÞD ðx1 Þ D ðx2 Þ
This is an example of the case discussed in condition (B). 2.2. Notations of fuzzy numbers ˜ is a special fuzzy subset on the set R of A fuzzy number m real numbers which satisfy the following conditions [2,4,5,16,32]. (1) There exists a x0 2 R at least so that the degree of its membership um˜ ðx0 Þ ¼ 1; (2) Membership function um˜ ðxÞ is left and right continuous. ˜ ¼ ða; b; c; dÞ be a trapezoid fuzzy number, where the Let m ˜ is given by membership function mm˜ of m 8xa > > > :d x dc
ða x bÞ ðb x cÞ ðc x dÞ
˜ and a and d are where [b, c] is called a mode interval of m, ˜ respectively. called the low and upper limits of m, ˜ ¼ ða; b; c; dÞ is called a triangular fuzzy IF b = c then m ˜ ¼ ða; m; dÞ, where m = b = c. Obviously, number, denoted m the membership function mm˜ of a triangular fuzzy number ˜ ¼ ða; m; dÞ is m 8 xa > < ða x mÞ m a mm˜ ðxÞ ¼ d x > : ðm x dÞ dm Therefore, a triangular fuzzy number is a special case of the trapezoid fuzzy number. In other words, a triangular fuzzy ˜ ¼ ða; m; dÞ can also be written as a trapezoid fuzzy number m ˜ ¼ ða; m; m; dÞ. number m
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D.-F. Li / Applied Soft Computing 7 (2007) 807–817
˜ ¼ It is easy to see that a trapezoid fuzzy number m ða; b; c; dÞ is reduced to a real number a if a = b = c = d. Conversely, a real number m can be written as a trapezoid fuzzy ˜ ¼ ðm; m; m; mÞ. number m For the sake of simplicity and without loss of generality, assume that all fuzzy numbers are trapezoid fuzzy numbers throughout the paper unless otherwise stated. ˜ ¼ ða; b; c; dÞ is called a positive trapezoid fuzzy number if m a 0, and a, b, c and d are not identical. ˜ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n˜ ¼ ðn1 ; n2 ; n3 ; n4 Þ Assume that m be two positive trapezoid fuzzy numbers, r > 0 is a positive real number. Then designate the following operators [32]: ð1Þ ð2Þ ð3Þ
˜ ¼ ðrm1 ; rm2 ; rm3 ; rm4 Þ rm ˜ 1 m1 m2 m3 m4 m ˜ ¼ ¼ m ; ; ; 2 r r r r r r ˜ n ¼ ðm1 n1 ; m2 n2 ; m3 n3 ; m4 n4 Þ m ˜
2.3. Linguistic variable is a variable whose values are linguistic terms. The concept of linguistic variable is very useful in situations where decision problems are too complex or too illdefined to be described properly using conventional quantitative expressions. For example, the performance ratings of alternatives on qualitative attributes could be expressed using linguistic variable such as very poor, poor, fair, good, very good, etc. Such linguistic values can be represented using positive trapezoid fuzzy numbers. For example, ‘‘poor’’ and ‘‘very good’’ can be represented by positive trapezoid fuzzy numbers (0.2, 0.3, 0.4, 0.5) and (0.8, 0.9, 1.0, 1.0), respectively.
For p = 2, Eq. (9) can be rewritten as follows: ˜ n˜ Þ ¼ d2 ðm; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm1 n1 Þ2 þ 2ðm2 n2 Þ2 þ 2ðm3 n3 Þ2 þ ðm4 n4 Þ2 6 (11) which is a weighted Euclidean distance. And if q ! 1, Eq. (9) can be rewritten as follows: m1 n1 m2 n2 m3 n3 m4 n4 ˜ n˜ Þ ¼ max d1 ðm; ; ; ; 6 3 3 6 (12) which is a weighted Chebyshev distance. ˜ and n˜ are real numbers then the distance Note that if both m ˜ n˜ Þ is identical to the Euclidean distance. In measurement d p ðm; ˜ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n˜ ¼ fact, suppose both m ðn1 ; n2 ; n3 ; n4 Þ are two real numbers and let m1 = m2 = m3 = m4 = m and n1 = n2 = n3 = n4 = n. The dis˜ n˜ Þ can be calculated as follows: tance measurement d p ðm; ˜ n˜ Þ d p ðm; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p p p p ðm1 n1 Þ þ 2ðm2 n2 Þ þ 2ðm3 n3 Þ þ ðm4 n4 Þ ¼ 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p p p p p ðm nÞ þ 2ðm nÞ þ 2ðm nÞ þ ðm nÞ ¼ 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ¼ ðm nÞ ¼ jm nj Furthermore, it is easily seen that two trapezoid fuzzy numbers ˜ and n˜ are identical if and only if the distance measurement m ˜ n˜ Þ ¼ 0. d p ðm;
2.4. Distance between trapezoid fuzzy numbers 2.5. The extension of the TOPSIS method for FMAGDM ˜ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n˜ ¼ ðn1 ; n2 ; n3 ; n4 Þ be two Let m trapezoid fuzzy numbers. Then the distance measure between them is defined using the Minkowski distance (or Lp-metric) as follows: ˜ n˜ Þ ¼ d p ðm; ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p p p p ðm1 n1 Þ þ 2ðm2 n2 Þ þ 2ðm3 n3 Þ þ ðm4 n4 Þ 6 (9) where p 1 is a distance parameter. It is easily seen that Eq. (9) is the weighted distance, which considers different important of the low limit, the left and right points of the mode interval, and the upper limits of the trapezoid fuzzy numbers. For p = 1, Eq. (9) can be rewritten as follows: ˜ n˜ Þ ¼ d1 ðm;
jm1 n1 j þ 2jm2 n2 j þ 2jm3 n3 j þ jm4 n4 j 6 (10)
which is a weighted Hamming distance.
In this paper, we discuss the following MAGDM in fuzzy environments. Suppose there exist n possible alternatives x1, x2, . . ., xn from which K decision makers Pk (k = 1, 2, . . ., K) have to choose on the basis of m attributes f1, f2, . . ., fm, both qualitative and quantitative. The importance weights of various attributes and ratings of qualitative attributes are considered as linguistic variables. These linguistic variables can be expressed in positive trapezoid fuzzy numbers in Tables 2 and 3, which corresponding membership functions are determined or extracted using a similar way to [1–3,11,22,28,30–33]. The important weight of each attribute can be obtained by either directly assign or indirectly using pairwise comparisons [4,16]. In here, it is suggested that the decision makers use the linguistic variables shown as Tables 2 and 3 to evaluate the importance of attributes and the ratings of alternatives with respect to various attributes. Suppose the rating of alternative xj (j = 1, 2, . . ., n) on attribute fi (i = 1, 2, . . ., m) given by decision maker Pk (k = 1, 2, k . . ., K) is f˜i j ¼ ðakij ; bkij ; ckij ; dikj Þ. Hence, a FMAGDM problem can be concisely expressed in matrix format as follows:
D.-F. Li / Applied Soft Computing 7 (2007) 807–817 Table 2 Linguistic variables for importance of each attribute Linguistic variables
Trapezoid fuzzy numbers
Extreme low (EL) Very low (LV) Low (L) Medium low (ML) Medium (M) Medium high (MH) High (H) Very high (VH) Extreme high (EH)
(0, 0, 0.1, 0.2) (0.1, 0.2, 0.3, 0.4) (0.2, 0.3, 0.4, 0.5) (0.3, 0.4, 0.5, 0.6) (0.4, 0.5, 0.6, 0.7) (0.5, 0.6, 0.7, 0.8) (0.6, 0.7, 0.8, 0.9) (0.7, 0.8, 0.9, 1.0) (0.8, 0.9, 1.0, 1.0)
Table 3 Linguistic variables for ratings of each alternative with respect to each attribute Linguistic variables
Trapezoid fuzzy numbers
Extreme poor (EP) Very poor (LP) Poor (P) Medium poor (MP) Fair (F) Medium good (MG) Good (G) Very good (VG) Extreme good (EG)
(0, (1, (2, (3, (4, (5, (6, (7, (8,
0, 2, 3, 4, 5, 6, 7, 8, 9,
1, 2) 3, 4) 4, 5) 5, 6) 6, 7) 7, 8) 8, 9) 9, 10) 10, 10)
811
v˜ i and f˜i j are denoted as v˜ i ¼ ðai ; bi ; g i ; di Þ and f˜i j ¼ ðai j ; bi j ; ci j ; di j Þ, respectively. As stated above, a FMAGDM problem can be concisely expressed as the following decision matrix:
The weight vector is v˜ ¼ ðv˜ 1 ; v˜ 2 ; . . . ; v˜ m ÞT . Y˜ is referred to as fuzzy decision matrix usually used to represent the FMAGDM problem. Since the m attributes may be measured in different ways, the decision matrix Y˜ needs to be normalized. The linear scale transformation is used here to transform the various attribute scales into a comparable scale [16]: r˜i j ¼
a i j bi j c i j di j ; ; ; dimax dimax dimax dimax
ð fi 2 F 1 Þ
(15)
and 8 min min min min a a a a > > < i ; i ; i ; i d c i j bi j ai j r˜i j ¼ i j di j ci j bi j ai j > > : 1 max ; 1 max ; 1 max ; 1 max di di di di
6¼ 0Þ ðamin i ¼ 0Þ ðamin i
ð fi 2 F 2 Þ where dimax ¼ max fdi j j f˜i j ¼ ðai j ; bi j ; ci j ; di j Þg 1 jn
which are referred to as fuzzy decision matrices usually used to represent the FMAGDM problem. Similarly, suppose the importance weight of attribute fi (i = 1, 2, . . ., m) given by decision maker Pk (k = 1, 2, . . ., K) is v˜ ki ¼ ðaki ; bki ; g ki ; dki Þ. Then the importance weights of the attributes and ratings of the alternatives with respect to each attribute can be calculated as follows: v˜ 1i v˜ 2i v˜ Ki K PK PK PK k PK k k k d k¼1 ai k¼1 bi k¼1 g i ; ; ; k¼1 i ¼ K K K K
