Journa of Uncertain Systems Vo.5, No.4, pp.27-285, 20 Onine at: www.jus.org.u Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution of Muti-objective Bi-Matri Games Prasun Kumar Naya,, Madhumanga Pa 2 Banura Christian Coege, Banura, 722 0, India 2 Department of Appied Mathematics with Oceanoogy and Computer Programming Vidyasagar University, Midnapore-72 02, India Emai: pamadumanga@ycos.com Received 2 December 2009; Revised 9 June 20 Abstract In this paper, we present an appication of intuitionistic fuzzy optimization mode to a two person mutiobjective bi-matri games (pair of pay-offs matrices) for the Nash equiibrium soution(nes) with mied strategies. We use inear membership and non-membership function for such computation. We introduce the intuitionistic fuzzy(if) goa for a choice of a strategy in a pay-off matri in order to incorporate ambiguity of human judgements and a payer wants to maimize his/her degree of attainment of the IF goa. It is shown that this NES is the optima soution of the mathematica programming probem, namey a quadratic programming probem. In addition, numerica eampe is aso presented to iustrate the methodoogy. c 20 Word Academic Press, UK. A rights reserved. Keywords: bi-matri game, Nash equiibrium, intuitionistic fuzzy sets, fuzzy optimization, intuitionistic fuzzy optimization, degree of attainment Introduction Fuzziness in bi-matri game theory was studied by various researchers [5, 7], where the goas are viewed as fuzzy sets, but they are very imited and in many cases they do not represent eacty the rea probems. In practica situation, due to insufficiency in the information avaiabe, it is not easy to describe the fuzzy constraint conditions by ordinary fuzzy sets and consequenty, the evauation of membership vaues is not aways possibe up to Decision Maer (DM) s satisfaction. Due to the same reason evauation of non-membership vaues is not aways possibe and consequenty there remains an indeterministic part of which hesitation survives. In such situation intuitionistic fuzzy set (IFS), Atanassov [2] serve better our required purpose. Aso, in reaistic modes, the more etensive appication of mutipe pay-off has been found to be more rather than the simpe pay-off []. No studies, however, have been made for NES of muti-objective bi-matri game with IF goa which wi be eamined in this paper. We introduce here a new approach to soving muti-objective bi-matri payoff matrices in IF environment. IF goa for the pay-off matrices has been formuated in order to incorporate the ambiguity of human judgement. We assume that each payer has a IF goa for the choice of the strategy and payers want to maimize the degree of attainment of the IF goa. For the purpose, this paper is organized as foows: In Section 2, we sha give some basic definitions, notations and optimization mode on IFS. In Section 3, a epected pay-off, IF goa for bi-matri game is defined and a degree of attainment of the IF goa is considered. The equiibrium soution with respect to a degree of attainment of a IF goa is aso defined. In Section 4, the methods of computing the NES of a singe-objective bi-matri game are proposed, when the membership, nonmembership functions of the IF goas are inear. In Section 5, the NES for muti-objective game is proposed. Lasty a numerica eampe is given in Section 6. Corresponding author. Emai: naya prasun@rediffmai.com (P.K. Naya).
272 P.K. Naya and M. Pa: Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution 2 Intuitionistic Fuzzy Sets The intuitionistic fuzzy set introduced by Atanassov [2] is characterized by two functions epressing the degree of beonging and the degree of non-beongingness respectivey. Definition Let U = {, 2,, n } be a finite universa set. An Atanassov s intuitionistic fuzzy set (IFS) A in a given universa set U is an object having the form where the functions A = { i, µ A ( i ), ν A ( i ) : i U} () µ A : U [0, ]; i.e., i U µ A ( i ) [0, ] and ν A : U [0, ] i.e., i U ν A ( i ) [0, ] define the degree of membership and the degree of nonmembership of an eement i U to the set A U, respectivey, such that they satisfy the foowing conditions : 0 µ A () + ν A (), i U which is nown as intuitionistic condition. The degree of acceptance µ A () and of nonacceptance ν A () can be arbitrary. Definition 2 For a A IF S(U), et π A ( i ) = µ A ( i ) ν A ( i ), (2) which is caed the the Atanassov s intuitionistic inde of the eement i in the set A or the degree of uncertainty or indeterministic part of i. Obviousy, 0 π A () ; for a i U. (3) Obviousy, when π A () = 0, U, i.e., µ A () + ν A () =, the set A is a fuzzy set as foows: Therefore fuzzy set is a especia IFS. A = {, µ A (), µ A () : U} = {, µ A () : U}. If an Atanassov s IFS C in U has ony an eement, then C is written as foows which is usuay denoted by C = { µ C ( ), ν C ( ) } in short. C = {, µ C ( ), ν C ( ) } (4) Definition 3 Let A and B be two Atanassov s IFS in the set U. A B if and ony if µ A ( i ) µ B ( i ) and ν A ( i ) ν B ( i ); for any i U. (5) Definition 4 Let A and B be two Atanassov s IFS in the set U. A = B if and ony if µ A ( i ) = µ B ( i ) and ν A ( i ) = ν B ( i ); for any i U. (6) Namey, A = B, if and ony if A B and B A. Definition 5 Let A and B be two Atanassov s IFS in the set U. The intersection of A and B is defined as A B = { i, min(µ A ( i ), µ B ( i )), ma(ν A ( i ), ν B ( i )) i U}. (7) Definition 6 Let A and B be two Atanassov s IFS in the set U and λ > 0 be a rea number. Designate: (i) A + B = { i, µ A ( i )µ B ( i ) µ A ( i )µ B ( i ), ν A ( i )ν B ( i ) U} (ii) AB = { i, µ A ( i )µ B ( i ), ν A ( i ) + ν B ( i ) ν A ( i )ν B ( i ) i U} (iii) λa = {< i, ( µ A ( i )) λ, (ν A ( i )) λ >: i U}. Note: From the above definitions we see that the numbers µ A () and ν A () refect respectivey the etent of acceptance and the degrees of rejection of the eement to the set A, and the numbers π A () is the etent of indeterminacy between both.
