Abstract
Fuzzy Inf. Eng. (2012) 4: 401-414 DOI 10.1007/s12543-012-0123-z ORIGINAL ARTICLE Application of Atanassov’s I-fuzzy Set Theory to Matrix Games with Fuzzy Goals and Fuzzy Payoffs A. Aggarwal · D. Dubey · S. Chandra · A. Mehra Received: 22 June 2011/ Revised: 12 August 2012/ Accepted: 27 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract We aim to extend some results in [6, 7, 8, 21] on two person zero sum matrix games (TPZSMG) with fuzzy goals and fuzzy payoffs to I-fuzzy scenario. Because the payoffs of the matrix game are fuzzy numbers, the aspiration levels of the players are fuzzy as well. It is reasonable to believe that there is some indeterminacy in estimating the aspiration levels of both players from their respective expected pay offs. This situation is modeled in the game using Atanassov’s I-fuzzy set theory. A new solution concept is proposed for such games and a procedure is outlined to obtain the degrees of suitability of the aspiration levels for each of the two players. Keywords Fuzzy matrix games · Fuzzy goals · Fuzzy payoffs · Fuzzy inequalities · I-fuzzy sets · I-fuzzy inequalities. 1. Introduction Fuzzy linear programming problems and fuzzy matrix games have been studied a great deal in the literature, e.g. Bector and Chandra [9], Nischzaki and Sakawa [19] and several references cited therein. The earliest study of TPZSMG with fuzzy pay- offs is due to Campos [11]. Later Bector et al [8] interpreted Compos’ model in the context of fuzzy linear programming duality and showed that solving a TPZSMG with fuzzy payoffs is equivalent to solving an appropriate pair of primal-dual fuzzy linear programming problems. Compos [11] and Bector et al [8] employed Yager’s A. Aggarwal () University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Delhi- 110403, India email: abhaaggarwal27@gmail.com D. Dubey ()· S. Chandra () · A. Mehra () Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India email: diptidubey@gmail.com chandras@maths.iitd.ac.in apmehra@maths.iitd.ac.in 402 A. Aggarwal· D. Dubey · S. Chandra · A. Mehra (2012) ranking function [23] for the purpose of defuzziﬁcation. In a related work, Li [15] presented a multiobjective linear programming approach to solve such fuzzy matrix games. In TPZSMG with fuzzy payoffs, the players can only estimate their aspiration lev- els (goals) and/or their values with some imprecision. It is therefore most likely that the players have some indeterminacy or hesitation about these approximations. The fuzzy set theory equips us with a membership function only and there is no means to incorporate the indeterminacy factor or hesitation degree into it. The membership degree only allows us to indicate the degree of belongingness to the fuzzy set under consideration while the degree of non-belongingness is taken as the complement of ‘one’. Fuzzy set theory is thus not enough to model the matrix game problems in- volving indeterminacy in aspiration levels of the players. But then, all is not lost. Atanassov [2-5] introduced an interesting generalization of the standard fuzzy set in which two membership functions, more or less independent, one for the degree of belongingness and the other for the degree of non-belongingness are deﬁned such that for each element of the universe their sum is less than or equal to one rather than being equal to one as is the case for the standard fuzzy sets. Atanassov [2-5] termed these sets as intuitionistic fuzzy sets. But this terminology has not found much fa- vor with many researchers. For details on this account, readers can refer to Dubois et al [13] and Grzeorzewshi and MrOwka [14]. The concern is mainly because the same nomenclature had earlier been used in the ﬁeld of intuitionistic logic. The An- tanassov’s intuitionistic fuzzy set theory and