v˜ i ¼
and ¼ min fai j j f˜i j ¼ ðai j ; bi j ; ci j ; di j Þg amin i 1 jn
Denote r˜i j as r˜i j ¼ ðmi j ; hi j ; ri j ; li j Þ. The normalization method mentioned above is to preserve the property that the range of a normalized trapezoid fuzzy number r˜i j belongs to the closed interval [0,1]. Then, the fuzzy decision matrix Y˜ ¼ ð f˜i j Þmn can be transformed into the normalized fuzzy decision matrix:
(13)
and 1 2 K f˜i j f˜i j f˜i j K PK k PK k PK k PK k k¼1 ai j k¼1 bi j k¼1 ci j k¼1 di j ; ; ; ¼ K K K K
f˜i j ¼
(14)
Considering the different importance of each attribute, R˜ is transformed into the weighted normalized fuzzy decision ˜ matrix V:
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D.-F. Li / Applied Soft Computing 7 (2007) 807–817
Hence Eq. (19) is reduced to the following formula: Pm Pm si j si j i¼1 ¼ i¼1 C ðx j Þ ¼ Pm Pm m m i¼1 s i j þ i¼1 s i j
where v˜ i j is a trapezoid fuzzy number: v˜ i j ¼ v˜ i ˜ri j ¼ ðai mi j ; bi hi j ; g i ri j ; di li j Þ
(16)
denote as v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ. Obviously, all v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ are normalized positive trapezoid fuzzy numbers and their ranges belong to the closed interval [0,1]. Then, we can define the fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x, whose ˜þ ˜þ weighted normalized fuzzy vectors are a˜ þ ¼ ð˜aþ mÞ 1 ;a 2 ;...;a þ and a˜ ¼ ð˜a1 ; a˜ 2 ; . . . ; a˜ m Þ, where a˜ i ¼ ð1; 1; 1; 1Þ ¼ 1 and a˜ i ¼ ð0; 0; 0; 0Þ ¼ 0. The distance of each alternative xj from the fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x can be calculated using Eq. (11) as follows: Dðx j ; xþ Þ ¼
m X
2.6. The further extension of the TOPSIS method for FMAGDM In the following, the distance of each alternative xj from the fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x is calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 þ ¯ j; x Þ ¼ ½d2 ð˜vi j ; a˜ þ Dðx i Þ i¼1
d2 ð˜vi j ; a˜ þ i Þ
i¼1
¼
which is the simple weighted average method with equal weights 1/m for all attributes. Therefore, the fuzzy extension of the TOPSIS method (Eq. (19)) can not reduced to the classical TOPSIS method for the crisp FMAGDM problems. In other words, the TOPSIS method developed by Huang and Yoon [12] is not really extended to the FMAGDM problems under fuzzy environments.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ð1 s i j Þ2 þ 2ð1 zi j Þ2 þ 2ð1 yi j Þ2 þ ð1 t i j Þ2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m uX ð1 s i j Þ2 þ 2ð1 zi j Þ2 þ 2ð1 yi j Þ2 þ ð1 ti j Þ2 ¼t 6 i¼1
6
i¼1
(20) (17)
and
and Dðx j ; x Þ ¼
¯ j ; x Þ ¼ Dðx
m X
d2 ð˜vi j ; a˜ i Þ
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m uX ðs i j Þ2 þ 2ðzi j Þ2 þ 2ðyi j Þ2 þ ðti j Þ2 ¼t 6 i¼1
i¼1
¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ðs i j Þ2 þ 2ðzi j Þ2 þ 2ðyi j Þ2 þ ðt i j Þ2 6
i¼1
(18)
respectively. The closeness coefficient of each alternative xj (j = 1, 2, . . ., n) with respect to the fuzzy positive ideal solution x+ is defined as Dðx j ; x Þ C ðx j Þ ¼ Dðx j ; xþ Þ þ Dðx j ; x Þ
(19)
The preference order of all alternatives xj (j = 1, 2, . . ., n) can be generated according to C*(xj). Analyzing Eqs. (17) and (18), however, we can find that if all v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ are real numbers (i.e., sij = zij = yij = tij) then Eqs. (17) and (18) are reduced to the following formulae: Dðx j ; xþ Þ ¼
m m m X X X d2 ð˜vi j ; a˜ þ ð1 s i j Þ ¼ m si j i Þ ¼ i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 ½d2 ð˜vi j ; a˜ i Þ
i¼1
respectively. The closeness coefficient of each alternative xj (j = 1, 2, . . ., n) with respect to the fuzzy positive ideal solution x+ is defined as ¯ j ; x Þ Dðx C¯ ðx j Þ ¼ ¯ ¯ j ; x Þ Dðx j ; xþ Þ þ Dðx
(22)
The preference order of all alternatives xj (j = 1, 2, . . ., n) can be generated according to C¯ ðx j Þ. Obviously, if all v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ are real numbers (i.e., sij = zij = yij = tij) then Eqs. (20) and (21) are reduced to the following formulae: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X 2 ¯ j ; xþ Þ ¼ ½d2 ð˜vi j ; a˜ þ ð1 s i j Þ2 Dðx i Þ ¼ i¼1
i¼1
i¼1
and
and m m X X Dðx j ; x Þ ¼ d2 ð˜vi j ; a˜ si j i Þ ¼
i¼1
i¼1
(21)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X 2 ¯ j; x Þ ¼ ¼ ½d2 ð˜vi j ; a˜ Þ ðs i j Þ2 Dðx i
i¼1
i¼1
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
813
And if q ! 1, Eq. (23) can be rewritten as follows:
Hence Eq. (22) is reduced to the following formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm 2 i¼1 ðs i j Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ðx j Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm Pm 2 2 i¼1 ð1 s i j Þ þ i¼1 ðs i j Þ which is the classical TOPSIS method developed by Huang and Yoon [12] for the crisp FMAGDM problems. Therefore, Eq. (22) is a real generalization of the classical TOPSIS method under crisp environments.
D1 ðx j ; xþ Þ ¼ max fd1 ð˜vi j ; a˜ þ i g 1im 1 s i j 1 z i j 1 yi j 1 t i j ; ; ; ¼ max max 1im 6 3 3 6 (26) The smaller Dp(xj, x+), the better xj. Rank the alternatives xj (j = 1, 2, . . ., n) by Dp(xj, x+) in increasing order. An alternative x j 2 X satisfying D p ðx j ; xþ Þ ¼ min fD p ðx j ; xþ Þg 1 jn
3. The compromise ratio method for FMAGDM The relative ratio method focuses on ranking and selecting from a set of alternatives in the presence of conflicting attributes. It determines the compromise ranking-list based on the concept that the chosen alternative should be as close as possible to the ideal solution and as far away from the negativeideal solution as possible simultaneously. Difference between each alternative and the ideal solution (or the negative-ideal solution) is measured with Lp-metric. The compromise ratio method introduces the multi-attribute ranking index based on the particular measure of both closing to the ideal solution and being far away from the negative-ideal solution. Difference between an alternative xj (j = 1, 2, . . ., n) and the ideal solution x+ can be measured using Eq. (9): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X p p D p ðx j ; x Þ ¼ ½d p ð˜vi j ; a˜ þ i Þ þ
i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ð1 s i j Þ p þ 2ð1 zi j Þ p þ 2ð1 yi j Þ p þ ð1 ti j Þ p p ¼ 6 i¼1 (23) For p = 1, Eq. (23) can be rewritten as follows: D1 ðx j ; xþ Þ ¼
m X
d1 ð˜vi j ; a˜ þ i Þ
i¼1
¼
m X ð1 s i j Þ þ 2ð1 zi j Þ þ 2ð1 yi j Þ þ ð1 t i j Þ
should be the best compromise solution, which has the shortest distance from the ideal solution. However, such a compromise solution may not always guarantee to have the longest distance from the negative-ideal solution. Similarly, difference between an alternative xj (j = 1, 2, . . ., n) and the negative-ideal solution x can be measured using Eq. (9): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X p p ½d p ð˜vi j ; a˜ D p ðx j ; x Þ ¼ i Þ i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ðs i j Þ p þ 2ðzi j Þ p þ 2ðyi j Þ p þ ðt i j Þ p p ¼ 6 i¼1
(24) For p = 2, Eq. (23) can be rewritten as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 þ ½d2 ð˜vi j ; a˜ þ D2 ðx j ; x Þ ¼ i Þ
The bigger Dp(xj, x), the better xj. Rank the alternatives xj (j = 1, 2, . . ., n) by Dp(xj, x) in decreasing order. An alternative x j 2 X satisfying D p ðx j ; x Þ ¼ max fD p ðx j ; x Þg 1 jn
should be the best compromise solution, which has the longest distance from the negative-ideal solution. However, such a compromise solution may not always guarantee to have the shortest distance from the ideal solution. In other words, it may not always x j ¼ x j . Let 8 < D p ðxþ Þ ¼ max fD p ðx j ; xþ Þg 1 jn (29) : D pþ ðxþ Þ ¼ min fD p ðx j ; xþ Þg and 8 D pþ ðxþ Þ ¼ min fD p ðx j ; xþ ÞgD pþ ðx Þ > > 1 jn > < ¼ max fD p ðx j ; x Þg 1 jn > > > : D p ðx Þ ¼ min fD p ðx j ; x Þg
(30)
1 jn
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m uX ð1 s i j Þ2 þ 2ð1 zi j Þ2 þ 2ð1 yi j Þ2 þ ð1 ti j Þ2 ¼t 6 i¼1
Define a compromise ratio for every alternative xj 2 X (j = 1, 2, . . ., n) as follows: j p ðx j Þ ¼ e
D p ðxþ Þ D p ðx j ; xþ Þ D p ðxþ Þ D pþ ðxþ Þ
(25) ¯ j ; xþ Þ. Obviously, D2 ðx j ; xþ Þ ¼ Dðx
