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닫기 International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 365-374

Published online September 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.3.365

© The Korean Institute of Intelligent Systems

## Toward Multi-Objective Optimization Approach for Solving Cooperative Continuous Static Games under Fuzzy Environment

Alhanouf Alburaikan1, Hamiden Abd El-Wahed Khalifa1,2, and Muhammad Saeed3

1Department of Mathematics, College of Science and Arts, Qassim University, Al-Badaya, Saudi Arabia
2Department of Operations and Management Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt
3Department of Mathematics, University of Management and Technology, Lahore, Pakistan

Correspondence to :
Hamiden Abd El-Wahed Khalifa (hamiden@cu.edu.eg)

Received: December 13, 2022; Revised: May 26, 2023; Accepted: June 9, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study focuses on the analysis of cooperative continuous static games involving multiple players (n-players) with fuzzy parameters present in both the cost functions and the right-hand side of constraints. The fuzzy parameters used in this study are described by piecewise quadratic fuzzy numbers that are known for their accurate representation. Specifically, we introduce close interval approximations as a reliable method for handling piecewise quadratic fuzzy numbers. The study defines and investigates the stability set of the second kind in a fuzzy environment, where differentiability is not assumed. The determination of this stability set is an essential aspect of the study. To illustrate the concepts presented, a numerical example is provided, offering a practical demonstration of the proposed framework.

Keywords: Optimization problems, Mathematical model, Cooperative continuous static games, Multi-objective decision making, Piecewise quadratic fuzzy number, Fuzzy logic, Close interval approximation, Weighting approach, α-Pareto optimal solution, Sensitivity analysis

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 , 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 . Elshafei  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  presented an interactive method to solve continuous cooperative static games with fuzzy parameters in objective function coefficients. Kacher and Larbain  introduced the concept of balance for non-cooperative games with fuzzy objectives, implying fuzzy parameters. Cruz and Simaan  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.  introduced a new game-theory approach to multi-response optimization problems, whereas Corley  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  proposed an interactive approach based on compromise programming and a compromise weighting method to resolve Nash continuous cooperative static games (NCCSTG). Khalifa  introduced an interactive approach to address multi-objective nonlinear programming problems, which was also applied to cooperative continuous static games. Silbermayr  conducted a comprehensive review of the utilization of non-cooperative game theory in inventory management. Shuler  studied cooperative games in which poor agents did not enjoy co-operation with rich agents. Bilal et al.  introduced a novel method called q-rung orthopair fuzzy and explored its application in topologies. Ayub et al.  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  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. , Khalifa et al. [22, 23], Xiao et al. , Rasoulzadeh et al. , Zanjani et al. , Das , Veeramani et al. , and Mao et al. . 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 , risk assessments , 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 (). piecewise quadratic fuzzy number (PQFN) is denoted by ãPQ = (a1, a2, a3, a4, a5), where a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 are real numbers, and its membership function μãPQ is given by (see, Figure 1)

Ma˜PQ={0,x<a1;121(a2-a1)2(x-a1)2,a1xa2;121(a3-a2)2(x-a3)2+1,a2xa3;121(a4-a3)2(x-a3)2+1,a3xa4;121(a5-a4)2(x-a5)2,a4xa5;0,x>a5.

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

aαL=inf{x:μA˜0.5},andaαU=sup{x:μA˜0.5}.

Definition 3 (Associated ordinary numbers, ). 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 (). Let [A]=[aαL,aαU] and [B]=[bαL,bαU] be two interval approximations of PQFN. Then, the arithmetic operations are

• 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+bαU0,[2(aαUbαL+bαU),2(aαLbαL+bαU)],[B]<0,bαL+bαU0..

• 6. The order relation: [A](≲)[B] if aαLbαL and aαUbαU or aαL+aαUbαL+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.

(F-CCSG)         G1(a,γ,b˜1P),G2(a,γ,b˜2PQ),,Gn(a,γ,b˜nPQ),

subject to

gi(a,γ)=0,i=1,2,,n,Φ(u˜PQ)={γt:hl(a,γ)u˜lPQ,l=1,2,,r}.

