Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—9

Contents lists available at ScienceDirect

ELSEVIER

Pacific Science Review A: Natural Science and Engineering

journal homepage: www.journals.elsevier.com/pacific-science-review-a-natural-science-and-engineering/

Aspiration level approach to solve matrix games with I-fuzzy goals and I-fuzzy pay-offs

Mijanur Rahaman Seikh a' *, Prasun Kumar Nayak b, Madhumangal Pal c

a Department of Mathematics, Kazi Nazrul University, Asansol, 713 340, India b Department of Mathematics, Midnapore College(Autonomous), Midnapore, 721 101, India

c Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721 102, India

ARTICLE INFO ABSTRACT

The objective of this paper is to develop a new solution methodology for matrix games, in which goals are viewed as intuitionistic fuzzy sets (IFSs) and the elements of the pay-off matrix are represented by triangular intuitionistic fuzzy numbers (TIFNs). In this methodology, a suitable ranking function is defined to establish an order relation between two TIFNs, and the concept of intuitionistic fuzzy (I-fuzzy) inequalities is interpreted. Utilizing these inequality relations and ranking functions, a pair of linear programming models is derived from a pair of auxiliary intuitionistic fuzzy programming models. Based on the aspiration levels, this pair of linear programming models is solved to determine the optimal strategies for both players of the game. The proposed method in this paper is illustrated with a voting share problem to demonstrate the validity and applicability of the method.

Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article history: Available online xxx

Keywords:

Intuitionistic fuzzy sets Triangular intuitionistic fuzzy numbers Fuzzy game theory Mathematical programming Ranking function

1. Introduction

In traditional game theory, the precise assessment information is difficult due to a lack of information about the exact values of certain parameters and the fuzzy understanding of various situations by players. Fuzzy set theory, which is a very useful tool in game theory, has achieved a substantial amount of success. (Campos [9], Nishizaki and Sakawa [27,28], Bector and Chandra [8], Nayak and Pal [21,22], Li [14,16], Vidyottama and Chandra [37], Vijay et al. [38,39], Liu and Kao [18], Cevikel and Ahlatcioglu [10], Kacher and Larbani [12], Seikh et al. [32,33]). However, a fuzzy set only employs a membership degree. The degree of non-membership is automatically equal to the complement to 1. In real situations, however, players/decision makers often do not express the degree of non-membership as the complement to 1 . Players/decision makers may exhibit some degree of hesitation. Therefore, a fuzzy set has no means of incorporating a degree of hesitation.

* Corresponding author. E-mail addresses: mrseikh@ymail.com (M.R. Seikh), prasunkrnayak@gmail.com (P.K. Nayak), mmpalvu@gmail.com (M. Pal).

Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University.

The intuitionistic fuzzy set (IFS), which is a generalization of fuzzy set theory, was introduced by Atanassov [4,5]; it is suitable for solving problems concerning vagueness. An IFS is characterized by two functions—a function that expresses the degree of membership and a function that expresses the degree of non-member-ship—thus, the sum of both values is less than or equal to 1. The degree of hesitation is equal to one minus the sum of the degree of membership and the degree of non-membership. Therefore, the concept of the IFS is considered to be an alternative approach for defining a fuzzy set in cases where available information is insufficient for defining an imprecise concept by conventional fuzzy sets. Therefore, an IFS can be employed to simulate the human decision-making process and any activity that requires human expertise and knowledge, which are inevitably imprecise or not completely reliable. Atanassov [6] discussed an open problem about the interpretation of an IFS in different optimization problems. Angelov [3] was the first researcher to answer this problem by implementing an optimization technique in an intuitionistic fuzzy environment. However, an IFS can be applied to game problems as the players have some degree of hesitation about appropriate pay-off values and the selection of a strategy for each of the pay-offs.

Intuitionistic fuzziness in matrix games is generally applied using two methods: in the first method, players have I-fuzzy goals; in the second method, the elements of the pay-off matrix

http://dx.doi.org/10.1016/j.psra.2016.03.001

2405-8823/Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

are represented by intuitionistic fuzzy numbers (Li [13], Seikh et al. [30]). Recent studies have focused on the application of an IFS to resolve game problems. Atanassov [6] described a game problem using an IFS. Li and Nan [17] developed a nonlinear programming approach to matrix games with pay-offs of Ata-nassov's IFS. Aggarwal et al. [1,2] extended the duality results for two-person zero-sum matrix games with fuzzy goals and fuzzy pay-offs to an 1-fuzzy scenario. Nan et al. [20] implemented a lexicographic method for matrix games in which pay-offs are represented by TIFNs. Li et al. [15] employed a bi-objective programming approach to solve a matrix game with pay-offs of TlFNs. Bandyopadhyay et al. [7] solved matrix games with intuitionistic fuzzy pay-offs using a score function. Nan and Li [19] outlined a linear programming approach to solve matrix games with 1-fuzzy goals. Seikh et al. [31,34,36] investigated matrix games in which goals are represented as an 1FS or elements of a pay-off matrix are represented by TIFNs. Nayak and Pal [23,24] constructed auxiliary linear programming models to solve bimatrix games with goals expressed by an 1FS. Seikh et al. [29,35] applied T1FNs to bi-matrix games. No study has investigated matrix games with 1-fuzzy goals and 1-fuzzy pay-offs. In this paper, a new methodology, in which goals are represented as an 1FS and elements of a pay-off matrix are represented by T1FNs, is introduced to solve matrix games. The idea of double 1-fuzzy inequalities, i.e., the 1-fuzzy inequalities and the parameters that are represented by 1-fuzzy numbers, is outlined. The concept of T1FNs and their arithmetic operations and cut sets are recalled. A new order relation is proposed to rank the two T1FNs. A pair of linear programming models is derived from a pair of auxiliary 1-fuzzy programming models using these ranking order relations. These two models are solved by aspiration levels, and the optimal strategies for both players are obtained.

