In the realm of real-world problems, the allocation of limited resources is often represented using mathematical programming models. However, the coefficients within these models, whether in the objective function, constraints, or both, are frequently unknown with precision. Consequently, many real-world problems fall into the realm of uncertainty, necessitating various approaches such as stochastic, fuzzy, and interval methods, all of which have been extensively explored in academic research. These approaches aim to provide decision-makers (DMs) with comprehensive insights into uncertain problems and recommendations for optimal decision making. The applications of game theory extend across diverse fields such as economics, engineering, and biology. The main classes of games include matrix, continuous static, and differential games. Continuous static games differ from their counterparts in that they involve decision possibilities that are not discrete but rather continuous, with decisions and costs linked in a continuous manner. Furthermore, these games are static, indicating that the relationship between costs and decisions lacks a time component. Fuzzy set theory, pioneered by Zadeh [1], is widely applied in solving practical problems across domains such as financial risk management, engineering, business, and natural sciences. This theory allows the description and treatment of imprecise and uncertain elements inherent in decision problems. Numerous studies have addressed fuzzy optimization problems, including those by Akram and his colleagues [2–7]. Elshafei [8] proposed an interactive approach to solve Nash cooperative continuous static games by determining the stability package of the first type matching the derived compromise solution. Khalifa and Zeineldin [9] presented an interactive method to solve continuous cooperative static games with fuzzy parameters in objective function coefficients. Kacher and Larbain [10] introduced the concept of balance for non-cooperative games with fuzzy objectives, implying fuzzy parameters. Cruz and Simaan [11] proposed the theory of ordinal games in which players can rank their decision choices against the choices made by other players instead of relying solely on payoff functions. Navidi et al. [12] introduced a new game-theory approach to multi-response optimization problems, whereas Corley [13] defined a mixed double at the Nash balance for n-person strategy games. In a Nash equilibrium, each player’s mixed strategy maximizes his/her own expected profit according to the strategies of the other n−1 players. The dual and its relationship with Nash’s mixed equilibrium are compared, incorporating both topological and algebraic requirements.

Additionally, this paper presents a comparison between the dual and the mixed Nash equilibrium by examining both topological and algebraic conditions. Sasikala and Kumaraghuru [14] proposed an interactive approach based on compromise programming and a compromise weighting method to resolve Nash continuous cooperative static games (NCCSTG). Khalifa [15] introduced an interactive approach to address multi-objective nonlinear programming problems, which was also applied to cooperative continuous static games. Silbermayr [16] conducted a comprehensive review of the utilization of non-cooperative game theory in inventory management. Shuler [17] studied cooperative games in which poor agents did not enjoy co-operation with rich agents. Bilal et al. [18] introduced a novel method called q-rung orthopair fuzzy and explored its application in topologies. Ayub et al. [19] established a robust fusion of binary relations and linear Diophantine fuzzy sets, introducing the concept of linear Diophantine fuzzy by employing reference parameters associated with membership and non-membership fuzzy relations. Farid and Riaz [20] developed new q-rung orthopair fuzzy aggregation operators based on Aczel–Alsina operations.

In recent years, numerous authors have studied games under uncertainty, including Garg et al. [21], Khalifa et al. [22, 23], Xiao et al. [24], Rasoulzadeh et al. [25], Zanjani et al. [26], Das [27], Veeramani et al. [28], and Mao et al. [29]. Osman [30,31] analyzed the solvability set, stability set of the first kind, and stability set of the second kind for parametric convex nonlinear programming. Various fuzzy structures have been applied to develop application-based algorithms for medical diagnosis [32–34], risk assessments [35], and operational research [36, 37].

This research report examines cooperative, continuous, static games within a hazy environment, with a specific focus on defining the stability set of the second kind, without assuming differentiability. Following are the notable contributions and innovations of this study.

• • Introducing appropriate terminologies and measures that consider the characteristics of a potential optimal solution.

• • Conducting a parametric study by solving a parametric problem and determining the stability set of the second kind, aiming to gather comprehensive information about the potential optimal solutions in uncertain scenarios.

• • Performing a multicriteria analysis through interactive engagement with the DM to select the most satisfactory optimal solution among the possibilities.

The remainder of the paper is structured as follows: Section 2 provides the basic information needed to understand the concepts used in this study. Section 3 presents the formulation of the fuzzy cooperative continuous static game. Section 4 outlines the approach for determining the stability set of the second kind. Section 5 presents a solution methodology for determining the stability set of the second kind corresponding to the obtained α-Pareto optimal solution. Section 6 includes an illustrative example to demonstrate the proposed concept. Finally, Section 7 provides the concluding remarks.

