Ambika approach for solving matrix games with payoffs of single-valued trapezoidal neutrosophic numbers

Abstract

The principal objective of this article is to develop an effective approach to solve matrix games with payoffs of single-valued trapezoidal neutrosophic numbers (SVTNNs). In this approach, the concepts and suitable ranking function of SVTNNs are defined. Hereby, the optimal strategies and game values for both players can be determined by solving the parameterized mathematical programming problems, which are obtained from two novel auxiliary SVTNNs programming problems based on the proposed Ambika approach. In this approach, it is verified that any matrix game with SVTNN payoffs always has a SVTNN game value. Moreover, an application example is examined to verify the effectiveness and superiority of the developed algorithm. Finally, a comparison analysis between the proposed and the existing approaches is conducted to expose the advantages of our work.

Keywords

Matrix games neutrosophic set mathematical programming trapezoidal neutrosophic number ambika approach

1 Introduction

Game theory [1] mainly concerns in competitive and skillful interaction between the decisions makers. In the real world, game theory is mainly used in military, finance, economic, strategic welfares, cartel behaviour, management models, social problems or auctions, political voting systems, races and development research. Usually, the two-person matrix game makes the assumption that the payoff values are described with crisp elements and exactly known by each player. However, players are not able to evaluate the games outcomes exactly due to the unavailability and ambiguity of information. To handle it, Zadeh [2] introduced the fuzzy set concept and since then various scholars have extend it to the different sets such as interval intuitionistic fuzzy set, intuitionistic fuzzy set (IFS), linguistic interval IFS and cubic IFS. Many researchers have studied various kinds of non-cooperative games under uncertainty. For instance, Li et al. [3] proposed a bilinear programming algorithm for solving bi-matrix games with intuitionistic fuzzy (IF) payoffs. Figueroa et al. [4] studied group matrix games with interval-valued fuzzy numbers payoffs. Jana et al. [5] introduced novel similarity measure to solve matrix games with dual hesitant fuzzy payoffs. Singh et al. [6] established 2-tuple linguistic matrix games.

Zhou et al. [7] constructed novel matrix game with generalized Dempster Shafer payoffs. Seikh et al. [8] solved matrix games with payoffs of hesitant fuzzy numbers. Han et al. [9] described new matrix game with Maxitive Belief information. Roy et al. [10] discussed Stackelberg game with payoffs of type-2 fuzzy numbers. Bhaumik et al. [11] solved Prisoners’ dilemma matrix game with hesitant interval-valued intuitionistic fuzzy-linguistic payoffs elements. Ammar et al. [12] stuided bi-matrix games with rough interval payoffs. Brikaa et al. [13] developed fuzzy multi-objective programming technique to solve fuzzy rough constrained matrix games. Bhaumik et al. [14] introduced multi-objective linguistic-neutrosophic matrix game with applications to tourism management. Brikaa et al. [15] applied resolving indeterminacy technique to find optimal solutions of multi-criteria matrix games with IF goals.

However, the IFS and fuzzy set theories are incapable to deal with inconsistent and indeterminate data correctly. To consider it, Smarandache [16] introduced the theory of neutrosophic set (NS) with defining the three components of indeterminacy, falsity and truth, all are lies in ] 0^-, 1⁺] and independent. As NS is difficult to implement on realistic applications, thus, Wang et al. [17] developed the single-valued neutrosophic set (SVNS) concept, which is an extension of the NS. Due to its importance, many scholars have applied the SVNS theory in various disciplines. For example, Garg [18] studied the analysis of decision making based on sine trigonometric operational laws for SVNSs. Murugappan [19] presented neutrosophic inventory problem with immediate return for deficient items. Garg [20] proposed new neutrality aggregation operators with multi-attribute decision making (MADM) approach for single-valued neutrosophic numbers (SVNNs). Abdel-Basset et al. [21] investigated resource levelling model in construction projects with neutrosophic information. Garai et al. [22] discussed variance, standard deviation and possibility mean of SVNNs with applications to MADM models. Broumi et al. [23] solved neutrosophic shortest path model by applying Bellman technique. Gaber et al. [24] solved constrained bi-matrix games with payoffs represented by single-valued trapezoidal neutrosophic numbers. Mullai et al. [25] presented inventory backorder model with neutrosophic environment. Leyva et al. [26] introduced a new problem of information technology project with neutrosophic information. Sun et al. [27] developed new SVNN decision-making algorithms based on the theory of prospect.

In the imprecise data game, players may encounter with some assessment data that cannot be represented as real numbers when estimating the utility functions or uncertain subjects. Since SVNS has great superiority and flexibility in describing many uncertainties with complex environments, it is effective and convenient to represent the matrix games with neutrosophic data. Due to decision makings growing requirements of expressing their judgments in a human friendly and neatly manner, it is important to extend the IF or fuzzy matrix games into neutrosophic environment. The SVNS is an effective tool to satisfy the increasing requirement of higher uncertain and complicated matrix game models. The fundamental targets of this article are listed as follows:

Develop a novel matrix game with SVTNNs payoffs;

Use Ambika approach for SVTNN matrix games to obtain the optimal strategies for both players;

Construct crisp linear optimization problems from the neutrosophic models based on the defined ambiguity and value indexes of SVTNN;

Solve an application example to demonstrate the effectiveness and applicability of the proposed method;

Compare our results with the result of the existing approaches.

The remainder of the manuscript is summarized as. Section 2 introduces the concept, cut sets and arithmetic operations of SVTNNs. Section 3 gives the concept of ambiguity and value indexes of SVTNNs and the ranking technique of SVTNNs. Section 4 formulates matrix games with SVTNNs payoffs. Section 5 describes Ambika approach to solve SVTNNs matrix games. The illustrative example with comparative analysis is discussed in Section 6. Lastly, a short conclusion is given in Section 7.

2 Preliminaries

In the following, we introduce the basic concepts of fuzzy sets, IFSs, NSs, SVNSs and SVTNNs.

Definition 1. [28] A fuzzy number $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ is said to be a trapezoidal fuzzy number (TFN), if its membership function $δ_{\tilde{B}} (y)$ is given by $δ_{\tilde{B}} (y) = {\begin{matrix} \frac{y - b_{1}}{b_{2} - b_{1}}, if b_{1} ⩽ y ⩽ b_{2}, \\ 1, if b_{2} ⩽ y ⩽ b_{3}, \\ \begin{matrix} \frac{b_{4} - y}{b_{4} - b_{3}}, {if b}_{3} ⩽ y ⩽ b_{4}, \\ 0, otherwise . \end{matrix} \end{matrix}$

Definition 2. [29] Suppose Y be a universal set, An IFS $\tilde{C}$ over Y is defined as follows: $\tilde{C} = {〈 y, δ_{\tilde{C}} (y), γ_{\tilde{C}} (y) 〉 : y \in Y}$ where $γ_{\tilde{C}} : Y \to [0, 1]$ and $δ_{\tilde{C}} : Y \to [0, 1]$ are the non-membership degree and the membership degree of y ∈ Y to the set $\tilde{C} \subseteq Y$ , such that $0 ⩽ δ_{\tilde{C}} (y) + γ_{\tilde{C}} (y) ⩽ 1, \forall y \in Y .$

Definition 3. [30] Let Y be a universe. A SVNS $\tilde{B}$ over Y is defined by $\tilde{B} = {〈 y, T_{\tilde{B}} (y), I_{\tilde{B}} (y), F_{\tilde{B}} (y) 〉 : y \in Y},$ where $T_{\tilde{B}} (y) : Y \to [0, 1]$ , $I_{\tilde{B}} (y) : Y \to [0, 1]$ and $F_{\tilde{B}} (y) : Y \to [0, 1]$ such that $0 ⩽ T_{\tilde{B}} (y) + I_{\tilde{B}} (y), + F_{\tilde{B}} (y) ⩽ 3, \forall y \in Y$ . The values $F_{\tilde{B}} (y), I_{\tilde{B}} (y) {andT}_{\tilde{B}} (y)$ respectively express the falsity membership, indeterminacy membership and truth membership degree of y to $\tilde{B}$ .

Definition 4. [30] An (α, β, γ)-cut set of SVNS $\tilde{B}$ , a crisp subset over the set of real numbers $ℝ$ , is given by: ${\tilde{B}}_{(α, β, γ)} = {y : T_{\tilde{B}} (y) ⩾ α, I_{\tilde{B}} (y) ⩽ β, F_{\tilde{B}} (y) ⩽ γ}$ where 0 ⩽ α ⩽ 1, 0 ⩽ β ⩽ 1, 0 ⩽ γ ⩽ 1 and 0 ⩽ α + β + γ ⩽ 3.

Definition 5. [31] A SVNS $\tilde{B} = {〈 y, T_{\tilde{B}} (y), I_{\tilde{B}} (y), F_{\tilde{B}} (y) 〉 : y \in Y}$ is called neutrosophic normal, if there exist at least three points y₁, y₂, y₃ ∈ Y such that $T_{\tilde{B}} (y_{1}) = I_{\tilde{B}} (y_{2}) = F_{\tilde{B}} (y_{3}) = 1$ .

Definition 6. [31] A SVNS $\tilde{B} = {〈 y, T_{\tilde{B}} (y), I_{\tilde{B}} (y), F_{\tilde{B}} (y) 〉 : y \in Y}$ is said to be neutrosophic convex, if ∀y₁, y₂ ∈ Y and ξ ∈ [0, 1], the following conditions are satisfied.

