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Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting
Knowledge-Based Systems 21 (2008) 833–836
Contents lists available at ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting Gui-Wu Wei Chongqing University of Arts and Sciences, Yongchuan City, Chongqing City, 402160 China
a r t i c l e
i n f o
Article history: Received 20 October 2007 Accepted 28 March 2008 Available online 4 April 2008 Keywords: Multiple attribute decision making Intuitionistic fuzzy number Intuitionistic fuzzy weighted averaging (IFWA) operator Weight information
a b s t r a c t With respect to multiple attribute decision making problems with intuitionistic fuzzy information, some operational laws of intuitionistic fuzzy numbers, score function and accuracy function of intuitionistic fuzzy numbers are introduced. An optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. For the special situations where the information about attribute weights is completely unknown, we establish another optimization model. By solving this model, we get a simple and exact formula, which can be used to determine the attribute weights. We utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Atanassov [1,2] introduced the concept of intuitionistic fuzzy set(IFS), which is a generalization of the concept of fuzzy set [3]. The intuitionistic fuzzy set has received more and more attention [4–10,18] since its appearance. Gau and Buehrer [4] introduced the concept of vague set. But Bustince and Burillo [5] showed that vague sets are intuitionistic fuzzy sets. In [6], Xu developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. In [7], Xu developed some arithmetic aggregation operators, such as the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid aggregation (IFHA) operator. In the process of MADM with intuitionistic fuzzy information, sometimes, the attribute values take the form of intuitionistic fuzzy numbers, and the information about attribute weights is incompletely known or completely unknown because of time pressure, lack of knowledge or data, and the expert’s limited expertise about the problem domain. All of the above methods, however, will be unsuitable for dealing with such situations. Therefore, it is necessary to pay attention to this issue. In [8], Xu investigated the intuitionistic fuzzy MADM with the information about attribute weights is incompletely known or completely unknown, a method E-mail address: [email protected] 0950-7051/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2008.03.038
based on the ideal solution was proposed. The aim of this paper is to develop another method, based on the maximizing deviation method, to overcome this limitation. The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to intuitionistic fuzzy sets. In Section 3 we introduce the MADM problem with intuitionistic fuzzy information, in which the information about attribute weights is incompletely known, and the attribute values take the form of intuitionistic fuzzy numbers. To determine the attribute weights, an optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. For the special situations where the information about attribute weights is completely unknown, we establish another optimization model. By solving this model, we get a simple and exact formula, which can be used to determine the attribute weights. We utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. In Section 4, an illustrative example is pointed out. In Section 5 we conclude the paper and give some remarks. 2. Preliminaries In the following, we introduce some basic concepts related to intuitionistic fuzzy sets. Definition 1. Let X be a universe of discourse, then a fuzzy set is defined as: A ¼ fhx; lA ðxÞijx 2 Xg
ð1Þ
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G.-W. Wei / Knowledge-Based Systems 21 (2008) 833–836
which is characterized by a membership function lA : X ? [0,1], where lA(x) denotes the degree of membership of the element x to the set A [3]. Atanassov extended the fuzzy set to the IFS, shown as follows: Definition 2. An IFS A in X is given by A ¼ fhx; lA ðxÞ; mA ðxÞijx 2 Xg
ð2Þ
where lA : X ? [0,1] and mA : X ? [0,1], with the condition 0 6 lA ðxÞ þ mA ðxÞ 6 1;
1 ðjl l2 j þ jm1 m2 jÞ 2 1
ð7Þ
3. Maximizing deviation method for intuitionistic fuzzy decision making problems with incomplete weight information The following assumptions or notations are used to represent the intuitionistic fuzzy MADM problems with incomplete weight information:
8x 2 X
The numbers lA(x) and mA(x) represent, respectively, the membership degree and non-membership degree of the element x to the set A [1,2]. Definition 3. For each IFS A in X, if pA ðxÞ ¼ 1 lA ðxÞ mA ðxÞ;
8x 2 X
ð3Þ
Then pA(x) is called the degree of indeterminacy of x to A [1,2]. ~ ¼ ðl; mÞ be an intuitionistic fuzzy number, a Definition 4. Let a score function S of an intuitionistic fuzzy value can be represented as follows [9]: ~Þ ¼ l m; Sða
~1 ; a ~2 Þ ¼ dða
~Þ 2 ½1; 1 Sða
ð4Þ