This section presents some basic concepts and results related to the neutrosophical numbers per chunk in the fuzzy quadratic direction, close-spaced reconciliation, and their arithmetic operations.

Definition 1 ([38]). piecewise quadratic fuzzy number (PQFN) is denoted by ã_PQ = (a₁, a₂, a₃, a₄, a₅), where a₁ ≤ a₂ ≤ a₃ ≤ a₄ ≤ a₅ are real numbers, and its membership function μ_ãPQ is given by (see, Figure 1)

Definition 2 ([38]). An interval approximation [A]=[aαL,aαU] of a PQFN Ã is called closed interval approximation if

Definition 3 (Associated ordinary numbers, [38]). If [A]=[aαL,aαU] is the close interval approximation of PQFN, the associated ordinary number of [A] is defined as A^=aαL+aαU2.

Definition 4 ([32]). Let [A]=[aαL,aαU] and [B]=[bαL,bαU] be two interval approximations of PQFN. Then, the arithmetic operations are

• 1. Addition: [A]⊕[B]=[aαL+bαL,aαU+bαU].

• 2. Subtraction: [A]⊖[B]=[aαL-bαU,aαU-bαL].

• 3. Scalar multiplication: α[A]={[αaαL, αaαU],α>0,[αaαU, αaαL],α<0..

• 4. Multiplication: [A]⊗[B]=[aαUbαL+aαLbαU2,aαLbαL+aαUbαU2].

• 5. Division: [A] ⊘ [B]={[2 (aαLbαL+bαU),2 (aαUbαL+bαU)],[B]>0, bαL+bαU≠0,[2 (aαUbαL+bαU),2 (aαLbαL+bαU)],[B]<0, bαL+bαU≠0..

• 6. The order relation: [A](≲)[B] if aαL≤bαL and aαU≤bαU or aαL+aαU≤bαL+bαU.

Notably, P(ℝ)⊂F(ℝ), where F(ℝ) and P(ℝ) represent the sets of all the PQFNs and close in interval approximation of the PQFN, respectively.

Consider the following fuzzy cooperative continuous static games (F-CCSGs) with n-players having piecewise quadratic fuzzy parameters in the cost functions of the players and the right-hand side of constraints. These players, respectively have the costs.

subject to

Here, G_j(a, γ, b̃_jPQ), j=1, n¯ are convex functions on ℝⁿ ×ℝ^t, h_l(a, γ), l=1, r¯ are concave functions on ℝⁿ ×ℝ^t, g_i(a, γ), i = 1, n are convex functions on ℝⁿ × ℝ^t, and b̃_PQ = (b̃_1PQ, b̃_2PQ, …, b̃_nPQ)^T and ũ_PQ = (ũ_1PQ, ũ_2PQ, …, ũ_rPQ)^T represent the vectors of PQFNs [38]. Assume that there exists a function a = f(γ). If function g_i(a, γ) = 0, i=1, n¯ is differentiable, then Jacobian |∂gi(a, γ)∂aq|≠0, i; q=1, n¯ in the neighborhood of a solution point (a, γ) to (2), and a = f(γ) is the solution to (2) generated by γ ∈ Φ(ũ); differentiability assumptions are not required here for all functions G_j(a, γ, b̃_jPQ), j=1, n¯ ; h_l(a, γ) and Φ(ũ_PQ) are regular sets.

Let b̃_jPQ, j=1, n¯, and ũ_lPQ, l=1, r¯ be the PQFN in the F-CCSG problem with the following convex membership functions: μ_b̃1PQ(b₁), b̃_2PQ(b₂), …, b̃_nPQ(b_n); μ_ũ1PQ (u₁), ũ_2PQ(u₂), …, ũ_rPQ(u_r); respectively.

For a certain degree of α and based on Definition 2, the F-CCSG problem can be rewritten as in the following fuzzy form (Sakawa and Yano [39]).

Subject to

Definition 5 ([39, 40]). Let a = f(γ) be the solution to (5) generated by γ ∈ Φ. A point γ̄ ∈ Φ is called an α-Pareto optimal solution to the α-CCSG problem, if and only if there does not exist γ ∈ Φ, bj∈[bjαL,bjαU],j=1, n¯;ul∈[ulαL,ulαU],l=1, r¯ such that Gj(f(γ),γ,bj)≤Gj(f(γ¯),γ¯,bj¯);∀j=1, n¯ and Gj(f(γ),γ,bj)<Gj(f(γ¯),γ¯,bj¯) for some j ∈ {1, 2, …, n}, where bj¯ and ul¯ are called α-level minimal parameters.