(28)
1 jn
6
i¼1
(27)
þ ð1 eÞ
D p ðx j ; x Þ D p ðx Þ D pþ ðx Þ D p ðx Þ
(31)
814
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
where e 2 [0,1] is a compromise coefficient, which is introduced as weight of the decision making strategy of ‘‘closeness to the positive ideal solution’’. The preference order of the alternative set X is generated only by the distances Dp(xj, x+) if e = 1; the preference order of the alternative set X is generated only by the distances Dp(xj, x) if e = 0; the preference order of the alternative set X is generated with considering equally importance of both the distances Dp(xj, x+) and Dp(xj, x) if e = 1/2. The index jp(xj) 2 [0,1] measures the compromise extent that the alternative xj 2 X closes to the positive ideal solution x+ and is far away from the negative-solution x. The bigger jp(xj), the better xj. The preference order of the alternatives xj (j = 1, 2, . . ., n) is generated according to jp(xj). An alternative x j0 2 X satisfying j p ðx j0 Þ ¼ max fj p ðx j Þg 1 jn
should be the best compromise solution, which has the best compromise level between the distance from the positive ideal solution x+ and the distance from the negative-ideal solution x. 4. Comparative analysis of the compromise ratio method and the fuzzy extension of the TOPSIS method The compromise ratio method introduces an aggregating function for ranking in Eq. (31), which reflects the extent that the alternative xj 2 X (j = 1, 2, . . ., n) closes to the ideal solution x+ and is far away from the negative-ideal solution x. It is only ideal goal that the best alternative is the one that has the shortest distance to the ideal solution and the longest distance from the negative-ideal solution. However, this goal is not always satisfied. In the compromise ratio method, the chosen alternative has the maximum value of jp(xj) (j = 1, 2, . . ., n) defined in Eq. (31), with the intention to minimize the distance Dp(xj, x+) from the ideal solution x+ and to maximize the distance Dp(xj, x) from the negative-ideal solution x simultaneously. The compromise ratio method considers the relative importance of distances Dp(xj, x+) (Eq. (23)) and Dp(xj, x) (Eq. (28)) within Eq. (31), which could be a major concern in decision making. In fact, jp(xj) represents the compromise satisfactory level which the decision maker considers the compromise extent between the shortest distance from the ideal solution x+ and the farthest distance from the negative-ideal solution x. And such chosen alternative could reflect the relative importance of distances Dp(xj, x+) and Dp(xj, x). The ranking order of all alternatives could realize the goal of the decision maker in real life situations. The fuzzy extension of the TOPSIS method introduces the ranking index defined in Eq. (19), including the distances from the ideal solution and from the negative-ideal solution. These distances in the fuzzy extended TOPSIS method are simply summed in Eq. (19). However, the reference point could be a major concern in decision making, and to be as close as possible to the ideal solution is the rationale of human choice. Being far away from a negative-ideal solution could be a goal only in a particular situation. The relative importance of distances D(xj,
x+) (Eq. (17)) and D(xj, x) (Eq. (18)) was not considered in the fuzzy extended TOPSIS method [4], although this could be a major concern in real life decision making. Lai et al. [15] considered this issue by introducing the satisfactory level for both criteria of the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution, and concluding ‘‘The compromise solution will exist at the point where the satisfactory levels of both criteria are the same. In future studies, applying compensatory operators should be emphasized’’. Thus, the relative importance of these distances D(xj, x+) and D(xj, x) remained an open question, which is considered in the compromise ratio method. 5. Numerical example In order to make comparison easily, we choose the numerical example from [4] (Note that a triangular fuzzy number is a special case of the trapezoid fuzzy number, so all stated as above are applicable to this example). Suppose that a software company desires to hire a system analysis engineer. After preliminary screening, three candidates (i.e., alternatives) x1, x2 and x3 remain for further evaluation. Denote the candidate set as X = {x1, x2, x3}. A committee of three decision makers P1, P2 and P3 has been formed to conduct the interview and to select the most suitable candidate. Five benefit attributes are considered, including emotional steadiness ( f1), oral communication skill ( f2), personality ( f3), past experience ( f4) and self-confidence ( f5). The three decision makers use the linguistic variables shown in Table 4 to assess the importance of the five attributes and present it in Table 5. The decision makers use the linguistic variables shown in Table 6 to evaluate the candidates with respect to each attribute and present it in Table 7. Converting the linguistic evaluation (shown in Tables 5 and 7) into trapezoid fuzzy numbers to construct the fuzzy decision Table 4 Linguistic variables for importance of each attribute Linguistic variables
Trapezoid fuzzy numbers
Very low (LV) Low (L) Medium low (ML) Medium (M) Medium high (MH) High (H) Very high (VH)
(0, 0, 0, 0.1) (0, 0.1, 0.1, 0.3) (0.1, 0.3, 0.3, 0.5) (0.3, 0.5, 0.5, 0.7) (0.5, 0.7, 0.7, 0.9) (0.7, 0.9, 0.9, 1.0) (0.9, 1.0, 1.0, 1.0)
Table 5 The importance of the attributes give by the decision makers Attributes
f1 f2 f3 f4 f5
Decision makers D1
D2
D3
H VH VH VH M
VH VH H VH MH
MH VH H VH MH
D.-F. Li / Applied Soft Computing 7 (2007) 807–817 Table 6 Linguistic variables for the ratings of each candidate with respect to each attribute Linguistic variables
Trapezoid fuzzy numbers
Very poor (LP) Poor (P) Medium poor (MP) Fair (F) Medium good (MG) Good (G) Very good (VG)
(0, (0, (1, (3, (5, (7, (9,
0, 0, 1) 1, 1, 3) 3, 3, 5) 5, 5, 7) 7, 7, 9) 9, 9, 10) 10, 10, 10)
815
Table 9 The normalized fuzzy decision matrix
f1 f2 f3 f4 f5
x1
x2
x3
(0.59, 0.79, 0.79, 0.96) (0.5, 0.7, 0.7, 0.9) (0.57, 0.77, 0.77, 0.9) (0.83, 0.97, 0.97, 1) (0.3, 0.5, 0.5, 0.7)
(0.65, 0.86, 0.86, 1) (0.9, 1, 1, 1) (0.83, 0.97, 0.97, 1) (0.9, 1, 1, 1) (0.7, 0.9, 0.9, 1)
(0.65, 0.82, 0.82, 0.93) (0.7, 0.9, 0.9, 1) (0.7, 0.9, 0.9,1) (0.7, 0.9, 0.9, 1) (0.63, 0.83, 0.83, 0.97)
Table 10 The weighted normalized fuzzy decision matrix x1 Table 7 The ratings of candidates given by the decision makers under all attributes Attributes
Candidates
Decision makers D1
D2
D3
f1
x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3
MG G VG G VG MG F VG G VG VG G F VG G
G G G MG VG G G VG MG G VG VG F MG G
MG MG F F VG VG G G VG VG VG MG F G MG
f2
f3
f4
f5
f1 f2 f3 f4 f5
(0.41, (0.45, (0.44, (0.75, (0.13,
0.71, 0.71, 0.96) 0.7, 0.7, 0.9) 0.72, 0.72, 0.9) 0.97, 0.97, 1) 0.32, 0.32, 0.58)
x2
x3
(0.46, 0.77, 0.77, 1) (0.81, 1, 1, 1) (0.64, 0.9, 0.9, 1) (0.81, 1, 1, 1) (0.3, 0.57, 0.57, 0.83)
(0.46, (0.63, (0.54, (0.63, (0.27,
0.74, 0.74, 0.93) 0.9, 0.9, 1) 0.84, 0.84, 1) 0.9, 0.9, 1) 0.52, 0.52, 0.81)
Chen [4] defined the distances and closeness coefficient as follows: dðx j ; xþ Þ ¼
m X dð˜vi j ; a˜ þ i Þ i¼1
¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ð1 s i j Þ2 þ ð1 yi j Þ2 þ ð1 ti j Þ2 3
i¼1
(32)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs i j Þ2 þ ðyi j Þ2 þ ðt i j Þ2 3 i¼1
m m X X dðx j ; x Þ ¼ dð˜vi j ; a˜ i Þ ¼
matrix and determine the fuzzy weight of each attribute as Table 8. Using Eq. (15) and Table 8, the fuzzy decision matrix can be transformed into the normalized fuzzy decision matrix as Table 9. Using Eq. (16) and Table 9, the normalized fuzzy decision matrix can be transformed into the weighted normalized fuzzy decision matrix as Table 10. The fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x can be constructed, whose weighted ˜þ ˜þ normalized fuzzy vectors are a˜ þ ¼ ð˜aþ 1 ;a 2 ;...;a 5 Þ and þ ˜ ˜ ˜ a˜ ¼ ð˜a ; a ; . . . ; a Þ, where a ¼ ð1; 1; 1; 1Þ ¼ 1 and i 1 2 5 a˜ ¼ ð0; 0; 0; 0Þ ¼ 0. Using Eqs. (17)–(23), (28) and (31), i the decision results are obtained as Table 11.