$T_{\tilde{B}} ({ξ y}_{1} + (1 - ξ) y_{2}) ⩾ \min (T_{\tilde{B}} (y_{1}), T_{\tilde{B}} (y_{2})),$

$I_{\tilde{B}} ({ξ y}_{1} + (1 - ξ) y_{2}) ⩽ \max (I_{\tilde{B}} (y_{1}), I_{\tilde{B}} (y_{2})),$

$F_{\tilde{B}} ({ξ y}_{1} + (1 - ξ) y_{2}) ⩽ \max (F_{\tilde{B}} (y_{1}), F_{\tilde{B}} (y_{2})),$

Definition 7. [31] A SVNS $\tilde{B} = {〈 y, T_{\tilde{B}} (y), I_{\tilde{B}} (y), F_{\tilde{B}} (y) 〉 : y \in Y}$ , is said to be single-valued neutrosophic number when

$\tilde{B}$ is neutrosophic normal,

$\tilde{B}$ is neutrosophic convex,

$T_{\tilde{B}} (y)$ is upper semi continuous, $I_{\tilde{B}} (y)$ is lower semi continuous, and $F_{\tilde{B}} (y)$ is lower semi continuous, and

The support of $\tilde{B}$ , i.e. $S (\tilde{B}) = {〈 T_{\tilde{B}} (y) > 0, I_{\tilde{B}} (y) < 1, F_{\tilde{B}} (y) < 1, \forall y \in Y 〉}$ is bounded.

Definition 8. [32] A SVTNN $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ is a special neutrosophic set on the set of real number $ℝ$ , whose truth-membership, indeterminacy-membership, and a falsity-membership are represented as: $μ_{\tilde{b}} (y) = {\begin{matrix} \frac{(y - k) u_{\tilde{b}}}{l - k}, if k ⩽ y < l, \\ u_{\tilde{b}}, if l ⩽ y ⩽ m, \\ \begin{matrix} \frac{(n - y) u_{\tilde{b}}}{n - m}, if m < y ⩽ n, \\ 0, otherwise . \end{matrix} \end{matrix}$ $δ_{\tilde{b}} (y) = {\begin{matrix} \frac{(l - y + (y - k) v_{\tilde{b}})}{l - k}, if k ⩽ y < l, \\ v_{\tilde{b}}, if l ⩽ y ⩽ m, \\ \begin{matrix} \frac{(y - m + (n - y) v_{\tilde{b}})}{n - m}, if m < y ⩽ n, \\ 0, otherwise . \end{matrix} \end{matrix}$

and $η_{\tilde{b}} (y) = {\begin{matrix} \frac{(l - y + (y - k) w_{\tilde{b}})}{l - k}, if k ⩽ y < l, \\ w_{\tilde{b}}, if l ⩽ y ⩽ m, \\ \begin{matrix} \frac{(y - m + (n - y) w_{\tilde{b}})}{n - m}, if m < y ⩽ n, \\ 0, otherwise . \end{matrix} \end{matrix}$ , respectively.

Definition 9. [32] Let $\tilde{c} = 〈 (k_{1}, l_{1}, m_{1}, n_{1});$ $u_{\tilde{c}}, v_{\tilde{c}}, w_{\tilde{c}} 〉$ , $\tilde{d} = 〈 (k_{2}, l_{2}, m_{2}, n_{2}); u_{\tilde{d}}, v_{d}, w_{\tilde{d}} 〉$ be two SVTNNs and λ ≠ 0 be any real number. Then,

$\tilde{c} + \tilde{d} = 〈 (k_{1} + k_{2}, l_{1} + l_{2}, m_{1} + m_{2}, n_{1} + n_{2}); u_{\tilde{c}} \land u_{d}, v_{\tilde{c}} \lor v_{\tilde{d}}, w_{\tilde{c}} \lor w_{\tilde{d}} 〉$

$\tilde{c} \tilde{d} = {\begin{matrix} 〈 (k_{1} k_{2}, l_{1} l_{2}, m_{1} m_{2}, n_{1} n_{2}); u_{\tilde{c}} \land u_{\tilde{d}}, v_{\tilde{c}} \lor v_{\tilde{d}}, w_{\tilde{c}} \lor w_{\tilde{d}} 〉 (n_{1} > 0, n_{2} > 0) \\ 〈 (k_{1} n_{2}, l_{1} m_{2}, m_{1} l_{2}, n_{1} k_{2}); u_{\tilde{c}} \land u_{\tilde{d}}, v_{\tilde{c}} \lor v_{\tilde{d}}, w_{\tilde{c}} \lor w_{\tilde{d}} 〉 (n_{1} < 0, n_{2} > 0) \\ 〈 (n_{1} n_{2}, m_{1} m_{2}, l_{1} l_{2}, k_{1} k_{2}); u_{\tilde{c}} \land u_{\tilde{d}}, v_{\tilde{c}} \lor v_{\tilde{d}}, w_{\tilde{c}} \lor w_{\tilde{d}} 〉 (n_{1} < 0, n_{2} > 0) \end{matrix}$

$λ \tilde{c} = {\begin{matrix} 〈 (λ k_{1}, λ l_{1}, λ m_{1}, λ n_{1},); u_{\tilde{c}}, v_{\tilde{c}}, w_{\tilde{c}}, 〉 (λ > 0) \\ 〈 (λ n_{1}, λ m_{1}, λ l_{1}, λ k_{1},); u_{\tilde{c}}, v_{\tilde{c}}, w_{\tilde{c}}, 〉 (λ < 0) \end{matrix}$

Definition 10. [32] Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN. Then 〈α, β, γ〉-cut set of the SVTNN $\tilde{b}$ in the set of real number $ℝ$ , represented by ${\tilde{b}}_{〈 α, β, γ 〉}$ , is given as: ${\tilde{b}}_{〈 α, β, γ 〉} = {y : μ_{\tilde{b}} (y) ⩾ α, δ_{\tilde{b}} (y) ⩽ β, η_{\tilde{b}} (y) ⩽ γ, y \in ℝ} .$ which satisfies the following conditions:

$0 ⩽ α ⩽ u_{\tilde{b}}, v_{\tilde{b}} ⩽ β ⩽ 1, w_{\tilde{b}} ⩽ γ ⩽ 1$ and 0 ⩽ α + β + γ ⩽ 3.

Obviously, any 〈α, β, γ〉-cut set ${\tilde{b}}_{〈 α, β, γ 〉}$ of a SVTNN $\tilde{b}$ is a crisp subset over the set of real number $ℝ$ .

Definition 11. [32] Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN. Then α-cut set of the SVTNN $\tilde{b}$ in the set of real number $ℝ$ , represented by ${\tilde{b}}_{α}$ , is given as: ${\tilde{b}}_{α} = {y : μ_{\tilde{b}} (y) ⩾ α, y \in ℝ} .$ where $α \in [0, u_{\tilde{b}}]$ .

Obviously, any α-cut set ${\tilde{b}}_{α}$ of a SVTNN $\tilde{b}$ is a crisp subset over the set of real number $ℝ$ .

In here, any α -cut set of a SVTNN $\tilde{b}$ for the truth membership function is a closed interval, represented by ${\tilde{b}}_{α} = [L^{α} (\tilde{b}), R^{α} (\tilde{b})]$ .

Definition 12. [32] Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN. Then β-cut set of the SVTNN $\tilde{b}$ in the set of real number $ℝ$ , represented by ${\tilde{b}}_{β}$ , is given as: ${\tilde{b}}_{β} = {y : δ_{\tilde{b}} (y) ⩽ β, y \in ℝ} .$ where $β \in [v_{\tilde{b}}, 1]$ .

Obviously, any β-cut set ${\tilde{b}}_{β}$ of a SVTNN $\tilde{b}$ is a crisp subset over the set of real number $ℝ$ .

In here, any β -cut set of a SVTNN $\tilde{b}$ for the indeterminacy membership function is a closed interval, represented by ${\tilde{b}}_{β} = [L^{β} (\tilde{b}), R^{β} (\tilde{b})]$ .

Definition 13. [32] Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN. Then γ-cut set of the SVTNN $\tilde{b}$ in the set of real number $ℝ$ , represented by ${\tilde{b}}_{γ}$ , is given as: ${\tilde{b}}_{γ} = {y : η_{\tilde{b}} (y) ⩽ γ, y \in ℝ} .$ where $γ \in [w_{\tilde{b}}, 1]$ .

Obviously, any γ-cut set ${\tilde{b}}_{γ}$ of a SVTNN $\tilde{b}$ is a crisp subset over the set of real number $ℝ$ .

In here, any γ-cut set of a SVTNN $\tilde{b}$ for the falsity membership function is a closed interval, represented by ${\tilde{b}}_{γ} = [L^{γ} (\tilde{b}), R^{γ} (\tilde{b})]$ .

3 The ranking approach for SVTNNs

3.1 Value and ambiguity of SVTNNs

Here, we introduce the basic definitions of value and ambiguity indices of SVTNNs.

Definition 14. [32] Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN and ${\tilde{b}}_{α} = [L^{α} (\tilde{b}), R^{α} (\tilde{b})]$ , ${\tilde{b}}_{β} = [L^{β} (\tilde{b}), R^{β} (\tilde{b})]$ and ${\tilde{b}}_{γ} = [L^{γ} (\tilde{b}), R^{γ} (\tilde{b})]$ be any α-cut set, β -cut set and γ-cut set of the SVTNN $\tilde{b}$ , respectively. Then,

The value of the SVTNN $\tilde{b}$ for α-cut set, represented by $V_{μ} (\tilde{b})$ , is given as:

$V_{μ} (\tilde{b}) = \int_{0}^{u_{\tilde{b}}} (L^{α} (\tilde{b}) + R^{α} (\tilde{b})) f (α) d α$ (1) where $f (α) \in [0, 1] (α \in [0, u_{\tilde{b}}])$ , f(0) = 0 and f(α) is monotonic and non-decreasing of $α \in [0, u_{\tilde{b}}]$ .

The value of the SVTNN $\tilde{b}$ for β-cut set, represented by $V_{δ} (\tilde{b})$ , is given as:

$V_{δ} (\tilde{b}) = \int_{v_{\tilde{b}}}^{1} (L^{β} (\tilde{b}) + R^{β} (\tilde{b})) g (β) d β$ (2) where $g (β) \in [0, 1] (β \in [v_{\tilde{b}}, 1])$ , g(1) = 0 and g(β) is monotonic and non-increasing of $β \in [v_{\tilde{b}}, 1]$ .