Journa of Uncertain Systems, Vo.5, No.4, pp.27-285, 20 273 2. Intuitionistic Fuzzy Optimization Mode Generay, an optimization probem incudes objective(s) and constraints. Let us consider an optimization probem f i () min; i =, 2,, p subject to, g j () 0; j =, 2,, q, where denotes the unnowns, f i () denotes the objective functions, g j () denotes constraints (non-equaities), p denotes the number of objectives and q denotes the number of constraints. The soution of this crisp optimization probem satisfies a constraints eacty. 2.. Fuzzy Optimization Mode In fuzzy optimization probem such as fuzzy mathematica programming, Zimmermann [7], fuzzy optima contro, Zadeh [6], the objective(s) and/or constraints or parameters and reations are represented by fuzzy sets. These fuzzy sets epain the degree of satisfaction of the respective condition and are epressed by their membership functions [8]. Anaogousy as in the crisp case, fuzzy optimization probem the degree of satisfaction of the objective(s) and the constraints is maimized: f i () min; i =, 2,, p subject to, g j () 0; j =, 2,, q, where min denotes fuzzy minimization and denotes fuzzy inequaity. According to Beman-Zadeh s approach [4] this probem can be transformed to the foowing optimization probem ma µ i (); R n ; i =, 2,, p + q subject to, 0 µ i (), where µ i () denotes degree of membership (acceptance) of to the respective fuzzy sets. This appication of Beman-Zadeh s approach [4] to sove such fuzzy optimization probem reaizes the min-aggregator. 2..2 Intuitionistic Fuzzy Optimization Mode Intuitionistic fuzzy optimization (IFO), a method of uncertainty optimization, is put forward on the basis of intuitionistic fuzzy sets, Atanassov [2]. It is an etension of fuzzy optimization in which the degrees of rejection of objective(s) and constraints are considered together with the degrees of satisfaction. This optimization probems simiar to fuzzy optimization probems can be represented as a two stage process which incudes (i) aggregation of constraints and objective(s) and (ii) defuzzification (maimization of aggregation function) According to IFO theory, we are to maimize the degree of acceptance of the IF objective(s) and constraints and to minimize the degree of rejection of IF objective(s) and constraints as ma R n{µ ()}; min{ν ()}; µ (), ν () 0; µ () ν (); 0 µ () + ν () ; =, 2,..., p + q where µ () denotes the degree of acceptance of from the th IFS and ν () denotes the degree of rejection of from the th IFS. According to Atanassov property of IFS, the conjunction of intuitionistic fuzzy objective(s) and constraints in a space of aternatives U is defined as A B = {, min{µ A (), µ B ()}, ma{ν A (), ν B ()} : U}, (8)
274 P.K. Naya and M. Pa: Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution which is defined as the intuitionistic fuzzy decision set (IFDS), where A denotes the integrated intuitionistic fuzzy objective/ goas and B denotes integrated intuitionistic fuzzy constraint set and they can be written as p A = {, µ A (), ν A () : U} = {, min µ i(), ma ν i() : U} (9) i= i= q B = {, µ B (), ν B () : U} = {, min µ j(), ma ν j() : U}. (0) j= j= Let the intuitionistic fuzzy decision set (8) be denoted by C, then min-aggregator is used for conjunction and ma operator for disjunction where, and p q C = A B = {, µ C (), ν C ()} U}, () µ C () = min{µ A (), µ B ()} = p+q min = µ () (2) ν C () = ma{ν A (), ν B ()} = p+q ma = ν (), (3) where µ C () denotes the degree of acceptance of IFDS and ν C () denotes the degree of rejection of IFDS. Therefore, µ C () µ (), ν C () ν (); p + q. (4) The formua can be transformed to the foowing system ma α, min β α µ (); =, 2,..., p + q β ν (); =, 2,..., p + q α β; and α + β ; α, β 0 where α denotes the minima acceptabe degree of objective(s) and constraints and β denotes the maima degree of rejection of objective(s) and constraints. The IFO mode can be changed into the foowing certainty (non-fuzzy) optimization mode as : ma(α β) α µ (); =, 2,..., p + q (5) β ν (); =, 2,..., p + q α β; and α + β ; α, β 0 which can be easiy soved by some simpe methods. 