INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 15, Issue: 3, Month: June, Year: 2020 Article Number: 3854, https://doi.org/10.15837/ijccc.2020.3.3854 CCC Publications Matrix Game with Payoffs Represented by Triangular Dual Hesitant Fuzzy Numbers Z. Yang, Y. Song Zhihui Yang* School of Science East China University of Technology Nanchang, 330013, China *Corresponding author: zhhyang@ecut.edu.cn Yang Song Zhumadian Technician College Zhumadian, 463000, China hnsongyang@126.com Abstract Due to the complexity of information or the inaccuracy of decision-makers’ cognition, it is dif- ficult for experts to quantify the information accurately in the decision-making process. However, the integration of the fuzzy set and game theory provides a way to help decision makers solve the problem. This research aims to develop a methodology for solving matrix game with payoffs represented by triangular dual hesitant fuzzy numbers (TDHFNs). First, the definition of TD- HFNs with their cut sets are presented. The inequality relations between two TDHFNs are also introduced. Second, the matrix game with payoffs represented by TDHFNs is investigated. More- over, two TDHFNs programming models are transformed into two linear programming models to obtain the numerical solution of the proposed fuzzy matrix game. Furthermore, a case study is given to to illustrate the efficiency and applicability of the proposed methodology. Our results also demonstrate the advantage of the proposed concept of TDHFNs. Keywords: matrix game, hesitant fuzzy set, triangular dual hesitant fuzzy numbers, multi- objective programming. 1 Introduction Game theory, an effective method to handle the problems in which two conflicting interests exist, has been extensively applied in some fields, such as marketing and finance. In classic game theory, payoffs are assumed to be precise. However, many problems cannot be explained in a simple crisp sense due to the complexity of the problem or a lack of adequate information. Generally, such game problems can be solved by combining game theory with fuzzy set (FS)[34]. Fuzzy logic-based method is important for application on many disciplines [22,25,28]. Decision makers sometimes have some https://doi.org/10.15837/ijccc.2020.3.3854 2 degree of hesitation in some situations. Their preference information is considered in terms of sup- port, opposition and their confidence and nonconfidence level about the approximation. The hesitant fuzzy set (HFS) was originally introduced as an extension of the FS by Torra [26]. The membership function of HFS is a set composed of several numbers in the interval [0,1], and HFS can describe the hesitation of the players. Nevertheless,HFS cannot reflect the degree of opposition expressed by the players. Atanassov proposed an intuitionistic fuzzy set (IFS) characterized by two functions: a mem- bership function and a non-membership function [5]. However, an IFS cannot describe the hesitation or difference in opinions of decision makers. Therefore, dual hesitant fuzzy set (DHFS) was proposed by Zhu et al. [35] to solve the above problems. DHFS consists of two components:a membership hes- itancy function and a non-membership hesitancy function.DHFS can effectively describe the different confidence and nonconfidence level about attitudes (support and opposition) of experts. After Aubin implemented fuzzy theory in a cooperative game model [6], many research works have been published on this topic. Borkotokey and Neog introduced a dynamic resource allocation mechanism within the framework of cooperative games with fuzzy coalitions [11]. Wu studied the cores and dominance cores of cooperative games with fuzzy payoffs [29]. Considering the solution concepts for fuzzy coalition games under a certain participation level, Yu et al. investigated the coincident fuzziness form for games with fuzzy coalition [33]. Yang et al. developed a methodology for solving bi-matrix games with intuitionistic fuzzy number payoffs [32]. Fei and Li proposed a bilinear programming method for solving interval bi-matrix games [13]. An and Li discussed methodologies for solving bi-matrix games with intuitionistic fuzzy information [3,4]. Two-person zero-sum games, which are often called matrix game for short, are an important type of non-cooperative game. Many research works have contributed to the research of the matrix game. Vijay et al. introduced a fuzzy relation approach to develop two-person zero-sum matrix game with fuzzy goals and fuzzy payoffs [27]. Collins and Hu considered interval-valued game matrices and extended the results of classic strictly determined matrix game to fuzzily determined interval games [12]. Gao et al. investigated two-player zero-sum games with fuzzy payoffs to define the best responses of the players[15]. They proposed three types of credibilistic equilibria and their existence theorem [15]. Nan and Li investigated methods for solving matrix game with intuitionistic fuzzy information [23,24]. Li developed a fast approach to compute fuzzy game values for two-person zero-sum games with triangular fuzzy number payoffs [19]. Li and Hong developed a methodology for solving constrained matrix game with trapezoidal fuzzy number payoffs [20]. Aggarwal et al. investigated a class of linear programming problems with fuzzy goals or fuzzy constraints to solve two-person zero-sum matrix game problems with IFSs [2]. Gao and Yu proposed three types of credibilistic equilibria and credibilistic subgame perfect equilibria for fuzzy extensive games [16]. Bhaumik et al. put forward the robust ranking technique to solve matrix game with triangular intuitionistic fuzzy number payoffs [9]. As deterministic payoffs are often unavailable, Figueroa-García et al. developed an uncertain based- matrix game model involving interval-valued fuzzy payoffs [14]. Jana and Roy discussed dual hesitant fuzzy matrix game based on the similarity measure between DHFS [18]. Bhaumik and Roy proposed a new aggregation operator for solving intuitionistic interval-valued hesitant fuzzy matrix games [8]. Bhaumik et al. investigated two-person non-zero-sum game with payoff values represented by the hesitant interval-valued intuitionistic fuzzy-linguistic term set [10].Furthermore, matrix game theory has various applications, including decision making and intelligent manufacturing [30,31]. Therefore, it is necessary to research matrix game with DHFS. In this paper, we propose the concept of triangular dual hesitant fuzzy numbers (TDHFNs), which can reflect the different level for decision-makers’ attitudes (support and opposition). Moreover, the cut set of TDHFNs is also investigated. Our work is focus on developing the matrix game with payoffs represented by TDHFNs. The remainder of this paper is organized as follows. In section 2, definitions and properties of TDHFNs are described. Section 3 proposes a programming method for matrix game with payoffs represented by TDHFNs. In section 4, an empirical case is given to illustrate the proposed method. Finally, section 5 draws conclusions. https://doi.org/10.15837/ijccc.2020.3.3854 3 2 Definition of TDHFNs and solution for matrix game with payoffs represented by TDHFNs In this section, the definition and some properties of TDHFNs are presented. Using the definitions and operations of TDHFNs, the ranking order of TDHFNs is described. Based on the ranking