VIEWS: 46 PAGES: 6 CATEGORY: Templates POSTED ON: 6/30/2011 Public Domain
International Journal of Computer Information Systems, Vol. 2, No. 5, 2011 Linear Multi-Objective Model for Solving Fuzzy Cooperative Games A.A.Tharwat O.S.Soliman, I.A.Elkhodary, M.M.Sabry Business Department Operations Research and Decision Support Department Canadian International College Faculty of Computers and Information, Cairo University Cairo, Egypt Cairo, Egypt E-mail: marwa_mostafa22@yahoo.com Abstract: In this paper, a linear multiobjective programming the characteristic value for each of the fuzzy coalitions is model is developed (LMO/FCGames) as a method for solving again a fuzzy quantity which maps the vague expectations cooperative games with fuzzy coalitions and fuzzy of the players in the model. characteristic function, in which the solution is always This paper is organized as follows: In section 2, a Pareto optimal. It is based on the idea of the core where the total worth of a fuzzy coalition will be allocated to the review of some basic definitions for fuzzy numbers is players whose participation rate is larger than zero. In the presented and the definitions of the fuzzy cooperative developed model, players do not need to know precise games are introduced. Section 3, presents the developed information about the payoff value or even to form a crisp linear multiobjective model, its solution methodology and coalition. The fuzzy set theory is then utilized to model the the proposed algorithm. Section 4 gives an applicable ambiguity of participation of each player in the formed example and a comparative study with the results obtained coalitions and representing the characteristic function value by the fuzzy Shapley value are shown in section 5. Finally (payoff) for these coalitions. This developed model is applied some conclusions will be discussed in section 6. to a joint production problem for the purpose of validation and the obtained results are compared with another common method for solving fuzzy games called fuzzy Shapley value. II. RELATED WORK These obtained results carried out the performance of the developed model which were more acceptable than the A. A Review of Fuzzy Sets compared ones. In this section some basic definitions, forms of membership functions and notions for the fuzzy sets are Key Words: Choquet Integral, Fuzzy Cooperative Games, introduced. Fuzzy Sets, Multi Objective, Shapley Value Basic Definitions Definition (1): Let X denotes a universal set. Then a I. INTRODUCTION fuzzy set of X is defined by its membership function: The basic concepts of cooperative game theory and : X [0, 1] their elementary properties were developed by Von Which assigns to each element x X a real number Neumann and Morgenstern [1]. In the game theory in the interval [0, 1], where the value of at x literature there are many concepts to solve cooperative represents the grade of membership of x in . games. The von Neumann stable sets, the core [2], the Definition (2): A fuzzy subset of X can be kernel, the Shapley value [3], the nucleolus, the characterized as a set of ordered pairs of elements x and lexicographical solution [4] are the most common grade and is often written as: concepts. Thereafter a lot of efforts have been made in ={(x, )xX}. developing and analyzing solution concepts of various types of cooperative games. Most of these literature deals Definition (3): The -level set of a fuzzy set is defined with cooperative games in characteristic function form as an ordinary set A for which the degree of its where the characteristic function of a game is a mapping membership function exceeds the level: that assigns to each subset (coalition) of the players‟ set a A = { x }, [ 0 , 1] . real number, called the worth of the coalition or payoff to Definition (4): A real fuzzy number is a convex the coalition. continuous fuzzy subset of the real line R whose However, there are some situations in which some membership function is defined as follows: players do not fully take part in a coalition, but they form 1. is a continuous mapping from R to the closed coalitions with a participation rate. A coalition in which interval [0, 1 ] . some players participate partially is called fuzzy coalition. 2. = 0 for all p (- , P1]. Also, the players can only know imprecise 3. is strictly increasing on [P1, P2]. information regarding the real outcome of the cooperation 4. = 1 for all p [P2, P3]. and the coalition values allocated to players are fuzzy 5. is strictly decreasing on [P3, P4] . numbers, i.e., fuzzy characteristic function. 