Here, Gj(a, γ, b̃jPQ), j=1,n¯ are convex functions on ℝn ×ℝt, hl(a, γ), l=1,r¯ are concave functions on ℝn ×ℝt, gi(a, γ), i = 1, n are convex functions on ℝn × ℝt, and PQ = (1PQ, 2PQ, …, nPQ)T and PQ = (1PQ, 2PQ, …, rPQ)T represent the vectors of PQFNs . Assume that there exists a function a = f(γ). If function gi(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(); differentiability assumptions are not required here for all functions Gj(a, γ, jPQ), j=1,n¯ ; hl(a, γ) and Φ(PQ) are regular sets.

Let jPQ, j=1,n¯, and lPQ, l=1,r¯ be the PQFN in the F-CCSG problem with the following convex membership functions: μ1PQ(b1), 2PQ(b2), …, nPQ(bn); μũ1PQ (u1), 2PQ(u2), …, rPQ(ur); 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 ).

(α-CCSG)         G1(a,γ,b1),G2(a,γ,b2),,Gn(a,γ,bn),

Subject to

gi(a,γ)=0,i=1,2,,n,Φ(u)={γt:hl(a,γ)ul,l=1,2,,r},bj[bjαL,bjαU],j=1,n¯;ul[ulαL,ulαU],l=1,r¯.

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

(α-MOP)         min(G1^(γ,b1),G2^(γ,b2),,Gn^(γ,bn)),

subject to

Φ(u)={γt:hl^(γ)ul,l=1,2,,r},bjLα(b˜j)=[bjαL,bjαU],j=1,n¯;ulLα(u˜l)=[ulαL,ulαU],l=1,r¯.

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

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

(αW)         minj=1nwjGj^(γ,bj),

subject to

Φ={γt:hl^(γ)ul,   l=1,2,,r},bj[bjαL,bjαU],j=1,n¯;ul[ulαL,ulαU],l=1,r¯.

Here, wj ≥ 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) ∈ ℝ2r, 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*)t+n+r:j=1nwjGj^(γ*,bj*)=min(γ,b,u)Φj=1nwjGj^(γ,bj)}, where

ΦLαbj˜,ul˜.

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 ).

Definition 6. The solvability set of the (αW) problem, denoted by D, is defined by D = {(w, c, d) ∈ ℝ3n+2r : α-MOP problem has α-Pareto optimal solutions}, where

wn,c=(c1,c2)2n,d=(d1,d2)2r.

Suppose that the (αW) problem is solvable for (w̄, c̄, d̄) ∈ ℝ3n+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

B={(γ,b,u)t+n+r:hl^(γ)=ul,bj-bjαU=0,bjαL-bj=0,ul-ulαU=0,ulαL-ul=0},C={(γ,b,u)t+n+r:hl^(γ)>ul,bj-bjαU<0,bjαL-bj<0,ul-ulαU<0,ulαL-ul<0}.

Let ψ(J, I) be the side (or sides) of Φ∩Lα(b̃j, ũl) or int(Φ∩ Lα(b̃j, ũl)) that is defined by

ψ(J,I)={(γ,b,u)t+n+r:(γ,b,u)B;(j,1)J×I;(γ,b,u)C;(j,1)J×I}.

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

H(ψ(J,I))={(w,c,d)3n+2r:ψ(J,I)containsα-Pareto optimal solution of the α-MOP problem}.

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

S(xk,bk,uk)={(w,c,d)3n+2r:(xk,bk,uk)is an α-Pareto optimal solution of the α-MOP problem},

and

K={(xk,bk,uk)ψ(J,I)is an α-Pareto optimal solution of α-MOP}.

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

S(xk,bk,uk)={(w,c,d)3n+2r:(xk,bk,uk)is an α-Pareto optimal solution of the (αW) problem}.