The value of the SVTNN $\tilde{b}$ for γ-cut set, represented by $V_{η} (\tilde{b})$ , is given as:

$V_{η} (\tilde{b}) = \int_{w_{\tilde{b}}}^{1} (L^{γ} (\tilde{b}) + R^{γ} (\tilde{b})) h (γ) d γ$ (3) where $h (γ) \in [0, 1] (γ \in [w_{\tilde{b}}, 1])$ , h(1) = 0 and h(γ) is monotonic and non-increasing of $γ \in [w_{\tilde{b}}, 1]$ .

Definition 15. [32] Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN and ${\tilde{b}}_{α} = [L^{α} (\tilde{b}), R^{α} (\tilde{b})]$ , ${\tilde{b}}_{β} = [L^{β} (\tilde{b}), R^{β} (\tilde{b})]$ and ${\tilde{b}}_{γ} = [L^{γ} (\tilde{b}), R^{γ} (\tilde{b})]$ be any α-cut set, β -cut set and γ-cut set of the SVNN $\tilde{b}$ , respectively. Then,

The ambiguities of the SVTNN $\tilde{b}$ for α-cut set, represented by $A_{μ} (\tilde{b})$ , is given as:

$A_{μ} (\tilde{b}) = \int_{0}^{u_{\tilde{b}}} (R^{α} (\tilde{b}) - L^{α} (\tilde{b})) f (α) d α$ (4) where $f (α) \in [0, 1] (α \in [0, u_{\tilde{b}}]$ ), f(0) = 0 and f(α) is monotonic and non-decreasing of $α \in [0, u_{\tilde{b}}]$ .

The ambiguities of the SVTNN $\tilde{b}$ for β-cut set, represented by $A_{δ} (\tilde{b})$ , is given as:

$A_{δ} (\tilde{b}) = \int_{v_{\tilde{b}}}^{1} (R^{β} (\tilde{b}) - L^{β} (\tilde{b})) g (β) d β$ (5) where $g (β) \in [0, 1] (β \in [v_{\tilde{b}}, 1])$ , g(1) = 0 and g(β) is monotonic and non-increasing of $β \in [v_{\tilde{b}}, 1]$ .

The ambiguities of the SVTNN $\tilde{b}$ for γ-cut set, represented by $A_{η} (\tilde{b})$ , is given as:

$A_{η} (\tilde{b}) = \int_{w_{\tilde{b}}}^{1} (R^{γ} (\tilde{b}) - L^{γ} (\tilde{b})) h (γ) d γ$ (6) where $h (γ) \in [0, 1] (γ \in [w_{\tilde{b}}, 1])$ , h(1) = 0 and h(γ) is monotonic and non-increasing of $γ \in [w_{\tilde{b}}, 1]$ .

In here, the weighting functions f(α), g(β), h(γ) can be supposed according as the nature of decision making model. Suppose f(α) = α,g(β) = 1 - β, h(γ) = 1 - γ.

Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN. Then the value and ambiguity indices, using the above descriptions, are constructed as: $\begin{matrix} V_{μ} (\tilde{b}) = \frac{(k + 21 + 2 m + n) u_{\tilde{b}}^{2}}{6}, A_{μ} (\tilde{b}) \\ = \frac{(n - k + 2 m - 2 l) u_{\tilde{b}}^{2}}{6} \\ V_{δ} (\tilde{b}) = \frac{(k + 2 l + 2 m + n) (1 - v_{\tilde{b}})^{2}}{6}, A_{δ} (\tilde{b}) \end{matrix}$ $\begin{matrix} = \frac{(n - k + 2 m + 2 l) {(1 - v_{\tilde{b}})}^{2}}{6} \\ V_{η} (\tilde{b}) = \frac{(k + 2 l + 2 m + n) (1 - w_{\tilde{b}})^{2}}{6}, A_{η} (\tilde{b}) \\ = \frac{(n - k + 2 m + 2 l) {(1 - w_{\tilde{b}})}^{2}}{6} \end{matrix}$ (7)

3.2 A ranking approach of a SVTNN based on value and ambiguity indices

This Subsection provides a ranking approach of SVTNNs based on the ambiguity and value indices of SVTNNs.

Definition 16. [32] Let $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ be a SVTNN. Then, for Ω ∈ [0, 1],

The Ω-weighted value of the SVTNN $\tilde{b}$ are described as;

$V_{Ω} (\tilde{b}) = Ω V_{μ} (\tilde{b}) + (1 - Ω) V_{δ} (\tilde{b}) + (1 - Ω) V_{η} (\tilde{b})$ (8)

The Ω-weighted ambiguity of the SVTNN $\tilde{b}$ are described as;

$A_{Ω} (\tilde{b}) = Ω A_{μ} (\tilde{b}) + (1 - Ω) A_{δ} (\tilde{b}) + (1 - Ω) A_{η} (\tilde{b})$ (9)

Definition 17. [32] Let $\tilde{c}$ and $\tilde{d}$ be two SVTNN and Ω ∈ [0, 1]. For the weighted ambiguity and values of the SVTNN $\tilde{c}$ and $\tilde{d}$ , the ranking order of $\tilde{c}$ and $\tilde{d}$ is described as;

If $V_{Ω} (\tilde{c}) >_{N} V_{Ω} (\tilde{d})$ , then $\tilde{c} >_{N} \tilde{d}$ ;

If $V_{Ω} (\tilde{c}) <_{N} V_{Ω} (\tilde{d})$ , then $\tilde{c} <_{N} \tilde{d}$ ;

If $V_{Ω} (\tilde{c}) =_{N} V_{Ω} (\tilde{d})$ , then

If $A_{Ω} (\tilde{c}) =_{N} A_{Ω} (\tilde{d})$ , then $\tilde{c} =_{N} \tilde{d}$ ;

If $A_{Ω} (\tilde{c}) >_{N} A_{Ω} (\tilde{d})$ , then $\tilde{c} >_{N} \tilde{d}$ ;

If $A_{Ω} (\tilde{c}) <_{N} A_{Ω} (\tilde{d})$ , then $\tilde{c} <_{N} \tilde{d}$ .

where >_N and <_N are neutrosophic versions of the order relations > and < in the real line, respectively.

4 Matrix games with SVTNNs payoffs

Let’s suppose a matrix game with SVTNNs payoffs. The pure strategies sets for both players I and II are represented by H₁ ={ ζ₁, ζ₂, … . , ζ_r } and H₂ ={ ς₁, ς₂, … . , ς_s }, respectively. The vectors z =(z₁, z₂, …, z_r) ^T and t =(t₁, t₂, …, t_s) ^T are mixed strategies for player I and II, respectively, where z_i(i = 1, 2, …, r) and t_j(j = 1, 2, …, s) are probabilities in which players I and II choose their pure strategies ζ_i ∈ H₁ and ς_j ∈ H₂, respectively. Sets of mixed strategies for two players are defined by Z and T, where $Z = {z = (z_{1}, z_{2}, \dots, z_{r}) | \sum_{i = 1}^{r} z_{i} = 1, z_{i} ⩾ 0}$ and $T = {t = (t_{1}, t_{2}, \dots, t_{s}) | \sum_{j = 1}^{s} t_{j} = 1, t_{j} ⩾ 0}$ , respectively.

Without loss of generality, suppose that the player I’s payoff matrix is described as $\tilde{B} = {({\tilde{b}}_{ij})}_{r \times s}$ , where each ${\tilde{b}}_{ij} = 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉$ (i = 1, 2, …, r ; j = 1, 2, …, s) is a SVTNN described as above. Thus, the matrix game with SVTNNs payoffs is expressed with the triplet $SVNG = (Z, T, \tilde{B})$ .

According to Definition 9, the player I’s expected payoff is obtained as follows:

$\begin{matrix} \tilde{E} (\tilde{B}) = z^{T} \tilde{B} t = \sum_{i = 1}^{r} \sum_{j = 1}^{s} {\tilde{b}}_{ij} z_{i} t_{j} \\ = 〈 (\sum_{i = 1}^{r} \sum_{j = 1}^{s} k_{ij} z_{i} t_{j}, \sum_{i = 1}^{r} \sum_{j = 1}^{s} l_{ij} z_{i} t_{j}, \sum_{i = 1}^{r} \sum_{j = 1}^{s} m_{ij} z_{i} t_{j}, \sum_{i = 1}^{r} \sum_{j = 1}^{s} n_{ij} z_{i} t_{j}) \\ ; min {u_{{\tilde{b}}_{ij}}}, max {v_{{\tilde{b}}_{ij}}}, max {w_{{\tilde{b}}_{ij}}} 〉 \end{matrix}$ (10) which is SVTNN.

Since the SVTNN matrix game is zero-sum game, from Definition 9, the player II’s expected payoff is obtained as follows:

$\begin{matrix} \tilde{E} (\tilde{B}) = z^{T} (- \tilde{B}) t = \sum_{i = 1}^{r} \sum_{j = 1}^{s} (- {\tilde{b}}_{ij}) z_{i} t_{j} \\ = 〈 (- \sum_{i = 1}^{r} \sum_{j = 1}^{s} n_{ij} z_{i} t_{j}, - \sum_{i = 1}^{r} \sum_{j = 1}^{s} m_{ij} z_{i} t_{j}, - \sum_{i = 1}^{r} \sum_{j = 1}^{s} l_{ij} z_{i} t_{j}, - \sum_{i = 1}^{r} \sum_{j = 1}^{s} k_{ij} z_{i} t_{j}) \\ ; min {u_{{\tilde{b}}_{ij}}}, max {v_{{\tilde{b}}_{ij}}}, max {w_{{\tilde{b}}_{ij}}} 〉 \end{matrix}$ (11)

It is customary to suppose that player II is a minimizing player and player I is a maximizing player. That is to say, player II is interested in obtaining a mixed strategy t ∈ T to minimize $\tilde{E} (z, t)$ , given by $\min_{t \in T} {\tilde{E} (z, t)}$ . So, player I should select a mixed strategy z ∈ Z that maximizes the minimum expected gain of player II, i.e., $\tilde{ρ} = max_{z \in Z} min_{t \in T} {\tilde{E} (z, t)},$ (12) which is called the gain-floor of player I.