The paper is organized as follows: 1n Section 2, some definitions and preliminaries about an 1FS and T1FNs are recalled and a ranking function is defined to establish an order relation between two T1FNs. Section 3 describes the application of an 1FS in optimization and the concept of double 1-fuzzy constraint conditions. The main problem about the matrix games with 1-fuzzy goals and 1-fuzzy pay-offs is formulated in Section 4. The solution procedure of these games is conceptualized by the degree of acceptance and the degree of rejection of the 1-fuzzy aspiration levels for two players. The results are illustrated by considering a voting share problem in Section 5. Section 6 reflects the conclusions of this paper.

2. Definitions and preliminaries

2.1. Intuitionistic fuzzy sets

The intuitionistic fuzzy set, which was introduced by Atanassov [5], is characterized by two functions that express the degree of belongingness and the degree of non-belongingness.

Definition 1. Let U = {x1,x2, •••,xng be a finite universal set. The Atanassov's intuitionistic fuzzy set (1FS) A in a given universal set U is an object with the form

Mb (x)

where the functions

degree of membership and

U/ [0,1] and gb : U/[0,1] define the

the degree of non-membership, respectively, of an element xeU to the set A4U such that they satisfy the following conditions:

0 < mA(x) + ^A(x)<1, V xeU

which is known as an intuitionistic condition. The degree of acceptance ^(x) and the degree of non-acceptance v~(x) can be arbitrary.

Definition 2. Let A and B be two 1FSs in the set U. Then, the intersection of A and B are defined as follows:

2.2. Triangular intuitionistic fuzzy number (TIFN)

1n this section, the definitions are derived from Li [13].

Definition 3. (T1FN) The T1FN a= ((al, a, ar); wa, ua > is a convex 1FS on the set < of real numbers with ai < a < ar, whose membership function and non-membership function are defined as

x - al ..

wa--: if al < x < a

a — a-

wa : if x = a

ar — x ..

wa-: if a < x < ar

ar — a

0 : if x < a- or x > ar

Ma(x) = '

'a - x + (x - a¡)ua

ná(x) =

a - a¡

ua : if x = a

x - a + (ar - x)ua

: if a¡ < x < a

: if a < x < ar

1 : if x < a¡ or x > ar

as depicted in Fig. 1.

The values wa and ua represent the maximum degree of membership and the minimum degree of non-membership, respectively, such that they satisfy the following conditions: 0 < wa < 1,0 < ua < 1 and 0< wa + ua < 1. Let pa (x) = 1 — Me{x)—n~(x), which is referred to as the 1-fuzzy index of an element x in the T1FNa. 1t is the degree of indeterminacy membership of the element x to the T1FN a.

1f mj(x)+n~(x) = 1 V x e U, then a = ((al, a, ar); wa, ua > is reduced to a = ((al, a, ar); wa,1 — wa >, which is a triangular fuzzy number (TFN). The definition of a T1FN is a generalization of the definition of the TFN, which was introduced by Dubois and Prade [11]. Two new parameters—wa and ua—are introduced to reflect the confidence level of a T1FN and the non-confidence level of a T1FN, respectively. Therefore, a T1FN may express more uncertainty compared with a TFN. The set of all T1FNs is denoted by ~(<).

___' ____ i \ i \ / i \ / 1 \ / 1

i / \ i / ■ /

i / i / N.. 1 i \ i i \ -►

Fig. 1. Triangular intuitionistic fuzzy number.

Definition 4. (Arithmetic Operations) Let a and ~ be two TIFNs denoted by a = ((al, a, ar); wa, ua > and ~ = ((bl, b,br); wb, ub>; the arithmetic operations are defined as follows: Addition:

a + b = ((at + b,a + b, ar + br); minfwa,wb}, maxfua, ub}>. Subtraction:

a - b = ((at - br, a - b, ar - bt); minfwa, wb}, maxfua, ub}>. Scalar Multiplication:

ka = ( ((kai'ka'kar); Wa'Ua>if k > 0, where k is a real number. \((kar, ka, kai); wa, Ua>if k < 0'

Definition 5. (Cut sets of TIFNs) For any a 2 [0,wa], the a-cut set ofTIFN a = ((ai,a, ar); wa, ua> can be expressed as a crisp subset of <, which is denoted by aa = {x| Ma(x) > a, x 2 < }. According to the definition of the TIFN, aa is a closed interval denoted by ~a = [La(~),Ra(a)]. From (2),

[La(~), Ra(~)] =

a(a - ai) _ a(ar - a} al +--^- ; ar -

Similarly, for any ß e[ua,1], a ß-cut set of the TIFN a = ((a^a, ar); wa, ua) can be expressed as a crisp subset of <, as denoted by aß = {x|na(x)< ß, x 2 < }. aß is a closed interval denoted

by aß = [Lß(a), Rß(~)]. From (3),

[Lb(a), Rß(a)j

(1 - ß)a + (ß - ua)ai (1 - ß)a + (ß - Ua)ar

1 - Ua

1 - Ua

In the next context, we discuss the value index and ambiguity index of a TIFN.

Definition 6. Let aa = [La(a), Ra(a)] and ab = [Lb(a), Rb(a)] be any a-cut set and a b-cut set, respectively, of the TIFN a = ((ai,a, ar); wa, ua>. Then, the values of the membership function and non-membership function are defined as.

Dm(a) -

La(a)+Ra(à)

f (a)da

and Dv(a) = / Ju.