i¼1
(33) and CCðx j Þ ¼
dðx j ; x Þ dðx j ; xþ Þ þ dðx j ; x Þ
(34)
where v˜ i j is a normalized triangular fuzzy number for each alternative xj with respect to attribute fi, v˜ i j ¼ ðs i j ; yi j ; ti j Þ; ˜ a˜ þ i ¼ ð1; 1; 1Þ ¼ 1 and a i ¼ ð0; 0; 0Þ ¼ 0 are triangular fuzzy numbers. The compromise solution obtained by the compromise ratio method and the TOPSIS method is the same alternative x2. Ranking results by these two methods are almost the
Table 8 The fuzzy decision matrix and fuzzy weights of three candidates
f1 f2 f3 f4 f5
x1
x2
x3
Weight
(5.7, 7.7, 7.7, 9.3) (5, 7, 7, 9) (5.7, 7.7, 7.7, 9) (8.3, 9.7, 9.7, 10) (3, 5, 5, 7)
(6.3, 8.3, 8.3, 9.7) (9, 10, 10, 10) (8.3, 9.7, 9.7, 10) (9, 10, 10, 10) (7, 9, 9, 10)
(6.3, 8, 8, 9) (7, 9, 9, 10) (7, 9, 9, 10) (7, 9, 9, 10) (6.3, 8.3, 8.3, 9.7)
(0.7, 0.9, 0.9, 1) (0.9, 1, 1, 1) (0.77, 0.93, 0.93, 1) (0.9, 1, 1, 1) (0.43, 0.63, 0.63, 0.83)
816
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
Table 11 Decision results obtained by the TOPSIS method and the compromise ratio method Parameter p
TOPSIS Chen [4]
Eqs. (17)–(19)
Eqs. (20)–(22)
CRM p=1
p=2
p ! +1
Candidates
d(xj, x) d(xj, x+) CC(xj) D(xj, x) D(xj, x+) C*(xj) ¯ j ; x Þ Dðx ¯ Dðx j ; xþ Þ C¯ ðx j Þ Dp(xj, Dp(xj, jp(xj) Dp(xj, Dp(xj, jp(xj) Dp(xj, Dp(xj, jp(xj)
x) x+) x) x+) x) x+)
Ranking order
x1
x2
x3
3.45 2.10 0.62 3.74 1.41 0.73 1.70 0.71 0.71
4.13 1.24 0.77 4.66 0.49 0.90 2.09 0.25 0.89
3.85 1.59 0.71 4.30 0.85 0.83 1.93 0.39 0.83
x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1
3.28 1.31 0 1.70 0.71 0 0.65 0.33 0.71(1 e)
4.65 0.35 1 2.09 0.25 1 0.67 0.09 1
4.13 0.72 0.62–0.01e 1.93 0.39 0.59+0.11e 0.6 0.12 0.88e
x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x1 x3 x2 x3 x1 e < 71/159 e = 71/159 e > 71/159
same except for the case p ! +1 in the compromise ratio method. 6. Short conclusions In general, multi-attribute decision making problems adhere to uncertain and imprecise data, and fuzzy set theory is adequate to deal with it. In this paper, using linguistic variables we develop the compromise ratio method for solving fuzzy multi-attribute group decision making, which is based on an aggregating function representing closeness to the ideal solution and being far away from the negative-ideal solution simultaneously, whereas the TOPSIS method is based on an aggregating function representing ‘‘closeness to the ideal’’ only. The compromise ratio method introduces the ranking index to reflect some balance between the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. The relative importance of the distances from the ideal solution and the negative-ideal solution, which has been considered in the compromise ratio method is a major concern in real life decision making. In contrast, the basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution, but such chosen alternative is not always the closest to the ideal solution (see the example in Section 2). The TOPSIS method introduces two reference points, but it does not consider the relative importance of the distances from these points. The compromise ratio method may use different distance measurements Lp-metric for real/fuzzy numbers, whereas the TOPSIS method uses only L2-metric or Euclidean
x2 x1 x3 x2 x3 x1 x2 x3 x1
distance. In fact, the fuzzy extension of the TOPSIS method (i.e., Eqs. (32)–(34)) is a simple weighted average method with equal weights. Under group decision-making process, it is not difficult to use other aggregation functions to pool the fuzzy ratings of decision makers in the proposed method. Although the proposed method presented in this paper is illustrated by a personnel selection problem, however, it can also be applied to problems such as information project selection, combat plan selection, material selection and many other areas of management decision problems. Obviously, how to determine membership functions of fuzzy numbers is a key and very difficult problem when the method proposed in this paper is applied to real life decision making problems under fuzzy environment. Recently, some methods were proposed [1,3,11,22,28,30–33]. A more effective method will be investigated on determining (learning or extracting) membership functions from the decision makers in the near future. Acknowledgments The author would like to thank the valuable reviews and also appreciate the constructive suggestions from the anonymous referees and Prof. Rajkumar Roy, the Editor in Chief for Applied Soft Computing Journal. This research was Sponsored by the Natural Science Foundation of China (No. 70571086), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No. 2005-546) and China Postdoctoral Science Foundation (No. 2005-38).
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
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Compromise ratio method for fuzzy multi-attribute group decision making Deng-Feng Li a,b,c,* a
Department of Sciences, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, Liaoning, China b Department Five, Dalian Naval Academy, No. 1, Xiaolong Street, Dalian 116018, Liaoning, China c The Missile Institute, Air Force Engineering University, Sanyuan, Shaanxi 713800, China Received 4 January 2005; received in revised form 7 February 2006; accepted 13 February 2006 Available online 2 May 2006
Abstract The aim of this paper is to develop a compromise ratio (CR) methodology for fuzzy multi-attribute group decision making (FMAGDM), which is an important part of decision support system. Owing to fuzziness being inherent in decision data and group decision making processes, the crisp values are inadequate to model real-life situations. In this paper, the weights of all attributes and the ratings of each alternative with respect to each attribute are described by linguistic terms which can be expressed in trapezoid fuzzy numbers. A fuzzy distance measure is developed to calculate difference between trapezoid fuzzy numbers. The compromise ratio method for FMAGDM is developed by introducing the ranking index based on the concept that the chosen alternative should be as close as possible to the ideal solution and as far away from the negative-ideal solution as possible simultaneously. The computation principle and procedure of the compromise ratio method are described in detail in this paper. Moreover the TOPSIS method which was developed for multi-attribute decision making (MADM) with crisp decision data is analyzed and extended to multiattribute group decision making (MAGDM) under fuzzy environments. A comparative analysis of the compromise ratio method and the extended fuzzy TOPSIS method is illustrated with a numerical example, showing their similarity and some differences. # 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy multi-attribute group decision making; Linguistic variable; Fuzzy number; Compromise ratio method; TOPSIS; Comparative analysis
1. Introduction Multi-attribute decision making (MADM) problems are wide spread in real life decision situations [6–10,12–21]. A MADM problem is to find a best compromise solution from all feasible alternatives assessed on multiple attributes, both quantitative and qualitative, and can be dealt with using several existing methods such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) developed by Huang and Yoon [12], which is one of the well-known MADM methods. The basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the positiveideal solution (PIS) and the farthest distance from the negativeideal solution (NIS). In the process of the TOPSIS method, the performance ratings and weights of the attributes are given as crisp values a priori. However, the highest ranked alternative by the TOPSIS method is the best in terms of the ranking index, * Tel.: +86 411 85856357; fax: +86 411 85856357. E-mail addresses: [email protected], [email protected]. 1568-4946/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2006.02.003
which does not mean that it is always the closest to the ideal solution (Section 2). Moreover, under many conditions, crisp data are inadequate or insufficient to model real-life decision problems [4,5,23,24]. Indeed, human judgments including preferences are vague or fuzzy in nature and as such it may not be appropriate to represent them by accurate numerical values. A more realistic approach could be to use linguistic variables to model human judgments [2,30], that is, to suppose that the ratings and weights of the attributes in the decision making problem are assessed by means of linguistic variables. In this paper, the TOPSIS method for solving MADM problems with crisp decision data is analyzed and further extended to multi-attribute group decision making (MAGDM) under fuzzy environments. A new compromise ratio methodology is developed to solve fuzzy multi-attribute group decision making (FMAGDM) problems. In this methodology, linguistic variables are used to capture fuzziness in decision information and decision making processes by means of a fuzzy decision matrix. The multiattribute ranking index is introduced based on the fuzzy
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distance measure, which is developed from the concept of Minkowski distance. The compromise ratio method is based on the concept that the chosen alternative should have the distance from the ideal solution as short as possible and the distance from the negative-ideal solution as far as possible simultaneously. The relative importance of these distances [8,9,25–27,29], which is a major concern in real-life decision situations, is considered in the developed method. A comparative analysis is illustrated with a numerical example, showing their similarity and some differences. The paper is organized as follows. A brief overview and analysis of the TOPSIS method is given in Section 2. A discussion on the TOPSIS method extended to the MAGDM problems in fuzzy environments is also presented in this section. The basic principle and procedure of the compromise ratio method are presented in Section 3. A comparative analysis of the compromise ratio method and the fuzzy extension of the TOPSIS method is given in Section 4. Section 5 gives a numerical example to show similarity and some differences of the compromise ratio method and the fuzzy extended TOPSIS method. A short conclusion is given in Section 6.
