Solving bi-matrix games with intuitionistic fuzzy goals and intuitionistic fuzzy payoffs

Abstract

The aim of this paper is to develop a new methodology for solving bi-matrix games in which goals are regarded as intuitionistic fuzzy (IF) sets (IFSs) and payoffs are expressed with triangular IF numbers (TIFNs). In this methodology, a new ranking method of TIFNs is proposed and the concept of IF inequalities is interpreted. An IF non-linear programming model is constructed to obtain the solution for such a type of bi-matrix games. Then utilizing these IF inequalities and the ranking method of TIFNs proposed in this paper, the solution of any bi-matrix game with goals of IFSs and payoffs of TIFNs can be transformed into a crisp non-linear programming problem. It is shown that the bi-matrix game with goals of IFSs and payoffs of TIFNs is a generalization of the bi-matrix game with goals of fuzzy sets and payoffs of triangular fuzzy numbers. The method proposed in this paper is demonstrated with a numerical example of commerce retailers’ strategy choice problem.

Keywords

Intuitionistic fuzzy set intuitionistic fuzzy number ranking method bi-matrix game intuitionistic fuzzy optimization

1 Introduction

1.1 Background

The bi-matrix game is an important type of two person non-cooperative games, which have been successfully applied to different areas such as politics, economics, and management. In classical bi-matrix games, payoffs of outcomes are made precise common knowledge to both players. In reality, however, players are not able to estimate exactly payoffs of outcomes in the game due to lack of adequate information. In order to make bi-matrix game theory more applicable to real competitive decision problems, the fuzzy set [1] introduced by Zadeh has been used to describe imprecise and uncertain information appearing in real game problems.

As we all know, fuzzy sets are designed to handle uncertainties by attributing a degree, called the membership degree, to which an object belongs to a set. The non-membership degree to which it does not belong to the same set is taken as 1 minus the membership degree. However, in real game problems, there are instances where players may have some hesitation degrees. The fuzzy set is no means to represent the hesitation degrees of players. Atanassov [2] extended the definition of the fuzzy set and introduced a concept of an intuitionistic fuzzy (IF) set (IFS) that assigns each element in the set to a membership degree and non-membership degree. The sum of a membership degree and a non-membership degree is less than or equal to 1 rather than being 1 as the fuzzy set. The idea of using IFSs to represent the uncertain information of bi-matrix games is useful. The reason is that IFSs may indicate players’ preferenceinformation in terms of favor, against, and neutral. IFSs may indicate uncertain information moreabundant and flexible than the fuzzy sets.

1.2 Literature review

Since Zadeh introduced the concept of fuzzy sets, fuzzy bi-matrix games have been extensively studied. Bi-matrix games in fuzzy scenario are considered in terms of either fuzzy goals or fuzzy payoffs. Using the possibility measure of fuzzy numbers, Maeda [3] introduced two concepts of equilibriums for the bi-matrix games with fuzzy payoffs. Using the ranking method of fuzzy numbers, Vijay et al. [4] studied the bi-matrix game with fuzzy goals and fuzzy payoffs, which is shown to be equivalent to a non-linear programming problem. Larbani [5] proposed an approach to solve fuzzy bi-matrix games based on the idea of introducing “nature” as a third player.

We all know that IFSs may indicate uncertain information more abundant and flexible than the fuzzy sets. However, there exists less investigation on bi-matrix games using the IFSs. Li and Yang [6] proposed a new order relation of trapezoidal IF numbers (TrIFNs) and develop a bi-linear programming method for solving bi-matrix games in which the payoffs are expressed with TrIFNs. Fan et al. [7] discussed two kinds of non-linear programming algorithms with the Nash equilibrium of bi-matrix games with payoffs of IFSs. The Nash equilibrium was proved by the fixed point theory and the algorithm was simplified by linear programming method. Seikh et al. [8] constructed an IF non-linear programming problem to conceptualize the term equilibrium solution for bi-matrix games with payoffs of TIFNs. Nayak and Pal [9] presented an application of IF programming to a bi-matrix game with IF goals. The above discussed bi-matrix games with IF information only taken into consideration the uncertainty in payoffs of players or the goals of players. However, in some situations, both payoffs and goals of players are imprecise. Therefore, the bi-matrix game with IF goals and IF payoffs is an important type of games. So far as we know, no study has yet been attempted for bi-matrix games with IF goals and IF payoffs.

In addition, the ranking of TIFNs is an important problem for finding the solutions of bi-matrix games with IF goals and IF payoffs. In last few years, different methods for ranking IFNs have been introduced [10 –16]. However, the existing ranking methods of IFNs have tedious calculations and also fail to provide reasonable order for some TIFNs.

1.3 The contribution and structure of this paper

In this paper, we consider bi-matrix games with IF goals and IF payoffs. In this methodology, players express their aspiration levels in form of uncertain goals that can be modeled using IFSs and the elements of the payoff matrix are represented by TIFNs. And a new ranking method of TIFNs is proposed and the concept of IF inequalities is interpreted. An IF non-linear programming model is constructed to obtain the solutions of any bi-matrix game with IF goals and IF payoffs. Utilizing the defined IF inequality relations and ranking method of TIFNs, the bi-matrix game with IF goals and IF payoffs is transformed into a crisp non-linear programming problem in which the objective function and all constraints are linear except four constraint functions, which arequadratic.

The rest of this paper is organized as follows. In Section 2, some definitions and preliminaries about IFSs and TIFNs are reviewed and a new ranking method of TIFNs based on weighted height index is proposed. Section 3 describes applications of IFSs in optimization and the concept of double IF constraint conditions. In Section 4, we formulate bi-matrix games with IF goals and IF payoffs and give the concept of solutions for such type of bi-matrix games. The auxiliary no-linear programming models are constructed to solve bi-matrix games with IF goals and IF payoffs. In Section 5, the proposed models and methods are illustrated with a numerical example of the commerce retailers’ strategy choice problem. Conclusion is given in Section 6.

2 TIFNs and the new ranking method

2.1 The definition and operations of TIFNs

The IFS introduced by Atanassov [2] is characterized by two functions expressing the membership degree and the non-membership degree, respectively.

Let U ={ x₁, x₂, …, x_n } be a finite universal set. An IFS $\tilde{A}$ in a given universal set U is an object having the form $\tilde{A} = {〈 x, μ_{\tilde{A}} (x), ν_{\tilde{A}} (x) 〉 | x \in U}$ , where $μ_{\tilde{A}} (x) \in [0, 1]$ and $v_{\tilde{A}} (x) \in [0, 1]$ are respectively the membership and non-membership degrees of the element x ∈ U to the set $\tilde{A} \subseteq U$ , such that they satisfy the condition: $0 \leq μ_{\tilde{A}} (x) + v_{\tilde{A}} (x) \leq 1$ for every x ∈ U.

Let $π_{\tilde{A}} (x) = 1 - μ_{\tilde{A}} (x) - v_{\tilde{A}} (x),$ which is called the intuitionistic index of the element x in the IFS $\tilde{A}$ . Obviously, $0 \leq π_{\tilde{A}} (x) \leq 1$ .