### 4. Determination of the Stability Set of the Second Kind without Differentiability

Given the close interval approximation of a PQFN jPQ, j=1,n¯; ũlPQ, l=1,r¯ (that is, bj[bjαL,bjαU],j=1,n¯;ul[ulαL,ulαU],l=1,r¯ ), assume that for (w̄, c̄, d̄) ∈ ℝ3n+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 ≠ ∈ ℝn, j ≥ 0, j=1,n¯ , ∈ ℝr, ζ̄, ξ̄ ∈ ℝn, ζ̄, ξ̄ ≥ 0; δ̄, ρ̄ ∈ ℝr, δ̄, ρ̄ ≥ 0 such that (γ̄, b̄, ū, , , ζ̄, ξ̄, δ̄, ρ̄) solves the Kuhn-Tucker saddle point problem, that is, χ(γ̄, b̄, ū, , v, ζ, ξ, δ, ρ) ≤ χ(γ̄, b̄, ū, , , ζ̄, ξ̄, δ̄, ρ̄) ≤ χ(γ, b, u, , , ζ̄, ξ̄, δ̄, ρ̄); ∀ (γ, b, u) ∈ ℝt+n+r; ∀ v ∈ ℝr, v ≥ 0, ζ, ξ ∈ ℝn; ζ, ξ ≥ 0; δ, ρ ∈ ℝr, 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) , wj ≥ 0, j=1nwj=1 . Let us formulate the Kuhn-Tucker saddle point conditions for the (αW) problem as follows:

j=1nwj¯Gj^(γ¯,bj¯)+l=1rvl(hl^(γ¯)-ul¯)+j=1nζj(bj¯-bjαU)+j=1nξj(bjαL-bj¯)+l=1rδl(ul¯-ulαU)+l=1rρl(ulαL-ul¯)j=1nwj¯Gj^(γ¯,bj¯)+l=1rvl¯(hl^(γ¯)-ul¯)+j=1nζj¯(bj¯-bjαU)+j=1nξj¯(bjαL-bj¯)+l=1rδl¯(ul¯-ulαU)+l=1rρl¯(ulαL-ul¯)j=1nwj¯Gj^(γ,bj)+l=1rvl¯(hl^(γ)-ul)+j=1nζj¯(bj-bjαU)×j=1nξj¯(bjαL-bj)+l=1rδl¯(ul-ulαU)+l=1rρl¯(ulαL-ul);(γ,b,u)t+n+r,vr,v0;ζ,ξn;ζ,η0;δ,ρr,δ,ρ0,hl^(γ¯)ul¯,l=1,2,,r,bj¯-bjαU0,j=1,2,,n,bjαL-bj¯0,j=1,2,,n,ul¯-ulαU0,1=1,2,,r,(ul)αL-u¯l0,l=1,2,,r,v¯l(hl^(γ¯)-ul¯)=0,l=1,2,,r,ζj¯(bj¯-bjαU)=0,j=1,2,,n,ξj¯(bjαL-bj¯)=0,j=1,2,,n,δl¯(ul¯-ulαU)=0,l=1,2,,r,ρl¯(ulαL-ul¯)=0,l=1,2,,r,ζj¯,ξj¯0,j=1,2,,n,v¯l,δl¯,ρl¯0,   l=1,2,,r.

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 → ℝm 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

H(ψ(J,I))={(w,c,d)B:hl^(p(w,b))=ul,for (j,l)J×I,hl^(p(w,b))<ul,for (j,l)J×I}.

Consider the following F-CCSG

(F-CCSG)         G1^(γ,b˜1PQ)=(γ1-b˜1PQ)2-8γ2,G2^(γ,b˜2PQ)=2γ1+(γ2+b˜2PQ)2,

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

γ1+γ2u˜1,2γ1+γ2u˜2,γ1,γ20.

Let 1PQ = (1, 1.36, 2, 4.64, 5), 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

min {(γ1-b1)2-8γ2,2γ1+(γ2+b2)2},

subject to

Φ={γ2:γ1+γ2u1,2γ1+γ2u2,γ1,γ20},b1[1.36,4.64],b2[3.72,9.64],u1[0.36,3.64],u2[1.36,5.64].