~ ¼ ðl; mÞ be an intuitionistic fuzzy number, a Definition 5. Let a accuracy function H of an intuitionistic fuzzy value can be represented as follows [10]: ~Þ ¼ l þ m; Hða ~Þ 2 ½0; 1 Hða ð5Þ to evaluate the degree of accuracy of the intuitionistic fuzzy value ~ ¼ ðl; mÞ, where Hða ~Þ 2 ½0; 1. The larger the value of Hða ~Þ, the more a ~. the degree of accuracy of the intuitionistic fuzzy value a As presented above, the score function S and the accuracy function H are, respectively, defined as the difference and the sum of ~A ðxÞ and the non-membership function the membership function l ~mA ðxÞ. Xu [6] showed that the relation between the score function S and the accuracy function H is similar to the relation between mean and variance in statistics. Based on the score function S and the accuracy function H, in the following, Xu give an order relation between two intuitionistic fuzzy values, which is defined as follows: ~2 ¼ ðl2 ; m2 Þ be two intuitionistic ~1 ¼ ðl1 ; m1 Þ and a Definition 6. Let a ~2 Þ ¼ l2 m2 be the scores of a ~ ~1 Þ ¼ l1 m1 and sða fuzzy values, sða ~ respectively, and let Hða ~2 Þ ¼ l2 þ m2 be ~1 Þ ¼ l1 þ m1 and Hða and b, ~ respectively, then if Sða ~ ~ and b, ~Þ < SðbÞ, the accuracy degrees of a ~ ~ ~ ~ ~ ~ then a is smaller than b, denoted by a < b; if SðaÞ ¼ SðbÞ, then ~ then a ~ represent the same information, ~Þ ¼ HðbÞ, ~ and b (1) if Hða ~ ~ denoted by a ¼ b; ~ a ~ denoted by a ~ [6]. ~Þ < HðbÞ, ~ is smaller than b, ~
Definition 7. Let aj = (lj,mj)(j = 1,2, . . ., n) be a collection of intuitionistic fuzzy values, and let IFWA: Qn ? Q , if ! n n n X Y Y x ~1 ; a ~2 ; . . . ; a ~n Þ ¼ ~j ¼ 1 ð6Þ xj a ð1 lj Þxj ; mj j IFWAx ða j¼1
j¼1
j¼1
T
~j ðj ¼ 1; 2; . . . ; nÞ, where x = (x1,x2, . . ., xn) be the weight vector of a P and xj > 0, nj¼1 xj ¼ 1, then IFWA is called the intuitionistic fuzzy weighted averaging (IFWA) operator [7]. ~1 ¼ ðl1 ; m1 Þ and a ~2 ¼ ðl2 ; m2 Þ be two intuitionistic Definition 8. Let a fuzzy numbers, then the normalized Hamming distance between ~2 ¼ ðl2 ; m2 Þ is defined as follows [8]: ~1 ¼ ðl1 ; m1 Þ and a a
(1) The alternatives are known. Let A = {A1,A2, . . ., Am} be a discrete set of alternatives; (2) The attributes are known. Let G = {G1,G2, . . ., Gn} be a set of attributes; (3) The information about attribute weights is incompletely known. Let w = (w1,w2, . . ., wn) 2 Hbe the weight vector of P attributes, wherewj P 0,j = 1,2, . . ., n, nj¼1 wj ¼ 1, H is a set of the known weight information, which can be constructed by the following forms [11–14], for i – j: Form 1. A weak ranking: wi P wj; Form 2. A strict ranking: wi wj P ai, ai > 0; Form 3. A ranking of differences:wi wj P wk wl, for j – k – l; Form 4. A ranking with multiples: wi P biwj, 0 6 bi 6 1; Form 5. An interval form: ai 6 wi 6 ai + ei, 0 6 ai < ai + ei 6 1. e ¼ ð~r ij Þ Suppose that R mn ¼ ðlij ; mij Þmn is the intuitionistic fuzzy decision matrix, where lij indicates the degree that the alternative Ai satisfies the attribute Gj given by the decision maker, mij indicates the degree that the alternative Ai does not satisfy the attribute Gj given by the decision maker, lij [0,1], mij [0,1],lij + mij 6 1, i = 1,2, . . ., m, j = 1,2, . . ., n. e ¼ ð~r ij Þ Definition 9. Let R mn ¼ ðlij ; mij Þmn be an intuitionistic fuzzy decision matrix, ~r i ¼ ð~ri1 ; ~r i2 ; . . . ; ~r in Þ be the vector of attribute values corresponding to the alternative Ai, i = 1,2, . . ., m, then we call ~ri ¼ ðlij ; mij Þ ¼ IFWAw ð~ri1 ; ~r i2 ; . . . ; ~rin Þ ! n n Y Y wj wj ¼ 1 ð1 lij Þ ; mij ; i ¼ 1; 2; . . . ; m j¼1
ð8Þ
j¼1
the overall value of the alternative Ai, where w = (w1,w2 , . . ., wn)T is the weight vector of attributes. In the situation where the information about attribute weights is completely known, i.e., each attribute weight can be provided by the expert with crisp numerical value, we can weight each attribute value and aggregate all the weighted attribute values corresponding to each alternative into an overall one by using Eq. (8). Based on the overall attribute values ~ri of the alternatives Ai(i = 1,2, . . ., m), we can rank all these alternatives and then select the most desirable one(s). The greater ~ri , the better the alternative Ai will be. The maximizing deviation method is proposed by Wang [15] to deal with MADM problems with numerical information. For a MADM problem, we need to compare the collective preference values to rank the alternatives, the larger the ranking value ~r i is, the better the corresponding alternative Ai is. If the performance values of each alternative have little differences under an attribute, it shows that such an attribute plays a small important role in the priority procedure. Contrariwise, if some attribute makes the performance values among all the alternatives have obvious differences, such an attribute plays an important role in choosing the best alternative. So to the view of sorting the alternatives, if one attribute has similar attribute values across alternatives, it should be assigned a small weight; otherwise, the attribute which makes larger deviations should be evaluated a bigger weight, in spite of
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the degree of its own importance. Especially, if all available alternatives score about equally with respect to a given attribute, then such an attribute will be judged unimportant by most decision makers. In other word, such an attribute should be assigned a very small weight. Wang [15] suggests that zero weight should be assigned to the corresponding attribute. The deviation method is selected here to compute the differences of the performance values of each alternative. For the attribute Gj 2 G,the deviation of alternative Ai to all the other alternatives can be defined as follows: Dij ðwÞ ¼
m X
tic fuzzy information. The method involves the following steps: Step 1.