Let $\tilde{A}$ and $\tilde{B}$ be two IFSs in the set U. The operations over IFSs are stipulated as follows [2]:

$\tilde{A} \cup \tilde{B} = {〈 x, \max (μ_{\tilde{A}} (x), μ_{\tilde{B}} (x)), \min (ν_{\tilde{A}} (x), v_{\tilde{B}} (x)) 〉 | x \in X}$ ;

$\tilde{A} \cap \tilde{B} = {〈 x, \min (μ_{\tilde{A}} (x), μ_{\tilde{B}} (x)), \max (ν_{\tilde{A}} (x), v_{\tilde{B}} (x)) 〉 | x \in X}$ ;

$\bar{\tilde{A}} = {〈 x, v_{\tilde{A}} (x), μ_{\tilde{A}} (x) 〉 | x \in U}$ ;

$\tilde{A} \subseteq \tilde{B}$ if and only if $μ_{\tilde{A}} (x) \leq μ_{\tilde{B}} (x)$ and $v_{\tilde{A}} (x) \geq v_{\tilde{B}} (x)$ .

A TIFN $\tilde{a} = 〈 (\underline{a}, a, \bar{a}); w_{\tilde{a}}, u_{\tilde{a}} 〉$ on the real number set R is a special IFS, whose membership and non-membership functions are defined as follows [12, 17]: $μ_{\tilde{a}} (x) = {\begin{matrix} 0 & (x < \underline{a}) \\ (x - \underline{a}) w_{\tilde{a}} / (a - \underline{a}) & (\underline{a} \leq x < a) \\ w_{\tilde{a}} & (x = a) \\ (\bar{a} - x) w_{\tilde{a}} / (\bar{a} - a) & (a < x \leq \bar{a}) \\ 0 & (x > \bar{a}) \end{matrix}$ (1)

and

$v_{\tilde{a}} (x) = {\begin{matrix} 1 & (x < \underline{a}) \\ [a - x + u_{\tilde{a}} (x - \underline{a})] / (a - \underline{a}) & (\underline{a} \leq x < a) \\ u_{\tilde{a}} & (x = a) \\ [x - a + u_{\tilde{a}} (\bar{a} - x)] / (\bar{a} - a) & (a < x \leq \bar{a}) \\ 1 & (x > \bar{a}), \end{matrix}$ (2)

respectively, where $w_{\tilde{a}}$ and $u_{\tilde{a}}$ represent the maximum membership degree and the minimum non-membership degree, respectively, such that they satisfy the following conditions: $0 \leq w_{\tilde{a}} \leq 1$ , $0 \leq u_{\tilde{a}} \leq 1$ , and $0 \leq w_{\tilde{a}} + u_{\tilde{a}} \leq 1$ .

$π_{\tilde{a}} (x) = 1 - μ_{\tilde{a}} (x) - v_{\tilde{a}} (x)$ is called the intuitionistic fuzzy index of an element x in the IFN $\tilde{a}$ . It is the degree of indeterminacy or hesitation membership of the element x to $\tilde{a}$ .

A TIFN $\tilde{a} = 〈 (\underline{a}, a, \bar{a}); w_{\tilde{a}}, u_{\tilde{a}} 〉$ is approximately equal to a. Namely, the ill-known quantity “approximate a” is expressed using any value between $\underline{a}$ and $\bar{a}$ with different membership and non-membership degrees. It is easily seen that $μ_{\tilde{a}} (x) + v_{\tilde{a}} (x) = 1$ for any x ∈ R if $w_{\tilde{a}} = 1$ and $u_{\tilde{a}} = 0$ . Hence, the TIFN $\tilde{a} = 〈 (\underline{a}, a, \bar{a}); w_{\tilde{a}}, u_{\tilde{a}} 〉$ degenerates to $\tilde{a} = 〈 (\underline{a}, a, \bar{a}); 1, 0 〉$ , which is just a triangular fuzzynumber.

The arithmetic operations of TIFNs are defined as follows:

Definition 1. Let $\tilde{a}$ and $\tilde{b}$ be two TIFNs, denoted by $\tilde{a} = 〈 (\underline{a}, a, \bar{a}); w_{\tilde{a}}, u_{\tilde{a}} 〉$ , $\tilde{b} = 〈 (\underline{b}, b, \bar{b}); w_{\tilde{b}}, u_{\tilde{b}} 〉$ with $w_{\tilde{a}} \neq w_{\tilde{b}}$ and $u_{\tilde{a}} \neq u_{\tilde{b}}$ , the arithmetic operations are defined as follows [12]:

$\begin{matrix} (1) \tilde{a} + \tilde{b} = 〈 (\underline{a} + \underline{b}, a + b, \bar{a} + \bar{b}); \\ min {w_{\tilde{a}}, w_{\tilde{b}}}, max {u_{\tilde{a}}, u_{\tilde{b}}} 〉; \\ (2) \tilde{a} - \tilde{b} = 〈 (\underline{a} - \bar{b}, a - b, \bar{a} - \underline{b}); \\ min {w_{\tilde{a}}, w_{\tilde{b}}}, max {u_{\tilde{a}}, u_{\tilde{b}}} 〉; \\ (3) k \tilde{a} = {\begin{matrix} 〈 (k \underline{a}, ka, k \bar{a}); w_{\tilde{a}}, u_{\tilde{a}} 〉 if k > 0 \\ 〈 (k \bar{a}, ka, k \underline{a}); w_{\tilde{a}}, u_{\tilde{a}} 〉 if k < 0 \end{matrix}, where \\ is any real number . \end{matrix}$

A α-cut set of a TIFN $\tilde{a}$ is a crisp subset of R, which can be expressed as ${\tilde{a}}_{α} = {x | μ_{\tilde{a}} (x) \geq α}$ , where $0 \leq α \leq w_{\tilde{a}}$ .

It directly follows that ${\tilde{a}}_{α}$ is a closed interval, denoted by ${\tilde{a}}_{α} = [L_{\tilde{a}} (α), R_{\tilde{a}} (α)]$ , which can becalculated as follows: ${\tilde{a}}_{α} = [[(w_{\tilde{a}} - α) \underline{a} + α a] / w_{\tilde{a}}, [(w_{\tilde{a}} - α) \bar{a} + α a] / w_{\tilde{a}}]$ (3)

A β-cut set of a TIFN $\tilde{a}$ is a crisp subset of R, which can be expressed as ${\tilde{a}}^{β} = {x | v_{\tilde{a}} (x) \leq β}$ , where $u_{\tilde{a}} \leq β \leq 1$ .

It directly follows that ${\tilde{a}}^{β}$ is a closed interval, denoted by ${\tilde{a}}^{β} = [L_{\tilde{a}} (β), R_{\tilde{a}} (β)]$ , which can be calculated as follows:

$\begin{matrix} {\tilde{a}}^{β} = [[(1 - β) a + (β - u_{\tilde{a}}) \underline{a}] / (1 - u_{\tilde{a}}), \\ [(1 - β) a + (β - u_{\tilde{a}}) \bar{a}] / (1 - u_{\tilde{a}})] \end{matrix}$ (4)

2.2 The new ranking method of TIFNs

In this subsection, inspired by the height ranking method of fuzzy numbers given by Choobineh [18], we propose a new ranking method of TIFNs based on $\tilde{a}$ -weighted height index of TIFNs.