intuitionistic logic differ completely in their mathematical structure and treatment, and hence it makes some what confusing to use the same terminology for two different concepts. Therefore it is suggested in [13] and [14] that the intuitionistic fuzzy sets are called Atanassov’s I-fuzzy sets or simply the I-fuzzy sets. From now onwards, without any ambiguity, we shall be call- ing them I-fuzzy sets to be understood in the sense of intuitionistic fuzzy sets deﬁned by Atanassov. Despite certain controversies on its name, the concepts and ideas from I-fuzzy set theory have increasingly been applied in several ﬁelds including pattern recognition, medical diagnosis, multiattribute decision making, to name a few. Though, in this context, the list of references is too long to be produced here, but we refer the readers to [12, 16, 20, 22], and references therein. In series of papers followed by a book [5], Atanassov explained various set theoretic operations on I-fuzzy sets. In [4], the author advanced the theory of operators and relations for I-fuzzy sets. Inspired by the work of Bellman and Zadeh [10] for fuzzy sets, Angelov [1] studied an optimization problem with I-fuzzy sets and illustrated its application in classical transportation problems. Further, Li and Nan [17] studied TPZSMG where entries of the payoffs matrix are I-fuzzy numbers. By using the I-fuzzy set inclusion relation, they obtained the optimum expected payoffs for the players. Recently, Nan and Li [18] studied a TPZSMG in which entries of the payoffs matrix are speciﬁed by triangular I-fuzzy numbers and employed a ranking method to the same. In this paper, we attempt to extend the results of Bector et al [8] and Vijay et al [21] to study matrix games with fuzzy goals and fuzzy payoffs by creating an I-fuzzy scenario. This new model provides the degree of acceptance as well as the degree of Fuzzy Inf. Eng. (2012) 4: 401-414 403 rejection to the fuzzy aspiration levels of the two players each. Our work differs from that of Li and Nan [17], and Nan and Li [18] because neither of these studies include I-fuzzy inequalities in their models although the payoffs are represented by I-fuzzy numbers. The paper is structured as follows. Section 2 provides a brief account of fuzzy inequalities and I-fuzzy sets. Section 3 explains the meaning of an I-fuzzy inequality from decision maker’s point of view. The discussion continues to explain the meaning of an I-fuzzy inequality between a pair of fuzzy numbers. The main problem of an I- fuzzy matrix game with fuzzy goals and fuzzy payoffs is formulated in Section 4. We ﬁrst conceptualize the meaning of a solution to such a game, and thereafter outline a procedure to obtain the degree of suitability of the fuzzy aspiration levels for two players each. The results are illustrated with a simple numerical example in Section 5. Some concluding remarks are furnished in Section 6. 2. Preliminaries In this section, we ﬁrst recall the notion of linear fuzzy inequalities and then present few deﬁnitions with regard to Atanassov’s I-fuzzy sets [2]. Further, we also present Angelov’s [1] model for decision making in I-fuzzy environment. 2.1. Linear Fuzzy Inequalities n n Let R denotes the n-dimensional Euclidean space and R be its non-negative orthant. Let a, b ∈ R. We now recall the meaning of the fuzzy inequality a b, to be read as “a is essentially greater than or equal to b” in the sense of Zimmermann [24]. To motivate a meaningful choice for the membership function corresponding to the inequality a b, it is argued in [24] that if a ≥ b, then the inequality is fully satisﬁed, while if a ≤ b− p, for a predeﬁned p > 0, the inequality is fully violated. Further for a ∈ (b − p, b), the membership function is monotonically increasing. Zimmermann [24] took this