Using the weighting method, the problem can be rewritten as

min{w(γ1-b1)2-8wγ2+2(1-w)γ1+(1-w)(γ2+b2)2},

subject to

Φ={γ2:γ1+γ2u1,2γ1+γ2u2,γ1,γ20},b1[1.36,4.64],b2[3.72,9.64],u1[0.36,3.64],u2[1.36,5.64],   w0.

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

(γ1¯+γ2¯u1,2γ1¯+γ2¯u2,-γ1¯<0,-γ2¯<0,b1¯]1.36,4.64[,b2¯]3.72,9.64[,u1¯]0.36,3.64[,u2¯]1.36,5.64[),

we get v̄l = 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:

w(γ1-b1)2-8wγ2+2(1-w)γ1+(1-w)(γ2+b2)2(w(γ1¯-b1¯)2-8wγ2¯+2(1-w)γ1¯+(1-w)(γ2¯+b2¯)2)0.

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

H(ψ(J1,I1))={(w,c,d)9:w=1,u¯1>b1¯,u¯2>2b2¯}{(w,c,d)9:w=0,u¯1>-b2¯,u¯2>-b2¯}{(w,c,d)9:w{0,1},u¯1>b1¯-b2¯+1-ww+4w1-w,u¯2>2b1¯-b2¯+2(1-w)w+4w1-w}.

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

H(ψ(J2,I2))={(w,c,d)9:w=1,u¯1=b1¯,u¯2>2b2¯}{(w,c,d)9:w=0,u¯1=-b2¯,u¯2>-b2¯}{(w,c,d)9:w{0,1},u¯1=b1¯-b2¯+1-ww+4w1-w,u¯2>2b1¯-b2¯+2(1-w)w+4w1-w}.

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

H(ψ(J3,I3))={(w,c,d)9:w=1,u¯1>b1¯,u¯2=2b2¯}{(w,c,d)9:w=0,u¯1>-b2¯,u¯2=-b2¯{(w,c,d)9:w{0,1},u¯1>b1¯-b2¯+1-ww+4w1-w,u¯2=2b1¯-b2¯+2(1-w)w+4w1-w}.

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

H(ψ(J4,I4))={(w,c,d)9:w=1,u¯1=b1¯,u¯2=2b2¯}{(w,c,d)9:w=0,u¯1=-b2¯,u¯2=-b2¯}{(w,c,d)9:w{0,1},u¯1=b1¯-b2¯+1-ww+4w1-w,u¯2=2b1¯-b2¯+2(1-w)w+4w1-w}.

### 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.

The researchers thank the Deanship of Scientific Research at Qassim University for funding this project. Fig. 1.

Graphical representation of a piecewise quadratic fuzzy number (PQFN).

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44. Stoer, J, and Witzgall, C (1970). Convexity and Optimization in Finite Dimensions I. Berlin, Germany: Springer-Verlag Alhanouf Alburikan received the Ph.D. in Mathematics from Howard University, Washington, DC, USA. She is affiliated with the Department of Mathematics, College of Science and Arts, Qassim University, Buraydah, Saudi Arabia. Her research interests include complex analysis, fuzzy mathematics, game theory, and nonlinear programming in real and complex spaces. She has published more than 15 articles in peer-reviewed journals. Hamiden Abd Ei-Wahed Khalifa received the Ph.D. from the Faculty of Science, Tanta University, Tanta, Gharbia Governorate, Egypt. She is a full-time professor of Operations Research (Mathematical Programming). She has recently been attached to the Operations Research Department, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt, as Professor. She is also attached to the Department of Mathematics, College of Science and Arts, Qassim University, Buraydah, Saudi Arabia, for the research projects. She has published more than 150 articles in peer-reviewed journals. Her research interests include game theory, multi-objective programming, fuzzy mathematics, rough sets, decision making, and optimization. Muhammad Saeed was born in Pakistan in July 1970. He received the Ph.D. in Mathematics from Quaid-i-Azam University, Islamabad, Pakistan, in 2012. He has taught mathematics at the intermediate and degree levels with exceptional results. He has been involved as a teacher trainer in professional development for more than five years. He has worked as Chairman of the Department of Mathematics at UMT, Lahore, from 2014 to 2021. Under his dynamic leadership, the Mathematics Department has produced more than 30 Ph.D.s. He has supervised 30 M.S. and 7 Ph.D. students and has published more than 150 articles in recognized journals. His research interests include fuzzy mathematics, rough sets, soft set theory, hypersoft sets, neutrosophic sets, algebraic and hybrid structures of soft sets and hypersoft sets, multicriteria decision making, optimizations, artificial intelligence, pattern recognition and optimization under convex environments, graph theory in fuzzy-, soft-, and hypersoft-like environments, and distance measures and their relevant operators in multipolar hybrid structures. He was awarded the “Best Teacher” in 1999, 2000, 2001, and 2002.