3 Bi-Matri Game A bi-matri game, Nash [8], can be considered as a natura etension of the matri game. Let I, II denote two payers and et M = {, 2,..., m} and N = {, 2,..., n} be the sets of a pure strategies avaiabe for payers I, II respectivey. By α ij and γ ij we denote the pay-offs that the payer I and II receive when payer I pays the pure strategy i and payer II pays the pure strategy j. Then we have the foowing pay-off matri A = α α 2 α n α 2 α 22 α 2n α m α m2 α mn ; B = γ γ 2 γ n γ 2 γ 22 γ 2n γ m γ m γ mn (6) where we assume that each of the two payers chooses a strategy, a pay-off for each of them is represented as a crisp number. We denote the game by Γ = {I, II}, A, B. For two person non zero sum muti objective game, mutipe pair of m n pay-off matrices can be written as A = α α 2 α n α 2 α 22 α 2n α m α m2 α mn ; A2 = α 2 α 2 2 α 2 n α 2 2 α 2 22 α 2 2n α 2 m α 2 m2 α 2 mn ;...
Journa of Uncertain Systems, Vo.5, No.4, pp.27-285, 20 275 B =... ; A n = β β2 βn β2 β22 β2n βm βm2 βmn... ; B n2 = α n α2 n αn n α2 n α22 n α2n n α n m α n m2 ; B2 = α n mn β n2 β2 n2 βn n2 β2 n2 β22 n2 β2n n2 β n2 m β n2 m2 β 2 β2 2 βn 2 β2 2 β22 2 β2n 2 βm 2 βm2 2 βmn 2 β n2 mn ;... where the payer I and the payer II have n and n2 number of objectives. Let the domain for the payer I be defined by D = D D 2... D n R n and that for the payer II be D 2 = D2 D2 2... D2 n2 R n2. 3. Nash Equiibrium Soution Nash [8] defined the concept of Nash equiibrium soutions (NES) in crisp bi-matri games for singe pair of payoff matrices and presented methodoogy for obtaining them. 3.. Pure Strategy Let I, II denote two payers and et M = {, 2,..., m} and N = {, 2,..., n} be the sets of a pure strategies avaiabe for payers I, II respectivey. Definition 7 A pair of strategies (row r, coumn s) is said to constitute a NES to a bi-matri game Γ if the foowing pair of inequaities is satisfied for a i =, 2,..., m and for a j =, 2,..., n: α is α rs ; γ rj γ rs (7) The pair (α rs, γ rs ) is nown as a Nash equiibrium outcome of the bi-matri game in pure strategies. A bi-matri game can admit more than one NES, with the equiibrium outcomes being different in each case. The NES concept for bi-matri games with singe pair of IF payoffs in pure strategy has been proposed by Naya and Pa []. 3..2 Mied Strategy In the previous section, we have encountered ony cases in which a given bi-matri game admits a unique or a mutipe of Nash equiibria. There are other cases, where the Nash equiibrium does not eist in pure strategies. We denote the sets of a mied strategies, caed strategy spaces, avaiabe for payers I, II by S I = {(, 2,..., m ) R m + : i 0; i =, 2,..., m and S II = {(y, y 2,..., y n ) R n + : y i 0; i =, 2,..., n and m i = } i= n y i = }, where R m + denotes the m dimensiona non negative Eucidean space. Since the payer is uncertain about what strategy he/she wi choose, he/she wi choose a probabiity distribution over the set of aternatives avaiabe to him/her or a mied strategy in terms of game theory. We sha denote this bi-matri game by Γ = {I, II}, S I S II, A, B. For S I, y S II, T Ay and T By are the epected pay-off to the payer I and payer II respectivey.the NES concept for bi-matri games with IF goas in mied strategies for singe pair of matrices has been proposed by Naya and Pa []. i=
276 P.K. Naya and M. Pa: Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution Definition 8 (Epected pay-off ): If the mied strategies and y are proposed by the payer I and payer II respectivey, then the epected pay-off of the payer I and payer II are defined by m n m n T Ay = i α ij y j and T By = i γ ij y j. (8) i= j= i= j= Definition 9 A pair ( S I, y S II ) is caed a NES to a bi-matri game Γ in mied strategies, if the foowing inequaities are satisfied. } T Ay T Ay ; S I T By T By (9) ; y S II and y are aso caed the optima strategies for the payer I and payer II respectivey. Then the pair of numbers V = T Ay, T By is said to be the Nash equiibrium outcome of Γ in mied strategies, and the tripet (, y, V ) is said to be a soution of bi-matri game. 4 Singe Objective Bi-Matri Game with IF Goa To derive the computationa procedure for NES with respect to degree of attainment of the IF goa in singe objective bi-matri games, first, we define some terms which are usefu in the soution procedure. Let the domain for the payer I be defined by and that for the payer II be defined by D = { T Ay : (, y) S I S II R m R n } R (20) D 2 = { T By : (, y) S I S II R m R n } R. (2) Definition 0 (IF goa ): A IF goa