For any TIFN $\tilde{a} = 〈 (\underline{a}, a, \bar{a}); w_{\tilde{a}}, u_{\tilde{a}} 〉$ , its height of the membership and non-membership functions are defined as $w_{\tilde{a}}$ and $1 - u_{\tilde{a}}$ , respectively. Its left area and right area of the membership function are defined as follows: $S_{f_{l}} (\tilde{a}) = \int_{0}^{w_{\tilde{a}}} (L_{\tilde{a}} (α) - m_{1}) d α = [(\underline{a} + a) / 2 - m_{1}] w_{\tilde{a}}$ (5)

and $S_{f_{r}} (\tilde{a}) = \int_{0}^{w_{\tilde{a}}} (m_{2} - R_{\tilde{a}} (α)) d α = [m_{2} - (a + \bar{a}) / 2] w_{\tilde{a}},$ (6)

respectively, where m₁ and m₂ are two real numbers called minimizing and maximizing barriers, which satisfy $m_{1} \leq \underline{a}$ and $m_{2} \geq \bar{a}$ . Hence, the height index of the membership function is expressed as follows: $\begin{matrix} H_{μ} (\tilde{a}) = [w_{\tilde{a}} - (S_{f_{r}} (\tilde{a}) - S_{f_{l}} (\tilde{a})) / (m_{2} - m_{1})] / 2 \\ = \frac{(\underline{a} + 2 a + \bar{a}) - 4 m_{1}}{4 (m_{2} - m_{1})} w_{\tilde{a}} \end{matrix}$ (7)

Likewise, the left area and right area of the non-membership function are defined as follows:

$\begin{matrix} S_{g_{l}} (\tilde{a}) = \int_{u_{\tilde{a}}}^{1} (L_{\tilde{a}} (β) - m_{1}) d β \\ = [(\underline{a} + a) / 2 - m_{1}] (1 - u_{\tilde{a}}) \end{matrix}$ (8)

and

$\begin{matrix} S_{g_{r}} (\tilde{a}) = \int_{u_{\tilde{a}}}^{1} (m_{2} - R_{\tilde{a}} (β)) d β \\ = [m_{2} - (a + \bar{a}) / 2] (1 - u_{\tilde{a}}), \end{matrix}$ (9)

respectively. Hence, the height index of the non-membership function is expressed as follows: $\begin{matrix} H_{υ} (\tilde{a}) = [(1 - u_{\tilde{a}}) - (S_{g_{r}} (\tilde{a}) - S_{g_{l}} (\tilde{a})) / (m_{2} - m_{1})] / 2 \\ = \frac{(\underline{a} + 2 a + \bar{a}) - 4 m_{1}}{4 (m_{2} - m_{1})} (1 - u_{\tilde{a}}) \end{matrix}$ (10)

The proposed ordering index for the TIFN $\tilde{a}$ is expressed as follows: $\begin{matrix} H_{λ} (\tilde{a}) = λ H_{μ} (\tilde{a}) + (1 - λ) H_{υ} (\tilde{a}) \\ = \frac{(\underline{a} + 2 a + \bar{a}) - 4 m_{1}}{4 (m_{2} - m_{1})} [λ w_{\tilde{a}} + (1 - λ) (1 - u_{\tilde{a}})] \end{matrix}$ (11)

where λ ∈ [0, 1] is a weight, which represents the decision maker’s preference information about the height index of the membership function $μ_{\tilde{a}} (x)$ and the height index of the non-membership function $v_{\tilde{a}} (x)$ for the TIFN $\tilde{a}$ . $H_{λ} (\tilde{a})$ is called λ-weighted height index of the TIFN $\tilde{a}$ . The proposed ranking method takes into consideration the two parameter called minimizing barrier m₁ and maximizing barrier m₂, which can reflect the risk attitude of decision makers.

After analysis, we easily see that the selection of m₁ and m₂ can reflect the risk attitude of decision makers. If m₂ becomes larger than $\bar{a}$ while m₁ is fixed, the aversion toward risk increases. This increasing in aversion toward risk is reflected in the decreasing values of $H_{μ} (\tilde{a})$ and $H_{υ} (\tilde{a})$ . Likewise, an increasing risk-taking attitude can be reflected in the values of $H_{μ} (\tilde{a})$ and $H_{υ} (\tilde{a})$ by moving m₁ to the left of $\underline{a}$ while maintaining m₂ is fixed.

Definition 2. Let $H_{λ} (\tilde{a})$ and $H_{λ} (\tilde{b})$ be λ-weighted height indexes of TIFNs $\tilde{a}$ and $\tilde{b}$ , respectively, λ ∈ [0, 1], then

$H_{λ} (\tilde{a}) > H_{λ} (\tilde{b})$ if and only if $\tilde{a}$ is larger than $\tilde{b}$ , denoted by $\tilde{a} >^{IF} \tilde{b}$ ;

$H_{λ} (\tilde{a}) = H_{λ} (\tilde{b})$ if and only if $\tilde{a}$ is equal to $\tilde{b}$ , denoted by $\tilde{a} =^{IF} \tilde{b}$ ;

$\tilde{a} \geq^{IF} \tilde{b}$ if and only if $\tilde{a} >^{IF} \tilde{b}$ or $\tilde{a} =^{IF} \tilde{b}$ .

The symbol “>^IF” is an IF version of the order relation “>” in the real number set and has the linguistic interpretation “essentially larger than”. The symbols“<^IF” and “= ^IF” are explained similarly.

Example 1. Let us consider three TIFNS $\tilde{a} = < (- 6, 1, 2); 0 . 7, 0 . 1 >$ , $\tilde{b} = < (- 8, 1, 4); 0 . 5, 0 . 3 >$ , and $\tilde{c} = < (- 2, 0, 2); 0.7, 0.2 >$ .

According to the above proposed λ-weighted height ranking method, and let the maximizing and minimizing barriers be m₁ = -6 and m₂ = 2.Using Equation (12), the ranking order of the three TIFNs is generated as follows: $\tilde{a} >^{IF} \tilde{c} >^{IF} \tilde{b}$ if λ ∈ [0, 0 . 506), and $\tilde{c} >^{IF} \tilde{a} >^{IF} \tilde{b}$ ifλ ∈ (0.506, 1].

However, according to the Li’s approach [13], the ratio value of TIFNs $\tilde{a}$ , $\tilde{b}$ , and $\tilde{c}$ can be calculated $R (\tilde{a}, λ) = R (\tilde{b}, λ) = R (\tilde{c}, λ) = 0$ for any λ ∈ [0, 1], then we have $\tilde{a} = \tilde{b} = \tilde{c}$ , which is obviously unreasonable. This indicates that the Li’s approach [13] fails to suggest an order among TIFNs $\tilde{a}$ , $\tilde{b}$ , and $\tilde{c}$ . Similarly, It is not difficult to see that the lexicographic ranking order of TIFNs given in [12] do not provide an order for TIFNs $\tilde{a}$ , $\tilde{b}$ ,and $\tilde{c}$ .