increase along a linear function and therefore choose the following membership function 1, a ≥ b, b− a 1− , b− p ≤ a < b, μ (a) = (1) S ⎪ 0, a < b− p, where S is the fuzzy set deﬁning the fuzzy inequality a b and p > 0 is the maximum tolerance away from b to be prescribed by the decision maker. For reasons that will be obvious from the later discussion, we shall be using the notation a b to represent the fuzzy inequality a b for the given tolerance level p. Our next task is to get familiar with an extension of the above approach for the case involving fuzzy numbers. Let F (R) denotes the set of all fuzzy numbers. We shall be representing a fuzzy number with a tilde overhead. For a ˜, b ∈F (R), Yager [23] ˜ ˜ interpreted an inequality of type a ˜ b as L(˜ a) ≥ L(b)− (1−λ)L(˜ p),λ ∈ [0, 1], where L : F (R) → R, is an appropriate linear ranking function, and p ˜ ∈F (R) denotes the measure of tolerance between the fuzzy numbers a ˜ and b. Yager [23] termed such an inequality as a double fuzzy inequality because not only the inequality ‘’ is fuzzy 404 A. Aggarwal· D. Dubey · S. Chandra · A. Mehra (2012) but also the numbers a ˜ and b are fuzzy. In the same spirit as above, now onwards, we ˜ ˜ shall be using the notation a ˜ b to represent the double fuzzy inequality a ˜ b in p ˜ the sense of Yager. Furthermore, in [23], Yager also suggested several linear ranking functions but the one deﬁned as xμ (x)dx L(˜ a) = (2) μ (x)dx became very popular in problems of engineering design and control that apply fuzzy logic, and of course also in fuzzy matrix games. For instance, we refer to Bector et al [8] and Campos [11] for the latter one. Above, a and a are the lower limit and l u upper limit of the support of the fuzzy number a ˜ deﬁning the fuzzy set A, that is a = inf cl{x ∈ R | μ (x) > 0}, a = sup cl{x ∈ R | μ (x) > 0}, l A u A and cl stands for closure of a set in R. In fact, the L function in (2) gains popularity for it measures the centroid of the area enclosed by the membership function μ (x) between x = a and x = a . In particular, if a ˜ = (a, a, a ) is a triangular fuzzy number l u l u a + a+ a l u (TFN) and L is as in (2), then L(˜ a) = , which is linear and preserves the inequality when the fuzzy numbers are multiplied by a non-negative constant. Consequently, for TFNs a ˜ = (a, a, a ), b = (b, b, b ), with a tolerance p ˜ = (p, p, p ) l u l u l u also a TFN, the inequality a ˜ b can be interpreted as p ˜ (a + a+ a ) ≥ (b + b+ b )− (1−λ)(p + p+ p ). l u l u l u The aforementioned approach of Yager [23] is sometimes termed as Yager’s res- olution method. Some authors, like Vijay et al [21], called L a defuzziﬁcation func- tion rather than a ranking function because it defuzziﬁes the double fuzzy inequality a ˜ b. p ˜ 2.2. I-fuzzy Sets The following deﬁnitions come from Atanassov [2] and Li [17]. Deﬁnition 2.1 (I-fuzzy set) Let X be a universal set. An I-fuzzy set (originally called an intuitionistic fuzzy set in [2]) A in X is described by A = {x,μ (x),ν (x)| x ∈ A A X}, where μ : X → [0, 1] and ν : X → [0, 1] deﬁne respectively, the degree of A A belongingness and the degree of non-belongingness of an element x ∈ X to the set A, with 0 ≤ μ (x)+ν (x) ≤ 1. A A Ifμ (x)+ν (x) = 1 for all x ∈ X, then A degenerates to a standard fuzzy set. A A Deﬁnition 2.2 (Union and intersection of I-fuzzy sets) Let A and B be two I-fuzzy sets in X. Then A∪ B is an I-fuzzy set C, written as C = A∪ B, and deﬁned as C = {x, max{μ (x),μ (x)}, min{ν (x),ν (x)} | x ∈ X}. A B A B Fuzzy Inf. Eng. (2012) 4: 401-414 405 Similarly, A∩ B is an I-fuzzy set D, written as D = A∩ B, and deﬁned as D = {x, min{μ (x),μ (x)}, max{ν (x),ν (x)} | x ∈ X}. A B A B Deﬁnition 2.3 (Score function) Let A be an I-fuzzy set in X. Then the function s(x) given by s(x) = μ (x)−ν (x), x ∈ X A A is