Ĝ for payer I is defined as a IFS in D characterized by the membership and nonmembership functions µĝ : D [0, ] and νĝ : D [0, ] or simpy, µ : D [0, ] and ν : D [0, ], (22) such that 0 µ () + ν (). Let Ĝ be the IF goa and Ĉ be the IF constraint in the space D, for the payer I, then according to Atanassov s property of IFS [2], the decision ˆD, which is a IFS, is a given by ˆD = Ĝ Ĉ = {, µ ˆD(), ν ˆD() : D } where, µ ˆD() = min{µĝ (), µĉ ()}; ν C () = ma{νĝ (), νĉ ()}, (23) where µ ˆD() denotes the degree of acceptance of IF decision set ˆD and ν ˆD() denotes the degree of rejection of IF decision set. Simiary, a IF goa for payer II is IFS on D 2 characterized by the membership function µĝ2 : D 2 [0, ] and nonmembership function νĝ2 : D 2 [0, ] such that 0 µĝ2 () + νĝ2 (). A membership, non-membership function vaue for a IF goa can be interpreted as the degree of attainment of the IF goa for a strategy of a payoff. Definition (Degree of attainment of IF goa): For any pair of mied strategies,(, y) S I S II, et the IF goa for payer I be denoted by Ĝ(, y), then the degree of attainment of the IF goa is defined as ma {µ Ĝ (, y)} and min{νĝ (, y)}. (24) The degree of attainment of the IF goa can be considered to be a concept of a degree of satisfaction of the fuzzy decision [2, 6], when the constraint can be repaced by epected pay-off. When the payer I and II choose strategies and y respectivey, the maimum degree of attainment of the IF goa is determined by the reation (24). Let the membership and non-membership function for the aggregated IF goa be µ (, y) and ν (, y) respectivey for the payer I and that for the payer II, they are µ 2 (, y) and ν 2 (, y) respectivey. Thus, when a payer has two different strategies, he/she prefers the strategy possessing the higher membership function vaue and ower nonmembership function vaue in comparison to the other.
Journa of Uncertain Systems, Vo.5, No.4, pp.27-285, 20 277 Definition 2 (Equiibrium soution of IF bi-matri game): Let A and B be the pay-off matrices of a singe objective bi-matri game Γ. If the payer I chooses a mied strategy, the payer II chooses a mied strategy y, the membership and non membership functions for payer I and payer II are µ (, y), ν (, y) and µ 2 (, y), ν 2 (, y) respectivey, and their epected pay-offs are T Ay and T By respectivey, then µ (, y) = µ ( T Ay); ν (, y) = ν ( T Ay) µ 2 (, y) = µ 2 ( T By); ν 2 (, y) = ν 2 ( T By). A pair (, y ) S I S II is caed a NES of the IF matri game, with respect to the degree of attainment of the IF goa in a singe objective bi-matri game if for any other mied strategies and y, the pair satisfies the foowing inequaities µ ( T Ay ) µ ( T Ay ) and ν ( T Ay ) ν ( T Ay ); S I µ 2 ( T By ) µ 2 ( T By ) and ν 2 ( T By ) ν 2 ( T By ); y S II. ( T Ay, T By ) is caed the Nash equiibrium outcome of the bi-matri game in mied strategies. Since the membership and non-membership functions are conve and continuous [6, 2], the soution with respect to the degree of attainment of the aggregated IF goa for Γ aways eist. Thus an equiibrium soution for IF games is defined with respect to the degree of attainment of the IF goas. 4. Optimization Probem for Payer I The inear membership and nonmembership functions [2] of the IF goa µ ( T Ay) and ν ( T Ay), for the payer I can mathematicay be represented as (Fig. ): µ ( T Ay) = ν ( T Ay) = ; T Ay a T Ay a a a ; a < T Ay < a 0; T Ay a ; T Ay a T Ay a a a ; a < T Ay < a 0; T Ay a where a and a are the toerances of the epected pay-off T Ay and µ ( T Ay) shoud be determined in objective aowabe region [a, a]. For payer I, a and a are the pay-off giving the worst and the best degree of satisfaction µ, ν ν a a µ T Ay Figure : Membership and nonmembership functions for payer I respectivey. Athough a and a woud be any scaars with a > a, Nishizai [4] suggested that, parameters a and a can be taen as a = ma ma T Ay = ma ma a ij y i j a = min min y T Ay = min min a ij. i j