order and the operations, the solution of the matrix game can be obtained by solving a pair of linear programming model. Considering a dual hesitant fuzzy set (DHFS) expressed as E = {< x,fE(x) > |x ∈ X}[35], where fE(x) =< at,at,at; µãt,νãt > (t = 1, · · · ,N) is a dual hesitant fuzzy number representing the possible membership and non-membership degrees of element x ∈ X to E, respectively. The possible membership functions are formulated as follows µãt (x) =   x−at at−at wãt, at ≤ x ≤ at, āt−x āt−at wãt, at < x ≤ āt, 0, x < at or x > āt. (1) The possible non-membership functions are formulated as follows vãt (x) =   at−x+uãt (x−at) at−at , at ≤ x ≤ at, x−at+uãt (āt−x) āt−at , at < x ≤ āt, 1, x < at or x > āt. (2) where wãt and uãt represent the maximum possible membership degree and the minimum possible non-membership degree of element x belonging to E, respectively. Moreover, these values satisfy the conditions 0 ≤ wãt ≤ 1, 0 ≤ uãt ≤ 1, and 0 ≤ wãt + uãt ≤ 1. Definition 1. ã =< (at,at, āt); wãt,uãt > is called a triangular dual hesitant fuzzy number (TDHFN). If at ≥ 0, āt > 0(t = 1, · · · ,N), then ã =< (a− t ,at, āt); wãt,uãt > is called a positive TDHFN, denoted by ã>d0. Similarly, if āt ≤ 0 , at < 0(t = 1, · · · ,N), then ã =< (at,at, āt); wãt,uãt > is called a negative TDHFN, denoted by ã d and = {x|µãt ≥ α,vãt ≤ β}, where α ∈ [0, min{wãt}], β ∈ [max{uãt}, 1] and 0 ≤ α + β ≤ 1. According to definition 2, the following theorem can be obtained. Theorem 1. Let ã be a TDHFN, ã<α,β> can be calculated as ã<α,β> = ãα ∩ ãβ. Definition 3. If ãα ≤ b̃α and ãβ ≤ b̃β, then ã is called not greater than b̃, which denoted as ã ≤d b̃. Similarly to the intuitionistic fuzzy number ranking relationship, let ãi =< (ai,ai, āi); wãi,uãi > and ãj =< (aj,aj, āj); wãj ,uãj >, if ãiα ≤ ãjα and ã β i ≤ ã β j , then ãi ≤ ãj(0 ≤ i,j ≤ N). Remark 1. In this paper, the elements of ã are arranged according to the increasing order. Definition 4. Let ã = (ã1, · · · , ãN ), then ξ = ςãN + (1 − ς)ã1(0 ≤ ς ≤ 1) is called an additional element of TDHFN ã. where ς can be decided according to the risk preference of decision maker. Definition 5. Let ã =< (at,at, āt); wãt,uãt > (t = 1, · · · ,N1) and b̃ =< (bt,bt, b̄t); wb̃t,ub̃t > (t = 1, · · · ,N2) be two TDHFNs. If N1 = N2, then the TDHFNs are called two normalized TDHFNs. Definition 6. Let ã =< (at,at, āt); wãt,uãt > and b̃ =< (bt,bt, b̄t); wb̃t,ub̃t > (t = 1, · · · ,N) be two normalized TDHFNs, and let λ be a real number. The arithmetic operations of TDHFNs are defined as follows. ã + b̃ =< (at + bt,at + bt, āt + b̄t); min{wãt,wb̃t}, max{uãt,ub̃t} > https://doi.org/10.15837/ijccc.2020.3.3854 4 ãb̃ =   < (atbt,atbt, ātb̄t); min{wãt,wb̃t}, max{uãt,ub̃t} >, ã >d 0, b̃ >d 0, < (atb̄t,atbt, ātbt); min{wãt,wb̃t}, max{uãt,ub̃t} >, ã >d 0, b̃ , ã , λ > 0, < (λāt,λat,λat); wãt,uãt >, λ < 0. ã−1 =< (1/at, 1/at, 1/āt); wãt,uãt > Let â = [aL,aU ] be an interval, where aL and aR are the left and right limits of the interval â , respectively. The maximization problem with an interval-valued objective is described as follows respectively. Ishibuchi and Tanaka [17] investigated the maximization problem with an interval-valued objective as max{â} ,s.t. â ∈ Ω1 which is equivalent to the following bi-objective programming problem max{aL, aL + aU 2 } s.t. â ∈ Ω1 (3) where Ω1 is the set of constraint conditions. The minimization problem with an interval-valued objective is described as min{â}, s.t. â ∈ Ω2 which is equivalent to the following bi-objective programming problem max{aU, aL + aU 2 } s.t. â ∈ Ω2 (4) where Ω2 is the set of constraint conditions. In the classical matrix game, the payoff function was often expressed by the crisp