6. = 0 for all p [P4, ) . Cooperative games with fuzzy coalitions and fuzzy characteristic function are called fuzzy cooperative games. Forms of Membership functions The purpose of this paper is to develop a linear The definition of fuzzy numbers represents multiobjective programming model for cooperative game heterogeneous data forms, including crisp data, fuzzy with fuzzy coalitions and fuzzy characteristic function numbers, interval values and linguistic variables. They are simultaneously, based on the idea of the core where the represented by different membership functions defined on total worth of a fuzzy coalition will be allocated to the their domains. players whose participation rate is larger than zero. Here, May Issue Page 41 of 53 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 2, No. 5, 2011 Once the fuzzy sets are chosen, a membership Empty: A fuzzy set ( ) on X is empty, denoted by , if function for each set should be created. A membership and only if function is a typical curve that converts the numerical = 0 for all x X value of input within a range from 0 to 1, indicating the Support: The support of a fuzzy subset of X denoted by belongingness of the input to a fuzzy set. This step is supp( ) , is the set of points at which is positive , known as „fuzzification‟. Membership function can have i.e , various forms: supp( ) = { x X >0} [1] Triangular, [2] Trapezoidal and [3] Gaussian. B. Fuzzy Cooperative Games [1] Triangular Membership Function Game theory includes not only strategic conflicts but also a possibility of cooperation and coalition forming. An important and simplest type of fuzzy numbers For each coalition, each player tries to maximize his/her in common use is the triangular fuzzy number. Its expected profit and join a coalition which promises the membership function has the form: best individual expectations. Each coalition is assumed to win some total profit that is distributed among its members, and the players tend to make this total payoff as high as possible. These kinds of games are known as coalitional or cooperative games. The classical theory of coalition forming is based on an assumption that the players, at the very beginning of the Where, m is the most promising value, L and R are the left negotiation process, know exactly the expected outcomes. and right spread (the smallest and largest values that x can But usually, realistic cooperative situations are subjected take), as shown in Fig. 1. to unpredictable worth (characteristic function) or payoff of the coalitions, which can be only vaguely incorporated [2] Trapezoidal Membership Function in the model. The trapezoidal membership curve has a flat top Such uncertainty can be modeled by fuzzy sets and and it is just a truncated triangle producing = 1. Its fuzzy quantities theory which reflects aspects that are not membership function has the form: incorporated in most game theoretical models. These games are known as cooperative games with fuzzy characteristic function. On the other hand, there are some situations in which some players do not fully take part in a coalition, but do to a certain extent. A coalition in which some players participate partially can be treated as a so-called fuzzy coalition. These games are known as cooperative games It is a function of a vector x and depends on the four scalar parameters a, b, c, d as shown in Fig. 2 [3] Gaussian Membership Function The Gaussian membership function depends on two parameters, standard deviation σ and mean μ. It is represented as shown in Fig III. Basic Notions for Fuzzy Sets 0 a b c d Height: The height of a fuzzy subset of X, denoted by Figure 2. The membership function of a trapezoidal fuzzy number hgt ( ), is the least upper bound of , i.e. , )= 1 1 σ 0 μ 0 al am ar Figure 3. The membership function of a Gaussian fuzzy number Figure 1. The membership function of a triangular fuzzy number with fuzzy coalition. A game with fuzzy coalitions and Normal: The fuzzy set on X is said to be normal if its fuzzy characteristic functions are called fuzzy games. height is unity, i.e., if there is x X such that = 1. Aubin [5] studied the problem at first by introducing If it is not normal, the fuzzy set is said to be subnormal. fuzzy coalition. A fuzzy coalition for n-person games is an May Issue Page 42 of 53 