Step 2. dð~rij ; ~rkj Þwj ;
i ¼ 1; 2; . . . ; m;
j ¼ 1; 2; . . . ; n
k¼1
P Pm Pm ~ ~ Let Dj ðwÞ ¼ m j ¼ 1; 2; . . . ; n i¼1 Dij ðwÞ ¼ i¼1 k¼1 dðr ij ; r kj Þwj ; Then Dj(w) represent the deviation value of all alternatives to other alternatives for the attribute Gj 2 G. Based on the above analysis, we have to choose the weight vector w to maximize all deviation values for all the attributes. To do so, we can construct a non-linear programming model as follows: 8 n P m n P m P m P P > > ~ ~ > < max DðwÞ ¼ j¼1 i¼1 Dij ðwÞ ¼ j¼1 i¼1 k¼1 dðr ij ; r kj Þwj ðM 1Þ n P > > > wj ¼ 1; wj P 0; j ¼ 1; 2; . . . ; n : Subject to w 2 H;
Step 3.
Step 4.
j¼1
1 ðjlij 2
where dð~r ij ; ~rkj Þ ¼ lkj j þ jmij mkj jÞ. By solving the model (M 1), we get the optimal solution w = (w1,w2, . . ., wn), which can be used as the weight vector of attributes. If the information about attribute weights is completely unknown, we can establish another programming model: 8 m n P m P m P P > > Di ðwÞ ¼ 12 wj ðjlij lkj j þ jmij mkj jÞ > < max DðwÞ ¼ i¼1
ðM 2Þ n P > > > w2j ¼ 1; wj P 0; : s:t:
Step 6.
4. Illustrative example
j¼1 i¼1 k¼1
j ¼ 1; 2; . . . ; n
j¼1
To solve this model, we construct the Lagrange function: ! n X m X m n 1X k X 2 wj ðjlij lkj j þ jmij mkj jÞ þ w 1 Lðw; kÞ ¼ 2 j¼1 i¼1 k¼1 4 j¼1 j ð9Þ where k is the Lagrange multiplier. Differentiating Eq. (9) with respect to wj(j = 1,2, . . ., n) and k, and setting these partial derivatives equal to zero, the following set of equations is obtained: 8 m P m P > oL > ¼ ðjlij lkj j þ jmij mkj jÞ þ kwj ¼ 0 > ow < j
Step 5.
e ¼ ð~r ij Þ Let R mn be an intuitionistic fuzzy decision matrix, where ~rij ¼ ðlij ; mij Þ, which is an attribute value, given by an expert, for the alternativeAi 2 A with respect to the attribute Gj 2 G, w = (w1,w2, . . ., wn) be the weight vector of attributes, wherewj 2 [0,1],j = 1,2, . . ., n, H is a set of the known weight information, which can be constructed by the forms 1–5. If the information about the attribute weights is partly known, then we solve the model (M 1) to obtain the attribute weights. If the information about the attribute weights is completely unknown, then we can obtain the attribute weights by using Eq. (11). Utilize the weight vector w = (w1,w2, . . ., wn) and by Eq. (8), we obtain the overall values ~ri of the alternative Ai(i = 1,2, . . ., m). Calculate the scores Sð~r i Þ of the overall intuitionistic fuzzy preference value ~r i ði ¼ 1; 2; . . . ; mÞ to rank all the alternatives Ai(i= 1,2, . . ., m) and then to select the best one(s) (if there is no difference between two scores Sð~ri Þ and Sð~r j Þ, then we need to calculate the accuracy degrees Hð~r i Þ and Hð~r j Þ of the overall intuitionistic fuzzy preference value ~r i and ~rj , respectively, and then rank the alternatives Ai and Aj in accordance with the accuracy degrees Hð~r i Þ and Hð~rj Þ. Rank all the alternatives Ai(i = 1,2, . . ., m) and select the best one(s) in accordance with Sð~r i Þ and Hð~ri Þði ¼ 1; 2; . . . ; mÞ. End.
i¼1 k¼1
n P > oL > > w2j 1 ¼ 0 : ok ¼ j¼1
By solving Eq. (10), we get a simple and exact formula for determining the attribute weights as follows: Pm Pm i¼1 k¼1 ðjlij lkj j þ jmij mkj jÞ wj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ 2 Pn Pm Pm j¼1 i¼1 k¼1 ðjlij lkj j þ jmij mkj jÞ By normalizing wj ðj ¼ 1; 2; . . . ; nÞ be a unit, we have Pm Pm ðjlij lkj j þ jmij mkj jÞ Pm k¼1 Pm wj ¼ Pn i¼1 j¼1 i¼1 k¼1 ðjlij lkj j þ jmij mkj jÞ
ð11Þ
Based on the above models, we develop a practical method for solving the MADM problems, in which the information about attribute weights is incompletely known or completely unknown, and the attribute values take the form of intuitionis-
Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from [16]). There is a panel with five possible alternatives to invest the money: rA1 is a car company; sA2 is a food company; tA3 is a computer company; uA4 is a arms company; vA5 is a TV company. The investment company must take a decision according to the following four attributes: rG1 is the risk analysis; sG2 is the growth analysis; tG3 is the social-political impact analysis; uG4 is the environmental impact analysis. The five possible alternatives Ai(i = 1,2,3,4,5) are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above four attributes, as listed in the following matrix. 3 2 ð0:5; 0:4Þ ð0:6; 0:3Þ ð0:3; 0:6Þ ð0:2; 0:7Þ 7 6 6 ð0:7; 0:3Þ ð0:7; 0:2Þ ð0:7; 0:2Þ ð0:4; 0:5Þ 7 7 6 e 6 R ¼ 6 ð0:6; 0:4Þ ð0:5; 0:4Þ ð0:5; 0:3Þ ð0:6; 0:3Þ 7 7 7 6 4 ð0:8; 0:1Þ ð0:6; 0:3Þ ð0:3; 0:4Þ ð0:2; 0:6Þ 5 ð0:6; 0:2Þ ð0:4; 0:3Þ ð0:7; 0:1Þ ð0:5; 0:3Þ Then, we utilize the approach developed to get the most desirable alternative(s). Case 1:
The information about the attribute weights is partly known and the known weight information is given as follows: ( H¼
0:15 6 w1 6 0:2; 0:16 6 w2 6 0:18; 0:30 6 w3 6 0:35; 0:30 6 w4 6 0:45; ) 4 P wj P 0; j ¼ 1; 2; 3; 4; wj ¼ 1 j¼1
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Step 1.