### Article

#### Original Article International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 365-374

Published online September 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.3.365

## Toward Multi-Objective Optimization Approach for Solving Cooperative Continuous Static Games under Fuzzy Environment

Alhanouf Alburaikan1, Hamiden Abd El-Wahed Khalifa1,2, and Muhammad Saeed3

1Department of Mathematics, College of Science and Arts, Qassim University, Al-Badaya, Saudi Arabia
2Department of Operations and Management Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt
3Department of Mathematics, University of Management and Technology, Lahore, Pakistan

Correspondence to:Hamiden Abd El-Wahed Khalifa (hamiden@cu.edu.eg)

Received: December 13, 2022; Revised: May 26, 2023; Accepted: June 9, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This study focuses on the analysis of cooperative continuous static games involving multiple players (n-players) with fuzzy parameters present in both the cost functions and the right-hand side of constraints. The fuzzy parameters used in this study are described by piecewise quadratic fuzzy numbers that are known for their accurate representation. Specifically, we introduce close interval approximations as a reliable method for handling piecewise quadratic fuzzy numbers. The study defines and investigates the stability set of the second kind in a fuzzy environment, where differentiability is not assumed. The determination of this stability set is an essential aspect of the study. To illustrate the concepts presented, a numerical example is provided, offering a practical demonstration of the proposed framework.

Keywords: Optimization problems, Mathematical model, Cooperative continuous static games, Multi-objective decision making, Piecewise quadratic fuzzy number, Fuzzy logic, Close interval approximation, Weighting approach, &alpha,-Pareto optimal solution, Sensitivity analysis

### 1. Introduction

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 , 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 . Elshafei  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  presented an interactive method to solve continuous cooperative static games with fuzzy parameters in objective function coefficients. Kacher and Larbain  introduced the concept of balance for non-cooperative games with fuzzy objectives, implying fuzzy parameters. Cruz and Simaan  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.  introduced a new game-theory approach to multi-response optimization problems, whereas Corley  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  proposed an interactive approach based on compromise programming and a compromise weighting method to resolve Nash continuous cooperative static games (NCCSTG). Khalifa  introduced an interactive approach to address multi-objective nonlinear programming problems, which was also applied to cooperative continuous static games. Silbermayr  conducted a comprehensive review of the utilization of non-cooperative game theory in inventory management. Shuler  studied cooperative games in which poor agents did not enjoy co-operation with rich agents. Bilal et al.  introduced a novel method called q-rung orthopair fuzzy and explored its application in topologies. Ayub et al.  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  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. , Khalifa et al. [22, 23], Xiao et al. , Rasoulzadeh et al. , Zanjani et al. , Das , Veeramani et al. , and Mao et al. . 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 , risk assessments , 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.

### 2. Preliminaries

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 (). piecewise quadratic fuzzy number (PQFN) is denoted by ãPQ = (a1, a2, a3, a4, a5), where a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 are real numbers, and its membership function μãPQ is given by (see, Figure 1)

$Ma˜PQ={0,xa5.$

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

$aαL=inf{x∈ℝ:μA˜≥0.5}, andaαU=sup{x∈ℝ:μA˜≥0.5}.$

Definition 3 (Associated ordinary numbers, ). 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 (). 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.

### 3. Problem Formulation and Solution Concepts

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.