1 Lß(~) + Rß(~)

g(ß)dß. where f (a) is a non-

negative and non-decreasing function on the interval [0, wa],

which should satisfy the conditions f(0) = 0 and / f (a)da = wa.

g(ß) is a non-negative and non-increasing function on the interval [ua,l], which should satisfy the conditions g(l) = 0 and

[ g(ß)dß = 1 - ua. The functions f(a) and g(ß) may be considered

as weighting functions and are expressed as

f (a)= W, a2[0, waj

1 - Ua

Thus, the values of membership function and non-membership function of the TIFN a = ((ai a, ar); wa, ua> are calculated as follows:

Dm(~) =

Dn(~)=

" a(a - al) _ a(ar - a)

ai +-----r ar -

a , (al + 4a + ar)

—da = ——F--Wa

(1 - ß)a + (ß - Ua)ai + (1 - ß)a + (ß - Ua)a

1 - Ua

(al + 4a + ar )

1 - ß 1- Ua

(1 - Ua)

Dm(à) and Dv(a) reflect information about the membership degrees and non-membership degrees.

Similarly, the ambiguities of the membership function and non-membership function for any TIFN a are defined by

Em (a) = J [Ra(a) - La (a)]f (a)da

and Ev(a)= / [Rb(a) -Lb(a)jg(b)db, respectively. Ra(a)-La(a) and

Rb(a)-Lb(~) represent the length of the interval ~a and the length of the interval ab, respectively. Therefore, Em(a) and En(a) represent the measure of uncertainty in a.

Using (4) and (5), the ambiguities of the membership function and the non-membership function of a TIFN aa are calculated as

Em(a) =(^-a>jWa and Ev(a) = (^j (1 - Ua)

Proposition 1. Let â ={(ai;a,ar);wa,uai and ~ = {{bip,br);wb,ubi be any two TIFNs with wa = Wb, ua = ub and let k be any nonnegative real number. Then, the following equalities are valid.

(i) Dm(k~ + b) = kDm(a) + DM(b)

(ii) Dv(kâ + ~) = kDv(â) + Dv(~)

(iii) Em(ka + b) = kEm(a) + EM(~)

(iv) Ev(kâ + ~) = kEv(â) + Ev(~).

Definition 7. (Value index and ambiguity index) The value index and ambiguity index of any TIFN âa are defined as follows:

value index Vx(a) = 1Dv(a) + (1 - 1)Dm(a) and

ambiguity index = !Em(a) + (1 - l)Ev(â),

where l 2 [0,1] represents the weight of the preference information about the players/decision makers (DMs). l 2 [0, 1/2 ) indicates the pessimistic attitude of the players/DMs towards uncertainty; l 2( 1/2,1] indicates the optimistic attitude of the players/DMs towards uncertainty, where l = 1/2 shows that players/DMs have indifferent attitude. Thus, the value index and ambiguity index may reflect the attitudes of the players/DMs about the TIFNs.

Proposition 2. Let a = ((a-,a,ar);wa,ua> and ~ = ((b-,b,br); Wb, ub> be any two T1FNs with wa = wb and ua = ub. Then, for any real number k e <, the following equalities are valid:

(i) vA (k~ + b) = kVi(a) + vA(a)

(ii) Ai(kb + ~) = kAi(a) + Ai(b).

which can be easily solved by various mathematical programming methods.

In the following context, we provide an interpretation of inequality relations in an I-fuzzy environment.

In the following definitions, a ranking function is defined based on the difference between the value index and the ambiguity index to obtain an order relation between two TIFNs.

Definition 8. A ranking function (or defuzzification function) is the function F: ~(<) /<, where ~(<) is a set of all TIFNs that are defined on <, which maps each TIFN into a real line. Let b be a TIFN. Then, Fx(a) = Vx(a)-Ax(a), where l 2 [0,1].

Assume that a and ~ be two TIFNs and l 2 [0,1 ] be any real number. Then, a new order relation between a and b is defined as follows:

(i) a7LFbiffFA(a)<FA(b)

(ii) adLFbiff FA(a) > FA(b)

(iii) a=LFbiff Fi(a) = FA(b)

The symbol' 7 I F' is an intuitionistic fuzzy version of the order relation '<' on the set of real numbers and has the linguistic interpretation as 'essentially less than or equal to'. Similarly, the symbols ' d I F' and '=I F' are the intuitionistic fuzzy versions of the order relations '>' and '=' on the set of real numbers and have the linguistic interpretations 'essentially greater than or equal to' and 'essentially equal to', respectively.

This proposed ranking method satisfies axioms, namely, the reasonable properties proposed by Wang and Kerre [40].

Proposition 3. Let a and b be any two TIFNs. Then, for any real number k, the following equality holds

Fi( ka + b) = kFi (a) + Fx( b).

This shows that the proposed ranking function is linear for these types of TIFNs.

3.1. Interpretation of I-fuzzy inequalities

Let x, a e < and let po, qo (>0) e Then the intuitionistic fuzzy statement 'xd p0 q0 a' to be read as " x is essentially greater than or equal to a with tolerances p0 and q0" is expressed in terms of the following membership and non-membership functions (depicted in Fig 2):

¡1; x > a

1 -^TT^; (a - P0) < x < a P0

0; x < (a - P0)

¡1; x < a

1 + (a - P0 - x); (a - P0) <x< (a - P0 + q0) q0

0; x > (a - P0 + q0)

The DM is completely satisfied if x > a and the tolerances p0, q0 exist, in which the DM is partially satisfied, i.e., for (a-p0) < x < (a-p0 + q0), the sum of the degrees of acceptance and the degree of rejection is less than or equal to one. He/she will not be satisfied if x < (a-p0). According to the IFO technique described in the I-fuzzy inequality relation 'x d p0 q0, a' is expressed as

r 1 a - x . „ a - p0 - x X1 < 1--and h1 < 1 +--—-

x > a - po(1 - Xi) and x > (a - po) + qo(1 - hi),

3. Application of IFS to optimization model

Any optimization model includes objective(s) and constraints.