Likewise, player I is interested in obtaining a mixed strategy z ∈ Z in order to maximize $\tilde{E} (z, t)$ , given by $\max_{z \in Z} {\tilde{E} (z, t)}$ . Therefore, player II should select a mixed strategy t ∈ T that minimizes the maximum expected loss of player I, i.e., $\tilde{σ} = \min_{t \in T} max_{z \in Z} {\tilde{E} (z, t)},$ (13) which is called the loss-ceiling of player II.

Clearly, the loss-ceiling of player II and the gain-floor of player I should be SVTNNs, denoted by $\tilde{σ} = 〈 (σ_{1}, σ_{2}, σ_{3}, σ_{4}); u_{\tilde{σ}}, v_{\tilde{σ}}, w_{\tilde{σ}} 〉$ and $\tilde{ρ} = 〈 (ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}); u_{\tilde{ρ}}, v_{\tilde{ρ}}, w_{\tilde{ρ}} 〉$ , respectively.

Definition 18. (Reasonable solution of a SVTNN matrix game) Let $\tilde{ρ} = 〈 (ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}); u_{\tilde{ρ}}, v_{\tilde{ρ}}, w_{\tilde{ρ}} 〉$ and $\tilde{σ} = 〈 (σ_{1}, σ_{2}, σ_{3}, σ_{4}); u_{\tilde{σ}}, v_{\tilde{σ}}, w_{\tilde{σ}} 〉$ be SVTNNs. Suppose that there exist z^* ∈ Z and t^* ∈ T. Then $(z^{*}, t^{*}, \tilde{ρ}, \tilde{σ})$ is called a reasonable solution of the SVTNN matrix game if for any z ∈ Z and t ∈ T, z^* and t^* satisfy the two conditions $z^{* T} \tilde{B} t ⩾_{N} \tilde{ρ}$ and $z^{T} \tilde{B} t^{*} ⩽_{N} \tilde{σ}$ . If $(z^{*}, t^{*}, \tilde{ρ}, \tilde{σ})$ is the reasonable solution of the SVTNN matrix game then z^* and t^* are called reasonable strategies for both players, $\tilde{ρ}$ and $\tilde{σ}$ are called reasonable game values of the two players.

Definition 19. (Solution of a SVTNN matrix game) Let $V$ and $W$ be the sets of all reasonable game values $\tilde{ρ}$ and $\tilde{σ}$ for both players. Suppose that there exist ${\tilde{ρ}}^{*} \in V$ and ${\tilde{σ}}^{*} \in W$ . If there do not exist any $\tilde{ρ} \in V$ and $\tilde{σ} \in W$ such that ${\tilde{ρ}}^{*} ⩽_{N} \tilde{ρ}$ and ${\tilde{σ}}^{*} ⩾_{N} \tilde{σ}$ are satisfied, then $(z^{*}, t^{*}, {\tilde{ρ}}^{*}, {\tilde{σ}}^{*})$ is called an efficient solution of the SVTNN matrix game, where t^* and z^* are called a minimax strategy and a maximin strategy for both players II and I, respectively, ${\tilde{σ}}^{*}$ and ${\tilde{ρ}}^{*}$ are called the loss-ceiling of player II and the gain-floor of player I, respectively, and $z^{* T} \tilde{B} t^{*}$ is called an efficient value of the SVTNN matrix game.

Theorem 1. For any z ∈ Z and t ∈ T, ${\tilde{ρ}}^{*} ⩽_{N} {\tilde{σ}}^{*}$ is valid.

Proof. For any z ∈ Z, we obtain $min_{t \in T} {\tilde{E} (z, t)} ⩽_{N} \tilde{E} (z, t)$

Likewise, for any t ∈ T, we have $max_{z \in Z} {\tilde{E} (z, t)} ⩾_{N} \tilde{E} (z, t)$

Thus, for any z ∈ Z and t ∈ T, we get $min_{t \in T} {\tilde{E} (z, t)} ⩽_{N} max_{z \in Z} {\tilde{E} (z, t)}$

Therefore, $min_{t \in T} {\tilde{E} (z, t)} ⩽_{N} \min_{t \in T} max_{z \in Z} {\tilde{E} (z, t)}$

Hence, $\begin{matrix} \max_{z \in Z} min_{t \in T} {\tilde{E} (z, t)} ⩽_{N} \min_{t \in T} max_{z \in Z} {E (z, t)} \\ i . e ., {\tilde{ρ}}^{*} ⩽_{N} {\tilde{σ}}^{*} \end{matrix}$

Theorem 2. For the SVTNN matrix game, we get $\max_{z \in Z} min_{t \in T} {\tilde{E} (z, t)} =_{N} \max_{z \in Z} min_{1 ⩽ j ⩽ s} {\sum_{i = 1}^{r} {\tilde{b}}_{ij} z_{i}}$ (14)

and $\min_{t \in T} max_{z \in Z} {\tilde{E} (z, t)} =_{N} \min_{t \in T} max_{1 ⩽ i ⩽ r} {\sum_{j = 1}^{s} {\tilde{b}}_{ij} t_{j}}$ (15)

In order to prove Theorem 2, Lemma 1 is given firstly as follows.

Lemma 1. (i) Suppose that there exists a set ${{\tilde{d}}_{1}, {\tilde{d}}_{2}, \dots, {\tilde{d}}_{s}}$ of s SVTNNs, where each ${\tilde{d}}_{j} (j = 1, 2, \dots, s)$ is a SVTNN. The following equality is valid $\min_{t \in T} {\sum_{j = 1}^{s} {\tilde{d}}_{j} t_{j}} =_{N} \min_{1 ⩽ j ⩽ s} {{\tilde{d}}_{j}}$ (16)

(ii) Suppose that there exists a set ${{\tilde{e}}_{1}, {\tilde{e}}_{2}, \dots, {\tilde{e}}_{r}}$ of r SVTNNs, where each ${\tilde{e}}_{i} (i = 1, 2, . ., r)$ is a SVTNN. The following equality is valid $max_{z \in Z} {\sum_{i = 1}^{r} {\tilde{e}}_{i} z_{i}} =_{N} \max_{1 ⩽ i ⩽ r} {{\tilde{e}}_{i}}$ (17)

Proof. (i) Based on the previous ranking order relation definition of SVTNNs, suppose that $\min_{1 ⩽ j ⩽ s} {{\tilde{d}}_{j}} =_{N} {\tilde{d}}_{ℓ}$ . Clearly, it follows that ${\tilde{d}}_{j} ⩾_{N} {\tilde{d}}_{ℓ} (j = 1, 2, \dots, s)$ . For any t_j ⩾ 0, we get ${\tilde{d}}_{j} t_{j} ⩾_{N} {\tilde{d}}_{ℓ} t_{j} (j = 1, 2, \dots, s)$ .

Summing the previous s inequalities, we obtain $\sum_{j = 1}^{s} {\tilde{d}}_{j} t_{j} ⩾_{N} \sum_{j = 1}^{s} {\tilde{d}}_{ℓ} t_{j} =_{N} {\tilde{d}}_{ℓ}$

Since t_j ⩾ 0(j = 1, 2, …, s) and $\sum_{j = 1}^{s} t_{j} = 1$ , we get $\min_{t \in T} {\sum_{j = 1}^{s} {\tilde{d}}_{j} t_{j}} ⩾_{N} {\tilde{d}}_{ℓ}$ (18)

Due to the fact that t =(0, 0, ·· · , 0, 1, 0, ·· ·0) ^T can be considered as a special mixed strategy, where t_ℓ = 1 and t_j = 0(j = 1, 2, …, s, j ≠ ℓ), we have

$\begin{matrix} {\tilde{d}}_{ℓ} =_{N} {\tilde{d}}_{1} \times 0 + {\tilde{d}}_{2} \times 0 + \dots + {\tilde{d}}_{ℓ} \times 1 + \dots + {\tilde{d}}_{s} \\ \times 0 ⩾_{N} \min_{t \in T} {\sum_{j = 1}^{s} {\tilde{d}}_{j} t_{j}} \end{matrix}$ (19)

Combining Equations (19), we obtain $\min_{t \in T} {\sum_{j = 1}^{s} {\tilde{d}}_{j} t_{j}} =_{N} {\tilde{d}}_{ℓ} =_{N} \min_{1 ⩽ j ⩽ s} {{\tilde{d}}_{j}} .$

(ii) Likewise, it is easily proved that $max_{z \in Z} {\sum_{i = 1}^{r} {\tilde{e}}_{i} z_{i}} =_{N} \max_{1 ⩽ i ⩽ r} {{\tilde{e}}_{i}} .$

Proof of Theorem 2. From Equations (17), it easily obtained that $\max_{z \in Z} min_{t \in T} {\tilde{E} (z, t)} =_{N} \max_{z \in Z} \min_{t \in T} {\sum_{j = 1}^{s} (\sum_{i = 1}^{r} {\tilde{b}}_{ij} z_{i}) t_{j}} =_{N} \max_{z \in Z} \min_{1 ⩽ j ⩽ s} {\sum_{i = 1}^{r} {\tilde{b}}_{ij} z_{i}}$

and $\min_{t \in T} \max_{z \in Z} {\tilde{E} (z, t)} =_{N} \min_{t \in T} \max_{z \in Z} {\sum_{i = 1}^{r} (\sum_{j = 1}^{s} {\tilde{b}}_{ij} t_{j}) z_{i}} =_{N} \min_{t \in T} \max_{1 ⩽ i ⩽ r} {\sum_{j = 1}^{s} {\tilde{b}}_{ij} t_{j}},$ which are just about the Equations (15), respectively. Therefore, the proof of Theorem 2 is completed.