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 2, No. 5, 2011 n-dimension vector. The ith component of it shows to and then the focus is shifted to illustrate the solution what extent the ith player participates in the coalition and methodology in order to propose a final algorithm to is called participation level (rate) or membership degree. apply the proposed model. The Shapley value [3] is one of the most appealing solution concepts in cooperative game theory. The A. Model Formulation Shapley value represents a vector whose elements are A cooperative game with crisp coalitions and crisp agents‟ shares derived from reasonable bases. Several characteristic function is called crisp cooperative game. A researchers who have investigated the Shapley value have crisp cooperative game is a pair (N,ν) where, N is the set focused their attention on fuzzy coalitions and real-valued of players with i ϵ N={1,…,n}and the characteristic characteristic functions. function v : S →R for all S N, where v(S) is the Butnariu [6, 7] defined a Shapley value and showed the characteristic function value that members of S will explicit form of the Shapley function on a limited class of receive if they play together such that v(φ) = 0. The class fuzzy games. of all crisp subsets of S is denoted by P(S). Tsurumi et al. [8] defined new Shapley axioms and a Taking imprecision of information in decision making new class of fuzzy games with Choquet integral form. problems into account, a fuzzy characteristic function This class of fuzzy games is both monotone non- (i.e., fuzzy coalition value) can be incorporated into a decreasing and continuous with respect to players' cooperative game, represented by fuzzy numbers ω. participation. Therefore, the characteristic function of such a game Branzei et al. [9] also give a concept of Shapley value associates a crisp coalition with a fuzzy number ω(S). for games with fuzzy coalition, which is defined by the Assessing such fuzzy numbers for any crisp coalition S, a associated crisp game corresponding to fuzzy game. In cooperative game with crisp coalitions and fuzzy that definition, only the case where the participation level characteristic function is defined by a pair (N, ω), where of each player is 1 is considered. The core for fuzzy the fuzzy characteristic function ω : P(N) → ={ | games is also studied and is focused on by Tijs et al. [2] ≥ 0} is such that ω(φ) = 0. On the other hand, Mares [10, 11] and Vlach [12] were A cooperative game with fuzzy coalitions and fuzzy concerned about the uncertainty in the value of the characteristic function is called fuzzy cooperative game characteristic function associated with a game. In their (FCGame). FCGame is a pair (N, ) where, N is the set of game models, the domain of the characteristic function of players with i ϵ N={1,…,n} and the fuzzy characteristic a game is still the class of crisp coalitions, but the function : U → for all U N, such that ω(φ) = 0. A coalition values allocated to players are fuzzy numbers. fuzzy coalition U is a fuzzy subset of N, which is a vector By introducing fuzzy information into cooperative games, U = {U(1),…,U(n)} with coordinates U(i) contained in the the implicit assumption that all players and coalitions interval [0,1]. The number U(i) indicates the membership know the expected payoffs even before the cooperation grade of i in U. The class of all fuzzy subsets of U is begins is unrealistic. There are many uncertain factors denoted by F(U). For a fuzzy subset U, the α–level set is during the process of negotiation and coalition forming, so defined as: [U]α = {i ϵ N | U(i) ≥ α }, and the support set is in most situations players can only know imprecise denoted by Supp(U) = {i ϵ N | U(i) > 0}. information regarding the real outcome of cooperation, Considering both fuzzy coalitions and fuzzy and this vague expectation can be modeled by characteristic functions, a cooperative game with fuzzy mathematical tools. coalitions and a fuzzy characteristic function is defined Borkotokey [13] undertook research on fuzzy games of by(N, ), in which the fuzzy characteristic function : which the coalition and the characteristic function are both F(N) → = { | ≥ 0} is such that (φ) = 0. fuzzy information. Hereinafter, a cooperative game with fuzzy coalitions and Also, Xiaohui Yu, Qiang Zhang [14] studied the games a fuzzy characteristic function will be called a “fuzzy with fuzzy coalitions and a fuzzy characteristic function. cooperative game” for short. They proposed the generalized form of games with fuzzy Let X(U) = [X1(U),…,Xn(U)] be the payoff vector, such characteristic functions and studied the Shapley function that Xi(U) is the payoff of player i of the fuzzy coalition U on this class of fuzzy games, called the Hukuhara– ϵ F(N) .A function X: F(N) → is said to be an Shapley function. They extended Tsurumi et al.