1ntuitionistic fuzzy optimization (1FO), is an extension of fuzzy

optimization, in which the degree of rejection of objective(s) and

constraints are considered with the degree of satisfaction (refer to

Angelov [3], Nayak and Pal [24]). Let U be the universal set. Let Gi,

i = 1,2, ...,r, be the set of r goals and Cj, j = 1,2, ...,m, be the set of m

constraints, each of which can be characterized by an 1FS on U. The

1-fuzzy decision D = (G1 n G2 n ... n Gr)n(C1 n C2 n . „ n Cm) is an 1FS

defined as D = {(x, mD(x), vD(x)|x eU>}, where

mD(x) = min{mGi(x), mq(x)} and vd(x) = max{VGi(x), Vq(x)} tj <J

According to 1FO theory, we can maximize the degree of acceptance of the 1F objective(s) and the constraints to minimize the degree of rejection of the 1F objective(s) and constraints. Let X, h denote the minimal degree of acceptance and the maximal degree of rejection, respectively, of the objective(s) and constraints. Then, the 1-fuzzy optimization problem is transformed into the following crisp optimization model—Angelov [3]—as

Subject to

maxfX - hg

X < mci(x); i = 1,2,..., r

h > vci(x); i = 1,2, X < mcj(x); j = 1,2, ...m h > vcj(x); j = 1, 2,

X > h;X + h < 1; X,h > o

where £i and hi denote the minimal degree of acceptance and the maximal degree of rejection, respectively, that satisfy the conditions 0 < X1 < 1 and 0 < h1 < 1.

Similarly, the I-fuzzy statement 'x 7 r0, s0 a' to be read as "x is essentially less than or equal to a with tolerances r0 and s0" is expressed in terms of the following membership and non-membership functions:

Fig. 2. Membership and non-membership functions for the statement x d p0, q0 a.

M2(x) =

v2(x) =

1; 1 -0;

(a - x) ;

a <x < (a + ro) x > (a + ro)

(a + ro - x) ; So ;

x > (a + ro)

(a + ro - so) <x < (a + ro) x < (a + ro - so)

The DM is completely satisfied if x < a and the tolerances r0 and s0 exist, in which the DM is partially satisfied, i.e., for (a + r0 — s0) < x < (a + r0), the sum of the degree of acceptance and the degree of rejection is less than or equal to one. He/she will not be satisfied ifx > (a + r0). In a similar manner, the I-fuzzy inequality relations x7 ro, s0 a is expressed as

x < a + r0(1 - X2) and x < (a + r0) - s0(1 - ^2) where 0 < X2 < 1 and 0 < r|2 < 1.

3.2. Interpretation of double I-fuzzy constraints

In this subsection, the concept of double I-fuzzy inequalities, i.e., the I-fuzzy constraints that involve I-fuzzy numbers, is interpreted. Based on the resolution method that was proposed in the previous section, we extend the interpretation of I-fuzzy inequalities to the case in which the parameters and the adequacies are also I-fuzzy numbers. Let S,B and C be the m x n matrix, the m x 1 vector and the n x 1 vector with entries from ~(<) and the double I-fuzzy constraints under consideration be expressed as S X d~F~ B and S Y7PlC, with the adequacies/tolerances p, ~ and S, s, respectively, which' are also I-fuzzy vectors. Therefore, the double I-fuzzy constraint conditions are expressed as

SXdLfß0-p,q

SjXdIFBj -p(1 - ?i), o < fi < 1

SjX dIF

Bj - Pj + 4j(1 - hi); o < hi < 1

S'Y 7lf

Si Y 7$ IFCi + ri(1 - ?2), o < f2 < 1

SiY <LF(Ci + ~rt) - Si (1 - h2), o < h2 < 1

where i = 1,2, ...,m and j = 1,2, ..., n. d I:F and 7I:F are the relations between two I-fuzzy numbers, which preserve the ranking when I-fuzzy numbers are multiplied by positive scalars. p~j and qj (j = 1, 2 ..., n) represent the jth component of the I-fuzzy vectors p and~, respectively. Similarly, n and s; (i = 1,2, ...,m) represent the ith component of I-fuzzy vectors r and ~s, respectively.

( an a12

a21 a22

V am1 am2

Bn a1n \

41. Mixed strategy

Consider the game G with no saddle point, i.e., maxfmin aij} s minfmaxa,j}. To solve these games, Neumann [25] introduced the accept of mixed strategy in classical form. We denote the sets of all mixed strategies, which are referred to as strategy spaces, that are available for players I and II by

Sj = { x = (x1,x2, xm)2<m

Sil H У = (У1; У 2 ; Уп)2<+

'■andj^xi = 1 ^ i=1 J

1,2, nand^yj j=1

where <m denotes the m-dimensional non-negative Euclidean space. Thus, by a crisp two-person zero-sum matrix game G, we indicate the triplet G = (Si, Sn, A). As each player is uncertain about which strategy he/she will choose, he/she will choose a probability distribution over the set of alternatives that are available to him/her or a mixed strategy in terms of game theory.

4.2. Matrix games with I-fuzzy goals and I-fuzzy pay-offs

Let SI, Sn be the strategy spaces for player I and player II, respectively, and let A = (âij)mxn be the pay-off matrix, where each aij = ((aiji; aij; aijr); Watj, ищ>(i = 1,2, ,„m;j = 1,2, ,„n) is the TIFN defined in Section 2.2. Let v and WW be two TIFNs that represent the aspiration levels of player I and player II, respectively. Then, a two-person zero-sum matrix game with I-fuzzy goals and I-fuzzy payoffs, as defined by the triplet (SI,Sn,A;V, dp'l;WW, ), where diF~ and 7 H, are explained in Section 3.2 and p, V, and r, S are the

p,q r,s f.i. >

adequacies/tolerance levels for player I and player II, respectively. In the following definition, we refer to a two-person zero-sum matrix game with I-fuzzy goals and I-fuzzy pay-offs as an intuitionistic fuzzy matrix game denoted by IFG.