From Definitions 18, 19 and Equations (12)–(13), the player I’s maximin strategy z^* ∈ Z and the player II’s minimax strategy t^* ∈ T can be constructed by solving a pair of SVTNN optimization problems given as follows: $\begin{matrix} max {\tilde{ρ}} \\ s . t . {\begin{matrix} \sum_{i = 1}^{r} {\tilde{b}}_{ij} z_{i} t_{j} ⩾_{N} \tilde{ρ}, j = 1, 2, \dots s; \\ \begin{matrix} \sum_{i = 1}^{r} z_{i} = 1; \\ z_{i} ⩾ 0, i = 1, 2, \dots r . \end{matrix} \end{matrix} \end{matrix}$ (20)

and $\begin{matrix} min {\tilde{σ}} \\ s . t . {\begin{matrix} \sum_{j = 1}^{s} {\tilde{b}}_{ij} z_{i} t_{j} ⩽_{N} \tilde{σ}, i = 1, 2, \dots r; \\ \begin{matrix} \sum_{j = 1}^{s} t_{j} = 1; \\ t_{j} ⩾ 0, j = 1, 2, \dots s . \end{matrix} \end{matrix} \end{matrix}$ (21) respectively, where $\tilde{ρ}$ and $\tilde{σ}$ are SVTNN variables. As Z and T are convex polytopes, only the extreme points of the sets Z and T are considered in the constraints conditions of Equations (21) since “⩾_N” and “⩽_N” preserve the ranking order relations when SVTNNs are multiplied by positive numbers according to Definition 9. Therefore, Equations (21) can be converted into the following linear SVTNN optimization problems,

$\begin{matrix} max {\tilde{ρ}} \\ s . t . {\begin{matrix} \sum_{i = 1}^{r} {\tilde{b}}_{ij} z_{i} ⩾_{N} \tilde{ρ}, j = 1, 2, \dots s; \\ \begin{matrix} \sum_{i = 1}^{r} z_{i} = 1; \\ z_{i} ⩾ 0, i = 1, 2, \dots r . \end{matrix} \end{matrix} \end{matrix}$ (22)

and $\begin{matrix} min {\tilde{σ}} \\ s . t . {\begin{matrix} \sum_{j = 1}^{s} {\tilde{b}}_{ij} t_{j} ⩽_{N} \tilde{σ}, i = 1, 2, \dots r; \\ \begin{matrix} \sum_{j = 1}^{s} t_{j} = 1; \\ t_{j} ⩾ 0, j = 1, 2, \dots s . \end{matrix} \end{matrix} \end{matrix}$ (23) respectively.

5 Ambika approach for solving SVTNN matrix game

In this section, we extend the Ambika approach [33] to solve matrix games with SVTNNs payoffs.

5.1 The player I’s minimum expected gain

The player I’s optimal strategies and the corresponding minimum expected gain can be computed as follows:

Step 1: Using the comparing technique Definition 17, to obtain the optimal solution ${t_{1}^{*}, t_{2}^{*}, \dots, t_{s}^{*}}$ such that $\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{j}$ is minimum for all (z₁, z₂, … . , z_r) ∈ Z, which it is equivalent to obtain ${t_{1}^{*}, t_{2}^{*}, \dots, t_{s}^{*}}$ such that $V_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{j})$ , is minimum for all (z₁, z₂, … . , z_r) ∈ Z i.e., obtain the optimal solution ${t_{j}^{*}, j = 1, 2, \dots, s}$ of Model 5.1.1.

Model 5.1.1 $\begin{matrix} min {V_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{j})} \\ s . t . {\begin{matrix} \sum_{i = 1}^{r} z_{i} = 1; \\ z_{i} ⩾ 0, i = 1, 2, \dots r; \\ \sum_{j = 1}^{s} t_{j} = 1; \\ t_{j} ⩾ 0, j = 1, 2, \dots s . \end{matrix} \end{matrix}$

Step 2: From the property, λ〈(k_i, l_i, m_i, n_i) ; $u_{{\tilde{c}}_{i}}, v_{{\tilde{c}}_{i}}, w_{{\tilde{c}}_{i}} 〉 = 〈 (λ k_{i}, λ l_{i}, λ m_{i}, λ n_{i}); u_{{\tilde{c}}_{i}}, v_{{\tilde{c}}_{i}}, w_{{\tilde{c}}_{i}} 〉, λ ⩾ 0$ and $\sum_{i = 1}^{r} 〈 (k_{i}, l_{i}, m_{i}, n_{i}); u_{{\tilde{c}}_{i}}, v_{{\tilde{c}}_{i}}, w_{{\tilde{c}}_{i}} 〉 = 〈 (\sum_{i = 1}^{r} k_{i},$ $\sum_{i = 1}^{r} l_{i}, \sum_{i = 1}^{r} m_{i}, \sum_{i = 1}^{r} n_{i}); min_{1 ⩽ i ⩽ r} u_{{\tilde{c}}_{i}}, max_{1 ⩽ i ⩽ r} v_{{\tilde{c}}_{i}}, max_{1 ⩽ i ⩽ r} w_{{\tilde{c}}_{i}} 〉$ , Model 5.1.1 can be converted into Model 5.1.2.

Model 5.1.2 $\begin{matrix} \min {V_{Ω} (〈 (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} k_{ij} t_{j}, \sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} l_{ij} t_{j}, \sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} m_{ij} t_{j}, \sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} n_{ij} t_{j},); \min_{\underset{1 \leq j \leq s}{1 \leq i \leq r}} u_{\tilde{b}}_{_{ij}}, \max_{\underset{1 \leq j \leq s}{1 \leq i \leq r}} v_{\tilde{b}}_{_{ij}}, \max_{\underset{1 \leq j \leq s}{1 \leq i \leq r}} w_{\tilde{b}}_{_{ij}} 〉)} \\ s .t . {\begin{matrix} \sum_{i = 1}^{r} z_{i} = 1; \\ z_{i} \geq 0, i = 1,2, \dots r; \\ \sum_{j = 1}^{s} t_{j} = 1; \\ t_{j} \geq 0, j = 1,2, \dots s . \end{matrix} \end{matrix}$

Step 3: Since, in Model 5.1.2, only t_j have been considered as decision variables. Thus, Model 5.1.2 is a linear programming model and therefore, the optimal solution of Model 5.1.2 will be equal to the optimal solution of its corresponding dual model i.e., Model 5.1.3.

Model 5.1.3 $s . t . {\begin{matrix} \max {ρ} \\ V_{Ω} (〈 (\sum_{i = 1}^{r} z_{i} k_{ij}, \sum_{i = 1}^{r} z_{i} l_{ij}, \sum_{i = 1}^{r} z_{i} m_{ij}, \sum_{i = 1}^{r} z_{i} n_{ij},); \underset{1 \leq j \leq s}{\min_{1 \leq i \leq r}} u_{{\tilde{b}}_{ij}}, \underset{1 \leq j \leq s}{\max_{1 \leq i \leq r}} v_{{\tilde{b}}_{ij}}, \underset{1 \leq j \leq s}{\max_{1 \leq i \leq r} w_{{\tilde{b}}_{ij}}} 〉) \\ \sum_{i = 1}^{r} z_{i} = 1; \\ z_{i} \geq 0, i = 1,2, \dots r . \end{matrix} \geq ρ, j = 1,2, \dots s;$

Step 4: Find the optimal value of Model 5.1.3.

Step 5: Substitute the optimal value ${z_{i}^{*}, i = 1, 2, \dots, r}$ of Model 5.1.3 in Model 5.1.1 and then obtain the optimal value ${t_{j}^{*}, j = 1, 2, \dots, s}$ of Model 5.1.1.

Case 1: If there exist a unique optimal value ${t_{j}^{*}, j = 1, 2, \dots, s}$ of Model 5.1.1 then the player I’s minimum expected gain is $\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij} 〉} t_{j}^{*}$ and the corresponding optimal strategy for player I will be ${z_{i}^{*}, i = 1, 2, \dots, r}$ , which is the optimal value of Model 5.1.3.

Case 2: If there exist more than one basic optimal value ${t_{j 1}^{*}, j = 1, 2, \dots, s}$ , ${t_{j 2}^{*}, j = 1, 2, \dots, s}$ , . . . , ${t_{jp}^{*}, j = 1, 2, \dots, s}$ of Model 5.1.1 then obtain $\min {A_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{jh}^{*}),$ h = 1, 2, . . , p}

If $A_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij} 〉} t_{jq}^{*})$ denotes the minimum value then the player I’s minimum expected gain is $\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{jq}^{*}$ and the corresponding optimal strategy for player I will be ${z_{i}^{*}, i = 1, 2, \dots, r}$ , which is the optimal value of Model 5.1.3.

5.2 The player II’s maximum expected loss

The player II’s optimal strategies and the corresponding maximum expected loss can be computed as follows:

Step 1: Using the comparing technique Definition 17, to obtain the optimal solution ${z_{1}^{*}, z_{2}^{*}, \dots, z_{r}^{*}}$ such that $\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij} 〉} t_{j}$ is maximum for all (t₁, t₂, … . , t_s) ∈ T, which it is equivalent to obtain ${z_{1}^{*}, z_{2}^{*}, \dots, z_{r}^{*}}$ such that $V_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij} 〉} t_{j})$ , is maximum for all (t₁, t₂, … . , t_s) ∈ T i.e., obtain the optimal solution ${z_{i}^{*}, i = 1, 2, \dots, r}$ of Model 5.2.1.