number. How- ever, there is often some hesitant fuzzy information about the payoffs. Based on the hesitant fuzzy information, we can develop the matrix game with payoffs represented by TDHFNs. Assume that S1 = {α1,α2, · · · ,αm} and S2 = {β1,β2, · · · ,βn} are the pure strategies for players p1 and p2, respec- tively. Without loss of generality, the payoff matrix for player p1 is given by à = (ãij)m×n, where ãij =< (aijt,aijt, āijt); wãijt,uãijt > (i = 1, · · · ,m; j = 1, · · · ,n; t = 1, · · · ,N). The sets of mixed strategies for players p1 and p2 are denoted by X and Y , respectively, where X = {x = (x1,x2, · · · ,xm)| m∑ i=1 xi = 1,xi ≥ 0}, Y = {y = (y1,y2, · · · ,yn)| n∑ j=1 yj = 1,yj ≥ 0}. If players p1 and p2 choose mixed strategies x ∈ X, y ∈ Y, respectively. According to definition 6, the expected payoff E(x, y) can be calculated as follows E(x, y) = xTÃy = m∑ i=1 n∑ j=1 ãijxiyj =< ( m∑ i=1 n∑ j=1 aijtxiyj, m∑ i=1 n∑ j=1 aijtxiyj, m∑ i=1 n∑ j=1 āijtxiyj); min{wãijt}, max{uãijt} > The interests of the two players are precisely opposite. The payoff of each player is not only related to the strategy he chooses, but also depends on the strategy chosen by the opponent. Therefore, each player considers the strategy chosen by the opponent when determining the strategy that is most beneficial to himself. Assume that the players are rational. Player p1 should choose the mixed strategy x ∈ X that maximizes E(x, y), while player p2 is interested in choosing the mixed strategy y ∈ Y that minimizes E(x, y). If player p1 chooses strategy x, then he could realize a payoff of at least min y∈Y {xTÃy} However, player p1 wants a larger payoff, so he chooses a strategy that ensures a https://doi.org/10.15837/ijccc.2020.3.3854 5 payoff of no less than max x∈X min y∈Y {xTÃy}. Similarly, if player p2 choose strategy y, then he could realize a payoff of at most max x∈X {xTÃy}. However, player p2 wants a smaller payoff, so he chooses a strategy to ensure a payoff of no more than want less payoff , so he would choose the strategy make the payoff is no more than min y∈Y max x∈X {xTÃy}. Player p1 chooses a strategy according to the maximin principle, while player p2 chooses his strategy according to the minimax principle. Therefore, if a pair of mixed strategies (x0, y0) satisfies x0TÃy0 = max x∈X min y∈Y {xTÃy} = min y∈Y max x∈X {xTÃy}, then x0 and y0 are called the optimal strategies for the two players. x0TÃy0 is the value of the mixed equilibrium situation. However, the expected payoff E(x, y) is a TDHFN with the membership degrees and non-memberships. Thus, there do not exist (x0, y0) meet the two scales at the same time. In fact, the problems max x∈X min y∈Y {xTÃy} and min y∈Y max x∈X {xTÃy} are bi-objective programming problems. Definition 7. Let ν̃ and ω̃ be TDHFNs. Assume that there exist (x∗, y∗, ν̃∗, ω̃∗), where x∗ ∈ X, y∗ ∈ Y , satisfying the following conditions (1) x∗TÃy ≥d ν̃∗ and xTÃy∗ ≤d ω̃∗, (2) There do not exist any ν̃ 6= ν̃∗ and ω̃ 6= ω̃∗ such that ν̃ ≥d ν̃∗ and ω̃ ≤d ω̃∗, then, (x∗, y∗, ν̃∗, ω̃∗) is the solution of the TDHFN-based matrix game, x∗ and y∗ are the optimal strategies of players p1 and p2, respectively, ν̃∗ is the gain-floor for player p1 and ω̃∗ is the loss-ceiling for player p2. Finally, x∗TÃy∗ is the payoff of the TDHFN-based matrix game. For the classic linear programming problem in the crisp environment, the aim is to maximize or minimize a linear objective function [7,21]. But in many actual situations, decision makers should specify the objective or constraint functions with fuzzy sets. Assume that each player chooses an optimal mixed strategy under the other player’s pure strategy. According to definition 7, the optimal strategy