The solution concept of IFG is defined as follows:

Definition 9. A point (x,y) the IF matrix game IFG if

eSi x Sn is referred to as a solution of

(i) b Aydpisí>' for all y 2 Sn and

(ii) xIAy7rJsW' for all x 2 SI.

Using this definition and employing a resolution procedure for the double I-fuzzy constraints proposed in Section 3, we can construct the following pair of I-fuzzy linear programming problems for player I and player II, respectively.

4. Mathematical model of a matrix game

Let {1, 2,..., m} be the set of pure strategies that are available to player I and let {1,2, ...,n} be the set of pure strategies that are available for player II. When player I chooses the pure strategy i and player II chooses the pure strategy j, then aij is a payoff for player I and -aij is a payoff for player II. The two-person zero-sum matrix and game G can be represented as a pay-off matrix, Owen [26]

Subject to

max{%1 - h1>

xTCydlF{v - p(1 - f1)> for all yeSn, xTAydI F{v - p + S(1 - h1)> for all yeSn, X2S1, f1 > h1, f1 + h1 < 1 x, ?1, h1 > o

Subject to

max{X2 - h2>

xTÄy7IF {w + r(1 - X2)} for all xeS,, xTÄy7IF{w + r - s(1 - h2)} for all xeSj, y^Sii , X2 > h2; X2 + h2 < 1 y, X2 ; h2 > 0

where dIF and 7IF are the relations between twoT1FNs. As S1 and S11 are convex polytopes, only the extreme points of the sets S1 and S11 are considered in the constraint conditions ofequations (11) and (12). Therefore, these problems will be converted as

max{?1 - h1}

xTAjdIF{v- pj(1- ^(j = 1,2,..., n),

Subject to

x^-d^v - j + ~(1 - hi)}(j = 1,2,..., n),

£x = 1, i=1

Xi > hi,Xi + hi < 1,

X, ?1, h1 > 0

Subject to

max {X2 - h2}

xTÄiy7IF{w + rf (1 - X2)}(i = 1,2,..., m), xT~iy7IF{(w + ri) - Si(1 - h2)}(i = 1,2,..., m),

^Vj = 1

X2 > h2, X2 + h2 < 1, y, X2, h2 > 0

max{X1 - h1}

Fl a ij xi > Fl

these problems can be transformed into linear programming problems as

max{^1 - h1}

Fi(Ä)jx > FM-Fj1 -X1 )(j = 1,2,..., n),

Fl(Ä)jx>Fi(y)-Fi (pj) + Fjj ( j (1 -h1)(j = 1,2,...,n),

Subject to

Y,xi = 1 i=1

X1 > h1, X1 + h1 < 1, x,?1,h1 > 0

max{(,2 - h2}

Fjj(Ä j < Fi(w)+Fi(rj)(1 - X2 )(i = 1,2,..., m),

FjjÄ .y <Fi(w)+Fi(rj) -FjjSj) (1 -h2)(i = 1,2,...,m),

Subjectto n j l l l

Y^yi = 1 j=1

X2 > h2,X2 + h2 < 1, y, X2, h2 > 0

Here Fi(A)i (Fi(A)j) denotes the ith row (jth column) of Fi(A), where i = 1,2, ...,m and j = 1,2, ...,n.

The two linear programming problems (15) and (16) can be written as

Subject to

£ [Fjt jxi > Fi(y) - Fifo) (1 - X1 )(j = 1, 2,..., n), P [Fi(~j)]xi > Fj(y) - FÀ(pjj + Fjj( j (1 - h1)(j = 1, 2,..., n),

^xi = 1 i=1

X1 > h1, X1 + h1 < 1, x, ?1, h1 > 0

Subject to

max{£2 - h2 }

£ [Fi(~ij)]yj < Fj(w) + Fj(ri)(1 - X2)(i = 1, 2,..., m),

P [Fjj (~j)]yj < Fj(w) + Fj(ri) - Fjj (Si) (1 - h2)(i = 1,2,..., m),

J=1 v 7

£yj = 1 j=1

X2 > h2, X2 + h2 < 1, y, X2, h2 > 0

where Ai (Aj) denotes the ith row (jth column) of A (i = 1,2, ...,m; j = 1,2, ...,n). By utilizing the ranking function Fi: ~ /<(l e[0,1]) (defined in Section 2.2, Definition 8), which preserves the ranking when 1-fuzzy numbers are multiplied by non-negative scalars,

To solve the I-fuzzy matrix game IFG, we have to solve the crisp linear programming problems (17) and (18) for player I and player II, respectively. If (b,X1,b1) is an optimal solution of (17), then x is an optimal strategy for player I, X1 is the minimal

degree of acceptance and hA1 is the maximal degree of rejection to the aspiration level v of player 1. A similar interpretation can also be given to the optimal solution (A, ?2, h2) of problem (18). The results discussed in this section are summarized in the following theorem.

v = ((160,170,180); 0.6, 0.2) and W = ((145,160,165); 0.6, 0.2).

The tolerance levels for player 1 and player 11 are

p-1 = p2 = p = <(14, 16, 20); 0.6, 0.2), S 1 = S2 = S = ((10,12,16); 0.6, 0.2) and S1 = ~2 = r<(13, 15, 22 ); 0.6, 0.2), also s1 = s2 = s = ((9, 11, 18 ); 0.6, 0.2).

Theorem 1. The 1-fuzzy matrix game IFG described by IFG =(S/, S//, A ; v, dfF~; W, 7—) is equivalent to the two crisp linear programming probfems (17)and (18), which can be easily solved by the simplex method.