3 Applications of IFSs to optimization

3.1 Decision making in intuitionistic fuzzy environment

Angelov [19] studied the problem of decision making in IF environment. Let U be the universal set. Let G_i (i = 1, 2, …, r) be the set of r goals and C_j (j = 1, 2, …, m) be the set of m constraints, each of which can be characterized by an IFS on U. The IF decision D = (G₁ ∩ G₂ ∩ ⋯ ∩ G_r) ∩ (C₁ ∩ C₂ ∩ ⋯ ∩ C_m) is an IFS defined as D ={ 〈 x, μ_D (x) , ν_D (x) |x ∈ U 〉 }, where $μ_{D} (x) = min_{i, j} {μ_{G_{i}} (x), μ_{C_{j}} (x)}$ and $ν_{D} (x) = max_{i, j} {v_{G_{i}} (x), v_{C_{j}} (x)}$ .

According to IF optimization, we are to maximize the degree of acceptance and also to minimize the degree of rejection of the IF objectives and constraints. Let ξ and η denote respectively the minimal degree of acceptance and maximal degree of rejection. The IF decision problem is transformed into the following crisp optimization problem [19], which can be easily solved by some mathematical programming methods. $\begin{matrix} max {ξ - η} \\ s . t . {\begin{matrix} μ_{G_{i}} (x) \geq ξ (i = 1, 2, \dots, r) \\ v_{G_{i}} (x) \leq η (i = 1, 2, \dots, r) \\ μ_{C_{j}} (x) \geq ξ (j = 1, 2, \dots, m) \\ v_{C_{j}} (x) \leq η (j = 1, 2, \dots, m) \\ ξ \geq η \geq 0 \\ ξ + η \leq 1 \end{matrix} \end{matrix}$ (12)

3.2 Interpretation of IF inequalities

Taking motivation from the work of Bector and Chandra [20], Aggarwal et al. [21] studied the inequality relations of IFSs in the pessimistic sense and optimistic sense. In this paper, the inequality relations of IFSs are interpreted in the pessimistic sense. In the pessimistic approach, the decision maker is presumably extra cautious for acceptance. That is, even if the degree of rejection of x is zero, the decision maker is not willing to accept it totally. To represent this situation, we assume that the tolerance p and q (0 < p < q) be known a priori, then the IF statement “ $x {\underline{≻}}_{p, q} a$ ” to be read as “x is essentially greater than or equal to a with tolerances p and q”, and is understood in terms of the following membership and non-membership functions: $μ (x) = {\begin{matrix} 1 & (x \geq a) \\ 1 - (a - x) / p & (a - p \leq x < a) \\ 0 & (x < a - p) \end{matrix}$ and $v (x) = {\begin{matrix} 1 & (x \leq a - p) \\ 1 - (x - a + p) / q & (a - p < x \leq a - p + q) \\ 0 & (x > a - p + q) \end{matrix} .$

It is important to note that there is an interval [a - p + q, a] in which the non-membership degree is equal to zero but the membership degree is not equal to 1, depicted as in Fig. 1.

Fig.1

Membership and non-membership functions for the statement $x {\underline{≻}}_{p, q} a$ .

Thus, according to IF optimization technique described in Subsection 3.1, the IF inequality relation “ $x {\underline{≻}}_{p, q} a$ ” is understood as ξ ≤ 1 - (a - x)/p and η ≥ 1 + (a - p - x)/q, i.e., x ≥ a - p (1 - ξ) and x ≥ (a - p) + q (1 - η), where ξ and η denote the minimal degree of acceptance and the maximal degree of rejection, which satisfy the conditions 0 ≤ ξ ≤ 1, 0 ≤ η ≤ 1, and 0 ≤ ξ + η ≤ 1.

Similarly, the IF statement “ $x {\underline{≺}}_{r, s} a$ ” is read as “x is essentially less than or equal to a with tolerances r and s”, and is understood in terms of the following membership and non-membership functions: $μ (x) = {\begin{matrix} 1 & (x \leq a) \\ 1 + (a - x) / r & (a < x \leq a + r) \\ 0 & (x \geq a + r) \end{matrix}$ $v (x) = {\begin{matrix} 1 & (x \leq a + r) \\ 1 - (a + r - x) / s & (a + r - s < x \leq a + r) \\ 0 & (x > a + r - s) \end{matrix} .$

It is important to note that there is an interval [a, a + r - s] in which the membership degree is equal to zero but the non-membership degree is not equal to 1, depicted as in Fig. 2.

Fig.2

Membership and non-membership functions for the statement $x {\underline{≺}}_{r, s} a$ .

In a similar way, the IF inequality relation “ $x {\underline{≺}}_{r, s} a$ ” is understood as ξ ≤ 1 + (a - x)/r and η ≥ 1 - (a + r - x)/s, i.e., x ≤ a + r (1 - ξ) and x ≤ (a - r) - s (1 - η), where 0 ≤ r ≤ 1, 0 ≤ s ≤ 1, and 0 ≤ r + s ≤ 1.

Lemma 1. Let α ≥ 0, β ≥ 0, and α + β = 1. Then, $x_{1} {\underline{≻}}_{p, q} a$ and $x_{2} {\underline{≻}}_{p, q} a$ imply $α x_{1} + β x_{2} {\underline{≻}}_{p, q} a$ .

Proof. Relations $x_{1} {\underline{≻}}_{p, q} a$ and $x_{2} {\underline{≻}}_{p, q} a$ can be written as $x_{1} \geq a - (1 - ξ) p, x_{1} \underline{≻} a + p - q (1 - η)$ and $x_{2} \geq a - (1 - ξ) p, x_{2} \underline{≻} a + p - q (1 - η),$ respectively. Then $α x_{1} + β x_{2} \underline{≻} (α + β) a - (α + β) (1 - ξ) p$ and $α x_{1} + β x_{2} \underline{≻} (α + β) (a + p) - (α + β) (1 - η) q .$

Due to α + β = 1, the above equations are equivalent to $α x_{1} + β x_{2} \underline{≻} a - (1 - ξ) p$ and $α x_{1} + β x_{2} \underline{≻} (a + p) - (1 - η) q,$ respectively. Then $α x_{1} + β x_{2} {\underline{≻}}_{p, q} a .$

3.3 Interpretation of double IF constraints

In this subsection, the concept of double IF constraints, i.e., constraints which are expressed as IF inequalities with TIFNs are introduced. The IF inequalities with the parameters and the adequacies being also TIFNs can be interpreted. For this, let N (R) be the set of all TIFNs. Also let $\tilde{A}$ , $\tilde{v}$ , and $\tilde{w}$ be m × n matrix, m × 1, and n × 1 vectors having entries from N (R), respectively, and the double IF constraints under consideration be given by $\tilde{Ax} {\underline{≻}}_{\tilde{p}, \tilde{q}}^{IF} \tilde{v}$ and $\tilde{Ay} {\underline{≺}}_{\tilde{r}, \tilde{s}}^{IF} \tilde{w}$ , with tolerances $\tilde{p}$ , $\tilde{q}$ , $\tilde{r}$ , and $\tilde{s}$ , which are also IF vectors, respectively.