‟s games imputation for such a game if: and introduced a new class of cooperative games with 1. Group rationality: fuzzy characteristic functions and fuzzy coalitions called = (U), i = 1,…,n fuzzy games with “indeterminate integral form”. Based on 2. Individual rationality: the Hukuhara–Shapley function for a game with a fuzzy characteristic function, they showed the explicit form of Xi(U) ≥ [U(i) . ({i})], i ϵ supp(U) the Shapley function for a fuzzy game with “indeterminate For any S ϵ P(N), where X = (X1,X2,…,Xn). integral form”. Thus, the payoff vector is not a reasonable candidate for a solution unless it satisfies: 1. Xi(U) = 0, i supp(U) 2. Group rationality: III. PROPOSED MODEL (LMO/FCGAMES) = (U), i = 1,…,n AND SOLUTION METHODOLOGY 3. Individual rationality: Xi(U) ≥ [U(i) . ({i})], i ϵ supp(U) In this section the formulation of the proposed Since, all players of coalition U cooperate together to model for solving fuzzy cooperative games is constructed maximize their profit, thus, a linear multiobjective May Issue Page 43 of 53 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 2, No. 5, 2011 programming model (LMO/FCGames) can be 2. If α ≠ 0, then, the α-cut analysis is applied to transform formulated for any fuzzy coalition U ϵ F(N), which is the fuzzy triangular worth into interval worth for each based on the idea of the core [15, 16] as follows: coalition U in ranges α-leftU, α-rightU as follows: Max [X1(U),…,Xn(U)] (K) = [α-leftK, α-rightK], K = 1,2,…m Subject to Where, α-leftK = α * ( mK – lK ) + lK = (U), i = 1,…,n (Problem 1) Xi(U) ≥ [U(i) . ({i})], i ϵ supp(U) α-rightK = uK – α * ( uK – mK ) Xi(U) = 0, i supp(U) These intervals values are converted into crisp value by applying the Lambda function based on the attitude of the B. Solution Methodology player (i.e., membership grade); his attitude may be Several multiobjective solution techniques are available. optimistic, moderate, or pessimistic. Decision maker who Some of these techniques are weighted objectives, has optimistic attitude will assigned the maximum hierarchal optimization, trade-off, global criterion and the lambda, while the moderate attitude will assigned the max-min method [17], the adopted technique in the medium lambda and the pessimistic attitude will assigned proposed model. the minimum lambda in the range of [0, 1] as follows: To solve the above formulated LMO/FCGames model Let: CWK = [λ * α-rightK + (1 – λ) * α-leftK], where CWK with fuzzy coalitions, the characteristic function should be is the crisp worth of coalition K. identified as well as the defuzzification of the characteristic function should be carried out, in which the Applying the maxmin method, the formulated fuzzy worth is converted into a crisp worth in a way that (LMO/FCGames) will be: this value intuitively represents this fuzzy set. Max λ Subject to Obtaining the characteristic function value for a fuzzy Xi(U) ≥ λ, i ϵ supp(U) coalition In general, in order to identify the characteristic = (U), i ϵ supp(U) (Problem 2) function of a game with fuzzy coalitions, it is often Xi(U) ≥ U(i) . ({i}), i ϵ supp(U) constructed on the basis of the characteristic function of Xi(U) = 0, i supp(U) the original crisp one. Extending the crisp game, the game with fuzzy coalitions can be represented by a mapping Where, from the characteristic function of the crisp game to that Ci = / (N) . of the game with fuzzy coalitions. Many extensions have been done, such as the Tsurumi et al.'s extension [16], in C. Proposed Algorithm which the characteristic function has the form: The proposed algorithm for solving cooperative games (Eq.1) with fuzzy coalitions and fuzzy characteristic function can Where, K F(N), Q(K) = {K(i) | K(i) > 0, i N}, be summarized into the following steps: q(K) is the cardinality of Q(K), i.e. q(K) = |Q(K)|, and Step 1: Initialize i = 1 rm(K) = {i | i N, K(i) = rm}. The elements in Q(K) are For ( i ≤ 2N – N ) written in the increasing order as r1 < … < rq(K), and let r0 do = 0. Step 2: Identify the fuzzy worth for the crisp coalition, . Handling