x∗ of player p1 and the optimal strategy y∗ of player p2 can be achieved by solving the following pair of mathematical programming models max{ν̃} s.t.   m∑ i=1 ãijxi ≥d ν̃ (j = 1, 2, · · · ,n) m∑ i=1 xi = 1 xi ≥ 0 (i = 1, 2, · · · ,m) (5) min{ω̃} s.t.   n∑ j=1 ãijyj ≤d ω̃ (i = 1, 2, · · · ,m) n∑ j=1 yj = 1 yj ≥ 0 (j = 1, 2, · · · ,n) (6) According to definition 3, model (5) can be transformed into the following programming model max{ν̃α, ν̃β} s.t.   m∑ i=1 (ãij)αxi ≥d ν̃α (j = 1, 2, · · · ,n) m∑ i=1 (ãij)βxi ≥d ν̃β (j = 1, 2, · · · ,n) m∑ i=1 xi = 1 xi ≥ 0 (i = 1, 2, · · · ,m) (7) https://doi.org/10.15837/ijccc.2020.3.3854 6 Let ν̃α = [νtLα,νtRα ], ν̃β = [νt β L,νt β R], (ãij)α = [Lãijt (α),Rãijt (α)], (ãij) β = [Lãijt (β),Rãijt (β)] be the α-cut and β-cut of TDHFNs ν̃ and ãij(i = 1, 2, · · · ,m; j = 1, 2, · · · ,n), respectively. According to the interval-valued rank relationship and definition 4, model (7) can be transformed into the following programming model max{[νtLα,νt R α ], [νt β L,νt β R]} s.t.   m∑ i=1 Lãijt (α)xi ≥ νt L α (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Rãijt (α)xi ≥ νt R α (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Lãijt (β)xi ≥ νt L β (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Rãijt (β)xi ≥ νt R β (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 xi = 1 xi ≥ 0 (i = 1, 2, · · · ,m) (8) The above proposed model can be transformed into the bi-objective linear programming model as follows max{ N∑ t=1 (νtLα + νt β L) 2N , N∑ t=1 (νtLα + νt β L + νt R α + νt β R) 4N } s.t.   m∑ i=1 Lãijt (α)xi ≥ νt L α (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Rãijt (α)xi ≥ νt R α (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Lãijt (β)xi ≥ νt L β (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Rãijt (β)xi ≥ νt R β (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 xi = 1 xi ≥ 0 (i = 1, 2, · · · ,m) (9) In the context of multi-objective linear programming, it is important to note that the objectives have different level of importance [17]. In such a situation, the linear weighted average method should be applied on the multi-objective linear programming problem. Objectives with more weight should be allocated more achievement levels in comparison to others [1]. Therefore, model (9) can be transformed into the following linear programming model max{λ( N∑ t=1 (νtLα + νt β L) 2N ) + (1 −λ)( N∑ t=1 (νtLα + νt β L + νt R α + νt β R) 4N )} s.t.   m∑ i=1 Lãijt (α)xi ≥ νt L α (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Rãijt (α)xi ≥ νt R α (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Lãijt (β)xi ≥ νt L β (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 Rãijt (β)xi ≥ νt R β (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 xi = 1 xi ≥ 0 (i = 1, 2, · · · ,m) (10) The parameter λ ∈ [0, 1] can be determined by the decision maker. https://doi.org/10.15837/ijccc.2020.3.3854 7 According to definitions 1 and 2, the the α-cut and β-cut set of ãij =< (aijt,aijt, āijt); wãijt,uãijt > (i = 1, 2, · · · ,m; j = 1, 2, · · · ,n; t = 1, · · · ,N) are represented as follows (ãijt)α = [ (wãijt −α)aijt + αaijt wãijt , (wãijt −α)āijt + αaijt wãijt ] (11) (ãijt)β = [ (1 −β)aijt + (β −uãijt )aijt 1 −uãijt , (1 −β)aijt + (β −uãijt )āijt 1 −uãijt ] (12) Therefore, model (10) can be transformed into the following model max{λ( N∑ t=1 (νtLα + νt β L) 2N ) + (1 −λ)( N∑ t=1 (νtLα + νt β L + νt R α + νt β R) 4N )} s.t.   m∑ i=1 (wãijt−α)aijt+αaijt wãij xi ≥ νtLα (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 (wãijt−α)āijt+αaijt wãijt xi ≥ νtRα (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 (1−β)aijt+(β−uãijt )aijt 1−uãijt xi ≥ νtLβ (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 (1−β)aijt+(β−uãijt )āijt 1−uãijt xi ≥ νtRβ (j = 1, 2, · · · ,n; t = 1, · · · ,N) m∑ i=1 xi = 1 xi ≥ 0 (i = 1, 2, · · · ,m) (13) where νtLα,νt β L,νt R α,νt β R and xi(i = 1, 2, · · · ,m) are decision variables, λ ∈ [0, 1],α ∈ [0, min{wãijt ] and β ∈ [max{uãijt, 1] (0 ≤ α + β ≤ 1)are parameters that can be determined according to the characteristics of the actual decision problem. For any given parameters λ,α and β,the optimal solution (x∗(α,β),νL∗α ,νR∗α ,ν β∗ L ,ν β∗ R ) can be obtained by solving model (13). (x∗(α,β),νL∗α ,νR∗α ,ν β∗ L ,ν β∗ R ) is the optimal solution of model (10). x∗(α,β) is the maximin strategy of player p1 at confidence level (α,β). The < α,β >-cut set of the minimum gain ν̃∗ of player p1, which is represented by ν̃∗<α,β>, can be obtained according to theorem 1. ν̃∗<α,β> denotes the possible range of ν̃ ∗ in the < α,β > confidence level. Specifically, if α = 0,β = 1, it means the largest scope of the minimum gain ν̃∗ of player p1. If α = min{wãijt}, β = max{uãijt}, then ν̃∗<α,β> denotes the minimum range of ν̃ ∗of player p1 . Let ω̃α = [ωtLα,ωtRα ],and ω̃β = [ωt β L,ωt β R] are the α-cut set and β-cut set of TDHFN ω̃, respectively. Therefore, model (6) can be transformed into the following model min{λ( N∑ t=1 (ωtLα + ωt β L) 2N ) + (1 −λ)( N∑ t=1 (ωtLα + ωt β L + ωt R α + ωt β R) 4N )} s.t.   n∑ j=1 (wãijt−α)aijt+αaijt wãij yj ≤ ωtLα (i = 1, 2, · · · ,m; t = 1, · · · ,N) n∑ j=1 (wãijt−α)āijt+αaijt wãijt yj ≤ ωtRα (i = 1, 2, · · · ,m; t = 1, · · · ,N) n∑ j=1 (1−β)aijt+(β−uãijt )aijt 1−uãijt yj ≤ ωtLβ (i = 1, 2, · · · ,m; t = 1, · · · ,N) n∑ j=1 (1−β)aijt+(β−uãijt )āijt 1−uãijt yj ≤ ωtRβ (i = 1, 2, · · · ,m; t = 1, · · · ,N) n∑ j=1 yj = 1 yj ≥ 0 (j = 1, 2, · · · ,n) (14) where ωtLα,ωt β L,ωt R α,ωt β R and yj(j = 1, 2, · · · ,n) are decision variables; λ ∈ [0, 1],α ∈ [0, min{wãijt}] and β ∈ [max{uãijt}, 1] ,0 ≤ α + β ≤ 1. https://doi.org/10.15837/ijccc.2020.3.3854 8 For any given parameters λ,α and β, the optimal solution (y∗(α,β),ωL∗α ,ωR∗α ,ω β∗ L ,ω β∗ R ) can be obtained by solving model (14). Similarly, (y∗(α,β),ωL∗α ,ωR∗α ,ω β∗ L ,ω β∗ R ) is the optimal solution of model (9). y∗(α,β) is the minimax strategy of player p2 at confidence level (α,β), ω̃∗<α,β> is the < α,β >-cut set of ω̃∗, and ω̃∗<α,β> denotes the possible range of the maximum loss ω̃ ∗ of player p2 in the < α,β > confidence level. 3 Illustrative case To illustrate the proposed model, we present an illustrative example considering a monopoly com- pany which plans to produce products with C1 and C2, where C1 and C2 are mutually replaceable products. In the current highly competitive business environment, it is very important for the enter- prise to capture the customer effectively. There are two types of customer group,A1 and A2 , in the market. As the demand for the product is basically fixed, an increase in the sales volume of one will inevitably lead to a decrease in the sales volume of the other. The company and the customer group can be regarded as two players, and the price selection for the company and the customer group can be viewed as a matrix strategy. The payoff matrix of the company is expressed as follows à = [ {< (150, 155, 160); 0.8, 0.1 >,< (150, 160, 170); 0.9, 0.1 >} {< (135, 138, 140); 0.7, 0.2 >,< (135, 140, 145); 0.8, 0.1 >} {< (75, 80, 85); 0.6, 0.2 >,< (80, 85, 90); 0.7, 0.1 >} {< (145, 150, 155); 0.6, 0.2 >,< (150, 155, 160); 0.7, 0.1 >} ] where < (150, 155, 160); 0.8, 0.1 > means that the most possible price of C1 is 155, the lowest possible price is 150, the highest possible price is 160, the maximum satisfaction of A1 is 0.8, and the minimum dissatisfaction is 0.1. The other values can be interpreted