278 P.K. Naya and M. Pa: Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution Therefore, the conditions a min a ij and a ma a ij hod. Thus payer I is not satisfied by the pay-off ess than a but is fuy satisfied by the pay-off greater than a. Now, T Ay a a a = ( a a a ) + A T ( a a )y = c() + T Ây, where c () = a a a and  = a aa. Simiary, if we define c(2) = a a a, then the membership and nonmembership functions can be written as µ ( T Ay) = ν ( T Ay) = 4.2 Optimization Probem for Payer II T Ay; a c () + T Ây; a < T Ay < a 0; T Ay a ; T Ay; a c (2) T Ây; a < T Ay < a 0; T Ay a Here we are consider payer II soution with respect to the degree of attainment of his fuzzy goa. The membership and nonmembership functions of the IF goa µ 2 ( T By) and ν 2 ( T By) can be represented as (Fig. 2): ; T By b µ 2 ( T By) = T By b ; b < T By < b b b 0; T By b ν 2 ( T By) = ; T By b T By b ; b b b < T By < b 0; T By b where b and b are the toerances of the epected pay-off T By and µ 2 ( T By) shoud be determined in objective µ 2, ν 2 aowabe region [b, b] and b min be written as where ˆB = B, c() b b 2 = b b b ν 2 b b µ 2 T By Figure 2: Membership and nonmembership functions for payer II b ij, b ma b ij. Aso, the membership and nonmembership functions can µ 2 ( T By) = ν 2 ( T By) = and c(2) 2 = b b b. ; T By; b c () 2 + T ˆBy; b < T By < b 0; T By b ; T By; b c (2) 2 T ˆBy; b < T By < b 0; T By b Since a the membership and non-membership functions of the IF goas are inear, the optima soution is equa to degree of attainment of the fuzzy goa for the matri game. Since S I and S II are conve poytopes,
Journa of Uncertain Systems, Vo.5, No.4, pp.27-285, 20 279 for the choice of such unimoda, inear membership and non-membership functions, the eistence of Nash equiibrium of the game is guaranteed. Thus the NES is equa to optima soution of the foowing mathematica probem P : s.t. and P 2 : s.t. ma µ ( T Ây ) and min ν ( T Ây ) m i = ; i 0; i =, 2,..., m. i= ma µ 2 ( T ˆBy) and min ν 2 ( T ˆBy) y y n y j = ; y i 0; i =, 2,..., n. j= Let (, y ) are the optima soution to P and P 2 respectivey. From the above two probems, we see that the constraints are separabe in the decision variabes, and y. Hence the two probems yied the foowing singe objective mathematica programming probem P 3 : s.t. ma {µ ( T Ây ) + µ 2 ( T ˆBy)} + min {ν ( T Ây ) + ν 2 ( T ˆBy)},y,y m n i = and y j = ; i 0; y i 0. i= j= Theorem If a the membership, non membership functions of the IF goas are inear, the NES with respect to the degree of attainment of the IF goa aggregated by a minimum component is equa to the optima soution of the non-inear programming probem P 4 : ma {λ + λ 2 + p 2 + q 2 β β 2 p q } (25),y,p,p 2,q,q 2,λ,λ 2,β,β 2 subject to Ây + c () em p e m ; Ây + c(2) em p 2 e m (26) ˆB + c () 2 en q e n ; ˆB + c (2) 2 en q 2 e n (27) T Ây + c () λ ; T Ây + c (2) β (28) T ˆBy + c () 2 λ 2 ; T ˆBy + c (2) 2 β 2 (29) λ + β ; λ 2 + β 2 ; λ β ; λ 2 β 2 (30) + 2 + 3 = ; y + y 2 + y 3 = (3) i, y i, λ i, β i, p i, q i 0, where e m = (,,..., ) T ; e n = (,,..., ) T. Theorem 2 If a the membership and non membership functions are inear, as introduced in Sections 4.4 and 4.5 and (, y ) is a NES for Γ = {I, II}, A, B, then (, y ) is a NES for Γ with respect to the degree of attainment of the IF goa. Coroary: Let the conditions a min α ij, a ma α ij and b min γ ij, b ma γ ij hod, then, (, y ) is a NES for Γ = {I, II}, A, B if and ony if (, y ) is a NES with respect to the degree of attainment of the IF goa. Thus, once an optima soution (, y, p, p 2, q, q 2, λ, λ 2, β, β 2) of the non-inear programming probem ( as in Theorem ) has been obtained, (, y ) gives an equiibrium soution of the bi-matri game. The degree of attainment of IF goas G and G 2 can then be determined by evauating T Ay and T By, then empoying the membership and non-membership function µ, ν and µ 2, ν 2. 5 Muti Objective IF Games For a singe objective bi-matri games, the NES is aready defined in Section 3 in this paper. Here we have deveoped the computationa procedure for NES with respect to degree of attainment of the IF goa in muti