G.-W. Wei / Knowledge-Based Systems 21 (2008) 833–836
Utilize the model (M 1) to establish the following single-objective programming model:
5. Conclusion
In this paper, we have investigated the problem of MADM with incompletely known information on attribute weights to which the attribute values are given in terms of intuitionistic fuzzy numbers. To determine the attribute weights, an optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. For the special situations where the information about attribute weights is completely unknown, we establish another optimization model. By solving this model, we get a simple and exact formula, which can be used to determine the attribute weights. We utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. Finally, an illustrative example is given. In the future, we shall continue working in the application of the intuitionistic fuzzy multiple attribute decision-making to other domains.
max DðwÞ ¼ 1:7w1 þ 1:4w2 þ 2:7w3 þ 3:1w4 s:t: w 2 H
Solving this model, we get the weight vector of attributes: w ¼ ð0:20 0:18 0:32 0:30ÞT Step 2.
Utilize the weight vector w = (w1,w2, . . ., wn) and by Eq. (8), we obtain the overall values ~ri of the alternatives Ai(i = 1,2, . . ., m). ~r 1 ¼ ð0:3841; 0:5115Þ; ~r 3 ¼ ð0:5528; 0:3347Þ; ~r 5 ¼ ð0:5804; 0:1946Þ
Step 3.
~r 2 ¼ ð0:6307; 0:2855Þ ~r 4 ¼ ð0:4872; 0:3251Þ
Calculate the scores Sð~r i Þof the overall intuitionistic fuzzy preference values ~ri ði ¼ 1; 2; . . . ; mÞ Sð~r 1 Þ ¼ 0:1274; Sð~r 2 Þ ¼ 0:3451 Sð~r 3 Þ ¼ 0:2181; Sð~r4 Þ ¼ 0:1621 Sð~r 5 Þ ¼ 0:3858
Step 4.
Rank all the alternatives Ai(i = 1,2,3,4,5) in accordance with the scoresSð~ri Þ ði ¼ 1; 2; . . . ; 5Þof the overall intuitionistic fuzzy preference values ~ri ði ¼ 1; 2; . . . ; mÞ: A5 A2 A3 A4 A1, and thus the most desirable alternative is A5.
Case 2:
If the information about the attribute weights is completely unknown, we utilize another approach developed to get the most desirable alternative(s).
Step 1.
Utilize the Eq. (11) to get the weight vector of attributes: w ¼ ð0:1910 0:1573 0:3034 0:3483ÞT
Step 2.
Utilize the weight vector w = (w1,w2, . . ., wn) and by Eq. (8), we obtain the overall values ~ri of the alternatives Ai(i = 1,2, . . ., m). ~r 1 ¼ ð0:3703; 0:5254Þ; ~r 3 ¼ ð0:5567; 0:3316Þ; ~r 5 ¼ ð0:5777; 0:1989Þ
Step 3.
~r2 ¼ ð0:6181; 0:2973Þ ~r 4 ¼ ð0:4714; 0:3379Þ
Calculate the scores Sð~ri Þði ¼ 1; 2; . . . ; mÞof the overall intuitionistic fuzzy preference values ~r i ði ¼ 1; 2; . . . ; mÞ Sð~r 1 Þ ¼ 0:1551; Sð~r 2 Þ ¼ 0:3207 Sð~r 3 Þ ¼ 0:2251; Sð~r4 Þ ¼ 0:1335 Sð~r 5 Þ ¼ 0:3788
Step 4.
Rank all the alternatives Ai(i = 1,2,3,4,5) in accordance with the scoresSð~ri Þ: A5 A2 A3 A4 A1, and thus the most desirable alternative is A5.