Therefore, the double IF constraints of the type $\tilde{Ax} {\underline{≻}}_{\tilde{p}, \tilde{q}}^{IF} \tilde{v}$ and ${\tilde{A}}^{T} y {\underline{≺}}_{\tilde{r}, \tilde{s}}^{IF} \tilde{w}$ are understood as $\tilde{Ax} {\underline{≻}}_{\tilde{p}, \tilde{q}}^{IF} \tilde{v} \Rightarrow {\begin{matrix} {\tilde{A}}_{i} x {\underline{≻}}^{IF} {\tilde{v}}_{i} - {\tilde{p}}_{i} (1 - ξ) \\ {\tilde{A}}_{i} x {\underline{≻}}^{IF} (\tilde{v} - {\tilde{p}}_{i}) + {\tilde{q}}_{i} (1 - η) \end{matrix}$ (13)

and

${\tilde{A}}^{T} y {\underline{≺}}_{\tilde{r}, \tilde{s}}^{IF} \tilde{w} \Rightarrow {\begin{matrix} {\tilde{A}}_{j} y {\underline{≺}}^{IF} {\tilde{w}}_{j} + {\tilde{r}}_{j} (1 - ξ) \\ {\tilde{A}}_{j} y {\underline{≺}}^{IF} ({\tilde{w}}_{j} + {\tilde{r}}_{j}) - {\tilde{s}}_{j} (1 - η) \end{matrix}$ (14)

respectively, where ${\tilde{A}}_{i}$ (respectively, ${\tilde{A}}_{j}$ ) denotes the i^th row (respectively, j^th column) of $\tilde{A} (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ . Also, ${\tilde{v}}_{i}$ , ${\tilde{p}}_{i}$ , and ${\tilde{q}}_{i}$ represent the i^th component of IF vectors $\tilde{v}$ , $\tilde{p}$ , and $\tilde{q}$ , respectively. Similarly, ${\tilde{w}}_{j}$ , ${\tilde{r}}_{j}$ , and ${\tilde{s}}_{j}$ represent the j^th component of IF vectors $\tilde{w}$ , $\tilde{r}$ , and $\tilde{s}$ , respectively. Here ${\underline{≻}}^{IF}$ and ${\underline{≺}}^{IF}$ are relations between TIFNs which preserves the ranking when TIFNs are multiplied by positive scalars.

4 Mathematical model of bi-matrix games with IF goals and IF payoffs

Vijay et al. [4] studied the bi-matrix game with fuzzy goals and fuzzy payoffs. However, the fuzzy set can only represent the degree of achieving the intended goal for players in a situation. Due to the subjective uncertainty of players, players have a certain degree of hesitation on degree of achieving the intended goal. An IFS can simultaneously indicate the degree to which a player has reached the intended goal in a situation and the degree of inability to reach the intended goal and the degree of hesitation to achieve the intended goal.

Let us consider any bi-matrix games with IF goals and IF payoffs as follows. Assume thatS₁ ={ δ₁, δ₂, …, δ_m } and S₂ ={ σ₁, σ₂, …, σ_n } are sets of pure strategies for players I and II, respectively. The vectors x = (x₁, x₂, …, x_m) ^T and y = (y₁, y₂, …, y_n) ^T are mixed strategies for players I and II, respectively, where x_i (i = 1, 2, …, m)and y_j (j = 1, 2, …, n) are probabilities in which players I and II choose their pure strategies δ_i ∈ S₁ (i = 1, 2, …, m) and σ_j ∈ S₂ (j = 1, 2, …, n), respectively. Sets of mixed strategies for players I and II are denoted by X and Y, where $X = {x | \sum_{i = 1}^{m} x_{i} = 1, x_{i} \geq 0 (i = 1, 2, \dots, m)}$ and Y = ${y | \sum_{j = 1}^{n} y_{j} = 1, y_{j} \geq 0 (j = 1, 2, \dots, n)}$ , respectively. If player I choose any pure strategy δ_i ∈ S₁ and player II choose any pure strategy σ_j ∈ S₂, then at the situation (δ_i, σ_j) players I and II gain payoffs, which are expressedwith TIFNs ${\tilde{A}}_{ij} = 〈 ({\underline{a}}_{ij}, a_{ij}, {\bar{a}}_{ij}); w_{{\tilde{a}}_{ij}}, u_{{\tilde{a}}_{ij}} 〉$ and ${\tilde{B}}_{ij} = 〈 ({\underline{b}}_{ij}, b_{ij}, {\bar{b}}_{ij}); w_{{\tilde{b}}_{ij}}, u_{{\tilde{b}}_{ij}} 〉$ (i = 1, 2, ⋯ , m ; j = 1, 2, …, n), respectively. Thus, the payoff matrices of players I and II are expressed as $\tilde{A} = {({\tilde{A}}_{ij})}_{m \times n}$ and $\tilde{B} = {({\tilde{B}}_{ij})}_{m \times n}$ , respectively.

Let $\tilde{v}$ and $\tilde{w}$ be TIFNs representing the aspiration level of players I and II, respectively. Then bi-matrix game with IF goals and IF payoffs, denoted by BGIFGIFP, is defined as $BGIFGIFP = (X, Y, \tilde{A}, \tilde{B}; \tilde{v}, \tilde{p}, \tilde{q}, {\tilde{p}}^{'}, {\tilde{q}}^{'}; \tilde{w}, \tilde{r}, \tilde{s}, {\tilde{r}}^{'}, {\tilde{s}}^{'}),$

where $\tilde{p}, \tilde{q}, {\tilde{p}}^{'}, {\tilde{q}}^{'}$ and $\tilde{r}, \tilde{s}, {\tilde{r}}^{'}, {\tilde{s}}^{'}$ are tolerance levels for players I and II, respectively. In the sequel, the bi-matrix game with IF goals and IF payoffs is often called as the IF bi-matrix game BGIFGIFP forshort.