280 P.K. Naya and M. Pa: Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution objective bi-matri games. Let us consider the muti objective bi-matri games (A, B ); =, 2,..., n and =, 2,..., n2. Let payer I s membership, non membership functions of the IF goa of the th pay-off matri be µ ( T  y); ν ( T  y); =, 2,..., n and that of payer II s membership, non membership functions of the IF goa of the th pay-off matri be µ 2( T ˆB y); ν 2( T ˆB y); =, 2,..., n2, where  = ˆB = a a A ; ĉ = a a ; =, 2,..., n a b b B ; ĉ = b b ; =, 2,..., n2. b Definition 3 (IF goa ) A IF goa Ĝ with respect to the th pay-off matri for payer I is defined as a IFS on D characterized by the membership and nonmembership functions µĝ : D [0, ] and νĝ : D [0, ] or simpy, µ : D [0, ] and ν : D [0, ] such that 0 µĝ () + νĝ (). Simiary, a IF goa Ĝ 2 with respect to the th pay-off matri for payer II is IFS on the set D 2 characterized by the membership function µĝ : D 2 2 [0, ] and nonmembership function νĝ : D 2 2 [0, ] such that 0 µĝ () + νĝ (). 2 2 For a bi matri game, if the payer I chooses the mied strategy and the payer II chooses the mied strategy y, the th pay-off for the payer I is represented by P = T A y and that the th pay-off for the payer II is represented by P 2 = T B y. 5. Aggregation Rue Let us consider the NES with respect to the degree of attainment of the IF goa aggregated by a minimum component, which is used in mutipe criteria decision maing, Beman and Zadeh [4]. A membership, nonmembership function vaue for a IF goa can be interpreted as the degree of attainment [4] of the IF goa for a strategy of a payoff. According to Atanassov s property [2] of IFS, the intersection of IF objective(s) and constraints is defined as (23). Basicay, in IF mutipe criteria decision maing the aggregation corresponds to the union of a the IFS intersection, and the soution is determined by maimizing the membership function and minimizing the non-membership function of aggregated function of the intersection. Thus the payer I and the payer II IF goas aggregated by minimum component are respectivey as µ (, y) = min µ ( T  y); ν (, y) = ma ν ( T  y) µ 2 (, y) = min µ 2( T ˆB y); ν 2 (, y) = ma ν( T ˆB y). A pair of strategies (, y ) is said to NES with respect to the degree of attainment o f the fuzzy goa aggregated by the minimum component for muti objective games (Â, ˆB ); =, 2,..., n and =, 2,..., n2, if for any other mied strategies and y, min µ ( T  y ) min min µ 2( T ˆB y ) min µ ( T  y ); ma µ 2( T ˆBy ); ma ν ( T  y ) ma ν ( T  y ) ν2( T ˆB y ) ma ν2( T ˆB y ). Thus the NES is equa to optima soution of the foowing mathematica probem P 5 : s.t. P 6 : s.t. ma m i= ma y n j= min µ ( T  y ) and min i = ; i 0; i =, 2,..., m. min µ 2( T ˆB y) and min y y j = ; y i 0; i =, 2,..., n. ma ν ( T  y ) ma ν2( T ˆB y)
Journa of Uncertain Systems, Vo.5, No.4, pp.27-285, 20 28 Therefore, the NES is equa to the optima soution of the foowing mathematica programming probem P7 : ma {min,y µ ( T Ây ) + min µ 2( T ˆBy) min ν ( T Ây ) min ν 2( T ˆBy)} s. t. i = ; y j = i i, y j 0, j where and y are the optima soutions. The process of soving IFO mode can be divided into two steps, which incude aggregation of IF objective(s) and constraints, aggregated by Atanassov s property [] of IFS AND defuzzification of the aggregated function so that the IFO mode is changed into crisp one. Let us consider the soution of the game with respect to the degree of attainment of the IF goa aggregated by minimum component [2]. Such rue is often used in decision maing [6]. Theorem 3 If a the membership functions of the IF goas are inear, the NES with respect to the degree of attainment of the IF goa aggregated by a minimum component is equa to the optima soution of the non-inear programming probem : subject to P 8 : ma {λ + λ 2 + p 2 + q 2 β β 2 p q } (32),y,p,p 2,q,q 2,λ,λ 2,β,β 2 Â y + c () e m p e m ; Â y + c (2) e m p 2 e m (33) ˆB + c () 2 e n q e n ; ˆB + c (2) 2 e n q 2 e n (34) T Â y + c () λ ; T Â y + c (2) β (35) T ˆB y + c () 2 λ 2 ; T ˆB y + c (2) 2 β 2 (36) λ + β ; λ 2 + β 2 ; λ β ; λ 2 β 2 (37) i = ; y j = (38) i i, y i, λ i, β i, p i, q i 0, i where e m = (,,..., ) T ; e n = (,,..., ) T, =, 2,..., n and =, 2,..., n2. Proof : The Kuhn-Tucer conditions to the probem is satisfied. Let S be the set of a feasibe soutions of the above non-inear programming probem P 8, then S φ, the nu set. From the first constraints (33), we have, Â y + c () e m p e m ; Â y + c (2) e m p 2 e m j T Â y + c () T e m p T e m ; T Â y + c (2) T e m p 2 T e m min[ T Â y + c () ] p ; ma [ T Â y + c (2) ] p 2, as T e m =. Simiary, from constraints (34) we can write, From constraints (35) and (36), we can write, min[ T ˆB y + c () 2 ] q ; ma[ T ˆB y + c (2) 2 ] q 2. min[ T Â y + c () ] λ ; ma [ T Â y + c (2) ] β min[ T ˆB y + c () 2 ] λ 2 ; ma[ T ˆB y + c (2) 2 ] β 2.