$(F-CCSG) G1(a,γ,b˜1P), G2(a,γ,b˜2PQ),…,Gn(a,γ,b˜nPQ),$

subject to

$gi(a,γ)=0,i=1,2,⋯,n,$$Φ(u˜PQ)={γ∈ℝt:hl(a,γ)≥u˜lPQ,l=1,2,…,r}.$

Here, Gj(a, γ, b̃jPQ), $j=1, n¯$ are convex functions on ℝn ×ℝt, hl(a, γ), $l=1, r¯$ are concave functions on ℝn ×ℝt, gi(a, γ), i = 1, n are convex functions on ℝn × ℝt, and PQ = (1PQ, 2PQ, …, nPQ)T and PQ = (1PQ, 2PQ, …, rPQ)T represent the vectors of PQFNs . Assume that there exists a function a = f(γ). If function gi(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(); differentiability assumptions are not required here for all functions Gj(a, γ, jPQ), $j=1, n¯$ ; hl(a, γ) and Φ(PQ) are regular sets.

Let jPQ, $j=1, n¯$, and lPQ, $l=1, r¯$ be the PQFN in the F-CCSG problem with the following convex membership functions: μ1PQ(b1), 2PQ(b2), …, nPQ(bn); μũ1PQ (u1), 2PQ(u2), …, rPQ(ur); 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 ).

$(α-CCSG) G1(a,γ,b1),G2(a,γ,b2),…,Gn(a,γ,bn),$

Subject to

$gi (a,γ)=0, i=1,2,…,n,$$Φ(u)={γ∈ℝt:hl(a,γ)≥ul, l=1,2,…,r},$$bj∈[bjαL,bjαU], j=1, n¯; ul∈[ulαL,ulαU], l=1, r¯.$

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) 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

$(α-MOP) min(G1^(γ, b1),G2^(γ, b2),…,Gn^(γ, bn)),$

subject to

$Φ(u)={γ∈ℝt:hl^(γ)≥ul, l=1,2,⋯,r},bj∈Lα(b˜j)=[bjαL, bjαU], j=1, n¯;ul∈Lα(u˜l)=[ulαL,ulαU],l=1, r¯.$

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

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

$(αW) min∑j=1nwjGj^ (γ,bj),$

subject to

$Φ={γ∈ℝt:hl^(γ)≥ul, l=1, 2,…,r},bj∈[bjαL,bjαU], j=1, n¯;ul∈[ulαL,ulαU], l=1, r¯.$

Here, wj ≥ 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) ∈ ℝ2r, 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

$Φ∨=Φ∩Lαbj˜,ul˜.$

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 ).

Definition 6. The solvability set of the (αW) problem, denoted by D, is defined by D = {(w, c, d) ∈ ℝ3n+2r : α-MOP problem has α-Pareto optimal solutions}, where

$w∈ℝn, c=(c1,c2)∈ℝ2n, d=(d1,d2)∈ℝ2r.$

Suppose that the (αW) problem is solvable for (w̄, c̄, d̄) ∈ ℝ3n+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

$B={(γ,b,u)∈ℝt+n+r:hl^(γ)=ul, bj-bjαU=0,bjαL-bj=0,ul-ulαU=0, ulαL-ul=0},C={(γ,b,u)∈ℝt+n+r:hl^(γ)>ul, bj-bjαU<0,bjαL-bj<0,ul-ulαU<0, ulαL-ul<0}.$

Let ψ(J, I) be the side (or sides) of Φ∩Lα(b̃j, ũl) or int(Φ∩ Lα(b̃j, ũl)) that is defined by

$ψ(J,I)={(γ,b,u)∈ℝt+n+r:(γ,b,u)∈B;∀(j,1)∈J×I;(γ,b,u)∈C;∀(j,1)∉J×I}.$

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

$H(ψ(J,I))={(w,c,d)∈ℝ3n+2r:ψ(J,I) contains α-Pareto optimal solution of the α-MOP problem}.$