Definition 3. (Equilibrium solution of BGIFGIFP) A point (x^*, y^*) ∈ (X, Y) is called an equilibrium solution to the IF bi-matrix game BGIF- GIFP if $\begin{matrix} x^{T} {\tilde{Ay}}^{*} {\underline{≺}}_{\tilde{p}, \tilde{q}}^{IF} \tilde{v} (x \in X), \\ x^{*^{T}} \tilde{By} {\underline{≺}}_{\tilde{r}, \tilde{s}}^{IF} \tilde{w} (y \in Y), \\ x^{*^{T}} {\tilde{Ay}}^{*} {\underline{≻}}_{{\tilde{p}}^{'}, {\tilde{q}}^{'}}^{IF} \tilde{v}, \\ x^{*^{T}} {\tilde{By}}^{*} {\underline{≻}}_{{\tilde{r}}^{'}, {\tilde{s}}^{'}}^{IF} \tilde{w} . \end{matrix}$

By using Definition 2, Equilibrium solution (x^*, y^*) of an IF bi-matrix game BGIFGIFP is equivalent to the following IF non-linear programming problem for players I and II:

Find (x, y) ∈ (X, Y) such that $\begin{matrix} x^{T} \tilde{Ay} {\underline{≺}}_{\tilde{p}, \tilde{q}}^{IF} \tilde{v} (x \in X), \\ x^{T} \tilde{By} {\underline{≺}}_{\tilde{r}, \tilde{s}}^{IF} \tilde{w} (y \in Y), \\ x^{T} \tilde{Ay} {\underline{≻}}_{{\tilde{p}}^{'}, {\tilde{q}}^{'}}^{IF} \tilde{v}, \\ x^{T} \tilde{By} {\underline{≻}}_{{\tilde{r}}^{'}, {\tilde{s}}^{'}}^{IF} \tilde{w} . \end{matrix}$ (15)

Now employing the resolution procedure for the double IF constraints proposed in Section 3.3 and Equation (13), the above IF non-linear programming problem (i.e., Equation (16)) is reduced to $\begin{matrix} max {ξ - η} \\ s . t . {\begin{matrix} x^{T} \tilde{Ay} {\underline{≺}}^{IF} \tilde{v} + \tilde{p} (1 - ξ) (x \in X) \\ x^{T} \tilde{Ay} {\underline{≺}}^{IF} \tilde{v} + \tilde{p} - \tilde{q} (1 - η) (x \in X) \\ x^{T} \tilde{By} {\underline{≺}}^{IF} \tilde{w} + \tilde{r} (1 - ξ) (y \in Y) \\ x^{T} \tilde{By} {\underline{≺}}^{IF} \tilde{w} + \tilde{r} - \tilde{s} (1 - η) (y \in Y) \\ x^{T} \tilde{Ay} {\underline{≻}}^{IF} \tilde{v} - {\tilde{p}}^{'} (1 - ξ) \\ x^{T} \tilde{Ay} {\underline{≻}}^{IF} \tilde{v} - {\tilde{p}}^{'} + {\tilde{q}}^{'} (1 - η) \\ x^{T} \tilde{By} {\underline{≻}}^{IF} \tilde{w} - {\tilde{r}}^{'} (1 - ξ) \\ x^{T} \tilde{By} {\underline{≻}}^{IF} \tilde{w} - {\tilde{r}}^{'} + {\tilde{s}}^{'} (1 - η) \\ ξ + η \leq 1 \\ x \in X, y \in Y, ξ \geq 0, η \geq 0 \end{matrix} \end{matrix}$ (16)

Using the above ranking method of TIFNs, i.e., the λ-weighted height index $H_{λ} (\tilde{a})$ proposed in Section 2.2, the IF non-linear programming problem (i.e., Equation (17)) is transformed into $\begin{matrix} max {ξ - η} \\ s . t . {\begin{matrix} H_{λ} (x^{T} \tilde{Ay}) \leq^{IF} H_{λ} (\tilde{v} + \tilde{p} (1 - ξ)) (x \in X) \\ H_{λ} (x^{T} \tilde{Ay}) \leq^{IF} H_{λ} (\tilde{v} + \tilde{p} - \tilde{q} (1 - η)) (x \in X) \\ H_{λ} (x^{T} \tilde{By}) \leq^{IF} H_{λ} (\tilde{w} + \tilde{r} (1 - ξ)) (y \in Y) \\ H_{λ} (x^{T} \tilde{By}) \leq^{IF} H_{λ} (\tilde{w} + \tilde{r} - \tilde{s} (1 - η)) (y \in Y) \\ H_{λ} (x^{T} \tilde{Ay}) \geq^{IF} H_{λ} (\tilde{v} - {\tilde{p}}^{'} (1 - ξ)) \\ H_{λ} (x^{T} \tilde{Ay}) \geq^{IF} H_{λ} (\tilde{v} - {\tilde{p}}^{'} + {\tilde{q}}^{'} (1 - η)) \\ H_{λ} (x^{T} \tilde{By}) \geq^{IF} H_{λ} (\tilde{w} - {\tilde{r}}^{'} (1 - ξ)) \\ H_{λ} (x^{T} \tilde{By}) \geq^{IF} H_{λ} (\tilde{w} - {\tilde{r}}^{'} + {\tilde{s}}^{'} (1 - η)) \\ ξ + η \leq 1 \\ x \in X, y \in Y, ξ \geq 0, η \geq 0 \end{matrix} \end{matrix}$ (17) Since X and Y are convex polytopes, it is sufficient to consider only the extreme points (i.e., pure strategies) of X and Y in the constraints of Equation (18). Then, the non-linear programming problem is reduced to the following non-linear programming problem for players I and II: $\begin{matrix} max {ξ - η} \\ s . t . {\begin{matrix} H_{λ} ({\tilde{A}}_{i} y) \leq^{IF} H_{λ} (\tilde{v} + \tilde{p} (1 - ξ)) \\ H_{λ} ({\tilde{A}}_{i} y) \leq^{IF} H_{λ} (\tilde{v} + \tilde{p} - \tilde{q} (1 - η)) \\ H_{λ} (x^{T} {\tilde{B}}_{j}) \leq^{IF} H_{λ} (\tilde{w} + \tilde{r} (1 - ξ)) \\ H_{λ} (x^{T} {\tilde{B}}_{j}) \leq^{IF} H_{λ} (\tilde{w} + \tilde{r} - \tilde{s} (1 - η)) \\ H_{λ} (x^{T} \tilde{Ay}) \geq^{IF} H_{λ} (\tilde{v} - {\tilde{p}}^{'} (1 - ξ)) \\ H_{λ} (x^{T} \tilde{Ay}) \geq^{IF} H_{λ} (\tilde{v} - {\tilde{p}}^{'} + {\tilde{q}}^{'} (1 - η)) \\ H_{λ} (x^{T} \tilde{By}) \geq^{IF} H_{λ} (\tilde{w} - {\tilde{r}}^{'} (1 - ξ)) \\ H_{λ} (x^{T} \tilde{By}) \geq^{IF} H_{λ} (\tilde{w} - {\tilde{r}}^{'} + {\tilde{s}}^{'} (1 - η)) \\ ξ + η \leq 1 \\ x \in X, y \in Y, ξ \geq 0, η \geq 0 \end{matrix} \end{matrix}$ (18)

From the above discussion, it is observed that for solving the IF bi-matrix game BGIFGIFP we have to solve the crisp non-linear programming problem (i.e., Equation (19)). The solution of the above non-linear programming problem (i.e., Equation 19) can be obtained by the Lemke-Howson’s algorithm [22], denoted by (x^*, y^*, ξ^*, η^*). Also, if (x^*, y^*, ξ^*, η^*) is an optimal solution of Equation (19), then (x^*, y^*) is an equilibrium strategies for players I and II of the IF bi-matrix game BGIFGIFP and ξ^* is the minimal degree of acceptance and η^* is maximal degree of rejection to aspiration levels $\tilde{v}$ and $\tilde{w}$ of playersI and II.