282 P.K. Naya and M. Pa: Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution Now, for arbitrary α = (, y, p, p 2, q, q 2, λ, λ 2, β, β 2 ) S, we have, λ + λ 2 + p 2 + q 2 β β 2 p q min[ T Â y + c () ] + min[ T ˆB y + c () 2 ] + ma [ T Â y + c (2) ] + ma[ T ˆB y + c (2) 2 ] β β 2 p q min[ T Â y + c () ] + min[ T ˆB y + c () 2 ] + ma [ T Â y + c (2) ] + ma[ T ˆB y + c (2) 2 ] ma [ T Â y + c (2) ] ma[ T ˆB y + c (2) 2 ] min[ T Â y + c () ] min[ T ˆB y c () 0, which shows that the optima vaue of the objective function is non positive. Thus, the soution (, y ), p = min [ T Â y + c () ]; p 2 = ma [ T Â y + c (2) ]; q = min [ T ˆB y + c () 2 ]; q 2 = ma[ T ˆB y + c (2) 2 ]; λ = min [ T Â y + c (2) ]; β = ma [ T Â y + c (2) ]; λ 2 = min [ T ˆB y + c (2) 2 ]; β 2 = ma[ T ˆB y + c (2) 2 ]; of the quadratic programming probem are feasibe and optima. Therefore α = (, y, p, p 2, q, q 2, λ, λ 2, β, β 2) S. The optima vaue of the objective function of the non-inear programming probem P 8 is 0. Conversey, et α S be the optima soution to the non-inear programming probem P 8. Since the optima vaue of the probem P 8 is zero, we get, From (38) and (39), we have, λ + λ 2 + p 2 + q 2 β β 2 p q = 0. (39) min[ Â y + c () ] p T e m = p ; ma [ T Â y + c (2) ] p 2 (40) min[ y + c () 2 ] q ; ma[ y + c (2) 2 ] q2 (4) and simiary, from (40) and (4), we have, Using (39),(42) and (43), we write, min[ Â y + c (2) ] λ ; ma [ T Â y + c (2) ] β (42) min[ y + c (2) 2 ] λ 2; ma[ y + c (2) 2 ] β2. (43) p + q p 2 q2 = λ + λ 2 β β2 min[ T Â y + c () ] + min[ T ˆB y + c (2) 2 ] ma [ T Â y + c (2) ] min[ T ˆB y + c (2) 2 ] min[ T ˆB y + c () 2 ] min[ T ˆB y + c (2) 2 ] min[ T Â y + c () ] + ma [ T Â y + c (2) ] + p + q p 2 q2. If we consider =, y = y as a particuar case in equations (40) and (4), we get, min[ T Â y + c () ] p ; ma [ T Â y + c (2) ] p 2 min[ T ˆB y + c () 2 ] q ; ma[ T ˆB y + c (2) 2 ] q2. 2 ]
Journa of Uncertain Systems, Vo.5, No.4, pp.27-285, 20 283 Therefore, q q2 min[ T Â y + c () ] + ma [ T Â y + c (2) ] + p + q p 2 q2 p p 2 min[ T Â y + c () ] ma [ T Â y + c (2) ] p min[ T Â y + c () ]; p 2 ma [ T Â y + c (2) ]. (44) Simiary, q min[ T ˆB y + c () 2 ]; q2 ma[ T ˆB y + c (2) 2 ]. (45) Hence from (40), (4), (44) and (45), we write, p = min[ T Â y + c () ]; p 2 = ma [ T Â y + c (2) ]; q = min[ T ˆB y + c () 2 ]; q 2 = ma[ T ˆB y + c (2) 2 ]. Finay, using (40) and (4) and above equations, we can write min[ T Â y + c () ] min[ T Â y + c () ]; min[ T ˆB y + c () 2 ] min[ T ˆB y + c () 2 ]. Therefore, we concude that the pair (, y ) is the NES with respect to the degree of attainment of the IF goa aggregated by the minimum component for a bi-matri game. Soution Procedure: The process of soving IFO mode can be divided into two steps, which incude aggregation of IF objective(s) and constraints, aggregated by Atanassov s property [] of IFS and defuzzification of the aggregated function so that the IFO mode is changed into crisp one. Let us consider the soution of the game with respect to the degree of attainment of the IF goa aggregated by minimum component. Basicay, in IF decision maing the aggregation corresponds to the union of a the IFS intersection, and the soution is determined by maimizing the membership function and minimizing the non-membership function of aggregated function of the intersection. 6 Iustrative Eampe Let us consider the payer I and payer II to have three strategies and three objectives which are defined in the foowing pair of matrices as A = B = 2 4 5 5 8 9 6 2 7 7 9 0 5 8 9 3 2 ; A 2 = ; B 2 = 8 0 3 2 4 6 8 5 0 9 2 4 2 8 4 ; A 3 = ; B 3 = 8 7 5 3 6 6 0 4 8 5 0 6 2 7 4 The toerances are arbitrary fied for each matrices for each membership and non-membership functions of the payers. Let the IF goas for the first objective of the payer I be represented by the foowing inear membership and non-membership functions, µ ( T A y) = ν( T A y) = ; T A y 9 T A y 2 7 ; 2 < T A y < 9 0; T A y 2 ; T A y 2 T A y 9 7 ; 2 < T A y < 9 0; T A y 9.