fuzziness in the above formulated model Step 3: Collect the degree of participation of each player, In order to use the maxmin method to solve the U(i). formulated (LMO/FCGames), defuzzification of the Step 4: Calculate the characteristic function value for the fuzzy worth should be performed, in which the fuzzy fuzzy coalition using (1) worth is converted into a crisp worth in a way that this Step 5: Formulate the fuzzy game Problem 1 value intuitively represents this fuzzy set. Step 6: Handle the fuzzification using α-cut technique There are many different techniques for defuzzification, Step 7: Formulate the crisp single objective Problem 2 such as the mean-of-maxima, the first-of-maxima and the Step 8: Solve the formulated crisp Model 2 using any generalized level set technique. The most common used suitable technique. de-fuzzification method is the α-cut method, in which the Step 9: Stop. fuzzy triangular values are converted into crisp one in the following steps [18]: Let α represent the degree of uncertainty in the IV. APPLICATION OF THE PROPOSED information (it is determined by the decision maker) MODEL which ϵ [0, 1] and lK, mK, uK are the lower, medium, upper values of the fuzzy characteristic function of the fuzzy Consider the joint production model, presented by Yu subset K ϵ F(N) respectively. and Zhang [14] in which three decision makers pool three 1. If α = 0, then the crisp worth for coalition U is defined resources to make seven finished products. Three decision as: makers, named 1, 2 and 3, possess three different initial (K) = ( λlK + λmK + λuK ) / 3. resources. Decision maker i has 10 tons of resource Ri and can produce ni tons of product Pii, i = 1, 2, 3. Let us consider that the decision makers decide to undertake a May Issue Page 44 of 53 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 2, No. 5, 2011 joint product: if decision makers i and j cooperate, they Subject to will produce nij tons of product Pij, and if all three X1(U) ≥ λ cooperate, n123 tons of product P123 can be produced. The effective output of each finished product is as shown in X2(U) ≥ λ Table I. X3(U) ≥ λ It is natural for the three decision makers to try to evaluate the revenue of the joint project in order to decide X1(U) + X2(U) + X3(U) = 41.07 whether the project can be realized or not. However the X1(U) ≥ 9.66 average profit per ton of each product is dependent on a X2(U) ≥ 5.37 number of factors such as product market price, product X3(U) ≥ 10.6 cost, consumer demand, the relation of commodity supply Where, and demand, etc. Hence, the average profit of each C1 = (ω{1,2,3} – ω{2,3}) / ω{1,2,3} = 0.51 product is an approximate evaluation, which is C2 = (ω{1,2,3} – ω{1,3}) / ω{1,2,3} = 0.29 represented by a triangular fuzzy numbers as shown in C3 = (ω{1,2,3} – ω{1,2}) / ω{1,2,3} = 0.57 Table 1. Solving the above linear model, an assessment of the Therefore the fuzzy worth of each of the crisp coalition payoff of each decision maker in the given fuzzy coalition could be calculated as follows: could be evaluated as follows: ω{1} = (14.4,16,17.6) ω{2} = (26.1,27,27.9) X1 = $15.29 X2 = $8.61 X3 = $17.18 ω{3} = (9,10,12) ω{1,2} = (52.2,55.8,59.4) V. COMPARATIVE STUDY ω{1,3} = (35,40.25,45.5) ω{2,3} = (54,57.6,61.2) In this section a comparative study of the obtained ω{1,2,3} = (89.6,98,106.4) results, using the LMO/FCGames with the one obtained During the early period of the joint project, every using the Hukuhara-Shapley function developed by Yu decision maker has to consider how many resources he or and Zhang [14] for the crisp and fuzzy coalitions, is she should provide in the cooperation. As we all know, carried and the mean absolute error (MAE) between the each decision maker does not need to supply all of his two methods is calculated where it is minimized, showing resources to cooperate in real life; it depends on individual how close the obtained results using the LMO/FCGames preference. are to those obtained using the Hukuhara-Shapley Suppose decision maker 1 would cooperate with function. decision maker 2 and 3, i.e. U={1,2,3}. In this coalition, Table II shows the payoff allocated to each DM 1 would supply 6 tons of resource R1 (i.e., decision maker (player) using the proposed membership grade of DM 1 is 0.6), DM 2 would supply 2 LMO/FCGames and the Hukuhara-Shapley function for tons of resource R2, and DM 3 would supply 10 tons of the fuzzy coalition {1, 2, 3}. It is clear that the proposed