It may be noted that when $\tilde{p} = \tilde{q}$ , $\tilde{r} = \tilde{s}$ , ${\tilde{p}}^{'} = {\tilde{q}}^{'}$ , ${\tilde{r}}^{'} = {\tilde{s}}^{'}$ , and η = 1 - ξ, the IF bi-matrix game BGIFGIFP is reduced to the fuzzy bi-matrix game with fuzzy goals and fuzzy payoffs BGFGFP (Vijay et al. [4]). Further, the IF non-linear programming problem (i.e., Equation (17)) is reduced to the following fuzzy non-linear programming problem (Vijayet al. [4]): $\begin{matrix} max {ξ} \\ s . t . {\begin{matrix} x^{T} \tilde{Ay} {\underline{≺}}^{F} \tilde{v} + \tilde{p} (1 - ξ) (x \in X) \\ x^{T} \tilde{By} {\underline{≺}}^{F} \tilde{w} + \tilde{r} (1 - ξ) (y \in Y) \\ x^{T} \tilde{Ay} {\underline{≻}}^{F} \tilde{v} - {\tilde{p}}^{'} (1 - ξ) \\ x^{T} \tilde{By} {\underline{≻}}^{F} \tilde{w} - {\tilde{r}}^{'} (1 - ξ) \\ 0 \leq ξ \leq 1 \\ x \in X, y \in Y, \end{matrix} \end{matrix}$ (19)

where the relations “ ${\underline{≺}}^{F}$ ” and “ ${\underline{≻}}^{F}$ ” are the relations between fuzzy numbers. Therefore, IF non-linear programming problem (i.e., Equation (17)) is a generalization of fuzzy non-linear programming problem (i.e., Equation (20)).

5 An example of two commerce retailers’ strategy choice problem

Let us consider the case of two commerce retailers P₁ and P₂ (i.e., players I and II) making a decision aiming to enhance the satisfaction degrees of customers. As players’ judgments for the satisfaction degrees of customers including preference and experience are often vague and players estimate them with their intuition. And assume that commerce retailers P₁ and P₂ are rational, i.e., they will choose optimal strategies to maximize their own profits without cooperation. Suppose that retailer P₁ has two pure strategies: establishing a scientific and rational service system δ₁ and providing customers with satisfaction products δ₂. Retailer P₂ possesses the same pure strategies, i.e., the options of retailer P₂ are: establishing a scientific and rational service system σ₁ and providing customers with satisfaction products σ₂.

Let us consider the following IF bi-matrix game BGIFGIFP for this scenario, where the payoff matrices of commerce retailers P₁ and P₂ are expressed with TIFNs as follows: $\tilde{A} = (\begin{matrix} 〈 (0 . 7, 0 . 8, 0.9); 0.8, 0.1 〉 & 〈 (0.3, 0.4, 0.5); 0.7, 0.2 〉 \\ 〈 (0.3, 0.4, 0.5); 0.7, 0.2 〉 & 〈 (0.4, 0.5, 0.6); 0.6, 0.2 〉 \end{matrix})$

and

$\tilde{B} = (\begin{matrix} 〈 (0 . 3, 0 . 36, 0 . 4); 0.8, 0.1 〉 & 〈 (0 . 2, 0 . 285, 0 . 3); 0.6, 0.3 〉 \\ 〈 (0 . 2, 0 . 25, 0 . 3); 0.7, 0.2 〉 & 〈 (0 . 35, 0 . 4, 0 . 45); 0.7, 0.1 〉 \end{matrix}),$

where the TIFN 〈 (0 . 7, 0 . 8, 0 . 9) ; 0.8, 0.1〉 in the payoff matrix $\tilde{A}$ indicates that the satisfaction degree of customers for commerce retailer P₁ is approximately 0.8 with the maximum satisfaction (or membership) degree 0.6 and the minimum dissatisfaction (or non-membership) degree 0.1 when P₁ and P₂ use strategy δ₁ simultaneously. Other entries in the TIFN payoff matrices $\tilde{A}$ and $\tilde{B}$ can be explainedsimilarly.

We assume that the aspiration levels for P₁ and P₂ are $\tilde{v} = 〈 (0.5, 0.6, 0.7); 0.8, 0.1 〉$ and $\tilde{w} =$ 〈 (0.4, 0.5, 0.6) ; 0.6, 0.2〉. The tolerance levels forP₁ and P₂ are ${\tilde{p}}_{1} = {\tilde{p}}_{2} = \tilde{p} = 〈 (0.1, 0.2, 0.3); 0.7, 0.2 〉$ , ${\tilde{q}}_{1} = {\tilde{q}}_{2} = \tilde{q} = 〈 (0.1, 0.15, 0.2); 0.6, 0.2 〉$ , ${\tilde{p}}^{'} = 〈 (0.2,$ 0 . 3, 0.4) ; 0.7, 0.2〉, ${\tilde{q}}^{'} = 〈 (0.1, 0.2, 0.3); 0.5, 0.2 〉$ , and

${\tilde{r}}_{1} = {\tilde{r}}_{2} = \tilde{r} = 〈 (0.1, 0.2, 0.3); 0.6, 0.3 〉$ , ${\tilde{s}}_{1} = {\tilde{s}}_{2}$ $= \tilde{s} = 〈 (0.1, 0.15, 0.2); 0.8, 0.1 〉$ , ${\tilde{r}}^{'} = 〈 (0.3, 0.4, 0.5);$ 0.6, 0.2〉, ${\tilde{s}}^{'} = 〈 (0.2, 0.3, 0.4); 0.8, 0.1 〉$ , respectively.