284 P.K. Naya and M. Pa: Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution Simiary, et the IF goas for the first objective of the payer II be represented by the foowing inear membership and non-membership functions, ; T B y 9 µ 2( T B y) = T B y 0 9 ; 0 < T B y < 9 0; T B y 0 ν2( T B y) = ; T B y 0 T B y 9 9 ; 0 < T B y < 9 0; T B y 9 Simiary for others. The soutions are obtained using LINGO software with formuating the probem by the hep of P 8 giving NES. The resuts for some combinations is shown in the foowing tabe: 7 Concusion NES 2 3 y y 2 y 3 V (A, B ) 0.8234 0.76 0.0590 0.88 0.0000 0.89 3., 5.52 (A 2, B 2 ) 0.0000 0.000 0.9000 0.0000 0.3925 0.6075 3.69, 5.33 (A 3, B 3 ) 0.0000.0000 0.0000 0.0000.0000 0.0000 6, 6 In this paper, we have presented a mode for studying muti-objective bi-matri games with IF goas. In this approach, the degree of acceptance and the degree of rejection of objective and constraints are introduced together, which both cannot be simpy considered as a compement each other and the sum of their vaue is ess than or equa to. Since the strategy spaces of payer I and payer II coud be poyhedra sets, we may aso conceptuaize constrained IF bi-matri game on the ines of crisp constrained bi-matri games. On the basis, we have defined the soution in terms of degree of attainment of IF goa in IFS environment, and find it by soving an mathematica programming probem. A numerica eampe has iustrated the proposed methods. This theory can be appied in decision maing theory such as economics, operation research, management, war science, etc. Acnowedgement The authors woud ie to than the anonymous referees for vauabe commends and aso epress appreciation of their constructive suggestions to improve the paper. References [] Angeov, P.P., Optimization in an intuitinistic fuzzy enviornment, Fuzzy Sets and Systems, vo.86, pp.299 306, 997. [2] Atanassov, K., Intuitionistic Fuzzy Sets: Theory and Appications, Physica-Verag, 999. [3] Basar, T., and G.J. Osder, Dynamic Non-Cooperative Game Theory, Acadenic Press, New Yor, 995. [4] Beman, R.E., and L.A. Zadeh, Decision maing under fuzzy Enviornment, Mangement Science, vo.7, pp.209 25, 970. [5] Campos, L., Fuzzy inear programming modes to sove fuzzy matri games, Fuzzy Sets and Systems, vo.32, pp.275 289, 989. [6] Dubois, D., and H. Prade, Fuzzy Sets and Systems, Academic Press, New Yor, 980. [7] Maeda, T., Characterization of the equiibrium strategy of two-person zero-sum games with fuzzy payoffs, Fuzzy Sets and Systems, vo.39, pp.283 296, 2003. [8] Nash, J.F., Non cooperative games, Annas of Mathematics, vo.54, pp.286 295, 95. [9] Naya, P.K., and M. Pa, Intuitionistic fuzzy bi-matri games, Notes on Intuitionistic Fuzzy Sets, vo.3, no.3, pp. 0, 2007. [0] Naya, P.K., and M. Pa, Bi-matri games with interva pay-offs and its Nash equyiibrium strategy, Journa of Fuzzy Mathematics, vo.7, no.2, pp.42 435, 2009.
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