Model 5.2.1 $\begin{matrix} max {V_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{j})} \\ s . t . {\begin{matrix} \sum_{i = 1}^{r} z_{i} = 1; \\ z_{i} ⩾ 0, i = 1, 2, \dots r; \\ \sum_{j = 1}^{s} t_{j} = 1; \\ t_{j} ⩾ 0, j = 1, 2, \dots s . \end{matrix} \end{matrix}$ Step 2: From the property, $λ 〈 (k_{i}, l_{i}, m_{i}, n_{i}); u_{{\tilde{c}}_{i}}, v_{{\tilde{c}}_{i}}, w_{{\tilde{c}}_{i}} 〉 = 〈 (λ k_{i}, λ l_{i}, λ m_{i}, λ n_{i});$ $u_{{\tilde{c}}_{i}}, v_{{\tilde{c}}_{i}}, w_{{\tilde{c}}_{i}} 〉, λ ⩾ 0$ and $\sum_{i = 1}^{r} 〈 (k_{i}, l_{i}, m_{i}, n_{i});$ $u_{{\tilde{c}}_{i}}, v_{{\tilde{c}}_{i}}, w_{{\tilde{c}}_{i}} 〉 = 〈 (\sum_{i = 1}^{r} k_{i}, \sum_{i = 1}^{r} l_{i}, \sum_{i = 1}^{r} m_{i}, \sum_{i = 1}^{r} n_{i}); min_{1 ⩽ i ⩽ r} u_{{\tilde{c}}_{i}},$ $max_{1 ⩽ i ⩽ r} v_{{\tilde{c}}_{i}}, max_{1 ⩽ i ⩽ r} w_{{\tilde{c}}_{i}} 〉$ , Model 5.2.1 can be converted into Model 5.2.2.

Model 5.2.2 $\begin{matrix} \max {V_{Ω} (〈 (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} k_{ij} t_{j}, \sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} l_{ij} t_{j}, \sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} m_{ij} t_{j}, \sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i} n_{ij} t_{j},); \underset{1 \leq j \leq s}{\min_{1 \leq i \leq r}} u_{\tilde{b}}_{_{ij}}, \underset{1 \leq j \leq s}{\max_{1 \leq i \leq r}} v_{\tilde{b}}_{_{ij}}, \underset{1 \leq j \leq s}{\max_{1 \leq i \leq r}} w_{\tilde{b}}_{_{ij}} 〉)} \\ s .t . {\begin{matrix} \sum_{i = 1}^{r} z_{i} = 1; \\ z_{i} \geq 0, i = 1,2, \dots r; \\ \sum_{j = 1}^{s} t_{j} = 1; \\ t_{j} \geq 0, j = 1,2, \dots s . \end{matrix} \end{matrix}$

Step 3: Since, in Model 5.2.2, only z_i have been considered as decision variables. Thus, Model 5.2.2 is a linear programming model and therefore, the optimal solution of Model 5.2.2 will be equal to the optimal solution of its corresponding dual model i.e., Model 5.2.3.

Model 5.2.3 $s .t . {\begin{matrix} \min {σ} \\ V_{Ω} (〈 (\sum_{j = 1}^{s} k_{ij} t_{j}, \sum_{j = 1}^{s} l_{ij} t_{j}, \sum_{j = 1}^{s} m_{ij} t_{j}, \sum_{i = 1}^{s} n_{ij} t_{j},); \underset{1 \leq j \leq s}{\min_{1 \leq i \leq r}} u_{\tilde{b}}_{_{ij}}, \underset{1 \leq j \leq s}{\max_{1 \leq i \leq r}} v_{\tilde{b}}_{_{ij}}, \underset{1 \leq j \leq s}{\max_{1 \leq i \leq r}} w_{\tilde{b}}_{_{ij}} 〉) \leq σ, i = 1,2, \dots r; \\ \sum_{j = 1}^{s} t_{j} = 1; \\ t_{j} \geq 0, j = 1,2, \dots s . \end{matrix}$

Step 4: Find the optimal value of Model 5.2.3.

Step 5: Substitute the optimal value ${t_{j}^{*}, j = 1, 2, \dots, s}$ of Model 5.2.3 in Model 5.2.1 and then obtain the optimal value ${z_{i}^{*}, i = 1, 2, \dots, r}$ of Model 5.2.1.

Case 1: If there exist a unique optimal value ${z_{i}^{*}, i = 1, 2, \dots, r}$ of Model 5.2.1 then the player II’s maximum expected loss is $\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{i}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{j}^{*}$ and the corresponding optimal strategy for player II will be ${t_{j}^{*}, j = 1, 2, \dots, s}$ , which is the optimal value of Model 5.2.3.

Case 2: If there exist more than one basic optimal value ${z_{i 1}^{*}, i = 1, 2, \dots, r}$ , ${z_{i 2}^{*}, i = 1, 2, \dots, r}$ , . . . , ${z_{ip}^{*}, i = 1, 2, \dots, r}$ of Model 5.2.1 then obtain $\max {A_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{ih}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij} 〉} t_{j}^{*}),$ h = 1, 2, . . , p)}

If $A_{Ω} (\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{iq}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{j}^{*})$ denotes the maximum value then the player II’s maximum expected loss is $\sum_{j = 1}^{s} \sum_{i = 1}^{r} z_{iq}^{*} 〈 (k_{ij}, l_{ij}, m_{ij}, n_{ij}); u_{{\tilde{b}}_{ij}}, v_{{\tilde{b}}_{ij}}, w_{{\tilde{b}}_{ij}} 〉 t_{j}^{*}$ and the corresponding optimal strategy for player II will be ${t_{j}^{*}, j = 1, 2, \dots, s}$ , which is the optimal value of Model 5.2.3.

6 Numerical example

This Section provides a numerical example to illustrate the solution procedure of a matrix game with payoffs of SVTNNs.

6.1 Application problem

“Suppose that there are two companies E₁ and E₁ aiming to enhance the market share of a product in a targeted market under the circumstance that the demand amount of the product in the targeted market basically is fixed. In other words, the market share of one company increases while the market share of another company decreases. The two companies are considering about two strategies to increase the market shares, i.e., strategies ζ₁(advertisement), ζ₂ (reduce the price). The above problem may be regarded as a matrix game. Namely, the companies E₁ and E₁ are regarded as players I and II, respectively. They may use the strategies ζ₁and ζ₂. Due to a lack of information or imprecision of the available information, the managers of the two companies usually are not able to exactly forecast the sales amount of the companies. In certain environment, they can estimate some approximate sales amounts, but it is possible that they are not so sure about them”. In order to handle the uncertain situation, SVTNNs are used to express the sales amounts of the product. The payoff matrix $\tilde{B}$ for the company E₁ is given as follows: $\tilde{B} = (\begin{matrix} 〈 (180, 185, 190, 200); 0.7, 0.4, 0.6 〉 & 〈 (160, 166, 168, 170); 0.6, 0.3, 0.5 〉 \\ 〈 (135, 138, 142, 150); 0.8, 0.1, 0.4 〉 & 〈 (185, 195, 205, 210); 0.5, 0.2, 0.7 〉 \end{matrix})$

6.2 The Solution procedure

6.2.1 The player I’s minimum expected gain

The player I’s optimal strategies and the corresponding minimum expected gain can be computed as follows:

Step 1. Find {t_j, j = 1, 2} ∈ T such that V_Ω(〈(180, 185, 190, 200) ;0.7, 0.4, 0.6〉z₁t₁ + 〈(160, 166, 168, 170) ;0.6, 0.3, 0.5〉z₁t₂ + 〈(135, 138, 142, 150) ;0.8, 0.1, 0.4〉z₂t₁ + 〈(185, 195, 205, 210) ;0.5, 0.2, 0.7〉z₂t₂) is minimum for all (z₁, z₂) ∈ Z i.e., find optimal solution {t_j, j = 1, 2} of Model 6.2.1.1.

Model 6.2.1.1 $\begin{matrix} \min {\begin{matrix} V_{Ω} (〈 (180, 185, 190, 200); 0.7, 0.4, 0.6 z_{1} t_{1} \\ + 〈 (160, 166, 168, 170); 0.6, 0.3, 0.5 z_{1} t_{2} \\ + 〈 (135, 138, 142, 150); 0.8, 0.1, 0.4 z_{2} t_{1} \\ + 〈 (185, 195, 205, 210); 0.5, 0.2, 0.7 z_{2} t_{2}) \end{matrix}} \\ s . t {\begin{matrix} z_{1} + z_{2} = 1; \\ t_{1} + t_{2} = 1; \\ z_{1} + z_{2} ⩾ 0; \\ t_{1} + t_{2} ⩾ 0 . \end{matrix} \end{matrix}$

Step 2: Since, in Model 6.2.1.1, only t_j have been considered as decision variables. Thus, Model 6.2.1.1 is a linear programming model and therefore, the optimal solution of Model 6.2.1.1 will be equal to the optimal solution of its corresponding dual model i.e., Model 6.2.1.2.

Model 6.2.1.2 $\begin{matrix} max {ρ} \\ s . t . {\begin{matrix} V_{Ω} (\begin{matrix} 〈 (180, 185, 190, 200); 0.5, 0.4, 0.7 z_{1} \\ + 〈 (135, 138, 142, 150); 0.5, 0.4, 0.7 z_{2} \end{matrix}) ⩾ ρ; \\ V_{Ω} (\begin{matrix} 〈 (160, 166, 168, 170); 0.5, 0.4, 0.7 z_{1} \\ + 〈 (185, 195, 205, 210); 0.5, 0.4, 0.7 z_{2} \end{matrix}) ⩾ ρ; \\ \begin{matrix} z_{1} + z = 1; \\ z_{1}, z_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$

Step 3: The optimal value of Model 6.2.1.2 is {z₁ = 0.717, z₂ = 0.283}.