Then, according to Definition 1, Equation (12), and Equation (19), the non-linear programming model can be constructed as follows: $\begin{matrix} max {ξ - η} \\ s . t {\begin{matrix} 0 {. 8 y}_{1} + 0.4 y_{2} + 0 . 3 ξ \leq 0.8 \\ 0.4 y_{1} + 0 . {5 y}_{2} + ξ (0 . 24 - 0 . 03 λ) / (0 . 8 - 0 . 2 λ) \leq \\ (0 . 64 - 0 . 08 λ) / (0 . 8 - 0 . 2 λ), \\ 0 {. 8 y}_{1} + 0.4 y_{2} - η (0.12 - 0.03 λ) / (0.8 - 0.1 λ) \leq \\ (0 . 52 - 0 . 13 λ) / (0 . 8 - 0 . 1 λ), \\ 0.4 y_{1} + 0.5 y_{2} - 0.15 η \leq 0.65, \\ 0.355 x_{1} + 0.25 x_{2} + ξ (0 . 14 - 0 . 02 λ) / (0 . 8 - 0 . 1 λ) \leq \\ (0 . 49 - 0 . 07 λ) / (0 . 8 - 0 . 1 λ), \\ 0 {. 2675 x}_{1} + 0.4 x_{2} + 0.2 ξ \leq 0.7, \\ 0.355 x_{1} + 0.25 x_{2} - η (0.0875 - 0.0125 λ) / \\ (0.8 - 0.1 λ) \leq (0 . 385 - 0 . 055 λ) / (0 . 8 - 0 . 1 λ), \\ 0 {. 2675 x}_{1} + 0.4 x_{2} - 0.125 η \leq 0.55, \\ 0.8 x_{1} y_{1} + 0.4 x_{1} y_{2} + 0.4 x_{2} y_{1} + 0.5 x_{2} y_{2} - ξ \\ (0 . 24 - 0 . 03 λ) / (0 . 8 - 0 . 2 λ) \geq (0 . 24 - 0 . 03 λ) / (0 . 8 - 0 . 2 λ), \\ 0.8 x_{1} y_{1} + 0.4 x_{1} y_{2} + 0.4 x_{2} y_{1} + 0.5 x_{2} y_{2} + \\ η (0.16 - 0.06 λ) / (0.8 - 0.2 λ) \geq (0 . 32 - 0 . 12 λ) / (0 . 8 - 0 . 2 λ), \\ 0 {. 355 x}_{1} y_{1} + 0.2675 x_{1} y_{2} + 0.25 x_{2} y_{1} + 0.4 x_{2} y_{2} - \\ ξ (0 . 32 - 0 . 08 λ) / (0 . 7 - 0 . 1 λ) \geq (0 . 08 - 0 . 02 λ) / (0 . 7 - 0 . 1 λ), \\ 0 {. 355 x}_{1} y_{1} + 0.2675 x_{1} y_{2} + 0.25 x_{2} y_{1} + 0.4 x_{2} y_{2} + η \\ (0.24 - 0.06 λ) / (0.7 - 0.1 λ) \geq (0 . 32 - 0 . 08 λ) / (0 . 7 - 0 . 1 λ), \\ x_{1} + x_{2} = 1, \\ y_{1} + y_{2} = 1, \\ ξ + η \leq 1, \\ x_{1}, x_{2}, y_{1}, y_{2}, ξ, η \geq 0 . \end{matrix} \end{matrix}$

Solving the above non-linear programming, we obtained the optimal strategies for P₁ and P₂ for different λ ∈ [0, 1], as depicted in Table 1.

Table 1
Optimal strategies for players I and II with different values of λ

λ x ^T y ^T ξ ^* η ^*

0 (1, 0) (0.65, 0.35) 0.46 0.39

0.1 (1, 0) (0.65, 0.35) 0.47 0.38

0.2 1, 0) (0.64, 0.36) 0.47 0.37

0.3 (1, 0) (0.64, 0.36) 0.48 0.36

0.4 (1, 0) (0.63, 0.37) 0.49 0.35

0.5 (1, 0) (0 . 63, 0 . 37) 0.50 0.34

0.6 (1, 0) (0.61, 0.40) 0.50 0.33

0.7 (0.44, 0.56) (0.40, 0.60) 0.52 0.30

0.8 (0.49, 0.51) (0.44, 0.56) 0.53 0.30

0.9 (0.53, 0.47) (0.48, 0.52) 0.53 0.29

1 (0.62, 0.38) (0.48, 0.52) 0.54 0.28

λ	x ^*T	y ^*T	ξ ^*	η ^*
0	(1, 0)	(0.65, 0.35)	0.46	0.39
0.1	(1, 0)	(0.65, 0.35)	0.47	0.38
0.2	1, 0)	(0.64, 0.36)	0.47	0.37
0.3	(1, 0)	(0.64, 0.36)	0.48	0.36
0.4	(1, 0)	(0.63, 0.37)	0.49	0.35
0.5	(1, 0)	(0 . 63, 0 . 37)	0.50	0.34
0.6	(1, 0)	(0.61, 0.40)	0.50	0.33
0.7	(0.44, 0.56)	(0.40, 0.60)	0.52	0.30
0.8	(0.49, 0.51)	(0.44, 0.56)	0.53	0.30
0.9	(0.53, 0.47)	(0.48, 0.52)	0.53	0.29
1	(0.62, 0.38)	(0.48, 0.52)	0.54	0.28

It can be easily seen from Table 1 that for different λ ∈ [0, 1], the optimal solutions (x^*, y^*, ξ^*, η^*) can be obtained for P₁ and P₂. For λ = 0.5, the optimal strategy for P₁ is obtained as x^* = (1, 0) ^T, the optimal strategy for P₂ is obtained as y^* = (0.63, 0.37) ^T, and the minimal degree of acceptance is ξ^* = 0.50, the maximal degree of rejection is η^* = 0.34 and the hesitation degree is 1 - ξ^* - η^* = 0.17 to aspiration levels $\tilde{v}$ and $\tilde{w}$ for P₁ and P₂.

It is easy to see from Table 1 that the optimal strategies, the degree of acceptance, and the degree of rejection to aspiration levels of the players P₁ and P₂ vary with the weight parameter λ. That is to say, the optimal solutions of the IF bi-matrix game vary with the players’ preference information about the membership function and the non-membership function. And strategy choice of a player is affected by other player’s preference. Compared with the reference [4] which studied bi-matrix games with fuzzy goals and fuzzy payoffs, it can only get the degree of acceptance, the degree of rejection is just automatically the complement to 1. However, it is possible that players may have a degree of hesitation to aspiration levels. In this methodology, the degree of acceptance and the degree of rejection of the objective and the constraints are considered together. These cannot be simply considered as a complement of each other and the sum of their values is less than or equal to 1. The result of this numerical example is consistent with theoretical analysis.

6 Conclusion

The bi-matrix game with IF goals and TIFN payoffs is studied and a solution methodology is proposed to solve such games. In this methodology, a new ranking function is defined based on height index of TIFNs. The concepts of double IF inequality relations are interpreted in the sense of aspiration levels and adequacies. Based on the definition of an equilibrium solution of the IF bi-matrix game, an IF programming model is established to obtain the solution of the IF bi-matrix game. Then, based on resolution method of double IF constraints and the proposed ranking function of height index of TIFNs, the IF programming model is transformed into a crisp non-linear programming problem which can be solved by the Lemke-Howson’s algorithm. In this methodology, the degree of acceptance and the degree of rejection of the objective and the constraints are considered together. These cannot be simply considered as a complement of each other and the sum of their values is less than or equal to 1. Further, it should be noted that the proposed IF bi-matrix game is a generalization of bi-matrix game with fuzzy goals and fuzzy payoffs.

Although the proposed method is illustrated with two commerce retailers’ strategy choice problem, it can be applied in competitive decision making fields such as economics, operations research,management, military, etc.

Footnotes

Acknowledgments

This research was supported by the key Program of National Natural Science Foundation of China (No. 71231003), the Natural Science Foundation of China (Nos. 71461005 and 71561008), and the Science Foundation of Guangxi Province in China (No. 2014GXNSFAA118010).

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