called the score function. It measures the degree of suitability with respect to a set of criteria represented by vague values. We conclude this section by taking a brief note on an application of I-fuzzy sets in optimization theory reported in [1]. Taking motivation from the work of Bellman and Zadeh [10], Angelov [1] studied a problem of decision making in I-fuzzy environment. We brieﬂy describe the model proposed by him in [1]. Let X be any set. Let G , i = 1, 2,··· , r, be the set of r goals and C , j = 1, 2,··· , m, be the set of m constraints, each of which can be characterized by an I-fuzzy set on X. Then, the I-fuzzy decision D = (G ∩ G ∩···∩ G )∩ (C ∩ 1 2 r 1 C ∩···∩ C ) is an I-fuzzy set deﬁned by D = {x,μ (x),ν (x)| x ∈ X}, where 2 m D D μ (x) = min μ (x),μ (x) and ν (x) = max ν (x),ν (x) . D G C D G C i j i j i, j i, j Let s(x) = μ (x) − ν (x), x ∈ X be the score function of the I-fuzzy set D. Then D D x ∈ X is deﬁned as an optimal decision in the I-fuzzy scenario if s(x) ≥ s(x), for all x ∈ X, i.e., s(x) = max s(x). Letα = μ (x) = min μ (x),μ (x) andβ = ν (x) = x∈X D G C D i j i, j max ν (x),ν (x) . Then the given I-fuzzy decision problem has been transformed G C i j i, j into the following crisp optimization problem in [1], max α−β subject to μ (x)≥ α, i = 1,··· , r, ν (x) ≤ β, i = 1,··· , r, μ (x)≥ α, j = 1,··· , m, ν (x)≤ β, j = 1,··· , m, α+β ≤ 1, α ≥ β ≥ 0, x ∈ X. 3. The I-fuzzy Inequalities In this section, we extend the concepts of fuzzy inequalities to the I-fuzzy inequalities. IF We begin by discussing the I-fuzzy inequality which we shall be denoting by a p,q b, where a, b ∈ R, and p > 0, q > 0 are the tolerances to be described shortly in the text to follow. This I-fuzzy inequality is read as “a is essentially more than or equal to b in I-fuzzy sense”. Though there is no general way to deﬁne this inequality, two approaches seem to be most natural depending upon the attitude of the decision 406 A. Aggarwal· D. Dubey · S. Chandra · A. Mehra (2012) maker. These are termed as the pessimistic approach and the optimistic approach, and are explained next. The pessimistic approach In this approach, a decision maker has a pessimistic attitude for acceptance. By this, we mean to say that if a ≥ b, the inequality is certainly satisﬁed, while for a ≤ b− p, the inequality is completely violated. However, there is a parameter 0 < q < p, supplied by the decision maker, such that the decision maker shows reluctancy in accepting the inequality if b− p+ q ≤ a ≤ b. To model this scenario, let the tolerances p, q, 0 < q < p be known priori and take the membership and the non-membership functions as shown in the Fig.1. ν μ b − p a b − p + q b Fig. 1 Membership and non-membership functions: pessimistic case Mathematically, the membership functionμ is the same as deﬁned in (1) while the non-membership function ν is as follows 1, a ≤ b− p, b− p− a 1+ , b− p < a ≤ b− p+ q, ν (a) = (3) A ⎪ 0, a > b− p+ q. Observe that there is an interval [b− p+ q, b] where the non-membership degree is zero but the membership degree is not equal to one, i.e., the decision maker is rejecting the inequality but not ready to accept it fully. Therefore in the pessimistic approach, the I-fuzzy statement can be written as a composition of two fuzzy state- ment, where the degree of membership and the degree of non-membership of I-fuzzy statement can be evaluated by the degree of membership of respective fuzzy state- ments. Equivalently, we mean ⎛ ⎞ ⎜ a b ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ IF ⎜ ⎟ ⎜ ⎟ a b ⇔ and , with b− p ≤ b− p+ q ≤ b. ⎜ ⎟ p,q ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ a b− p To make it more clear, the membership function and the non-membership function of IF an I-fuzzy inequality a b are respectively obtained by the membership