Step 4: Substituting the optimal value {z₁ = 0.717, z₂ = 0.283} of Model 6.2.1.2 in Model 6.2.1.1, the basic optimal solutions of Model 6.2.1.1 are ${t_{1}^{1 *} = 1, t_{2}^{1 *} = 0}$ and ${t_{1}^{2 *} = 0, t_{2}^{2 *} = 1}$ . Since, there exist more than one optimal solution of Model 6.2.1.1 so obtain the minimum of { $A_{Ω} (〈 (180, 185, 190, 200); 0.7, 0.4, 0.6 〉 z_{1}^{*} t_{1}^{h *} + 〈 (160,$ $166, 168, 170); 0.6, 0.3, 0.5 〉 z_{1}^{*} t_{2}^{h *} + 〈 (135, 138, 142, 150); 0.8,$ $0.1, 0.4 〉 z_{2}^{*} t_{1}^{h *} + 〈 (185, 195, 205, 210); 0.5, 0.2, 0.7 〉 z_{2}^{*} t_{2}^{h *}, h = 1, 2)$ . $Since, { t12 * = 0, t22 * = 1 }$ denotes the minimum value thus, player I’s minimum expected gain is $〈 (180, 185, 190, 200); 0.7, 0.4, 0.6 〉 z_{1}^{*} t_{1}^{2 *} + 〈 (160, 166,$ $168, 170); 0.6, 0.3, 0.5 〉 z_{1}^{*} t_{2}^{2 *} + 〈 (135, 138, 142, 150);$ $0.8, 0.1, 0.4 〉 z_{2}^{*} t_{1}^{2 *} + 〈 (185, 195, 205, 210); 0.5, 0.2, 0.7 〉$ $z_{2}^{*} t_{2}^{2 *} = 〈 (167.069, 174.2004, 178.463, 181.311), 0.5, 0.4, 0.7 〉$ which is a SVTNN and the corresponding optimal strategy for player I will be {z₁ = 0.717, z₂ = 0.283}, which is the optimal value of Model 6.2.1.2.

6.2.2 The player II’s maximum expected loss

The player II’s optimal strategies and the corresponding maximum expected loss can be computed as follows:

Step 1. Find {z_i, i = 1, 2} ∈ Z such that V_Ω(〈(180, 185, 190, 200) ;0.7, 0.4, 0.6〉z₁t₁ + 〈(160, 166, 168, 170) ;0.6, 0.3, 0.5〉z₁t₂ + 〈(135, 138, 142, 150) ;0.8, 0.1, 0.4〉z₂t₁ + 〈(185, 195, 205, 210) ;0.5, 0.2, 0.7〉z₂t₂) is maximum for all (t₁, t₂) ∈ T i.e., obtain optimal solution {z_i, i = 1, 2} of Model 6.2.2.1.

Model 6.2.2.1 $\begin{matrix} \max {\begin{matrix} V_{Ω} (〈 (180, 185, 190, 200); 0.7, 0.4, 0.6 z_{1} t_{1} \\ + 〈 (160, 166, 168, 170); 0.6, 0.3, 0.5 z_{1} t_{2} \\ + 〈 (135, 138, 142, 150); 0.8, 0.1, 0.4 z_{2} t_{1} \\ + 〈 (185, 195, 205, 210); 0.5, 0.2, 0.7 z_{2} t_{2}) \end{matrix}} \\ s . t {\begin{matrix} z_{1} + z_{2} = 1; \\ t_{1} + t_{2} = 1; \\ z_{1} + z_{2} ⩾ 0; \\ t_{1} + t_{2} ⩾ 0 . \end{matrix} \end{matrix}$

Step 2: Since, in Model 6.2.2.1, only z_i have been considered as decision variables. Thus, Model 6.2.2.1 is a linear programming model and therefore, the optimal solution of Model 6.2.2.1 will be equal to the optimal solution of its corresponding dual model i.e., Model 6.2.2.2.

Model 6.2.2.2 $\begin{matrix} min {σ} \\ s . t . {\begin{matrix} \begin{matrix} V_{Ω} (〈 (180, 185, 190, 200); 0.5, 0.4, 0.7 〉 t_{1} + 〈 (160, 166, 168, 170); 0.5, 0.4, 0.7 〉 t_{2}) ⩽ σ; \\ V_{Ω} (〈 (135, 138, 142, 150); 0.5, 0.4, 0.7 〉 t_{1} + 〈 (185, 195, 205, 210); 0.5, 0.4, 0.7 〉 t_{2}) ⩽ σ; \end{matrix} \\ \begin{matrix} t_{1} + t_{2} = 1; \\ t_{1}, t_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$

Step 3: The optimal value of Model 6.2.2.2 is {t₁ = 0.408, t₂ = 0.592}.

Step 4: Substituting the optimal value {t₁ = 0.408, t₂ = 0.592} of Model 6.2.2.2 in Model 6.2.2.1, the basic optimal solutions of Model 6.2.2.1 is ${z_{1}^{*} = 1, z_{2}^{*} = 0}$ . Since, there exist a unique optimal value of Model 6.2.2.1 thus, player II’s maximum expected loss is $〈 (180, 185, 190, 200); 0.7, 0.4, 0.6 〉 z_{1}^{*} t_{1}^{*} + 〈 (160, 166, 168, 170); 0.6, 0.3, 0.5 〉 z_{1}^{*} t_{2}^{*} + 〈 (135, 138, 142, 150); 0.8, 0.1, 0.4 〉 z_{2}^{*} t_{1}^{*} + 〈 (185, 195, 205, 210); 0.5, 0.2, 0.7 〉 z_{2}^{*} t_{2}^{*} = 〈 (168.165, 173.757, 176.981, 182.247); 0.5, 0.4, 0.7 〉$ which is a SVTNN and the corresponding optimal strategy foe player II will be {t₁ = 0.408, t₂ = 0.592}, which is the optimal value of Model 6.2.2.2.

6.3 Comparison analysis

In this Subsection, the proposed ranking approach is compared with other four approaches that were introduced in Refs. Jun Ye [34], Liang et al. [35], H. A. Khalifa [36] and Abdel-Basset et al. [21].

We compare our results with Jun Ye [34], where the score function is described by: $S (\tilde{b}) = \frac{1}{12} (k + l + m + n) (2 + u_{\tilde{b}} - v_{\tilde{b}} - w_{\tilde{b}})$

Here, $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ expresses a SVTNN. Based on this ranking approach, we obtain set of linear optimization models, as follows: $\begin{matrix} max {ρ} \\ s . t . {\begin{matrix} \begin{matrix} 88.0838 z_{1} + 65.9162 z_{2} ⩾ ρ; \\ 77.462 z_{1} + 92.75 z_{2} ⩾ ρ; \end{matrix} \\ \begin{matrix} z_{1} + z_{2} = 1; \\ z_{1}, z_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$ $\begin{matrix} min {σ} \\ s . t . {\begin{matrix} \begin{matrix} 88.0838 t_{1} + 77.462 t_{2} ⩽ σ; \\ 65.9162 t_{1} + 92.75 t_{2} ⩽ σ; \end{matrix} \\ \begin{matrix} t_{1} + t_{2} = 1; \\ t_{1}, t_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$

We solve the above linear optimization problems, and then obtain player I’s optimal solutions (z^*, S(ρ^*)), in which z^* =(0.716, 0.284), S(ρ^*) = 81.797 and the player II’s optimal solutions (t^*, S(σ^*)), in which t^* =(0.408, 0.592) and S(σ^*) = 81.797. Though this approach gives optimal strategies similar to our results, it considers no parameters. But, in our approach parameters play important roles toward optimal solutions.

We compare our results with Liang et al. [35], where the score function is described by: $S (\tilde{b}) = \frac{(k + 2 l + 2 m + n)}{6} \times \frac{(2 + u_{\tilde{b}} - v_{\tilde{b}} - w_{\tilde{b}})}{3}$

Here, $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ expresses a SVTNN. Based on this ranking approach, we obtain set of linear optimization models, as follows: $\begin{matrix} max {ρ} \\ s . t . {\begin{matrix} \begin{matrix} 87.8952 z_{1} + 65.7269 z_{2} ⩾ ρ; \\ 77.6278 z_{1} + 92.9511 z_{2} ⩾ ρ; \end{matrix} \\ \begin{matrix} z_{1} + z_{2} = 1; \\ z_{1}, z_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$ $\begin{matrix} min {σ} \\ s . t . {\begin{matrix} \begin{matrix} 87.8952 t_{1} + 77.6278 t_{2} ⩽ σ; \\ 65.7269 t_{1} + 92.9511 t_{2} ⩽ σ; \end{matrix} \\ \begin{matrix} t_{1} + t_{2} = 1; \\ t_{1}, t_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$

We solve the above linear optimization problems, and then obtain player I’s optimal solutions (z^*, S(ρ^*)), in which z^* =(0.726, 0.274), S(ρ^*) = 81.824 and the player II’s optimal solutions (t^*, S(σ^*)), in which t^* =(0.409, 0.591) and S(σ^*) = 81.824. Though this approach gives optimal strategies similar to our results, it considers no parameters. But, in our approach parameters play important roles toward optimal solutions.

We compare our results with H. A. Khalifa [36], where the score function is described by: $S (\tilde{b}) = \frac{1}{16} (k + l + m + n) (u_{\tilde{b}} + (1 - v_{\tilde{b}}) + (1 - w_{\tilde{b}}))$

Here, $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ represents a SVTNN. Based on this ranking approach, we obtain set of linear optimization models, as follows: $\begin{matrix} max {ρ} \\ s . t . {\begin{matrix} \begin{matrix} 66.0625 z_{1} + 49.4375 z_{2} ⩾ ρ; \\ 58.1 z_{1} + 69.5625 z_{2} ⩾ ρ; \end{matrix} \\ \begin{matrix} z_{1} + z_{2} = 1; \\ z_{1}, z_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$ $\begin{matrix} min {σ} \\ s . t . {\begin{matrix} \begin{matrix} 66.0625 t_{1} + 58.1 t_{2} ⩽ σ; \\ 49.4375 t_{1} + 69.5625 t_{2} ⩽ σ; \end{matrix} \\ \begin{matrix} t_{1} + t_{2} = 1; \\ t_{1}, t_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$

We solve the above linear optimization problems, and then obtain player I’s optimal solutions (z^*, S(ρ^*)), in which z^* =(0.717, 0.283), S(ρ^*) = 61.349 and the player II’s optimal solutions (t^*, S(σ^*)), in which t^* =(0.408, 0.592) and S(σ^*) = 61.349. Though this approach gives optimal strategies similar to our results, it considers no parameters. But, in our approach parameters play important roles toward optimal solutions.