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

$S(xk,bk,uk)={(w,c,d)∈ℝ3n+2r:(xk,bk,uk) is an α-Pareto optimal solution of the α-MOP problem},$

and

$K={(xk,bk,uk)∈ψ(J,I) is an α-Pareto optimal solution of α-MOP}.$

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

$S(xk,bk,uk)={(w,c,d)∈ℝ3n+2r:(xk,bk,uk) is an α-Pareto optimal solution of the (αW) problem}.$

### 4. Determination of the Stability Set of the Second Kind without Differentiability

Given the close interval approximation of a PQFN jPQ, $j=1, n¯$; ũlPQ, $l=1, r¯$ (that is, $bj∈[bjαL, bjαU], j=1, n¯;ul∈[ulαL, ulαU], l=1, r¯$ ), assume that for (w̄, c̄, d̄) ∈ ℝ3n+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 ≠ ∈ ℝn, j ≥ 0, $j=1, n¯$ , ∈ ℝr, ζ̄, ξ̄ ∈ ℝn, ζ̄, ξ̄ ≥ 0; δ̄, ρ̄ ∈ ℝr, δ̄, ρ̄ ≥ 0 such that (γ̄, b̄, ū, , , ζ̄, ξ̄, δ̄, ρ̄) solves the Kuhn-Tucker saddle point problem, that is, χ(γ̄, b̄, ū, , v, ζ, ξ, δ, ρ) ≤ χ(γ̄, b̄, ū, , , ζ̄, ξ̄, δ̄, ρ̄) ≤ χ(γ, b, u, , , ζ̄, ξ̄, δ̄, ρ̄); ∀ (γ, b, u) ∈ ℝt+n+r; ∀ v ∈ ℝr, v ≥ 0, ζ, ξ ∈ ℝn; ζ, ξ ≥ 0; δ, ρ ∈ ℝr, δ, ρ ≥ 0, where $χ(γ,b,u,w,v,ζ,ξ,δ,ρ)=∑j=1nwjGj^(γ,bj)+∑l=1rvl(hl^(γ)-ul)+∑j=1nζj(bj-bjαU)+∑j=1nξj(bjαL-bj)+∑l=1rδl(ul-ulαU)+∑l=1rρl(ulαL-ul)$ , wj ≥ 0, $∑j=1nwj=1$ . Let us formulate the Kuhn-Tucker saddle point conditions for the (αW) problem as follows:

$∑j=1nwj¯Gj^(γ¯,bj¯)+∑l=1rvl(hl^(γ¯)-ul¯)+∑j=1nζj(bj¯-bjαU)+∑j=1nξj(bjαL-bj¯)+∑l=1rδl(ul¯-ulαU)+∑l=1rρl(ulαL-ul¯)≤∑j=1nwj¯Gj^(γ¯,bj¯)+∑l=1rvl¯(hl^(γ¯)-ul¯)+∑j=1nζj¯(bj¯-bjαU)+∑j=1nξj¯(bjαL-bj¯)+∑l=1rδl¯(ul¯-ulαU)+∑l=1rρl¯(ulαL-ul¯)≤∑j=1nwj¯Gj^(γ,bj)+∑l=1rvl¯(hl^(γ)-ul)+∑j=1nζj¯(bj-bjαU)×∑j=1nξj¯(bjαL-bj)+∑l=1rδl¯(ul-ulαU)+∑l=1rρl¯(ulαL-ul);∀ (γ,b,u)∈ℝt+n+r, v∈ℝr, v≥0;∀ζ, ξ∈ℝn; ζ, η≥0; ∀δ, ρ∈ℝr, δ, ρ≥0,hl^(γ¯)≤ul¯, l=1,2,…,r,bj¯-bjαU≤0, j=1,2,…,n,bjαL-bj¯≤0, j=1,2,…,n,ul¯-ulαU≤0, 1=1,2,…,r,(ul)αL-u¯l≤0, l=1,2,…,r,v¯l(hl^(γ¯)-ul¯)=0, l=1,2,…,r,ζj¯(bj¯-bjαU)=0, j=1,2,…,n,ξj¯(bjαL-bj¯)=0, j=1,2,…,n,δl¯(ul¯-ulαU)=0, l=1,2,…,r,ρl¯(ulαL-ul¯)=0, l=1,2,…,r,ζj¯,ξj¯≥0, j=1,2,…,n,v¯l, δl¯, ρl¯≥0, l=1,2,…,r.$