function p,q Fuzzy Inf. Eng. (2012) 4: 401-414 407 F F of fuzzy inequality a b and the membership function of fuzzy inequality a p q b− p. The optimistic approach In the optimistic approach, the decision maker is assumed to take a liberal view on IF rejection. For I-fuzzy inequality a b,if a ≥ b, then the inequality is certainly p,q satisﬁed. The decision maker is able to identify a parameter q > 0 such that if a ≤ b− p−q, then the inequality is completely rejected. There is an interval [b− p−q, b− p] in which the decision maker is not accepting the inequality but not in the position of fully rejecting it. To model this situation, let the tolerances p and q be known priori and take the membership and non-membership functions as shown in the Fig.2. b − p − q b b − p a Fig. 2 Membership and non-membership functions: optimistic case Mathematically, the membership function μ is same as deﬁned in (1) while the non-membership function ν is 1, a ≤ b− p− q, b− p− q− a ν (a) = 1+ , b− p− q < a ≤ b, (4) ⎪ p+ q 0, a > b. Again note that there is an interval [b− p−q, b− p] in which the membership degree is zero but the non-membership degree is not zero. Hence, in the optimistic approach an I-fuzzy statement can be written as a composition of two fuzzy statement, where the degree of membership and non-membership of above I-fuzzy statement can be evaluated by the degree of membership of respective fuzzy statements. ⎛ ⎞ ⎜ ⎟ a b ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ IF ⎜ ⎟ ⎜ ⎟ a b ⇔ , with b− p− q ≤ b− p ≤ b. and ⎜ ⎟ p,q ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ a b− p− q p+q In other words, the membership function and the non-membership function of an I- IF fuzzy inequality a b are respectively obtained by the membership function of p,q F F fuzzy inequality a b and the membership function of fuzzy inequality a p p+q b− p− q. 408 A. Aggarwal· D. Dubey · S. Chandra · A. Mehra (2012) We are now in a position to describe what we mean by double I-fuzzy inequality. For this, we shall be extending the aforementioned interpretation of I-fuzzy inequality IF a b to the case when a, b are fuzzy numbers a ˜ and b, respectively, in F (R), p,q and the tolerances p, q are also fuzzy numbers p ˜, q ˜, respectively. Thus we wish to IF interpret the I-fuzzy inequality a ˜ b. The forgoing discussion suggests that for p ˜,q ˜ the pessimistic case ⎛ ⎞ ⎜ ⎟ a ˜ b ⎜ ⎟ ⎜ p ˜ ⎟ ⎜ ⎟ IF ⎜ ⎟ ˜ ⎜ ⎟ ˜ ˜ ˜ a ˜ b ⇔ ⎜ and ⎟ , where L(b− p ˜) ≤ L(b− p ˜ + q ˜) ≤ L(b). p ˜,q ˜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ a ˜ b− p ˜ q ˜ Similarly, for optimistic case ⎛ ⎞ ⎜ ⎟ a ˜ b ⎜ ⎟ p ˜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ IF ⎜ ⎟ ˜ ˜ ˜ ˜ ⎜ ⎟ a ˜ b ⇔ , where L(b− p ˜ − q ˜) ≤ L(b− p ˜ ) ≤ L(b). ⎜ and ⎟ p ˜,q ˜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ a ˜ b− p ˜ p ˜+q ˜ Here we must note that each of the double fuzzy inequality appearing above is to be understood as per the resolution method of Yager [23] discussed earlier in Subsection 2.1. IF IF ˜ ˜ The inequality a ˜ b is equivalent to (−a ˜ ) (−b) with the understanding that p ˜,q ˜ p ˜,q ˜ every where the same sense (pessimistic/optimistic) is adhered to. 4. Matrix Games: An I-fuzzy Interpretation m×n T LetA∈ R , A = (a ), be m × n matrix, and e = (1,··· , 1) be a vector of ones ij whose dimension is speciﬁed in the speciﬁc context. By a TPZSMG G, we mean a m n m m T n n T triplet G = (S , S , A), where S = {x ∈ R , e x = 1} and S = {x ∈ R , e y = 1} + + denote the sets of all mixed strategies for player 1 and player 2 respectively, and A is the payoff matrix of player 1. The symbol T is the transpose of a vector/matrix. If player 1 chooses the mixed strategy x ∈ S and player 2 chooses mixed strategy m n n T y ∈ S , then the expected payoff of player 1 is the scalar x Ay = x