References [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87– 96. [2] K. Atasov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 37–46. [3] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–356. [4] W.L. Gau, D.J. Buehrer, Vague sets, IEEE Transactions on Systems, Man and Cybernetics 23 (2) (1993) 610–614. [5] H. Bustine, P. Burillo, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems 79 (1996) 403–405. [6] Z.S. Xu, R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General System 35 (2006) 417–433. [7] Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transations on Fuzzy Systems 15 (2007) 1179–1187. [8] Z.S. Xu, Models for multiple attribute decision-making with intuitionistic fuzzy information, International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems 15 (3) (2007) 285–297. [9] S.M. Chen, J.M. Tan, Handling multicriteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems 67 (1994) 163–172. [10] D.H. Hong, C.H. Choi, Multicriteria fuzzy problems based on vague set theory, Fuzzy Sets and Systems 114 (2000) 103–113. [11] K.S. Park, S.H. Kim, Tools for interactive multi-attribute decision making with incompletely identified information, European Journal of Operational Research 98 (1997) 111–123. [12] S.H. Kim, S.H. Choi, J.K. Kim, An interactive procedure for multiple attribute group decision making with incomplete information: range-based approach, European Journal of Operational Research 118 (1999) 139–152. [13] S.H. Kim, B.S. Ahn, Interactive group decision making procedure under incomplete information, European Journal of Operational Research 116 (1999) 498–507. [14] K.S. Park, Mathematical programming models for charactering dominance and potential optimality when multicriteria alternative values and weights are simultaneously incomplete, IEEE Transactions on Systems, Man, and Cybernetics – Part A, Systems and Humans 34 (2004) 601–614. [15] Y.M. Wang, Using the method of maximizing deviations to make decision for multi-indices, System Engineering and Electronics 7 (1998) 24–26. 31. [16] F. Herrera, E. Herrera-Viedma, Linguistic decision analysis: steps for solving decision problems under linguistic information, Fuzzy Sets and Systems 115 (2000) 67–82. [18] D.F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets, Journal of Computer and systems Sciences 70 (2005) 73–85.
Contents lists available at ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting Gui-Wu Wei Chongqing University of Arts and Sciences, Yongchuan City, Chongqing City, 402160 China
a r t i c l e
i n f o
Article history: Received 20 October 2007 Accepted 28 March 2008 Available online 4 April 2008 Keywords: Multiple attribute decision making Intuitionistic fuzzy number Intuitionistic fuzzy weighted averaging (IFWA) operator Weight information
a b s t r a c t With respect to multiple attribute decision making problems with intuitionistic fuzzy information, some operational laws of intuitionistic fuzzy numbers, score function and accuracy function of intuitionistic fuzzy numbers are introduced. An optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. For the special situations where the information about attribute weights is completely unknown, we establish another optimization model. By solving this model, we get a simple and exact formula, which can be used to determine the attribute weights. We utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Atanassov [1,2] introduced the concept of intuitionistic fuzzy set(IFS), which is a generalization of the concept of fuzzy set [3]. The intuitionistic fuzzy set has received more and more attention [4–10,18] since its appearance. Gau and Buehrer [4] introduced the concept of vague set. But Bustince and Burillo [5] showed that vague sets are intuitionistic fuzzy sets. In [6], Xu developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. In [7], Xu developed some arithmetic aggregation operators, such as the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid aggregation (IFHA) operator. In the process of MADM with intuitionistic fuzzy information, sometimes, the attribute values take the form of intuitionistic fuzzy numbers, and the information about attribute weights is incompletely known or completely unknown because of time pressure, lack of knowledge or data, and the expert’s limited expertise about the problem domain. All of the above methods, however, will be unsuitable for dealing with such situations. Therefore, it is necessary to pay attention to this issue. In [8], Xu investigated the intuitionistic fuzzy MADM with the information about attribute weights is incompletely known or completely unknown, a method E-mail address: [email protected] 0950-7051/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2008.03.038
based on the ideal solution was proposed. The aim of this paper is to develop another method, based on the maximizing deviation method, to overcome this limitation. The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to intuitionistic fuzzy sets. In Section 3 we introduce the MADM problem with intuitionistic fuzzy information, in which the information about attribute weights is incompletely known, and the attribute values take the form of intuitionistic fuzzy numbers. To determine the attribute weights, an optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. For the special situations where the information about attribute weights is completely unknown, we establish another optimization model. By solving this model, we get a simple and exact formula, which can be used to determine the attribute weights. We utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. In Section 4, an illustrative example is pointed out. In Section 5 we conclude the paper and give some remarks. 2. Preliminaries In the following, we introduce some basic concepts related to intuitionistic fuzzy sets. Definition 1. Let X be a universe of discourse, then a fuzzy set is defined as: A ¼ fhx; lA ðxÞijx 2 Xg
ð1Þ
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which is characterized by a membership function lA : X ? [0,1], where lA(x) denotes the degree of membership of the element x to the set A [3]. Atanassov extended the fuzzy set to the IFS, shown as follows: Definition 2. An IFS A in X is given by A ¼ fhx; lA ðxÞ; mA ðxÞijx 2 Xg
ð2Þ
where lA : X ? [0,1] and mA : X ? [0,1], with the condition 0 6 lA ðxÞ þ mA ðxÞ 6 1;
1 ðjl l2 j þ jm1 m2 jÞ 2 1
ð7Þ