From the α-Pareto optimal solution to the α-CCSG problem concept, one can show that a point γ̄ ∈ Φ is an α-Pareto minimal solution to the α-CCSG problem if and only if γ̄ is an α-Pareto minimal solution to the following α-multi-objective optimization problem

subject to

Here, Gj^(γ,bj),j=1, n¯ are convex functions on ℝⁿ × ℝ^t, hl^(γ), l = 1, 2, …, r are concave functions on ℝ^t, and Gj^(γ, bj)=Gj(f(γ),γ,bj), hl^(γ)=hl(f(γ),γ).

Assume that the α-MOP must be stable [41]. By applying the weighting method, the α-MOP becomes

subject to

Here, w_j ≥ 0, j = 1, 2, …, n, and ∑j=1nwj=1.

The stability of the α-MOP implies the stability of the αW problem for all w ∈ ℝⁿ. Let c = (c₁, c₂) ∈ ℝ²ⁿ, where c1=(c1αL,c2αL,…,cnαL)T,c2=(c1αU,c2αU,…,cnαU)T, and d = (d₁, d₂) ∈ ℝ^2r, where d1=(d1αL,d2αL,…,drαL)T and d2=(d1αU,d2αU,…,drαU)T. The set of α–Pareto optimal solutions of the (αW) problem is defined as follows: Q(w,b,u)={(γ*,b*,u*)∈ℝt+n+r:∑j=1nwjGj^(γ*,bj*)=min(γ,b,u)∈Φ∑j=1nwjGj^(γ,bj)}, where

Clearly, (γ^*, b^*) is an α-Pareto optimal solution, and (b^*, u^*) are the α-level optimal parameters of the α-MOP problem if there exists w^* ≠ 0, wj*≥0,j=1, n¯ such that (γ^*, b^*) is the unique optimal solution of the (αW) problem, that is,

Q(w*,b*,u*)={(w*,b*,u*)}(Chankong and Haimes [42]).

Definition 6. The solvability set of the (αW) problem, denoted by D, is defined by D = {(w, c, d) ∈ ℝ^3n+2r : α-MOP problem has α-Pareto optimal solutions}, where

Suppose that the (αW) problem is solvable for (w̄, c̄, d̄) ∈ ℝ^3n+2r with a corresponding α-Pareto optimal solution (γ̄, b̄, ū), and let I ⊆ {1, 2, …, r}, J ⊆ {1, 2, …, n}. The following two sets are defined

Let ψ(J, I) be the side (or sides) of Φ∩L_α(b̃_j, ũ_l) or int(Φ∩ L_α(b̃_j, ũ_l)) that is defined by

Definition 7 ([30, 31]). The stability set of the second kind of the (αW) problem corresponding to ψ(J, I) is denoted by H(ψ(J, I)) and is defined as

Remark 1. By definition 6, it follows that H(ψ(J, I)) = ⋃_k∈K S(x^k, b^k, u^k) where S(x^k, b^k, u^k) is the solvability set of the first kind of the (αW) problem corresponding to (x^k, b^k, u^k) that is defined as

and

Remark 2 (Osman and Dauer, 1983). The relation between the α-MOP problem and (αW) problem implies that

Given the close interval approximation of a PQFN b̃_jPQ, j=1, n¯; ũ_lPQ, l=1, r¯ (that is, bj∈[bjαL, bjαU], j=1, n¯;ul∈[ulαL, ulαU], l=1, r¯ ), assume that for (w̄, c̄, d̄) ∈ ℝ^3n+2r, an α-Pareto optimal solution (γ̄, b̄, ū) exists. Because α-MOP is stable, the αW problem is also stable, and it follows from the Kuhn-Tucker saddle-point optimality theorem [43, 44] that there exists 0 ≠ w̄ ∈ ℝⁿ, w̄_j ≥ 0, j=1, n¯ , v̄ ∈ ℝ^r, ζ̄, ξ̄ ∈ ℝⁿ, ζ̄, ξ̄ ≥ 0; δ̄, ρ̄ ∈ ℝ^r, δ̄, ρ̄ ≥ 0 such that (γ̄, b̄, ū, w̄, v̄, ζ̄, ξ̄, δ̄, ρ̄) solves the Kuhn-Tucker saddle point problem, that is, χ(γ̄, b̄, ū, w̄, v, ζ, ξ, δ, ρ) ≤ χ(γ̄, b̄, ū, w̄, v̄, ζ̄, ξ̄, δ̄, ρ̄) ≤ χ(γ, b, u, w̄, v̄, ζ̄, ξ̄, δ̄, ρ̄); ∀ (γ, b, u) ∈ ℝ^t+n+r; ∀ v ∈ ℝ^r, v ≥ 0, ζ, ξ ∈ ℝⁿ; ζ, ξ ≥ 0; δ, ρ ∈ ℝ^r, δ, ρ ≥ 0, where χ(γ,b,u,w,v,ζ,ξ,δ,ρ)=∑j=1nwjGj^(γ,bj)+∑l=1rvl(hl^(γ)-ul)+∑j=1nζj(bj-bjαU)+∑j=1nξj(bjαL-bj)+∑l=1rδl(ul-ulαU)+∑l=1rρl(ulαL-ul) , w_j ≥ 0, ∑j=1nwj=1 . Let us formulate the Kuhn-Tucker saddle point conditions for the (αW) problem as follows:

In this section, a solution method for determining the stability set of the second kind is introduced.