The value indexes and ambiguity indexes of = ((ajjl, ay, j; waij, uaiji (i = 1,2; j = 1,2) can be obtained as follows:

5. Application to voting share problem

1n this section, a voting share problem in a particular region is considered to demonstrate the application of the proposed methodology.

Assume that the two major political parties P1 and P2 participate in a parliamentary constituency election in a particular region. Let the secretaries of both the parties P1 and P2 aim to enhance their expected votes under the circumstance that the total number of voters in that region is fixed. The voting share percentage of one party increases while the voting share percentage of another party decreases. Assume that the secretaries of both parties are rational, i.e., they will choose optimal strategies to maximize their expected votes without cooperation. Assume that P1 has two strategies to increase their voting share: strategy e1 (campaigning by media), and e2 (campaigning by hosting rallies). Let P2 possess the same strategies as P1. This problem may be regarded as a matrix game, namely, the parties P1 and P2 are regarded as player 1 and player 11, respectively. They may use strategies e1 and e2. Due to a lack of information or imprecise available information, the secretaries of the two parties are usually not able to precisely forecast the voting percentage. They estimate the voting percentage with a certain degree of confidence but they may hesitate regarding the estimation of expected votes. To address the uncertainty, T1FNs are employed to express the voting share for a particular political party. The pay-off matrix A for both the parties is expressed as

Vjj(ay) = IDv (S j + (1 - 1 )Dm (â y)

ain + 4ay + ayr r , , )

aÉ-Jl—SL [wa,j+ 1(1 - wa,j- j

Äl a ij

■'ijr

1 ("ijj = iEm j + (1 - 1 ) Em (~ij) ^ [(1 - ua.) - X(1 - wa,j - ua,j)]

Additionally,

Fj ij) = äy) - Ä^ày) Therefore,

V1(a11) = 108.5 + 36.161 V,(a12) = 93.2 + 31.061 V (a21 ) = 54 + 181 V;(a22) = 108.5 + 36.161

Äi (an) =4-1 Ä1 (a12) = 2.13-Ä1 (a21) = 5.33-Ä1 (a22) =4-1

F1(a11) = 104.5 + 37.161 0.531 F1(a12) =91.07 + 31.591 1.331 F1(a21) = 48.67 + 19.331 F1 (a22) = 104.5 + 37.161

campaigning by media hosting rallies

/ campaigningbymedia

<(175,180,190); 0.6, 0.2) <(80,90,100); 0.6, 0.2)

hosting rallies \

<(150,156,158); 0.6, 0.2) <(175,180,190); 0.6, 0.2)

where ((175, 180, 190); 0.6, 0.2) in the matrix A is a T1FN, which indicates that the expected votes in favour of P1 is "approximately 180" when parties P1 and P2 simultaneously apply the strategy e1 (campaigning by media). The maximum degree of confidence of the secretary is 0.6, whereas the minimum degree of non-confidence is 0.2. His/her degree of hesitation is 0.2. Other elements in the matrix A are similarly explained.

We assume that the aspiration levels for player 1 and player 11 are

and similarly,

V(v) = 102+341 Äl(v = 5.33 - 1.331 F1( V) = 96:67 + 35:33l

V(w) = 95 + 31:67l Äl(w 5:33 - 1.331 F1(w) = = 89.67 + 331

V(p) = 9:8+3:3l Äl(p = : 1.6 - 0:4l F1(p) = 8:2+3:7l

V® = 7:4+2:5l Äl(q = 1.6 - 0:4l F1(S) = 5:8+2:9l

V1(r) = 9:5+3:2l Äl(r = 2:4- 0:6l F1( = 7.1 + 3.71

V(s) = 7.1 + 2.361 Äl(s = 2:4- 0:6l F1(s) = 4.7 + 2.961

According to equations (17) and (18), the linear programming 1 - b - j = 0.2239 is obtained. A similar interpretation can be models for players I and II are obtained as follows: given for player II.

max{f1 - h1>

(Ю4.5 - Ь 37.161)x1 + Ь (48.67 + 19 : 33 1 ) x 2 > (96.67 + 35:331) - (8.2 + Ь 3.71)(1 - f1)

(91:o7 b31.591)x1 - + (Ю4.5- h37.161)x2 > (96.67 - 35:331) - (8.2- Ь 3.71)(1 - f1)

(Ю4.5- b37.161)x1 - + (48.67 - 19 : 33 1 ) x 2 > (96.67 - 35:331) - (8.2- Ь 3.71) + (5.8 + 2:91)(1 - h1)

(91:o7 b31.591)x1 - + (Ю4.5- h37.161)x2 > (96.67 - 35:331) - (8.2- Ь 3.71) + (5.8 + 2:91)(1 - h1)

Subject to (91.o7 + 31.591)x1 + (Ю4.5 + 37.161)x2 > (96.67 + 35.331) - (8.2 + 3.71) + (5.8 + 2.91)(1 - h1) (19)

x1 + x2 = 1 f1 > h1 f1 + h1 < 1 x1, x2, ft, h1 > o

max {f2 - h2>

(Ю4.5 + Ь 37.161)y1 + Ь (91.o7 + h 31.591)y2 < (89.67 + 1-331) + h (7.1 - f 3.761)(1 - f2 )

(48:67 b 19.331)y1 - + (Ю4.5- h37.161)y2 < (89.67 - 1-331) + h (7.1 - f3.761)(1 - f2 )

(Ю4.5- b37.161)y1 - + (91.o7- h31.591)y2 < (89.67 - 1-331) + h (7.1 - f 3.761) - (4.7 + 2:961)(1 - h2)

(48:67 b 19.331)y1 - + (Ю4.5- h37.161)y2 < (89.67 - 1-331) + h (7.1 - f 3.761) - (4.7 + 2:961)(1 - h2)

Subject to (48.67 + 19.331)y1 + (Ю4.5 + 37.161)y2 < (89.67 + 331) + (7.1 + 3.761) - (4.7 + 2.961)(1 - h2) (2o)

У1 + У2 = 1

f2 > h2 ?2 + h2 < 1 У1,У2, ?2, h2 > o

Solving (19) and (20) with the help of LINGO software, we obtained the optimal strategies for players I and II for differ-entl2 [0,1], as depicted in Table 1.