resource R3. Thus the membership grades for the decision model showed more promising results, when considering makers for this fuzzy coalition U = {1, 2, 3} are as the third scenario, than the Hukuhara-Shapley function follows: U(1) = 0.6 U(2) = 0.2 U(3) = 1 proposed by Yu and Zhang [14] resulting in a small mean Using the proposed (LMO/FCGames) with fuzzy absolute error. coalitions and solving for the grand coalition U = {1, 2, III. CONCLUSION 3}, an assessment of the payoff of each decision maker in the given fuzzy coalition could be evaluated as follows: Game theoretic approaches to cooperative TABLE I. The effective output and the average profit of each finished situations in fuzzy environments have given rise to several product kinds of fuzzy games. In this paper, the games with fuzzy Product coalitions and fuzzy characteristic functions are P11 P22 P33 P12 considered. Our study proposed a new fuzzy game model that Output of Product(tons) 8 9 10 18 is a linear multiobjective programming problem. This Average (1.8,2.0,2 (2.9,3.0,3 (0.9,1.0,1 (2.9,3.1,3 approach is very simple; it needs the solution of a simple Profit(1000$) .2) .1) .2) .3) mathematical programming problem. P13 P23 P123 Also a numerical example is presented to illustrate Output of the construction of the developed fuzzy cooperative game 17.5 18 28 Product(tons) model and the obtained results were compared with those Average (2.0,2.3,2 (3.0,3.2,3 (3.2,3.5,3 obtained using the fuzzy Shapley value. A comparative Profit(1000$) .6) .4) .8) study was carried out and the performance of the Max λ developed model was more acceptable. May Issue Page 45 of 53 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 2, No. 5, 2011 TABLE II. Decision makers‟ profit share using Hukuhara-Shapley Value Vs. LMO/FCGames Shapley Value Fuzzy Coalition {1,2,3} LMO / MAE Interval Valued Fuzzy Value FCGame (Based on α-cut) Crisp Value s LO MED UP LO UP DM 1 13.15 14.98 16.57 14.43 15.46 15.15 15.29 0.26 DM 2 8.14 8.56 8.95 8.43 8.68 8.60 8.61 DM 3 14.23 16.16 18.75 15.58 16.94 16.53 17.18 REFERENCES [1] J.V. Neumann, D. Morgenstern, Theory of Games and Economic Behaviour, third ed., Princeton University Press, Princeton, NJ, 1953. [2] S. Tijs, R. Branzei, S. Ishihara, S. Muto, On cores and stable sets for fuzzy games, Fuzzy Sets and Systems 146 (2004) 285–296. [3] L.S. Shapley, A value for n-person game, in: H. Kuhn, A. Tucker (Eds.), Contributions to the Theory of Games. Vol. 2, Princeton University Press, Princeton, NJ, 1953, pp. 307_317. [4] M. Sakawa, I. Nishizaki, A lexicographical solution concept in an n- person cooperative fuzzy game, Fuzzy Sets and Systems 61 (1994) 265_275. [5] J.-P. Aubin, Mathematical Methods of Game and Economic Theory Rev. ed., North-Holland, Amsterdam, 1982. [6] D. Butnariu, Fuzzy games: a description of the concept, Fuzzy Sets and Systems 1 (1978) 181–192. [7] D. Butnariu, T. Kroupa, Shapley mappings and the cumulative value for n-person games with fuzzy coalition, European Journal of Operational Research 186 (2008) 288_299. [8] M. Tsurumi, T. Tanino, M. Inuiguchi, A Shapley function on a class of cooperative fuzzy games, European Journal of Operational Research 129 (2001) 596–618. [9] R. Branzei, D. Dimitrov, S. Tijs, Models in Cooperative Game Theory: Crisp, Fuzzy and Multichoice Games, Springer-Verlag, 2005. [10] M. Mares, Fuzzy coalition structures, Fuzzy Sets and Systems 114 (2000) 23–33. [11] M. Mares, Fuzzy Cooperative Games: Cooperation with Vague Expectations, Physica-Verlag, New York, 2001. [12] M. Mares, M. Vlach, Linear coalition games and their fuzzy extensions, International Journal of Uncertainty Fuzziness Knowledge- Based Systems 9 (2001) 341–354. [13] S. Borkotokey, Cooperative games with fuzzy coalitions and fuzzy characteristic functions, Fuzzy Sets and Systems 159 (2008) 138–151. [14] X. Yu, Q. Zhang, An extension of cooperative fuzzy games, Fuzzy Sets and Systems 161(2010)1614–1634. [15] Y. Maali, A multiobjective approach for solving cooperative n-person games, Electrical Power and Energy Systems 31 (2009) 608–610. [16] M. Tsurumi, T. Tanino, M. Inuiguchi, A Shapley function on a class of cooperative fuzzy games, European Journal of Operational Research 129 (2001) 596–618. [17] Y. Collette, P. Siarry, Multiobjective Optimization, Springer, 2003. [18] Q. Song,R.P. Leland, Defuzzification techniques and applications, Fuzzy sets and systems 81 (1996) 321-329. May Issue Page 46 of 53 ISSN 2229 5208