3. Maximizing deviation method for intuitionistic fuzzy decision making problems with incomplete weight information The following assumptions or notations are used to represent the intuitionistic fuzzy MADM problems with incomplete weight information:
8x 2 X
The numbers lA(x) and mA(x) represent, respectively, the membership degree and non-membership degree of the element x to the set A [1,2]. Definition 3. For each IFS A in X, if pA ðxÞ ¼ 1 lA ðxÞ mA ðxÞ;
8x 2 X
ð3Þ
Then pA(x) is called the degree of indeterminacy of x to A [1,2]. ~ ¼ ðl; mÞ be an intuitionistic fuzzy number, a Definition 4. Let a score function S of an intuitionistic fuzzy value can be represented as follows [9]: ~Þ ¼ l m; Sða
~1 ; a ~2 Þ ¼ dða
~Þ 2 ½1; 1 Sða
ð4Þ
~ ¼ ðl; mÞ be an intuitionistic fuzzy number, a Definition 5. Let a accuracy function H of an intuitionistic fuzzy value can be represented as follows [10]: ~Þ ¼ l þ m; Hða ~Þ 2 ½0; 1 Hða ð5Þ to evaluate the degree of accuracy of the intuitionistic fuzzy value ~ ¼ ðl; mÞ, where Hða ~Þ 2 ½0; 1. The larger the value of Hða ~Þ, the more a ~. the degree of accuracy of the intuitionistic fuzzy value a As presented above, the score function S and the accuracy function H are, respectively, defined as the difference and the sum of ~A ðxÞ and the non-membership function the membership function l ~mA ðxÞ. Xu [6] showed that the relation between the score function S and the accuracy function H is similar to the relation between mean and variance in statistics. Based on the score function S and the accuracy function H, in the following, Xu give an order relation between two intuitionistic fuzzy values, which is defined as follows: ~2 ¼ ðl2 ; m2 Þ be two intuitionistic ~1 ¼ ðl1 ; m1 Þ and a Definition 6. Let a ~2 Þ ¼ l2 m2 be the scores of a ~ ~1 Þ ¼ l1 m1 and sða fuzzy values, sða ~ respectively, and let Hða ~2 Þ ¼ l2 þ m2 be ~1 Þ ¼ l1 þ m1 and Hða and b, ~ respectively, then if Sða ~ ~ and b, ~Þ < SðbÞ, the accuracy degrees of a ~ ~ ~ ~ ~ ~ then a is smaller than b, denoted by a < b; if SðaÞ ¼ SðbÞ, then ~ then a ~ represent the same information, ~Þ ¼ HðbÞ, ~ and b (1) if Hða ~ ~ denoted by a ¼ b; ~ a ~ denoted by a ~ [6]. ~Þ < HðbÞ, ~ is smaller than b, ~
Definition 7. Let aj = (lj,mj)(j = 1,2, . . ., n) be a collection of intuitionistic fuzzy values, and let IFWA: Qn ? Q , if ! n n n X Y Y x ~1 ; a ~2 ; . . . ; a ~n Þ ¼ ~j ¼ 1 ð6Þ xj a ð1 lj Þxj ; mj j IFWAx ða j¼1
j¼1
j¼1
T
~j ðj ¼ 1; 2; . . . ; nÞ, where x = (x1,x2, . . ., xn) be the weight vector of a P and xj > 0, nj¼1 xj ¼ 1, then IFWA is called the intuitionistic fuzzy weighted averaging (IFWA) operator [7]. ~1 ¼ ðl1 ; m1 Þ and a ~2 ¼ ðl2 ; m2 Þ be two intuitionistic Definition 8. Let a fuzzy numbers, then the normalized Hamming distance between ~2 ¼ ðl2 ; m2 Þ is defined as follows [8]: ~1 ¼ ðl1 ; m1 Þ and a a
(1) The alternatives are known. Let A = {A1,A2, . . ., Am} be a discrete set of alternatives; (2) The attributes are known. Let G = {G1,G2, . . ., Gn} be a set of attributes; (3) The information about attribute weights is incompletely known. Let w = (w1,w2, . . ., wn) 2 Hbe the weight vector of P attributes, wherewj P 0,j = 1,2, . . ., n, nj¼1 wj ¼ 1, H is a set of the known weight information, which can be constructed by the following forms [11–14], for i – j: Form 1. A weak ranking: wi P wj; Form 2. A strict ranking: wi wj P ai, ai > 0; Form 3. A ranking of differences:wi wj P wk wl, for j – k – l; Form 4. A ranking with multiples: wi P biwj, 0 6 bi 6 1; Form 5. An interval form: ai 6 wi 6 ai + ei, 0 6 ai < ai + ei 6 1. e ¼ ð~r ij Þ Suppose that R mn ¼ ðlij ; mij Þmn is the intuitionistic fuzzy decision matrix, where lij indicates the degree that the alternative Ai satisfies the attribute Gj given by the decision maker, mij indicates the degree that the alternative Ai does not satisfy the attribute Gj given by the decision maker, lij [0,1], mij [0,1],lij + mij 6 1, i = 1,2, . . ., m, j = 1,2, . . ., n. e ¼ ð~r ij Þ Definition 9. Let R mn ¼ ðlij ; mij Þmn be an intuitionistic fuzzy decision matrix, ~r i ¼ ð~ri1 ; ~r i2 ; . . . ; ~r in Þ be the vector of attribute values corresponding to the alternative Ai, i = 1,2, . . ., m, then we call ~ri ¼ ðlij ; mij Þ ¼ IFWAw ð~ri1 ; ~r i2 ; . . . ; ~rin Þ ! n n Y Y wj wj ¼ 1 ð1 lij Þ ; mij ; i ¼ 1; 2; . . . ; m j¼1
ð8Þ
j¼1
the overall value of the alternative Ai, where w = (w1,w2 , . . ., wn)T is the weight vector of attributes. In the situation where the information about attribute weights is completely known, i.e., each attribute weight can be provided by the expert with crisp numerical value, we can weight each attribute value and aggregate all the weighted attribute values corresponding to each alternative into an overall one by using Eq. (8). Based on the overall attribute values ~ri of the alternatives Ai(i = 1,2, . . ., m), we can rank all these alternatives and then select the most desirable one(s). The greater ~ri , the better the alternative Ai will be. The maximizing deviation method is proposed by Wang [15] to deal with MADM problems with numerical information. For a MADM problem, we need to compare the collective preference values to rank the alternatives, the larger the ranking value ~r i is, the better the corresponding alternative Ai is. If the performance values of each alternative have little differences under an attribute, it shows that such an attribute plays a small important role in the priority procedure. Contrariwise, if some attribute makes the performance values among all the alternatives have obvious differences, such an attribute plays an important role in choosing the best alternative. So to the view of sorting the alternatives, if one attribute has similar attribute values across alternatives, it should be assigned a small weight; otherwise, the attribute which makes larger deviations should be evaluated a bigger weight, in spite of
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the degree of its own importance. Especially, if all available alternatives score about equally with respect to a given attribute, then such an attribute will be judged unimportant by most decision makers. In other word, such an attribute should be assigned a very small weight. Wang [15] suggests that zero weight should be assigned to the corresponding attribute. The deviation method is selected here to compute the differences of the performance values of each alternative. For the attribute Gj 2 G,the deviation of alternative Ai to all the other alternatives can be defined as follows: Dij ðwÞ ¼
m X
tic fuzzy information. The method involves the following steps: Step 1.