As shown in Table 1, for different l 2 [0, 1], different optimal strategies can be obtained for the players. For l = 1/2, the optimal solution for player I is

â = (0.7997, 0.2003)T and ~ 1 = 0.5799, h = 0.1962, and the optimal solution for player II is y = (0.2003, 0.7997)T and ~2 = 0.5609, h2 = 0.1835.

Therefore, the optimal strategy for player I is (0.7997, 0.2003)T, and the aspiration level v is accepted with a minimal degree 0.5799 and rejected with a maximal degree 0.1962. Note that the sum of two optimal degrees is not 1. A degree of hesitation of

6. Conclusions

In this paper, matrix games with I-fuzzy goals and the pay-offs represented by TIFNs are explored. A solution methodology is proposed to solve these games. First, the concept of double I-fuzzy inequality constraints is interpreted with regard to aspiration levels and adequacies. Second, a defuzzification function (ranking function) is defined based on the value index and ambiguity index of TIFNs. A pair of I-fuzzy programming models is established for two players. Based on the resolution method of double I-fuzzy constraints and by applying the ranking functions, these two models are transformed into two crisp linear programming models that can be solved by a simplex method to obtain the optimal strategies. The numerical example indicates that the optimal strategies for both players are dependent on their subjective preference information.

The main advantage of this method is that it provides not only the degree of acceptance but also the degree of rejection of the

Table 1

Optimal strategies for players I and II for different values of l.

Л У 1 У 2 f 1 y1 У 1 У 2 f 2 h 2

o.o o.8o61 o.1939 o.6346 o.1o27 o.1939 o.8o61 o.436o o.3399

o.1 o.8o47 o.1953 o.6217 o.125o o.1953 o.8o47 o.4661 o.3o11

o.2 o.8o33 o.1967 o.6o99 o.1453 o.1967 o.8o33 o.4932 o.2667

o.3 o.8o2o o.198o o.599o o.1637 o.198o o.8o2o o.5179 o.236o

o.4 o.8oo8 o.1992 o.589o o.18o6 o.1992 o.8oo8 o.54o3 o.2o84

0.5 0.7997 0.2003 0.5799 0.1962 0.2003 0.7997 0.5609 0.1835

o.6 o.7987 o.2o13 o.5713 o.21o4 o.2o13 o.7987 o.5798 o.16o9

o.7 o.7977 o.2o23 o.5633 o.2237 o.2o23 o.7977 o.5972 o.14o4

o.8 o.7967 o.2o33 o.5559 o.2359 o.2o33 o.7967 o.6133 o.8ooo

o.9 o.7958 o.2o42 o.549o o.2473 o.2o42 o.7958 o.6282 o.1o42

1.o o.7949 o.2o51 o.5425 o.2579 o.2o51 o.7949 o.6421 o.o883

aspiration levels for both players. The proposed methodology in this paper differs from the proposed methodology of Nan et al. [ 19,20], Seikh et al. [31,34,36], Aggarwal et al. [ 1], Bandyopadhyay et al. [ 7] and Li et al. [ 15], as they employed an IFS to express goals or only considered pay-offs as I-fuzzy numbers. However, neither of these studies include I-fuzzy inequalities in their models with the pay-offs that are represented by I-fuzzy numbers. In our methodology, the inequalities are I-fuzzy, whereas the pay-offs are also described by I-fuzzy numbers.

The major limitation of this proposed methodology is that it is dependent on the ranking function. Different types of ranking functions yield different types of solution. A more general methodology will be investigated in the future.

Although the proposed method is illustrated with the voting share problem, it can be applied in decision-making theory in areas such as economics, operations research, management, and war science.

Acknowledgements

The authors thank the editor Dr. Seungho Cho and the anonymous learned referees for their valuable comments and appreciate their constructive suggestions.

References

[ 1] A. Aggarwal, D. Dubey, S. Chandra, A. Mehra, Application of Atanassov's I-fuzzy set theory to matrix games with fuzzy goals and fuzzy payoffs, Fuzzy Inf. Eng. 4 (2012) 401-414.

2] A. Aggarwal, A. Mehra, S. Chandra, Application of linear programming with I-fuzzy sets to matrix games with I-fuzzy goals, Fuzzy Optim. Decis. Mak. 11 (no. 4) (2012) 465-480.

[ 3] P.P. Angelov, Optimization in an intuitinistic fuzzy enviornment, Fuzzy Sets Syst. 86 (No. 3) (1997) 299-306.

[4] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (No. 1) (1986) 87-96.

[ 5] K. Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Physica-Verlag, 1999.

6] K. Atanassov, Ideas for intuitionistic fuzzy equations, inequalities and optimization, Notes Intuitionistic Fuzzy Sets 1 (No. 1) (1995) 17-24.

7] S. Bandyopadhyay, P.K. Nayak, M. Pal, Solution of matrix game with triangular intuitionistic fuzzy pay-off using score function, Open J. Optim. 2 (2013) 9-15.

[ 8] C.R. Bector, S. Chandra, Fuzzy Mathematical Programming and Fuzzy Matrix Games, Springer Verlag, Berlin, Germany, 2005.

9] L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets Syst. 32 (No. 3) (1989) 275-289.