Finally, we compare our results with Abdel-Basset et al. [21], where the score function is described by: $S (\tilde{b}) = \frac{1}{16} (k + l + m + n) (2 + u_{\tilde{b}} - v_{\tilde{b}} - w_{\tilde{b}})$

Here, $\tilde{b} = 〈 (k, l, m, n); u_{\tilde{b}}, v_{\tilde{b}}, w_{\tilde{b}} 〉$ expresses a SVTNN. Based on this ranking approach, we obtain set of linear optimization models, as follows: $\begin{matrix} max {ρ} \\ s . t . {\begin{matrix} \begin{matrix} 66.0625 z_{1} + 49.4375 z_{2} ⩾ ρ; \\ 58.1 z_{1} + 69.5625 z_{2} ⩾ ρ; \end{matrix} \\ \begin{matrix} z_{1} + z_{2} = 1; \\ z_{1}, z_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$ $\begin{matrix} min {σ} \\ s . t . {\begin{matrix} \begin{matrix} 66.0625 t_{1} + 58.1 t_{2} ⩽ σ; \\ 49.4375 t_{1} + 69.5625 t_{2} ⩽ σ; \end{matrix} \\ \begin{matrix} t_{1} + t_{2} = 1; \\ t_{1}, t_{2} ⩾ 0 . \end{matrix} \end{matrix} \end{matrix}$

6.4 Advantages and disadvantages

The advantages of this method can be pointed out as follows:

The proposed method is based on a SVTNN, and it is owing to the fact that SVTNN suitably reflects the uncertainty and hesitation in game theory.

The decision maker should know about the ranking method and arithmetic operations of SVTNNs which is very easy to learn for a new decision maker.

The value of any matrix game with payoffs of SVTNNs is also an SVTNN, which is always obtained by the proposed method.

The main advantage of this method is that it provides not only the degree of acceptance but also the degree of rejection and degree of indeterminacy to judge the object’s behavior.

The proposed models and method may be applied to solving many competitive decision problems in similar fields such as management, supply chain, economics, operation research, war science and advertising.

The disadvantages of this method can be written as follows:

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.

7 Conclusion

In this article, we formulate matrix games with payoffs of SVTNNs and propose corresponding parameterized linear programming Ambika technique. The highlights include:

The reasonable solution of a SVTNN matrix game and some important theorems are presented.

A pair of parameterized linear optimization problems is derived from the SVTN mathematical programming models to obtain the optimal strategies for both players.

Proposing an effective approach based on the rank order relations of the ambiguity and value of SVTNN and Tina et al. [33] Ambika approach.

Conducting a numerical example to prove the feasibility of the proposed Ambika approach.

The numerical study shows that the SVNS outperform IFS when studying matrix games under uncertainty.

To conclude, the approach discussed in this manuscript is applicable to various decision-making models with SVTN information. For future research, we will study the application of the proposed approach to solve Stackelberg security games, n-person games, constrained games and nonzero-sum games with SVTNNs parameters.

Footnotes

Acknowledgments

The authors would like to thank the valuable reviews and also appreciate the constructive suggestions from the anonymous referees. The researcher Mohamed Gaber Brikaa is funded by a scholarship A13585134 under the joint Executive Program between the Arab Republic of Egypt and China. This work was partly supported by the National Key Research an Development Program of China (No.2017YFB0305601, the National Key Research an Development Program of China (No. 2017YFB0701700).

References

Owen

, Game theory, 2nd Ed. Acad. Press. New York, (1982).

Zadeh

L.A.

, Fuzzy sets, Inf. Control 8(3) (1965), 338–353.

D.F.

, Bi-matrix games with pay-offs of intuitionistic fuzzy sets and bilinear programming method, Decis. Game Theory Manag. with Intuitionistic Fuzzy Sets. Springer, Berlin, Heidelb. 308 (2014), 421–441.

Figueroa-García

, Carlos

A.M.

and Chandra

, Optimal solutions for group matrix games involving interval-valued fuzzy numbers, Fuzzy Sets Syst 362 (2019), 55–70.

Jana

and Roy

S.K.

, Dual hesitant fuzzy matrix games: based on new similarity measure, Soft Comput. 23(18) (2018), 8873–8886.

Singh

, Gupta

and Mehra

, Matrix games with 2-tuple linguistic information, Ann. Oper. Res. 287(2) (2020), 895–910.

Zhou

and Xiao

, A new matrix game with payoffs of generalized Dempster-Shafer structures, Int. J. Intell. Syst. 34(9) (2019), 2253–2268.

Seikh

M.R.

, Karmakar

and Xia

, Solving matrix games with hesitant fuzzy pay-offs, Iran. J. Fuzzy Syst. 17(4) (2020), 25–40.

Han

and Deng

, A novel matrix game with payoffs of Maxitive Belief Structure, Int. J. Intell. Syst. 34(4) (2019), 690–706.

10.

Roy

S.K.

and Maiti

S.K.

, Reduction methods of type-2 fuzzy variables and their applications to Stackelberg game, Appl. Intell. 50 (2020), 1398–1415.

11.

Bhaumik

, Roy

S.K.

and Weber

G.W.

, Hesitant interval-valued intuitionistic fuzzy-linguistic term set approach in Prisoners’ dilemma game theory using TOPSIS: a case study on Human-trafficking, Cent. Eur. J. Oper. Res. 28(2) (2020), 797–816.

12.

Ammar

and Brikaa

M.G.

, Solving bi-matrix games in tourism planning management under rough interval approach, I.J. Math. Sci. Comput. 5(4) (2019), 44–62.

13.

Brikaa

M.G.

, Zheng

and Ammar

E.-S.

, Fuzzy Multi-objective Programming Approach for Constrained Matrix Games with Payoffs of Fuzzy Rough Numbers, Symmetry (Basel). 11 (2019), 702.

14.

Bhaumik

, Roy

S.K.

and Weber

G.W.

, Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management, J. Dyn. Games 8(2) (2021), 101–118.

15.

Brikaa

M.G.

, Zheng

and Ammar

E.-S.

, Resolving Indeterminacy Approach to Solve Multi-criteria Zero-sum Matrix Games with Intuitionistic Fuzzy Goals, Mathematics 8 (2020), 305.

16.

Smarandache

, Neutrosophic set a generalization of the intuitionistic fuzzy set, Int. J. Pure Appl. Math. 24 (2005), 287–297.

17.

Wang

, Smarandache

, Zhang

Y.Q.

and Sunderraman

, Single valued neutrosophic sets, Multisp. Multistructure 4 (2010), 410–413.

18.

Garg

, Decision making analysis based on sine trigonometric operational laws for single-valued neutrosophic sets and their applications, Appl. Comput. Math. 19(2) (2020), 255–276.

19.

Surya

and Murugappan

, Neutrosophic Inventory Model under Immediate Return for Deficient Items, Ann. Optim. Theory Pract. 3(4) (2020), 1–9.

20.

Harish

, Novel neutrality aggregation operators-based multiattribute group decision making method for single-valued neutrosophic numbers, Soft Comput 24 (2020), 10327–10349.

21.

Abdel-Basset

, Ali

and Atef

, Resource levelling problem in construction projects under neutrosophic environment, J. Supercomput. 76(2) (2020), 964–988.

22.

Garai

, Dalapati

, Garg

and Roy

T.K.

, Possibility mean, variance and standard deviation of single-valued neutrosophic numbers and its applications to multi-attribute decision-making problems, Soft Comput. 24(24) (2020), 18795–18809.

23.

Broumi

, Dey

, Talea

, Bakali

, Smarandache

, Nagarajan

and Kumar

, Shortest path problem using Bellman algorithm under neutrosophic environment, Complex Intell. Syst. 5(4) (2019), 409–416.

24.

Gaber

, Alharbi

M.G.

, Dagestani

A.A.

and Ammar

E.S.

, Optimal Solutions for Constrained Bimatrix Games with Payoffs Represented by Single-Valued Trapezoidal Neutrosophic Numbers, J. Math. 2021 (2021).

25.

Mullai

and Surya

, Neutrosophic Inventory Backorder Problem Using Triangular Neutrosophic Numbers, Neutrosophic Sets Syst. 31 (2020), 148–155.

26.

Leyva-Vázquez

, Quiroz-Martínez

M.A.

, Portilla-Castell

, Hechavarría-Hernández

and González-Caballero

J.R.

, A new model for the selection of informationtechnology project in a neutrosophic environment, Neutrosophic Sets Syst. 32(1) (2020), 344–360.

27.

Sun

and Junhua Hu

X.C.

, Novel single-valued neutrosophic decision-making approaches based on prospect theory and their applications in physician selection, Soft Comput. 23(1) (2019), 211–225.

28.

Dubois

and Prade

, Fuzzy sets and systems: theory and applications, Acad. Press. New York 144 (1980).

29.

Atanassov

, Intuitionistic Fuzzy Sets, Theory Appl. Verlag, Heidelb., (1999), 1–137.

30.

Jun

, A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, J. Intell. Fuzzy Syst. 26(5) (2014), 2459–2466.

31.

Bhaumik

, Roy

S.K.

and Li

, (α,β,γ) -cut set based ranking approach to solving bi-matrix games in neutrosophic environment, Soft Comput. 25(4) (2021), 2729–2739.

32.

Deli

and Şubaş

, A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems, Int. J. Mach. Learn. Cyber. 8 (2017), 1309–1322.

33.

Verma

and Kumar

, Ambika methods for solving matrix games with Atanassov’s intuitionistic fuzzy payoffs, IEEE Trans Fuzzy Syst 26(1) (2018), 270–283.

34.

, Some weighted aggregation operators of trapezoidal neutrosophic numbers and their multiple attribute decision making method, Informatica 28(2) (2017), 387–402.

35.

Liang

, Wang

and Li

, Multi-criteria group decision-making method based on interdependent inputs of single-valued trapezoidal neutrosophic information, Neural Comput. Appl. 30(1) (2018), 241–260.

36.

Khalifa

H.A.

, An approach for solving two-person zero-sum matrix games in neutrosophic environment, J. Ind. Syst. Eng. 12(2) (2019), 186–198.