similarly. For the given λ = 0.5 (the other values are calculated similarly) and select < α,β >, where α ∈ [0, 0.6], β ∈ [0.2, 1], models (13) and (14) can be solved using Lingo11, and the optimal strategy x∗T (α,β) of p1 and the optimal strategy y∗T (α,β) of p2 can be obtained. The < α,β > -cut set of the minimum gain of player p1 and the maximum loss of player p2 can be calculated according to theorem 1. The results are shown in table 1 and table 2. Table 1 The max-min strategy and gain-floor cut sets of player p1 < α,β > x∗T (α,β) ν̃∗<α,β> < 0, 1 > (0.8235, 0.1765) {[136.76, 142.65], [137.65, 147.65]} < 0.1, 0.8 > (0.8216, 0.1784) {[137.62, 142.04], [138.79, 146.56]} < 0.2, 0.7 > (0.8197, 0.1803) {[138.06, 141.75], [139.37, 146.06]} < 0.3, 0.6 > (0.8179, 0.1821) {[138.50, 141.45], [139.95, 145.51]} < 0.4, 0.5 > (0.8160, 0.1840) {[138.95, 141.17], [140.54, 144.98]} < 0.5, 0.3 > (0.8141, 0.1859) {[139.81, 140.55], [141.68, 143.90]} < 0.6, 0.2 > (0.8123, 0.1877) {140.25, [142.26, 143.37]} Table 2 The min-max strategy and loss-ceiling cut sets of player p2 < α,β > y∗T (α,β) ω̃∗<α,β> < 0, 1 > (0.1765, 0.8235) {[137.65, 143.53], [137.65, 149.41]} < 0.1, 0.8 > (0.1742, 0.8258) {[138.43, 142.88], [138.92, 148.05]} < 0.2, 0.7 > (0.1731, 0.8269) {[138.81, 142.55], [139.55, 147.37]} < 0.3, 0.6 > (0.1720, 0.8280) {[139.20, 142.23], [140.18, 146.70]} < 0.4, 0.5 > (0.1709, 0.8291) {[139.59, 141.91], [141.90, 146.02]} < 0.5, 0.3 > (0.1688, 0.8312) {[140.37, 141.26], [142.08, 144.67]} < 0.6, 0.2 > (0.1677, 0.8323) {[140.76, 140.94], [142.71, 144.00]} According to table 1 and table 2, if < α,β >=< 0, 1 >, then ν̃∗<α,β> = {[136.76, 142.65], [137.65, 147.65]} which denotes the greatest possible range of minimum gain ν̃∗ of player p1, and ω̃∗<α,β> = {[137.65, 143.53], [137.65, 149.41]} denotes the greatest possible range of maximum loss ω̃∗ of player p2. When α is larger and β is smaller, the lengths of ν̃∗<α,β> and ω̃ ∗ <α,β> are smaller . https://doi.org/10.15837/ijccc.2020.3.3854 9 From the above discussion, the concept of TDHFNs proposed in the paper has the advantage that it can take into account the uncertainty and hesitation inherent in the assessment of decision makers, and it can also simulate the psychological behavior of decision makers. The proposed matrix games with payoff represented by TDHFNs makes full use of information from both the market and consumers. 4 Conclusion In this paper, a new integrated concept of the triangular fuzzy number and dual hesitant fuzzy set enables decision makers to express psychological behaviour. The matrix game with payoffs represented by TDHFNs is investigated. Based on the cut set and operational rules of TDHFNs, the matrix game with payoffs represented by TDHFNs is transformed into a multi-objective linear programming problem. 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Fuzzy sets, Information and Control, 8(3), 338-353, 1965. [35] Zhu, B.; Xu, Z.; Xia, M. (2012). Dual hesitant fuzzy sets, Journal of Applied Mathematics, 13 pages, 2012. Copyright c©2020 by the authors. Licensee Agora University, Oradea, Romania. This is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International License. Journal’s webpage: http://univagora.ro/jour/index.php/ijccc/ This journal is a member of, and subscribes to the principles of, the Committee on Publication Ethics (COPE). https://publicationethics.org/members/international-journal-computers-communications-and-control Cite this paper as: Yang, Z.; Song, Y. (2020). Matrix Game with Payoffs Represented by Triangular Dual Hesitant Fuzzy Numbers, International Journal of Computers Communications & Control, 15(3), 3854, 2020. https://doi.org/10.15837/ijccc.2020.3.3854