10] A.C. Cevikel, M. Ahlatcioglu, Solutions for fuzzy matrix games, Comput. Math. Appl. 60 (No. 3) (2010) 399-410.

11] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Application, Academic Press, New York, 1980.

12] F. Kacher, M. Larbani, Existence of equilibrium solution for a non-cooperative game with fuzzy goals and parameters, Fuzzy Sets Syst. 159 (No. 3) (2008) 164-176.

13] D.F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Comput. Math. Appl. 60 (6) (2010b) 1557-1570.

14] D.F. Li, A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers, Eur. J. Operational Res. 223 (2012a) 421—429.

15] D.F. Li, J.X. Nan, Z.P. Tang, K.J. Chen, X.D. Xiang, F.X. Hong, A bi-objective programming approach to solve matrix games with payoffs of Atanassov's triangular intuitionistic fuzzy numbers, Iran. J. Fuzzy Syst. 9 (No. 3) (2012b) 93—110.

16] D.F. Li, An effective methodology for solving matrix games with fuzzy payoffs, IEEE Trans. Cybern. 43 (No. 2) (2013) 610—621.

17] D.F. Li, J.X. Nan, A nonlinear programming approach to matrix games with payoffs of Atanassov's intuitionistic fuzzy sets, Int. J. Uncertain. Fuzziness Knowl. Based Syst. 17 (No. 4) (2009) 585—607.

18] S.T. Liu, C. Kao, Solution of fuzzy matrix games: an application of the extension principle, Int. J. Intell. Syst. 22 (No. 8) (2007) 891—903.

19] J.X. Nan, D.F. Li, Linear Programming Approach to matrix games with intui-tionistic fuzzy goals, Int. J. Comput. Intell. Syst. 6 (No. 1) (2013) 186—197.

20] J.X. Nan, D.F. Li, M.J. Zhang, A Lexicographic method for matrix games with Pay-offs of triangular intuitionistic fuzzy numbers, Int. J. Comput. Intell. Syst. 3 (No. 3) (2010) 280—289.

21] P.K Nayak, M. Pal, Solution of interval games using graphical method, Tamsui Oxf. J. Math. Sci. 22 (No. 1) (2006) 95—115.

22] P.K. Nayak, M. Pal, Linear programming technique to solve two person matrix games with interval pay-offs, Asia-Pacific J. Operational Res. 26 (No. 2) (2009) 285—305.

23] P.K. Nayak, M. Pal, Bi-matrix games with intuitionistic fuzzy goals, Iran. J. Fuzzy Syst. 7 (no. 1) (2010) 65—79.

24] P.K. Nayak, M. Pal, Intuitionistic fuzzy optimization technique for Nash equilibrium solution of multi-objective bi-matrix game, J. Uncertain Syst. 5 (No. 4) (2011) 271—285.

25] J.V. Neumann, O. Morgenstern, Theory of Games and Economic Behaviour, Princeton University Press, Princeton, New Jersey, 1947.

26] G. Owen, Game Theory, Academic Press, San Diego, 1995.

27] M. Sakawa, I. Nishizaki, Max-min solution for fuzzy multi-objective matrix games, Fuzzy Sets Syst. 67 (No. 1) (1994) 53—59.

28] I. Nishizaki, M. Sakawa, Fuzzy and Multi-objective Games for Conflict Resolution, Physica-Verlag, Berlin, Germany, 2001.

29] M.R. Seikh, M. Pal, P.K. Nayak, Application of triangular intuitionistic fuzzy numbers in bi-matrix games, Int. J. Pure Appl. Math. 79 (no. 2) (2012) 235—247.

30] M.R. Seikh, P.K. Nayak, M. Pal, Notes on triangular intuitionistic fuzzy numbers, Int. J. Math. Operational Res. 5 (No. 4) (2013a) 446—465.

31] M.R. Seikh, P.K. Nayak, M. Pal, Matrix games in intuitionistic fuzzy environment, Int. J. Math. Operational Res. 5 (No. 6) (2013b) 693—708.

32] M.R. Seikh, P.K. Nayak, M. Pal, Solving interval valued matrix games by using fuzzy programming approach, Wesley. J. Res. 7 (No. 1) (2014) 188—199.

33] M.R. Seikh, P.K. Nayak, M. Pal, An alternative approach for solving fuzzy matrix games, Int. J. Math. Soft Comput. 5 (No. 1) (2015a) 79—92.

34] M.R. Seikh, P.K. Nayak, M. Pal, Application of intuitionistic fuzzy mathematical programming with exponential membership and quadratic non-membership functions in matrix games, Ann. Fuzzy Math. Inf. 9 (No. 2) (2015b) 183—195.

35] M.R. Seikh, P.K. Nayak, M. Pal, Solving Bi-matrix games with pay-offs of triangular intuitionistic fuzzy numbers, Eur. J. Pure Appl. Math. 8 (No. 2) (2015c) 153—171.

36] M.R. Seikh, P.K. Nayak, M. Pal, Matrix games with intuitionistic fuzzy pay-offs, J. Inf. Optim. Sci. 36 (No. 1—2) (2015d) 159—181.

37] V. Vidyottama, S. Chandra, Bi-matrix games with fuzzy goals and fuzzy payoffs, Fuzzy Optim. Decis. Mak. 3 (2004) 327—344.

38] V. Vijay, S. Chandra, C.R. Bector, Matrix games with fuzzy goals and fuzzy Payoffs, Omega 33 (No. 5) (2005) 425—429.

39] V. Vijay, A. Mehra, S. Chandra, C.R. Bector, Fuzzy matrix games via a fuzzy relation approach, Fuzzy Optim. Decis. Mak. 6 (No. 4) (2007) 299—314.

40] X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets Syst. 118 (2001) 375—385.