a y . i ij j i=1 j=1 Since the game is a zero sum, the payoff of player 2 is −x Ay. We now introduce the I-fuzzy matrix game to be studied in the sequel. Let A be the payoff matrix with entries as fuzzy numbers. Because the payoffs of the matrix ˜ ˜ game are fuzzy, the aspiration levels are fuzzy as well. Let U , V ∈F (R) be the 0 0 aspiration levels of player 1 and player 2 respectively. Consider the following I-fuzzy matrix game m n IF IF ˜ ˜ ˜ (IFG):≡ (S , S , A, U , , V , ), 0 0 p ˜ ,q ˜ ˜ 0 0 s ˜ ,t 0 0 where p ˜ ∈F (R) (respectively q ˜ ∈F (R)) is the fuzzy tolerance associated with the 0 0 acceptance (respectively rejection) of the aspiration level U for player 1. Similarly, s ˜ ∈F (R) (respectively t ∈F (R)) is the fuzzy tolerance associated with acceptance 0 0 (respectively rejection) of the aspiration level V for player 2. The game (IFG)isa two person matrix game with fuzzy goals and fuzzy payoffs, involving inequalities that are to be interpreted in the I-fuzzy sense described in Section 3. Thus the game Fuzzy Inf. Eng. (2012) 4: 401-414 409 (IFG) extends the fuzzy game model studied by Vijay et al [21] so as to provide more meaningful interpretations to the results obtained therein. m n Deﬁnition 4.1 (Solution of the game (IFG)) A point (x, y) ∈ S × S is called a solution of the (IFG) if T IF n ˜ ˜ (I) x Ay U , ∀ y ∈ S , p ˜ ,q ˜ 0 0 and T IF m (II) x Ay V , ∀ x ∈ S . s ˜ ,t 0 0 Here x is called an optimal strategy for player 1 and y is called an optimal strategy for player 2. In view of the above deﬁnition, it is natural to construct the following pair of fuzzy linear programming problems for player 1 and player 2 respectively (IFP1) Find x ∈ S such that T IF n ˜ ˜ x Ay U , ∀ y ∈ S p ˜ ,q ˜ 0 0 and (IFP2) Find y ∈ S such that T IF m x Ay V , ∀ x ∈ S . s ˜ ,t 0 0 To carry forward the discussion, from now onwards, we shall assume that both players ˜ ˜ have pessimistic view points for their fuzzy aspiration levels U and V respectively. 0 0 The optimistic case can be dealt analogously. For the case of pessimistic scenario, we ˜ ˜ ˜ ˜ ˜ need to have L(U − p ˜ ) ≤ L(U − p ˜ + q ˜ ) ≤ L(U ), and L(V ) ≤ L(V + s ˜ − t ) ≤ 0 0 0 0 0 0 0 0 0 0 L(V + s ˜ ), and also interpret I-fuzzy inequalities in (IFP1) and (IFP2) as per our 0 0 discussion in Section 3 and Yager’s [23] resolution method discussed in Subsection 2.1. Thus the tolerances p ˜ , q ˜ , s ˜ , and t have to satisfy 0 ≤ L(˜ q ) ≤ L(˜ p ) and 0 ≤ 0 0 0 0 0 0 L(t ) ≤ L(˜ s ). 0 0 Let α and β respectively denotes the minimal degree of acceptance and the maxi- mal degree of rejection of the constraints of (IFP1). Then solving problem (IFP1) is the same as solving the following equivalent crisp problem (ECP1) for player 1. (ECP1) max α−β subject to T n L(x Ay) ≥ L(U )− (1−α)L(˜ p ), ∀ y ∈ S , 0 0 T n ˜ ˜ L(x Ay) ≥ L(U − p ˜ )+ (1−β)L(˜ q ),∀ y ∈ S , 0 0 0 α+β ≤ 1,α ≥ β ≥ 0, e x = 1, x ≥ 0. Similarly, let δ denotes the minimal degree of acceptance and η denotes the maxi- mal degree of rejection of the m constraints of (IFP2). Then solving problem (IFP2) is the same as solving the following equivalent crisp problem (ECP2) for player 2. 410 A. Aggarwal· D. Dubey · S. Chandra · A. Mehra (2012) (ECP2) max δ−η subject to T m ˜ ˜ L(x Ay)≤ L(V )+ (1−δ)L(˜ s ), ∀ x ∈ S , 0 0 T m ˜ ˜ L(x Ay)≤ L(V + s ˜ )− (1−η)L(t ),∀ x ∈ S , 0 0 0 δ+η ≤ 1,δ ≥ η ≥ 0, e y = 1, y ≥ 0. As mentioned earlier, the defuzziﬁcation function L preserves ranking when fuzzy numbers are multiplied by non-negative scalars; problems (ECP1) and (ECP2) can respectively be rewritten as follows: (ECP1) max α−β subject to T n ˜ ˜ x L(A)y ≥ L(U )− (1−α)L(˜ p ), ∀ y ∈ S , 0 0 T n ˜ ˜ x L(A)y ≥ L(U − p ˜ )+ (1−β)L(˜ q ),∀ y ∈ S , 0 0 0 α+β ≤ 1,α ≥ β ≥ 0, e x = 1, x ≥ 0, and (ECP2) max δ−η subject to T m x L(A)y≤ L(V )+ (1−δ)L(˜ s ), ∀ x ∈ S , 0 0 T m ˜ ˜ x L(A)y≤ L(V + s ˜ )− (1−η)L(t ),∀ x ∈ S , 0 0 0 δ+η ≤ 1,δ ≥ η ≥ 0, e y = 1, y ≥ 0. Above, L(A) is a crisp m × n matrix having entries as L(˜ a ), i = 1,··· , m, j = ij 1,··· , n. m n Moreover, since S and S are convex polytopes, it is sufﬁcient to consider only m n the extreme points (i.e., pure strategies) of S and S in the constraints of (ECP1) and (ECP2) . This observation leads to the following linear programming problems for player 1 and player 2 respectively. (ECP1) max α−β subject to ˜ ˜ x L(A ) ≥ L(U )− (1−α)L(˜ p ), j = 1,··· , n, j 0 0 ˜ ˜ x L(A ) ≥ L(U − p ˜ )+ (1−β)L(˜ q ), j = 1,··· , n, j 0 0 0 α+β ≤ 1,α ≥ β ≥ 0, e x = 1, x ≥ 0, and Fuzzy Inf. Eng. (2012) 4: 401-414 411 (ECP2) max δ−η subject to ˜ ˜ L(A )y ≤ L(V )+ (1−δ)L(˜ s ), i = 1,··· , m, i 0 0 ˜ ˜ L(A )y ≤ L(V + s ˜ )− (1−η)L(t ), i = 1,··· , m, i 0 0 0 δ+η ≤ 1,δ ≥ η ≥ 0, e y = 1, y ≥ 0. th th ˜ ˜ Here L(A ) (respectively L(A )) denotes the j column (respectively the i row) of j i L(A), j = 1,··· , n (respectively i = 1,··· , m). Thus we observe that solving the fuzzy matrix game (IFG) is equivalent to solving the crisp linear programming problems (ECP1) and (ECP2) for player 1 and player 2 2 ∗ ∗ ∗ ∗ 2 respectively. Also, if (x ,α ,β ) is an optimal solution of (ECP1) , then x is an ∗ ∗ ∗ ∗ optimal strategy for player 1, and α ,β and α − β are respectively the minimum degree of acceptance, the maximum degree of rejection and the degree of suitability for the aspiration level U . Similar interpretation can also be given to an optimal ∗ ∗ ∗ solution (y ,δ ,η ) to problem (ECP2) for player 2. Remark 4.1 An obvious problem at this stage is to specify the fuzzy aspiration ˜ ˜ levels U and V for player 1 and player 2 respectively. In practice, the two are to 0 0 be prescribed respectively by player 1 and player 2. For this, they can either depend on their own judgement or use some experts’ knowledge. However in absence of any ˜ ˜ such information, their aspiration levels U and V can also be obtained by solving 0 0 m n the fuzzy matrix game (FG):≡ (S , S , A ), by the procedure presented by Bector et al in [8]. It requires solving the following crisp linear programming problems for player 1 and player 2 respectively: (CP1) max L(U ) subject to ˜ ˜ x L(A )≥ L(U )− (1−α)L(˜ p), j = 1,··· , n, α ≤ 1, e x = 1, x,α ≥ 0 and (CP2) max L(V ) subject to ˜ ˜ L(A )y ≤ L(V )+ (1−δ)L(˜ s), i = 1,··· , m, δ ≤ 1, e y = 1, y,δ ≥ 0. ∗ ∗ ˜ ˜ Let L(U ) and L(V ) be the optimal values of (CP1) and (CP2) respectively. The ˜ ˜ ˜ ˜ players can take their aspiration levels U and V for which L(U ) and L(V ) is close 0 0 0 0 ∗ ∗ ˜ ˜ to L(U ) and L(V ) respectively. 5. Numerical Example 412 A. Aggarwal· D. Dubey · S. Chandra · A. Mehra (2012) Consider the two person zero sum matrix game (IFG) whose fuzzy payoff matrix is ⎛ ⎞ ⎜ ˜ ˜ ⎟ ⎜ 180 156⎟ ⎜ ⎟ ⎜ ⎟ A = ⎜ ⎟ , ⎝ ⎠ ˜ ˜ 90 180 ˜ ˜ ˜ where 180 = (175, 180, 190), 156 = (150, 156, 158) and 90 = (80, 90, 100) are TFNs of the form (a, a, a ). l u Now if we employ the procedure presented by Bector et al [8] and solve the crisp linear programs (CP1) and (CP2) for player 1 and player 2 respectively, we get ∗ ∗ ˜ ˜ ˜ L(U ) = 159.71 and L(V ) = 157.47. Therefore player 1 may aspire for a value U for ˜ ˜ ˜ which L(U ) is close to 159.71, while player 2 may aspire V for which L(V ) is close 0 0 0 to 157.47. Let us assume that player 1 and player 2 have their aspiration levels U ˜ ˜ ˜ and V for which L(U ) = 160 and L(V ) = 158 respectively. Let the corresponding 0 0 0 tolerances be (p ˜ = (7, 20, 23), q ˜ = (13, 15, 17)) and (s ˜ = (8, 17, 20), t = (5, 8, 17)) 0 0 0 0 respectively. To solve the game (IFG), we need to solve the following pair of linear programming problems (P1) max α−β subject to 545x + 270x ≥ 480− 50(1−α), 1 2 464x + 545x ≥ 480− 50(1−α), 1 2 545x + 270x ≥ 430+ 45(1−β), 1 2 464x + 545x ≥ 430+ 45(1−β), 1 2 x + x = 1,α+β ≤ 1, 1 2 x , x ≥ 0,α ≥ β ≥ 0 1 2 and (P2) max δ−η subject to 545y + 464y ≤ 474+ (1−δ)45, 1 2 270y + 545y ≤ 474+ (1−δ)45, 1 2 545y + 464y ≤ 519− (1−η)30, 1 2 270y + 545y ≤ 519− (1−η)30, 1 2 y + y = 1,δ+η ≤ 1, 1 2 y , y ,≥ 0,δ ≥ η ≥ 0. 