Step 2. dð~rij ; ~rkj Þwj ;
i ¼ 1; 2; . . . ; m;
j ¼ 1; 2; . . . ; n
k¼1
P Pm Pm ~ ~ Let Dj ðwÞ ¼ m j ¼ 1; 2; . . . ; n i¼1 Dij ðwÞ ¼ i¼1 k¼1 dðr ij ; r kj Þwj ; Then Dj(w) represent the deviation value of all alternatives to other alternatives for the attribute Gj 2 G. Based on the above analysis, we have to choose the weight vector w to maximize all deviation values for all the attributes. To do so, we can construct a non-linear programming model as follows: 8 n P m n P m P m P P > > ~ ~ > < max DðwÞ ¼ j¼1 i¼1 Dij ðwÞ ¼ j¼1 i¼1 k¼1 dðr ij ; r kj Þwj ðM 1Þ n P > > > wj ¼ 1; wj P 0; j ¼ 1; 2; . . . ; n : Subject to w 2 H;
Step 3.
Step 4.
j¼1
1 ðjlij 2
where dð~r ij ; ~rkj Þ ¼ lkj j þ jmij mkj jÞ. By solving the model (M 1), we get the optimal solution w = (w1,w2, . . ., wn), which can be used as the weight vector of attributes. If the information about attribute weights is completely unknown, we can establish another programming model: 8 m n P m P m P P > > Di ðwÞ ¼ 12 wj ðjlij lkj j þ jmij mkj jÞ > < max DðwÞ ¼ i¼1
ðM 2Þ n P > > > w2j ¼ 1; wj P 0; : s:t:
Step 6.
4. Illustrative example
j¼1 i¼1 k¼1
j ¼ 1; 2; . . . ; n
j¼1
To solve this model, we construct the Lagrange function: ! n X m X m n 1X k X 2 wj ðjlij lkj j þ jmij mkj jÞ þ w 1 Lðw; kÞ ¼ 2 j¼1 i¼1 k¼1 4 j¼1 j ð9Þ where k is the Lagrange multiplier. Differentiating Eq. (9) with respect to wj(j = 1,2, . . ., n) and k, and setting these partial derivatives equal to zero, the following set of equations is obtained: 8 m P m P > oL > ¼ ðjlij lkj j þ jmij mkj jÞ þ kwj ¼ 0 > ow < j
Step 5.
e ¼ ð~r ij Þ Let R mn be an intuitionistic fuzzy decision matrix, where ~rij ¼ ðlij ; mij Þ, which is an attribute value, given by an expert, for the alternativeAi 2 A with respect to the attribute Gj 2 G, w = (w1,w2, . . ., wn) be the weight vector of attributes, wherewj 2 [0,1],j = 1,2, . . ., n, H is a set of the known weight information, which can be constructed by the forms 1–5. If the information about the attribute weights is partly known, then we solve the model (M 1) to obtain the attribute weights. If the information about the attribute weights is completely unknown, then we can obtain the attribute weights by using Eq. (11). Utilize the weight vector w = (w1,w2, . . ., wn) and by Eq. (8), we obtain the overall values ~ri of the alternative Ai(i = 1,2, . . ., m). Calculate the scores Sð~r i Þ of the overall intuitionistic fuzzy preference value ~r i ði ¼ 1; 2; . . . ; mÞ to rank all the alternatives Ai(i= 1,2, . . ., m) and then to select the best one(s) (if there is no difference between two scores Sð~ri Þ and Sð~r j Þ, then we need to calculate the accuracy degrees Hð~r i Þ and Hð~r j Þ of the overall intuitionistic fuzzy preference value ~r i and ~rj , respectively, and then rank the alternatives Ai and Aj in accordance with the accuracy degrees Hð~r i Þ and Hð~rj Þ. Rank all the alternatives Ai(i = 1,2, . . ., m) and select the best one(s) in accordance with Sð~r i Þ and Hð~ri Þði ¼ 1; 2; . . . ; mÞ. End.