Step 1: Consider bj∈[bjαL,bjαU],j=1, n¯;ul∈[ulαL,ulαU],l=1, r¯ .

Step 2: Find the Lagrange function of the (αW) problem and formulate the Kuhn-Tucker saddle point (KTSP) conditions illustrated previously.

Step 3: Find function p : ℝ²ⁿ → ℝ^m such that γ̄ = p(w, b) and satisfies the KTSP condition in Step 2.

Step 4: Determine the stability set of the second kind as

Consider the following F-CCSG

where player I controls γ₁ ∈ ℝ and player II controls γ₁ ∈ ℝ with

Let b̃_1PQ = (1, 1.36, 2, 4.64, 5), b̃_2PQ = (3, 3.72, 5, 9.64, 10), ũ₁ = (0, 0.36, 1, 3.64, 4), and ũ₂ = (1, 1.36, 2, 5.64, 6). Then, the nonfuzzy α-MOP problem becomes

subject to

Using the weighting method, the problem can be rewritten as

subject to

We have J ⊆ {1, 2, 3, 4}, I ⊆ {1, 2, 3, 4}. Let I₁ = ∅, J₁ = ∅, for the interior side.

we get v̄_l = 0, δl¯=0 , l = 1, 2, 3, 4; ζj¯=0, j = 1, 2, 3, 4.

The first condition of the KTSP conditions take the following form:

Consequently, the stability set of the second type corresponding to that side is

For side J₂ = {1} and I₂ = {1}, the corresponding stability set of the second kind is

For side J₃ = {2} and I₃ = {2}, the corresponding stability set of the second kind is

For side J₄ = {1, 2} and I₄ = {1, 2}, the corresponding stability set of the second kind is

6.1 Advantages/Limitations of the Proposed Algorithm

The main advantage of the proposed algorithm lies in its unique combination of a parametric study, multicriteria analysis, and integration of the perspective of the DM. This innovative approach leverages the benefits of a parametric study that intelligently explores the search space, as well as the advantages of multicriteria analysis that ranks alternative solutions based on the vision of the DM. Moreover, it actively incorporates the perspective of the DM throughout the process.

Nevertheless, applying the proposed algorithm to real-life problems may result in certain limitations, including the following:

• • Incomplete consideration of the entire parametric space that comprises an infinite number of possible scenarios. However, no existing technique can effectively handle such situations with infinite scenarios.

• • Lack of a unified technique for assigning interesting scenarios to the DM. This approach does not involve a standardized method, as the vision and weighting of the DM may vary from one individual to another.

• • Several factors must be considered such as (i) the feasibility of formulating the problem as a nonlinear multi-objective (NLMO) problem, (ii) the potential for formulating and solving the Karush–Kuhn–Tucker (KKT) conditions, and (iii) the capability of solving selected scenarios of the parametric multi-objective nonlinear programming (PMONLP) problem and obtaining their exact optimal solutions.

This article presents continuous and fuzzy cooperative static games involving n players in which cost functions and constraints integrate fuzzy quadratic parameters using chunks. The stability set of the second kind, with no differentiability, was defined and determined. In addition, the proposed approach integrated the perspective of the DM with the process of determining the best solution. A utility function was used to classify different options, such that a satisfactory optimal solution could be easily identified. Furthermore, an illustrative example was presented to demonstrate the strength of the proposed attack. The results from a genetic algorithm (GA), a major scalable algorithm, were also compared to validate the accuracy and reliability of the simulation results. Future work may involve extending this field to encompass other fuzzy-like structures, such as neutrosophic, interval-valued fuzzy, spherical fuzzy, Pythagorean fuzzy, and linear Diophantine fuzzy sets. Additionally, new fuzzy systems, such as interval type-2, interval type-3, possibility interval-valued intuitionistic fuzzy sets, possibility neutrosophic sets, possibility interval-valued neutrosophic sets, possibility interval-valued fuzzy sets, and possibility fuzzy expert sets can be weighed, with possible applications in decision making.

All the data used to support the findings of this study are included in this article.