1 2 ∗ ∗ ∗ The optimal solutions of (P1) and (P2) are (x = 0.7725, x = 0.2275,α = 1 2 ∗ ∗ ∗ ∗ ∗ 0.6486,β = 0.2793) and (y = 0.2036, y = 0.7963,δ = 0.6666,η = 0) respec- 1 2 tively. ∗ ∗ Therefore the optimal strategy for player 1 is (x = 0.7725, x = 0.2275), and the 1 2 above set aspiration level for player 1 is accepted with a minimum degree 0.6486 and rejected with a maximum degree 0.2793. It is important to note that the sum of two ∗ ∗ optimal degrees is not one. In fact, there is an indeterminacy factor of 1−α −β = 0.0721. Furthermore, the degree of suitability of the aspiration level of player 1 is Fuzzy Inf. Eng. (2012) 4: 401-414 413 ∗ ∗ 0.3693. Similarly, the optimal strategy for player 2 is (y = 0.2036, y = 0.7963), 1 2 and player 2 aspiration level is accepted with a minimum degree 0.6666 and rejected with a maximum degree 0. Note that although player 2 aspiration has a rejection degree zero but acceptance degree is not 1. This clearly indicates his pessimistic approach towards setting tolerances for aspiration level V . Moreover, the sum of two optimal degrees is not 1 again. The player 2 has indeterminacy factor 0.3334, and the degree of suitability of aspiration level of player 2 is 0.6666. In particular, if β = 1 − α, δ = 1 − η, p ˜ = q ˜ and s ˜ = t , the I-fuzzy linear 0 0 0 0 programs (P1) and (P2) subsume to the fuzzy linear programs (GFP8) and (GFD8), introduced in [21], for player 1 and player 2 respectively. In such a case, the optimal ∗ ∗ ∗ solutions for player 1 and player 2 are respectively (x = 0.8024, x = 0.1975,α = 1 2 ∗ ∗ ∗ 1.0) and (y = 0.6253, y = 0.2275,η = 0.7724). Observe that the maximum degrees 1 2 of rejection of the aspiration levels for player 1 and player 2 are respectively β = 0 andδ = 0.2276. 6. Conclusion We have extended the Yager’s resolution method [23] to provide an I-fuzzy interpre- tation of the double fuzzy inequality, which is subsequently used to study a class of I-fuzzy matrix games (IFG). Though the games of type (IFG) are very much in the sprit of the fuzzy matrix games (FG) studied by Bector et al [8] and Vijay et al [21], who provided an additional information about the solution to the game because of the I-fuzzy interpretation of the same. The main advantage of the proposed approach is that it not only provides the degrees of acceptance but also the degrees of rejections of the aspiration levels for both the players. It would be interesting to extend this methodology to the fuzzy games in which not only the inequalities are I-fuzzy but the payoffs are also described by I-fuzzy numbers as well. Acknowledgements The authors would like to thank the esteemed referees for their valuable comments and suggestions. References 1. Angelov P P (1997) Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems 86: 299-306 2. Atanassov K T (1986) Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20: 87-96 3. Atanassov K T (1989) More on intuitionistic fuzzy sets. Fuzzy Sets and Systems 33: 37-45 4. Atanassov K T (1994) New operations deﬁned over the intuitionistic fuzzy sets. Fuzzy Sets and Systems 61: 137-142 5. Atanassov K T (1999) Intuitionistic fuzzy sets, Springer-Verlag, Heidelberg 6. Bector C R, Chandra S (2002) On duality in linear programming under fuzzy environment. Fuzzy Sets and Systems 125: 317-325 7. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Dec 1, 2012
Keywords: Fuzzy matrix games; Fuzzy goals; Fuzzy payoffs; Fuzzy inequalities; I-fuzzy sets; I-fuzzy inequalities