i¼1 k¼1
n P > oL > > w2j 1 ¼ 0 : ok ¼ j¼1
By solving Eq. (10), we get a simple and exact formula for determining the attribute weights as follows: Pm Pm i¼1 k¼1 ðjlij lkj j þ jmij mkj jÞ wj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ 2 Pn Pm Pm j¼1 i¼1 k¼1 ðjlij lkj j þ jmij mkj jÞ By normalizing wj ðj ¼ 1; 2; . . . ; nÞ be a unit, we have Pm Pm ðjlij lkj j þ jmij mkj jÞ Pm k¼1 Pm wj ¼ Pn i¼1 j¼1 i¼1 k¼1 ðjlij lkj j þ jmij mkj jÞ
ð11Þ
Based on the above models, we develop a practical method for solving the MADM problems, in which the information about attribute weights is incompletely known or completely unknown, and the attribute values take the form of intuitionis-
Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from [16]). There is a panel with five possible alternatives to invest the money: rA1 is a car company; sA2 is a food company; tA3 is a computer company; uA4 is a arms company; vA5 is a TV company. The investment company must take a decision according to the following four attributes: rG1 is the risk analysis; sG2 is the growth analysis; tG3 is the social-political impact analysis; uG4 is the environmental impact analysis. The five possible alternatives Ai(i = 1,2,3,4,5) are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above four attributes, as listed in the following matrix. 3 2 ð0:5; 0:4Þ ð0:6; 0:3Þ ð0:3; 0:6Þ ð0:2; 0:7Þ 7 6 6 ð0:7; 0:3Þ ð0:7; 0:2Þ ð0:7; 0:2Þ ð0:4; 0:5Þ 7 7 6 e 6 R ¼ 6 ð0:6; 0:4Þ ð0:5; 0:4Þ ð0:5; 0:3Þ ð0:6; 0:3Þ 7 7 7 6 4 ð0:8; 0:1Þ ð0:6; 0:3Þ ð0:3; 0:4Þ ð0:2; 0:6Þ 5 ð0:6; 0:2Þ ð0:4; 0:3Þ ð0:7; 0:1Þ ð0:5; 0:3Þ Then, we utilize the approach developed to get the most desirable alternative(s). Case 1:
The information about the attribute weights is partly known and the known weight information is given as follows: ( H¼
0:15 6 w1 6 0:2; 0:16 6 w2 6 0:18; 0:30 6 w3 6 0:35; 0:30 6 w4 6 0:45; ) 4 P wj P 0; j ¼ 1; 2; 3; 4; wj ¼ 1 j¼1
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Utilize the model (M 1) to establish the following single-objective programming model:
5. Conclusion
In this paper, we have investigated the problem of MADM with incompletely known information on attribute weights to which the attribute values are given in terms of intuitionistic fuzzy numbers. To determine the attribute weights, an optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. For the special situations where the information about attribute weights is completely unknown, we establish another optimization model. By solving this model, we get a simple and exact formula, which can be used to determine the attribute weights. We utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. Finally, an illustrative example is given. In the future, we shall continue working in the application of the intuitionistic fuzzy multiple attribute decision-making to other domains.
max DðwÞ ¼ 1:7w1 þ 1:4w2 þ 2:7w3 þ 3:1w4 s:t: w 2 H
Solving this model, we get the weight vector of attributes: w ¼ ð0:20 0:18 0:32 0:30ÞT Step 2.
Utilize the weight vector w = (w1,w2, . . ., wn) and by Eq. (8), we obtain the overall values ~ri of the alternatives Ai(i = 1,2, . . ., m). ~r 1 ¼ ð0:3841; 0:5115Þ; ~r 3 ¼ ð0:5528; 0:3347Þ; ~r 5 ¼ ð0:5804; 0:1946Þ
Step 3.
~r 2 ¼ ð0:6307; 0:2855Þ ~r 4 ¼ ð0:4872; 0:3251Þ
Calculate the scores Sð~r i Þof the overall intuitionistic fuzzy preference values ~ri ði ¼ 1; 2; . . . ; mÞ Sð~r 1 Þ ¼ 0:1274; Sð~r 2 Þ ¼ 0:3451 Sð~r 3 Þ ¼ 0:2181; Sð~r4 Þ ¼ 0:1621 Sð~r 5 Þ ¼ 0:3858
Step 4.
Rank all the alternatives Ai(i = 1,2,3,4,5) in accordance with the scoresSð~ri Þ ði ¼ 1; 2; . . . ; 5Þof the overall intuitionistic fuzzy preference values ~ri ði ¼ 1; 2; . . . ; mÞ: A5 A2 A3 A4 A1, and thus the most desirable alternative is A5.
Case 2:
If the information about the attribute weights is completely unknown, we utilize another approach developed to get the most desirable alternative(s).
Step 1.
Utilize the Eq. (11) to get the weight vector of attributes: w ¼ ð0:1910 0:1573 0:3034 0:3483ÞT
Step 2.
Utilize the weight vector w = (w1,w2, . . ., wn) and by Eq. (8), we obtain the overall values ~ri of the alternatives Ai(i = 1,2, . . ., m). ~r 1 ¼ ð0:3703; 0:5254Þ; ~r 3 ¼ ð0:5567; 0:3316Þ; ~r 5 ¼ ð0:5777; 0:1989Þ
Step 3.
~r2 ¼ ð0:6181; 0:2973Þ ~r 4 ¼ ð0:4714; 0:3379Þ
Calculate the scores Sð~ri Þði ¼ 1; 2; . . . ; mÞof the overall intuitionistic fuzzy preference values ~r i ði ¼ 1; 2; . . . ; mÞ Sð~r 1 Þ ¼ 0:1551; Sð~r 2 Þ ¼ 0:3207 Sð~r 3 Þ ¼ 0:2251; Sð~r4 Þ ¼ 0:1335 Sð~r 5 Þ ¼ 0:3788
Step 4.
Rank all the alternatives Ai(i = 1,2,3,4,5) in accordance with the scoresSð~ri Þ: A5 A2 A3 A4 A1, and thus the most desirable alternative is A5.
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