Document Sample
Paper-8 Powered By Docstoc
					                                                                   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

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,          )xX}.
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. ,


                                                                                         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:
                                                                                            = (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)
   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
                                                                                 kinds of fuzzy games. In this paper, the games with fuzzy
                                                  Product                        coalitions and fuzzy characteristic functions are
                             P11            P22             P33         P12
                                                                                         Our study proposed a new fuzzy game model that
Output of
                         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
   {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

[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)
[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

Shared By: