Orlovsky [17] proposed the fuzzy equilibrium solution of games with possible sets of strategies for each player and a crisp payoff matrix. Ragade [19] proposed ...
International Journal of Uncertainty, Puzziness and Knowledge-Based Systems Vol. 10, No. 4 (2002) 385-400 © World Scientific Publishing Company
FUZZY MULTIOBJECTIVE PROGRAMMING METHODS FOR FUZZY CONSTRAINED MATRIX GAMES WITH FUZZY NUMBERS
Li DENGFENG* and CHENG CHUNTIAN Department Two, Dalian Naval Academy, Dalian 116018, Liaoning, China Department of Civil Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China dengfengli @ sina. com
Received February 2001 Revised April 2002 The purpose of the paper is to introduce a new type of fuzzy matrix games: fuzzy constrained matrix games. A computational method for its solution based on establishment of the auxiliary fuzzy linear programming for each player is proposed. The approach based on the multiobjective programming is established to solve these fuzzy linear programming. Effectiveness is illustrated with a numerical example. Keywords: Game theory; mathematical programming; multicriteria analysis; fuzzy sets.
1. Introduction Game Theory has a remarkable importance in the field of Operations Research, Decision Theory and Systems Engineering due to its great applicability. Many real problems can be modeled as games. Furthermore, it is very common, when real situations are concerned, to deal with elements or data defined in an approximate or vague way. These problems are classically modeled, however, without taking into account such lacks of precision and this leads to less suitability of the model for the proposed problem. In these cases, the theoretical support provided by the fuzzy subsets can become a very useful tool to model Corresponding author.
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these problems properly. From this point of view, Fuzzy Games study and solve those real problems that can be modeled as a game with some kind of vagueness in some of their elements. Fuzzy Game Theory is an active research field in Operations Research and Systems Engineering. It is a very efficient tool to solve real conflict problems with fuzzy information. The research on fuzzy games has been developed by Aubin[l] and Butnariu [3,4]. Recently, an increasing literature has appeared on this topic in which several types of fuzzy games have been investigated (see Chapter 9 of [9] for references). Campos [5] solved a matrix game with imprecise payoffs by deriving two auxiliary fuzzy linear programming models from the game and obtaining a fuzzy value and optimal strategies from their solutions. Campos et al [6,7] discussed the fuzzy matrix game by using subjective ranking functions. Dubois and Prade [9] described the fuzzy game with crisp sets of strategies and fuzzy payoffs due to some lack of precision on the knowledge of the associated payoffs. Li [10] studies several types of fuzzy multiobjective matrix games. Kandel and Zhang [12] develop the basic principles of the theory of fuzzy moves based on the theory of moves and game theory. Mare [13] discusses fuzzy coalition structures. Nishizaki and M.Sakawa [14-16] discuss bimatrix games with fuzzy goals. Orlovsky [17] proposed the fuzzy equilibrium solution of games with possible sets of strategies for each player and a crisp payoff matrix. Ragade [19] proposed the concept about outcome set of X -equilibrium solution. Sakawa and Nishizaki [20] propose a lexicographical solution concept for an n-person cooperative fuzzy game. Sakawa and Nishizaki [21] propose max-min solutions for two-person zero-sum multiobjective fuzzy matrix games with fuzzy goals. However, in some real game problems, choice of strategies for each player is constrained due to some practical reason why this should be (see Chapter 5 of Dresher [8] and page 58-59 of Owen [18] for references), i.e., not all mixed strategies in a game are permitted for each player. Such a two-person zero-sum finite game is called a matrix game with sets of constraint strategies, briefly, a constrained matrix game. In other words, by a constrained matrix game we mean that it is a matrix game in which choice of strategies for each player is constrained. Dresher [8] (see Chapter 5 of [8] for references) gives a real example of the constrained matrix game. A constrained matrix game with fuzzy payoffs or data is called a fuzzy constrained matrix game. So far as we know, no studies have yet been attempted for fuzzy constrained matrix games. In this paper, we study the fuzzy constrained matrix game with payoffs of triangular fuzzy numbers. Using operations of triangular fuzzy numbers, we propose new multiobjective programming models and solution methods for the fuzzy constrained matrix game. Our study is remarkably different from others (Campos et al. [5,6,7]) since constraint strategies are considered for each player. On the other hand, the value of the fuzzy constrained matrix game is considered as a triangular fuzzy number and both the players do not need to use subjective ranking functions in our models. In Section 2, we give the definition and linear programming models of the
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constrained matrix game. In Section 3, the concept of the fuzzy constrained matrix game is presented and the auxiliary fuzzy linear programming models are proposed for its solution. In Section 4, the approach based on the multiobjective programming is established to solve these fuzzy linear programmings. An illustrative numerical example is given in Section 5. The paper ends with a short concluding remarks in Section 6.
2. Preliminaries 2.1. Notations and operations of fuzzy numbers For the sake of simplicity and without loss of generality, along this paper, assume that all fuzzy numbers be triangular fuzzy numbers (see Chapter 2 of Dubois and Prade [9] for references). Let a = (a,a,a) be a triangular fuzzy number, where the membership function //~ of a is I x-a + a
a-
a0, BTy* 0, Az < Bs\ d < Ez Therefore, using Corollary 1 and 2, we obtain dTx
< zT(ETx*)
< (Azfi
< s*T(BTy*) < s*Tc = cTs*
i.e. U* < CO*
Theorem 2 means that I's gain-floor cannot exceed IFs loss-ceiling.
4. Fuzzy Multiobjective Programming Technique for Fuzzy Constrained Matrix Games In this Section, we will study how to solve Eqs. (6) and (7). Firstly, we consider solution of Eq. (6). By Corollary 2, let us reconsider Eq. (6) as mnx{((d - d)Tx,dTx,(d + d)T x)}
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where d = (dl,d2,---,dq)T ,d = (dl,d2,---,dq)T and (A'p + 4A + Aup)z and (Elp + 4E + Eup)z>dlp+4d + d"p
,
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Thus, we have the crisp constraint sets of players I and II as follow Yp={y\(Elp + 42* +E;)Tx)}
(k = 1,2,3)
(17)
By computing, we have gk. = r^{gk(^)9gkOvlgk(y'*)}
(£ = 1,2,3)
(18)
Then, the positive ideal solutions and negative ideal solutions of Eq. (15) are g*=(gl*,g2\g*) and g*=(gl*,g2.,g3.), respectively. The relative membership function of these objective functions can be computed as follows |iif
gkCy)^gk
gk{y)
gk
gk
* if gk. < gkG) < gk* (* = 1,2,3)
(19)
gk*
[o if gkO))) A general fuzzy mathematical programming can be defined by maxjum(y)
(20)
yeY
Observe that the value of jum(y) can be interpreted as the overall degree of satisfaction of the player Ps fuzzy payoff. The fuzzy decision or the minimum operator of Bellman and Zadel [2],
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™n{Mgl(y),Mg20lMg3(y)} can be viewed only as one special example of jum(y). In the conventional fuzzy approaches it has been implicitly assumed that the minimum operator is the proper representation of a human player fuzzy preferences, and hence Eq. (20) has been interpreted as follows majcmin{//gi (y),Mg2 G),Mg3 (.?)} yeY
However, in general decision situations, players do not always use the minimum operator when they combine the three fuzzy objective functions ju (y), // (y) and ju (j)), and all of the three objectives are treated partially. Probably the most crucial problem in the fuzzy multiobjective decision problem is the identified of an appropriate aggregation function which represents well the human players' fuzzy preferences. In our study, we choose Moi(y) = m i n ( M n 0),A2jug2 G)9^Mg3 G)} where Xk > 0 ( k = 1,2,3 ) is weight of the objective gk , respectively, and Xx + /L2 + A3 — 1. Hence, Eq. (20) has been interpreted as follows maxminlV^ (j)),^// (y)^Mg3 (})} (21) yeY
Set ju = min{A1//gi (y),A2jug2 (y)>A3/ug3 (y)} .Then, Eq. (21) is equivalent to the following linear programming max//
Jj>4
(22)
The optimal solution of Eq. (22) will provide player I a satisfying max-min strategy y* and the fuzzy value u* = dJx*. Using a similar approach (as previously described ), we can solve Eq. (7). In fact, a satisfying min-max strategy z and the fuzzy value co* = cTs* of player II will be provided by solving the following linear programming max X
£el
(23)
4 ' [5kXhk(z)> A,* = 1,2,3 where X = mm{SlAh(z),S2All2(z),S,Alli(z)} , Sk>0
is weight of the objective hk and