P attribute f i, where vi 0 (i = 1, 2, . . ., m) and m i¼1 vi ¼ 1. Denote a weight vector by v = (v1, v2, . . ., vm)T. The above MADM problem can be dealt with using the TOPSIS method developed by Huang and Yoon [12]. The basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. The procedure of the TOPSIS method consists of the following steps: (1) Calculate the normalized decision matrix. The normalized value rij (i = 1, 2, . . ., m; j = 1, 2, . . ., n) of f ij is calculated as fi j ffi ri j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 f j¼1 i j
(1)
(2) Calculate the weighted normalized decision matrix. The weighted normalized value vi j (i = 1, 2, . . ., m; j = 1, 2, . . ., n) of rij is calculated as vi j ¼ v i r i j
(2) *+
2. The TOPSIS method and its extension to FMAGDM 2.1. The TOPSIS method A MADM problem is to find a best compromise solution from all feasible alternatives assessed on multiple attributes [5,12,14,16]. Suppose the decision maker has to choose one of or rank n efficient/non-inferior alternatives xj (j = 1, 2, . . ., n) based on m attributes f i (i = 1, 2, . . ., m). Denote an alternative set by X = {x1, x2, . . ., xn} and an attribute set by F = {f 1, f 2, . . ., f m). In general, attributes can be classified into two types: benefit attributes and cost attributes. In other words, the attribute set F can be divided into two subsets: F 1 and F 2, where F k (k = 1, 2) is the subset of benefit attributes and cost attributes, respectively. Furthermore, F = F 1 [ F 2 and F 1 \ F 2 = 1, where 1 is an empty set. Then the MADM model can be built as follows: maxf fi ðx j Þj fi 2 F 1 g minf fi ðx j Þj fi 2 F 2 g s:t: x j 2 X The alternative set X = {x1, x2, . . ., xn} and the attribute set F = {f 1, f 2, . . ., f m} are finite, so it is very convenient to denote the rating of alternative xj (j = 1, 2, . . ., n) on attribute f i (i = 1, 2, . . ., m) by f ij, i.e., f ij = f i(xj). Then a MADM problem can be concisely expressed as the following decision matrix:
(3) Determine the ideal solution x and the negative-ideal solution x*, whose weighted normalized value vectors are þ þ denoted by vþ ¼ ðvþ ¼ ðv 1 ; v2 ; 1 ; v2 ; . . . ; vm Þ and v . . . ; vm Þ, respectively. For every ith attribute function (i = 1, 2, . . ., m), 8 < max fvi j g ð fi 2 F 1 g 1 jn þ (3) vi ¼ : min fvi j g ð fi 2 F 2 g 1 jn
and v i
¼
8 < min fvi j g
ð fi 2 F 1 g
: max fvi j g
ð fi 2 F 2 g
1 jn 1 jn
(4)
(4) Calculate the separation measures, using the m-dimensional Euclidean distance. The separation of each alternative xj (j = 1, 2, . . ., n) from the ideal solution x*+ is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 þ D ðx j Þ ¼ (5) ðvi j vþ i Þ i¼1
Similarly, the separation of each alternative xj (j = 1, 2, . . ., n) from the negative-ideal solution x* is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 D ðx j Þ ¼ (6) ðvi j v i Þ i¼1
(5) Calculate the relative closeness to the ideal solution. The relative closeness of each alternative xj (j = 1, 2, . . ., n) with respect to x*+ is defined as C ðx j Þ ¼ In decision making process, different attributes have different importance. Suppose vi (i = 1, 2, . . ., m) is the relative weight of
D ðx j Þ Dþ ðx j Þ þ D ðx j Þ
(7)
(6) Rank the preference order of all alternatives xj (j = 1, 2, . . ., n) by the C*(xj) in decreasing order.
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
According to the formulation of the ranking index C*(xj) (j = 1, 2, . . ., n) in Eq. (7), alternative xj is better than xk if C*(xj) > C*(xk), i.e., D ðx j Þ D ðxk Þ > þ D ðxk Þ þ D ðxk Þ j Þ þ D ðx j Þ
Dþ ðx j Þ < Dþ ðxk Þ and
ðBÞ
Alternatives
Ranking order
x1
x2
x3
0.114 0.365 0.762
0.108 0.280 0.722
0.365 0.114 0.238
x2 x1 x3 x1 x2 x3 x1 x2 x3
D ðx j Þ > D ðxk Þ;
or þ
Decision matrix
D (xj) D*(xj) C*(xj)
which will hold if ðAÞ
Table 1 Decision results obtained by the TOPSIS method
*+
Dþ ðx
809
However, x1 is not the closest to the ideal solution x*+, i.e., D*+(x1) = 0.114, D*+(x2) = 0.108. According to condition (B) as stated above, in this case, x1 is ranked best by the TOPSIS method, although it is not the closest to the ideal solution, because
þ
D ðx j Þ > D ðxk Þ and
D ðx j Þ > D ðxk Þ; but Dþ ðx j Þ <
Dþ ðxk ÞD ðx j Þ : D ðxk Þ
Condition (A) shows the ‘‘regular’’ situation, when alternative xj is better than xk because it is closer to the ideal solution x*+ and farther from the negative-ideal solution x*. On the contrary, condition (B) shows that an alternative xj could be better than xk even though xj is farther from ideal solution x*+ than xk. Let xk be the alternative with D*+(xk) = D*(xk) and C*(xk) = 0.5. In this case, all alternatives xj with D*+(xj) > D*+(xk) and D*+(xj) < D*(xj) are better ranked than xk, although xk is closer to the ideal solution x*+. For example, a mountain climber (beginner) must choose an alternative (destination) from a set of three alternatives, i.e., the alternative set X = {x1, x2, x3}. The alternatives are evaluated according to two attributes: risk and altitude [26]. Risk and altitude are denoted as f 1 and f 2, respectively. Risk f 1 is a cost attribute and evaluated using subjective scales: 1–5. Altitude f 2 is a benefit attribute and evaluated in meters above the sea (m.a.s). Denote the attribute set as F = {f 1, f 2}. Attribute values f ij of alternatives are given as the following decision matrix:
(8)
Let us suppose that both attributes f 1 and f 2 are equally important, i.e., the weights of attributes are vi = 1/2 (i = 1, 2). Using Eq. (1), the normalized decision matrix of the decision matrix Yf is computed as follows:
The decision results obtained by the TOPSIS method are presented in Table 1 using Eqs. (5)–(7). Analyzing the decision results of the TOPSIS method, we find that the decision results from the TOPSIS method are interesting due to the following. According to C*(xj) (j = 1, 2, 3) as in Table 1, the best solution is alternative x1, i.e., C*(x1) = 0.762, C*(x1) = 0.722; and it is the best according to D*(xj) (j = 1, 2, 3), i.e., D*(x1) = 0.365, D*(x2) = 0.280.
Dþ ðx1 Þ ¼ 0:114 < 0:141 ¼ ¼
0:108 0:365 0:280
Dþ ðx2 ÞD ðx1 Þ D ðx2 Þ
This is an example of the case discussed in condition (B). 2.2. Notations of fuzzy numbers ˜ is a special fuzzy subset on the set R of A fuzzy number m real numbers which satisfy the following conditions [2,4,5,16,32]. (1) There exists a x0 2 R at least so that the degree of its membership um˜ ðx0 Þ ¼ 1; (2) Membership function um˜ ðxÞ is left and right continuous. ˜ ¼ ða; b; c; dÞ be a trapezoid fuzzy number, where the Let m ˜ is given by membership function mm˜ of m 8xa > > > :d x dc
ða x bÞ ðb x cÞ ðc x dÞ
˜ and a and d are where [b, c] is called a mode interval of m, ˜ respectively. called the low and upper limits of m, ˜ ¼ ða; b; c; dÞ is called a triangular fuzzy IF b = c then m ˜ ¼ ða; m; dÞ, where m = b = c. Obviously, number, denoted m the membership function mm˜ of a triangular fuzzy number ˜ ¼ ða; m; dÞ is m 8 xa > < ða x mÞ m a mm˜ ðxÞ ¼ d x > : ðm x dÞ dm Therefore, a triangular fuzzy number is a special case of the trapezoid fuzzy number. In other words, a triangular fuzzy ˜ ¼ ða; m; dÞ can also be written as a trapezoid fuzzy number m ˜ ¼ ða; m; m; dÞ. number m
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˜ ¼ It is easy to see that a trapezoid fuzzy number m ða; b; c; dÞ is reduced to a real number a if a = b = c = d. Conversely, a real number m can be written as a trapezoid fuzzy ˜ ¼ ðm; m; m; mÞ. number m For the sake of simplicity and without loss of generality, assume that all fuzzy numbers are trapezoid fuzzy numbers throughout the paper unless otherwise stated. ˜ ¼ ða; b; c; dÞ is called a positive trapezoid fuzzy number if m a 0, and a, b, c and d are not identical. ˜ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n˜ ¼ ðn1 ; n2 ; n3 ; n4 Þ Assume that m be two positive trapezoid fuzzy numbers, r > 0 is a positive real number. Then designate the following operators [32]: ð1Þ ð2Þ ð3Þ
˜ ¼ ðrm1 ; rm2 ; rm3 ; rm4 Þ rm ˜ 1 m1 m2 m3 m4 m ˜ ¼ ¼ m ; ; ; 2 r r r r r r ˜ n ¼ ðm1 n1 ; m2 n2 ; m3 n3 ; m4 n4 Þ m ˜
2.3. Linguistic variable is a variable whose values are linguistic terms. The concept of linguistic variable is very useful in situations where decision problems are too complex or too illdefined to be described properly using conventional quantitative expressions. For example, the performance ratings of alternatives on qualitative attributes could be expressed using linguistic variable such as very poor, poor, fair, good, very good, etc. Such linguistic values can be represented using positive trapezoid fuzzy numbers. For example, ‘‘poor’’ and ‘‘very good’’ can be represented by positive trapezoid fuzzy numbers (0.2, 0.3, 0.4, 0.5) and (0.8, 0.9, 1.0, 1.0), respectively.
For p = 2, Eq. (9) can be rewritten as follows: ˜ n˜ Þ ¼ d2 ðm; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm1 n1 Þ2 þ 2ðm2 n2 Þ2 þ 2ðm3 n3 Þ2 þ ðm4 n4 Þ2 6 (11) which is a weighted Euclidean distance. And if q ! 1, Eq. (9) can be rewritten as follows: m1 n1 m2 n2 m3 n3 m4 n4 ˜ n˜ Þ ¼ max d1 ðm; ; ; ; 6 3 3 6 (12) which is a weighted Chebyshev distance. ˜ and n˜ are real numbers then the distance Note that if both m ˜ n˜ Þ is identical to the Euclidean distance. In measurement d p ðm; ˜ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n˜ ¼ fact, suppose both m ðn1 ; n2 ; n3 ; n4 Þ are two real numbers and let m1 = m2 = m3 = m4 = m and n1 = n2 = n3 = n4 = n. The dis˜ n˜ Þ can be calculated as follows: tance measurement d p ðm; ˜ n˜ Þ d p ðm; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p p p p ðm1 n1 Þ þ 2ðm2 n2 Þ þ 2ðm3 n3 Þ þ ðm4 n4 Þ ¼ 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p p p p p ðm nÞ þ 2ðm nÞ þ 2ðm nÞ þ ðm nÞ ¼ 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ¼ ðm nÞ ¼ jm nj Furthermore, it is easily seen that two trapezoid fuzzy numbers ˜ and n˜ are identical if and only if the distance measurement m ˜ n˜ Þ ¼ 0. d p ðm;
2.4. Distance between trapezoid fuzzy numbers 2.5. The extension of the TOPSIS method for FMAGDM ˜ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n˜ ¼ ðn1 ; n2 ; n3 ; n4 Þ be two Let m trapezoid fuzzy numbers. Then the distance measure between them is defined using the Minkowski distance (or Lp-metric) as follows: ˜ n˜ Þ ¼ d p ðm; ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p p p p ðm1 n1 Þ þ 2ðm2 n2 Þ þ 2ðm3 n3 Þ þ ðm4 n4 Þ 6 (9) where p 1 is a distance parameter. It is easily seen that Eq. (9) is the weighted distance, which considers different important of the low limit, the left and right points of the mode interval, and the upper limits of the trapezoid fuzzy numbers. For p = 1, Eq. (9) can be rewritten as follows: ˜ n˜ Þ ¼ d1 ðm;
jm1 n1 j þ 2jm2 n2 j þ 2jm3 n3 j þ jm4 n4 j 6 (10)
which is a weighted Hamming distance.
In this paper, we discuss the following MAGDM in fuzzy environments. Suppose there exist n possible alternatives x1, x2, . . ., xn from which K decision makers Pk (k = 1, 2, . . ., K) have to choose on the basis of m attributes f1, f2, . . ., fm, both qualitative and quantitative. The importance weights of various attributes and ratings of qualitative attributes are considered as linguistic variables. These linguistic variables can be expressed in positive trapezoid fuzzy numbers in Tables 2 and 3, which corresponding membership functions are determined or extracted using a similar way to [1–3,11,22,28,30–33]. The important weight of each attribute can be obtained by either directly assign or indirectly using pairwise comparisons [4,16]. In here, it is suggested that the decision makers use the linguistic variables shown as Tables 2 and 3 to evaluate the importance of attributes and the ratings of alternatives with respect to various attributes. Suppose the rating of alternative xj (j = 1, 2, . . ., n) on attribute fi (i = 1, 2, . . ., m) given by decision maker Pk (k = 1, 2, k . . ., K) is f˜i j ¼ ðakij ; bkij ; ckij ; dikj Þ. Hence, a FMAGDM problem can be concisely expressed in matrix format as follows:
D.-F. Li / Applied Soft Computing 7 (2007) 807–817 Table 2 Linguistic variables for importance of each attribute Linguistic variables
Trapezoid fuzzy numbers
Extreme low (EL) Very low (LV) Low (L) Medium low (ML) Medium (M) Medium high (MH) High (H) Very high (VH) Extreme high (EH)
(0, 0, 0.1, 0.2) (0.1, 0.2, 0.3, 0.4) (0.2, 0.3, 0.4, 0.5) (0.3, 0.4, 0.5, 0.6) (0.4, 0.5, 0.6, 0.7) (0.5, 0.6, 0.7, 0.8) (0.6, 0.7, 0.8, 0.9) (0.7, 0.8, 0.9, 1.0) (0.8, 0.9, 1.0, 1.0)
Table 3 Linguistic variables for ratings of each alternative with respect to each attribute Linguistic variables
Trapezoid fuzzy numbers
Extreme poor (EP) Very poor (LP) Poor (P) Medium poor (MP) Fair (F) Medium good (MG) Good (G) Very good (VG) Extreme good (EG)
(0, (1, (2, (3, (4, (5, (6, (7, (8,
0, 2, 3, 4, 5, 6, 7, 8, 9,
1, 2) 3, 4) 4, 5) 5, 6) 6, 7) 7, 8) 8, 9) 9, 10) 10, 10)
811
v˜ i and f˜i j are denoted as v˜ i ¼ ðai ; bi ; g i ; di Þ and f˜i j ¼ ðai j ; bi j ; ci j ; di j Þ, respectively. As stated above, a FMAGDM problem can be concisely expressed as the following decision matrix:
The weight vector is v˜ ¼ ðv˜ 1 ; v˜ 2 ; . . . ; v˜ m ÞT . Y˜ is referred to as fuzzy decision matrix usually used to represent the FMAGDM problem. Since the m attributes may be measured in different ways, the decision matrix Y˜ needs to be normalized. The linear scale transformation is used here to transform the various attribute scales into a comparable scale [16]: r˜i j ¼
a i j bi j c i j di j ; ; ; dimax dimax dimax dimax
ð fi 2 F 1 Þ
(15)
and 8 min min min min a a a a > > < i ; i ; i ; i d c i j bi j ai j r˜i j ¼ i j di j ci j bi j ai j > > : 1 max ; 1 max ; 1 max ; 1 max di di di di
6¼ 0Þ ðamin i ¼ 0Þ ðamin i
ð fi 2 F 2 Þ where dimax ¼ max fdi j j f˜i j ¼ ðai j ; bi j ; ci j ; di j Þg 1 jn
which are referred to as fuzzy decision matrices usually used to represent the FMAGDM problem. Similarly, suppose the importance weight of attribute fi (i = 1, 2, . . ., m) given by decision maker Pk (k = 1, 2, . . ., K) is v˜ ki ¼ ðaki ; bki ; g ki ; dki Þ. Then the importance weights of the attributes and ratings of the alternatives with respect to each attribute can be calculated as follows: v˜ 1i v˜ 2i v˜ Ki K PK PK PK k PK k k k d k¼1 ai k¼1 bi k¼1 g i ; ; ; k¼1 i ¼ K K K K
v˜ i ¼
and ¼ min fai j j f˜i j ¼ ðai j ; bi j ; ci j ; di j Þg amin i 1 jn
Denote r˜i j as r˜i j ¼ ðmi j ; hi j ; ri j ; li j Þ. The normalization method mentioned above is to preserve the property that the range of a normalized trapezoid fuzzy number r˜i j belongs to the closed interval [0,1]. Then, the fuzzy decision matrix Y˜ ¼ ð f˜i j Þmn can be transformed into the normalized fuzzy decision matrix:
(13)
and 1 2 K f˜i j f˜i j f˜i j K PK k PK k PK k PK k k¼1 ai j k¼1 bi j k¼1 ci j k¼1 di j ; ; ; ¼ K K K K
f˜i j ¼
(14)
Considering the different importance of each attribute, R˜ is transformed into the weighted normalized fuzzy decision ˜ matrix V:
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D.-F. Li / Applied Soft Computing 7 (2007) 807–817
Hence Eq. (19) is reduced to the following formula: Pm Pm si j si j i¼1 ¼ i¼1 C ðx j Þ ¼ Pm Pm m m i¼1 s i j þ i¼1 s i j
where v˜ i j is a trapezoid fuzzy number: v˜ i j ¼ v˜ i ˜ri j ¼ ðai mi j ; bi hi j ; g i ri j ; di li j Þ
(16)
denote as v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ. Obviously, all v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ are normalized positive trapezoid fuzzy numbers and their ranges belong to the closed interval [0,1]. Then, we can define the fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x, whose ˜þ ˜þ weighted normalized fuzzy vectors are a˜ þ ¼ ð˜aþ mÞ 1 ;a 2 ;...;a þ and a˜ ¼ ð˜a1 ; a˜ 2 ; . . . ; a˜ m Þ, where a˜ i ¼ ð1; 1; 1; 1Þ ¼ 1 and a˜ i ¼ ð0; 0; 0; 0Þ ¼ 0. The distance of each alternative xj from the fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x can be calculated using Eq. (11) as follows: Dðx j ; xþ Þ ¼
m X
2.6. The further extension of the TOPSIS method for FMAGDM In the following, the distance of each alternative xj from the fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x is calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 þ ¯ j; x Þ ¼ ½d2 ð˜vi j ; a˜ þ Dðx i Þ i¼1
d2 ð˜vi j ; a˜ þ i Þ
i¼1
¼
which is the simple weighted average method with equal weights 1/m for all attributes. Therefore, the fuzzy extension of the TOPSIS method (Eq. (19)) can not reduced to the classical TOPSIS method for the crisp FMAGDM problems. In other words, the TOPSIS method developed by Huang and Yoon [12] is not really extended to the FMAGDM problems under fuzzy environments.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ð1 s i j Þ2 þ 2ð1 zi j Þ2 þ 2ð1 yi j Þ2 þ ð1 t i j Þ2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m uX ð1 s i j Þ2 þ 2ð1 zi j Þ2 þ 2ð1 yi j Þ2 þ ð1 ti j Þ2 ¼t 6 i¼1
6
i¼1
(20) (17)
and
and Dðx j ; x Þ ¼
¯ j ; x Þ ¼ Dðx
m X
d2 ð˜vi j ; a˜ i Þ
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m uX ðs i j Þ2 þ 2ðzi j Þ2 þ 2ðyi j Þ2 þ ðti j Þ2 ¼t 6 i¼1
i¼1
¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ðs i j Þ2 þ 2ðzi j Þ2 þ 2ðyi j Þ2 þ ðt i j Þ2 6
i¼1
(18)
respectively. The closeness coefficient of each alternative xj (j = 1, 2, . . ., n) with respect to the fuzzy positive ideal solution x+ is defined as Dðx j ; x Þ C ðx j Þ ¼ Dðx j ; xþ Þ þ Dðx j ; x Þ
(19)
The preference order of all alternatives xj (j = 1, 2, . . ., n) can be generated according to C*(xj). Analyzing Eqs. (17) and (18), however, we can find that if all v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ are real numbers (i.e., sij = zij = yij = tij) then Eqs. (17) and (18) are reduced to the following formulae: Dðx j ; xþ Þ ¼
m m m X X X d2 ð˜vi j ; a˜ þ ð1 s i j Þ ¼ m si j i Þ ¼ i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 ½d2 ð˜vi j ; a˜ i Þ
i¼1
respectively. The closeness coefficient of each alternative xj (j = 1, 2, . . ., n) with respect to the fuzzy positive ideal solution x+ is defined as ¯ j ; x Þ Dðx C¯ ðx j Þ ¼ ¯ ¯ j ; x Þ Dðx j ; xþ Þ þ Dðx
(22)
The preference order of all alternatives xj (j = 1, 2, . . ., n) can be generated according to C¯ ðx j Þ. Obviously, if all v˜ i j ¼ ðs i j ; zi j ; yi j ; ti j Þ are real numbers (i.e., sij = zij = yij = tij) then Eqs. (20) and (21) are reduced to the following formulae: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X 2 ¯ j ; xþ Þ ¼ ½d2 ð˜vi j ; a˜ þ ð1 s i j Þ2 Dðx i Þ ¼ i¼1
i¼1
i¼1
and
and m m X X Dðx j ; x Þ ¼ d2 ð˜vi j ; a˜ si j i Þ ¼
i¼1
i¼1
(21)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X 2 ¯ j; x Þ ¼ ¼ ½d2 ð˜vi j ; a˜ Þ ðs i j Þ2 Dðx i
i¼1
i¼1
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
813
And if q ! 1, Eq. (23) can be rewritten as follows:
Hence Eq. (22) is reduced to the following formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm 2 i¼1 ðs i j Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ðx j Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm Pm 2 2 i¼1 ð1 s i j Þ þ i¼1 ðs i j Þ which is the classical TOPSIS method developed by Huang and Yoon [12] for the crisp FMAGDM problems. Therefore, Eq. (22) is a real generalization of the classical TOPSIS method under crisp environments.
D1 ðx j ; xþ Þ ¼ max fd1 ð˜vi j ; a˜ þ i g 1im 1 s i j 1 z i j 1 yi j 1 t i j ; ; ; ¼ max max 1im 6 3 3 6 (26) The smaller Dp(xj, x+), the better xj. Rank the alternatives xj (j = 1, 2, . . ., n) by Dp(xj, x+) in increasing order. An alternative x j 2 X satisfying D p ðx j ; xþ Þ ¼ min fD p ðx j ; xþ Þg 1 jn
3. The compromise ratio method for FMAGDM The relative ratio method focuses on ranking and selecting from a set of alternatives in the presence of conflicting attributes. It determines the compromise ranking-list based on the concept that the chosen alternative should be as close as possible to the ideal solution and as far away from the negativeideal solution as possible simultaneously. Difference between each alternative and the ideal solution (or the negative-ideal solution) is measured with Lp-metric. The compromise ratio method introduces the multi-attribute ranking index based on the particular measure of both closing to the ideal solution and being far away from the negative-ideal solution. Difference between an alternative xj (j = 1, 2, . . ., n) and the ideal solution x+ can be measured using Eq. (9): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X p p D p ðx j ; x Þ ¼ ½d p ð˜vi j ; a˜ þ i Þ þ
i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ð1 s i j Þ p þ 2ð1 zi j Þ p þ 2ð1 yi j Þ p þ ð1 ti j Þ p p ¼ 6 i¼1 (23) For p = 1, Eq. (23) can be rewritten as follows: D1 ðx j ; xþ Þ ¼
m X
d1 ð˜vi j ; a˜ þ i Þ
i¼1
¼
m X ð1 s i j Þ þ 2ð1 zi j Þ þ 2ð1 yi j Þ þ ð1 t i j Þ
should be the best compromise solution, which has the shortest distance from the ideal solution. However, such a compromise solution may not always guarantee to have the longest distance from the negative-ideal solution. Similarly, difference between an alternative xj (j = 1, 2, . . ., n) and the negative-ideal solution x can be measured using Eq. (9): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X p p ½d p ð˜vi j ; a˜ D p ðx j ; x Þ ¼ i Þ i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ðs i j Þ p þ 2ðzi j Þ p þ 2ðyi j Þ p þ ðt i j Þ p p ¼ 6 i¼1
(24) For p = 2, Eq. (23) can be rewritten as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X 2 þ ½d2 ð˜vi j ; a˜ þ D2 ðx j ; x Þ ¼ i Þ
The bigger Dp(xj, x), the better xj. Rank the alternatives xj (j = 1, 2, . . ., n) by Dp(xj, x) in decreasing order. An alternative x j 2 X satisfying D p ðx j ; x Þ ¼ max fD p ðx j ; x Þg 1 jn
should be the best compromise solution, which has the longest distance from the negative-ideal solution. However, such a compromise solution may not always guarantee to have the shortest distance from the ideal solution. In other words, it may not always x j ¼ x j . Let 8 < D p ðxþ Þ ¼ max fD p ðx j ; xþ Þg 1 jn (29) : D pþ ðxþ Þ ¼ min fD p ðx j ; xþ Þg and 8 D pþ ðxþ Þ ¼ min fD p ðx j ; xþ ÞgD pþ ðx Þ > > 1 jn > < ¼ max fD p ðx j ; x Þg 1 jn > > > : D p ðx Þ ¼ min fD p ðx j ; x Þg
(30)
1 jn
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m uX ð1 s i j Þ2 þ 2ð1 zi j Þ2 þ 2ð1 yi j Þ2 þ ð1 ti j Þ2 ¼t 6 i¼1
Define a compromise ratio for every alternative xj 2 X (j = 1, 2, . . ., n) as follows: j p ðx j Þ ¼ e
D p ðxþ Þ D p ðx j ; xþ Þ D p ðxþ Þ D pþ ðxþ Þ
(25) ¯ j ; xþ Þ. Obviously, D2 ðx j ; xþ Þ ¼ Dðx
(28)
1 jn
6
i¼1
(27)
þ ð1 eÞ
D p ðx j ; x Þ D p ðx Þ D pþ ðx Þ D p ðx Þ
(31)
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D.-F. Li / Applied Soft Computing 7 (2007) 807–817
where e 2 [0,1] is a compromise coefficient, which is introduced as weight of the decision making strategy of ‘‘closeness to the positive ideal solution’’. The preference order of the alternative set X is generated only by the distances Dp(xj, x+) if e = 1; the preference order of the alternative set X is generated only by the distances Dp(xj, x) if e = 0; the preference order of the alternative set X is generated with considering equally importance of both the distances Dp(xj, x+) and Dp(xj, x) if e = 1/2. The index jp(xj) 2 [0,1] measures the compromise extent that the alternative xj 2 X closes to the positive ideal solution x+ and is far away from the negative-solution x. The bigger jp(xj), the better xj. The preference order of the alternatives xj (j = 1, 2, . . ., n) is generated according to jp(xj). An alternative x j0 2 X satisfying j p ðx j0 Þ ¼ max fj p ðx j Þg 1 jn
should be the best compromise solution, which has the best compromise level between the distance from the positive ideal solution x+ and the distance from the negative-ideal solution x. 4. Comparative analysis of the compromise ratio method and the fuzzy extension of the TOPSIS method The compromise ratio method introduces an aggregating function for ranking in Eq. (31), which reflects the extent that the alternative xj 2 X (j = 1, 2, . . ., n) closes to the ideal solution x+ and is far away from the negative-ideal solution x. It is only ideal goal that the best alternative is the one that has the shortest distance to the ideal solution and the longest distance from the negative-ideal solution. However, this goal is not always satisfied. In the compromise ratio method, the chosen alternative has the maximum value of jp(xj) (j = 1, 2, . . ., n) defined in Eq. (31), with the intention to minimize the distance Dp(xj, x+) from the ideal solution x+ and to maximize the distance Dp(xj, x) from the negative-ideal solution x simultaneously. The compromise ratio method considers the relative importance of distances Dp(xj, x+) (Eq. (23)) and Dp(xj, x) (Eq. (28)) within Eq. (31), which could be a major concern in decision making. In fact, jp(xj) represents the compromise satisfactory level which the decision maker considers the compromise extent between the shortest distance from the ideal solution x+ and the farthest distance from the negative-ideal solution x. And such chosen alternative could reflect the relative importance of distances Dp(xj, x+) and Dp(xj, x). The ranking order of all alternatives could realize the goal of the decision maker in real life situations. The fuzzy extension of the TOPSIS method introduces the ranking index defined in Eq. (19), including the distances from the ideal solution and from the negative-ideal solution. These distances in the fuzzy extended TOPSIS method are simply summed in Eq. (19). However, the reference point could be a major concern in decision making, and to be as close as possible to the ideal solution is the rationale of human choice. Being far away from a negative-ideal solution could be a goal only in a particular situation. The relative importance of distances D(xj,
x+) (Eq. (17)) and D(xj, x) (Eq. (18)) was not considered in the fuzzy extended TOPSIS method [4], although this could be a major concern in real life decision making. Lai et al. [15] considered this issue by introducing the satisfactory level for both criteria of the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution, and concluding ‘‘The compromise solution will exist at the point where the satisfactory levels of both criteria are the same. In future studies, applying compensatory operators should be emphasized’’. Thus, the relative importance of these distances D(xj, x+) and D(xj, x) remained an open question, which is considered in the compromise ratio method. 5. Numerical example In order to make comparison easily, we choose the numerical example from [4] (Note that a triangular fuzzy number is a special case of the trapezoid fuzzy number, so all stated as above are applicable to this example). Suppose that a software company desires to hire a system analysis engineer. After preliminary screening, three candidates (i.e., alternatives) x1, x2 and x3 remain for further evaluation. Denote the candidate set as X = {x1, x2, x3}. A committee of three decision makers P1, P2 and P3 has been formed to conduct the interview and to select the most suitable candidate. Five benefit attributes are considered, including emotional steadiness ( f1), oral communication skill ( f2), personality ( f3), past experience ( f4) and self-confidence ( f5). The three decision makers use the linguistic variables shown in Table 4 to assess the importance of the five attributes and present it in Table 5. The decision makers use the linguistic variables shown in Table 6 to evaluate the candidates with respect to each attribute and present it in Table 7. Converting the linguistic evaluation (shown in Tables 5 and 7) into trapezoid fuzzy numbers to construct the fuzzy decision Table 4 Linguistic variables for importance of each attribute Linguistic variables
Trapezoid fuzzy numbers
Very low (LV) Low (L) Medium low (ML) Medium (M) Medium high (MH) High (H) Very high (VH)
(0, 0, 0, 0.1) (0, 0.1, 0.1, 0.3) (0.1, 0.3, 0.3, 0.5) (0.3, 0.5, 0.5, 0.7) (0.5, 0.7, 0.7, 0.9) (0.7, 0.9, 0.9, 1.0) (0.9, 1.0, 1.0, 1.0)
Table 5 The importance of the attributes give by the decision makers Attributes
f1 f2 f3 f4 f5
Decision makers D1
D2
D3
H VH VH VH M
VH VH H VH MH
MH VH H VH MH
D.-F. Li / Applied Soft Computing 7 (2007) 807–817 Table 6 Linguistic variables for the ratings of each candidate with respect to each attribute Linguistic variables
Trapezoid fuzzy numbers
Very poor (LP) Poor (P) Medium poor (MP) Fair (F) Medium good (MG) Good (G) Very good (VG)
(0, (0, (1, (3, (5, (7, (9,
0, 0, 1) 1, 1, 3) 3, 3, 5) 5, 5, 7) 7, 7, 9) 9, 9, 10) 10, 10, 10)
815
Table 9 The normalized fuzzy decision matrix
f1 f2 f3 f4 f5
x1
x2
x3
(0.59, 0.79, 0.79, 0.96) (0.5, 0.7, 0.7, 0.9) (0.57, 0.77, 0.77, 0.9) (0.83, 0.97, 0.97, 1) (0.3, 0.5, 0.5, 0.7)
(0.65, 0.86, 0.86, 1) (0.9, 1, 1, 1) (0.83, 0.97, 0.97, 1) (0.9, 1, 1, 1) (0.7, 0.9, 0.9, 1)
(0.65, 0.82, 0.82, 0.93) (0.7, 0.9, 0.9, 1) (0.7, 0.9, 0.9,1) (0.7, 0.9, 0.9, 1) (0.63, 0.83, 0.83, 0.97)
Table 10 The weighted normalized fuzzy decision matrix x1 Table 7 The ratings of candidates given by the decision makers under all attributes Attributes
Candidates
Decision makers D1
D2
D3
f1
x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3
MG G VG G VG MG F VG G VG VG G F VG G
G G G MG VG G G VG MG G VG VG F MG G
MG MG F F VG VG G G VG VG VG MG F G MG
f2
f3
f4
f5
f1 f2 f3 f4 f5
(0.41, (0.45, (0.44, (0.75, (0.13,
0.71, 0.71, 0.96) 0.7, 0.7, 0.9) 0.72, 0.72, 0.9) 0.97, 0.97, 1) 0.32, 0.32, 0.58)
x2
x3
(0.46, 0.77, 0.77, 1) (0.81, 1, 1, 1) (0.64, 0.9, 0.9, 1) (0.81, 1, 1, 1) (0.3, 0.57, 0.57, 0.83)
(0.46, (0.63, (0.54, (0.63, (0.27,
0.74, 0.74, 0.93) 0.9, 0.9, 1) 0.84, 0.84, 1) 0.9, 0.9, 1) 0.52, 0.52, 0.81)
Chen [4] defined the distances and closeness coefficient as follows: dðx j ; xþ Þ ¼
m X dð˜vi j ; a˜ þ i Þ i¼1
¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X ð1 s i j Þ2 þ ð1 yi j Þ2 þ ð1 ti j Þ2 3
i¼1
(32)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs i j Þ2 þ ðyi j Þ2 þ ðt i j Þ2 3 i¼1
m m X X dðx j ; x Þ ¼ dð˜vi j ; a˜ i Þ ¼
matrix and determine the fuzzy weight of each attribute as Table 8. Using Eq. (15) and Table 8, the fuzzy decision matrix can be transformed into the normalized fuzzy decision matrix as Table 9. Using Eq. (16) and Table 9, the normalized fuzzy decision matrix can be transformed into the weighted normalized fuzzy decision matrix as Table 10. The fuzzy positive ideal solution x+ and the fuzzy negative ideal solution x can be constructed, whose weighted ˜þ ˜þ normalized fuzzy vectors are a˜ þ ¼ ð˜aþ 1 ;a 2 ;...;a 5 Þ and þ ˜ ˜ ˜ a˜ ¼ ð˜a ; a ; . . . ; a Þ, where a ¼ ð1; 1; 1; 1Þ ¼ 1 and i 1 2 5 a˜ ¼ ð0; 0; 0; 0Þ ¼ 0. Using Eqs. (17)–(23), (28) and (31), i the decision results are obtained as Table 11.
i¼1
(33) and CCðx j Þ ¼
dðx j ; x Þ dðx j ; xþ Þ þ dðx j ; x Þ
(34)
where v˜ i j is a normalized triangular fuzzy number for each alternative xj with respect to attribute fi, v˜ i j ¼ ðs i j ; yi j ; ti j Þ; ˜ a˜ þ i ¼ ð1; 1; 1Þ ¼ 1 and a i ¼ ð0; 0; 0Þ ¼ 0 are triangular fuzzy numbers. The compromise solution obtained by the compromise ratio method and the TOPSIS method is the same alternative x2. Ranking results by these two methods are almost the
Table 8 The fuzzy decision matrix and fuzzy weights of three candidates
f1 f2 f3 f4 f5
x1
x2
x3
Weight
(5.7, 7.7, 7.7, 9.3) (5, 7, 7, 9) (5.7, 7.7, 7.7, 9) (8.3, 9.7, 9.7, 10) (3, 5, 5, 7)
(6.3, 8.3, 8.3, 9.7) (9, 10, 10, 10) (8.3, 9.7, 9.7, 10) (9, 10, 10, 10) (7, 9, 9, 10)
(6.3, 8, 8, 9) (7, 9, 9, 10) (7, 9, 9, 10) (7, 9, 9, 10) (6.3, 8.3, 8.3, 9.7)
(0.7, 0.9, 0.9, 1) (0.9, 1, 1, 1) (0.77, 0.93, 0.93, 1) (0.9, 1, 1, 1) (0.43, 0.63, 0.63, 0.83)
816
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
Table 11 Decision results obtained by the TOPSIS method and the compromise ratio method Parameter p
TOPSIS Chen [4]
Eqs. (17)–(19)
Eqs. (20)–(22)
CRM p=1
p=2
p ! +1
Candidates
d(xj, x) d(xj, x+) CC(xj) D(xj, x) D(xj, x+) C*(xj) ¯ j ; x Þ Dðx ¯ Dðx j ; xþ Þ C¯ ðx j Þ Dp(xj, Dp(xj, jp(xj) Dp(xj, Dp(xj, jp(xj) Dp(xj, Dp(xj, jp(xj)
x) x+) x) x+) x) x+)
Ranking order
x1
x2
x3
3.45 2.10 0.62 3.74 1.41 0.73 1.70 0.71 0.71
4.13 1.24 0.77 4.66 0.49 0.90 2.09 0.25 0.89
3.85 1.59 0.71 4.30 0.85 0.83 1.93 0.39 0.83
x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1
3.28 1.31 0 1.70 0.71 0 0.65 0.33 0.71(1 e)
4.65 0.35 1 2.09 0.25 1 0.67 0.09 1
4.13 0.72 0.62–0.01e 1.93 0.39 0.59+0.11e 0.6 0.12 0.88e
x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x1 x3 x2 x3 x1 e < 71/159 e = 71/159 e > 71/159
same except for the case p ! +1 in the compromise ratio method. 6. Short conclusions In general, multi-attribute decision making problems adhere to uncertain and imprecise data, and fuzzy set theory is adequate to deal with it. In this paper, using linguistic variables we develop the compromise ratio method for solving fuzzy multi-attribute group decision making, which is based on an aggregating function representing closeness to the ideal solution and being far away from the negative-ideal solution simultaneously, whereas the TOPSIS method is based on an aggregating function representing ‘‘closeness to the ideal’’ only. The compromise ratio method introduces the ranking index to reflect some balance between the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. The relative importance of the distances from the ideal solution and the negative-ideal solution, which has been considered in the compromise ratio method is a major concern in real life decision making. In contrast, the basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution, but such chosen alternative is not always the closest to the ideal solution (see the example in Section 2). The TOPSIS method introduces two reference points, but it does not consider the relative importance of the distances from these points. The compromise ratio method may use different distance measurements Lp-metric for real/fuzzy numbers, whereas the TOPSIS method uses only L2-metric or Euclidean
x2 x1 x3 x2 x3 x1 x2 x3 x1
distance. In fact, the fuzzy extension of the TOPSIS method (i.e., Eqs. (32)–(34)) is a simple weighted average method with equal weights. Under group decision-making process, it is not difficult to use other aggregation functions to pool the fuzzy ratings of decision makers in the proposed method. Although the proposed method presented in this paper is illustrated by a personnel selection problem, however, it can also be applied to problems such as information project selection, combat plan selection, material selection and many other areas of management decision problems. Obviously, how to determine membership functions of fuzzy numbers is a key and very difficult problem when the method proposed in this paper is applied to real life decision making problems under fuzzy environment. Recently, some methods were proposed [1,3,11,22,28,30–33]. A more effective method will be investigated on determining (learning or extracting) membership functions from the decision makers in the near future. Acknowledgments The author would like to thank the valuable reviews and also appreciate the constructive suggestions from the anonymous referees and Prof. Rajkumar Roy, the Editor in Chief for Applied Soft Computing Journal. This research was Sponsored by the Natural Science Foundation of China (No. 70571086), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No. 2005-546) and China Postdoctoral Science Foundation (No. 2005-38).
D.-F. Li / Applied Soft Computing 7 (2007) 807–817
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