### 5. Decision-Making Process

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 → ℝm 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

$H(ψ(J,I))={(w,c,d)∈B:hl^(p(w,b))=ul, for (j,l)∈J×I, hl^(p(w,b))

### 6. Numerical Example

Consider the following F-CCSG

$(F-CCSG) G1^(γ,b˜1PQ)=(γ1-b˜1PQ)2-8γ2, G2^(γ,b˜2PQ)=2γ1+(γ2+b˜2PQ)2,$

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

$γ1+γ2≤u˜1, 2γ1+γ2≤u˜2, γ1,γ2≥0.$

Let 1PQ = (1, 1.36, 2, 4.64, 5), 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

$min {(γ1-b1)2-8γ2, 2γ1+(γ2+b2)2},$

subject to

$Φ={γ∈ℝ2:γ1+γ2≤u1, 2γ1+γ2≤u2, γ1,γ2≥0},b1∈[1.36, 4.64], b2∈[3.72, 9.64],u1∈[0.36, 3.64], u2∈[1.36, 5.64].$

Using the weighting method, the problem can be rewritten as

$min{w(γ1-b1)2-8wγ2+2(1-w)γ1+(1-w)(γ2+b2)2},$

subject to

$Φ={γ∈ℝ2:γ1+γ2≤u1, 2γ1+γ2≤u2, γ1,γ2≥0},b1∈[1.36, 4.64], b2∈[3.72, 9.64],u1∈[0.36, 3.64], u2∈[1.36, 5.64], w≥0.$

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

$(γ1¯+γ2¯≤u1, 2γ1¯+γ2¯≤u2,-γ1¯<0, -γ2¯<0,b1¯∈]1.36, 4.64[, b2¯∈]3.72, 9.64[,u1¯∈]0.36, 3.64[, u2¯∈]1.36, 5.64[),$

we get v̄l = 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:

$w(γ1-b1)2-8wγ2+2(1-w)γ1+(1-w)(γ2+b2)2(w(γ1¯-b1¯)2-8wγ2¯+2(1-w)γ1¯+(1-w)(γ2¯+b2¯)2)≥0.$

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

$H(ψ(J1,I1))={(w,c,d)∈ℝ9:w=1, u¯1>b1¯, u¯2>2b2¯}∪{(w,c,d)∈ℝ9:w=0, u¯1>-b2¯, u¯2>-b2¯}∪{(w,c,d)∈ℝ9:w∉{0, 1}, u¯1>b1¯-b2¯+1-ww+4w1-w, u¯2>2b1¯-b2¯+2(1-w)w+4w1-w}.$

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

$H(ψ(J2,I2))={(w,c,d)∈ℝ9:w=1, u¯1=b1¯, u¯2>2b2¯}∪{(w,c,d)∈ℝ9:w=0, u¯1=-b2¯, u¯2>-b2¯}∪{(w,c,d)∈ℝ9:w∉{0, 1}, u¯1=b1¯-b2¯+1-ww+4w1-w, u¯2>2b1¯-b2¯+2(1-w)w+4w1-w}.$

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

$H(ψ(J3,I3))={(w,c,d)∈ℝ9:w=1, u¯1>b1¯, u¯2=2b2¯}∪{(w,c,d)∈ℝ9:w=0, u¯1>-b2¯, u¯2=-b2¯∪{(w,c,d)∈ℝ9:w∉{0, 1}, u¯1>b1¯-b2¯+1-ww+4w1-w, u¯2=2b1¯-b2¯+2(1-w)w+4w1-w}.$

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

$H(ψ(J4,I4))={(w,c,d)∈ℝ9:w=1, u¯1=b1¯, u¯2=2b2¯}∪{(w,c,d)∈ℝ9:w=0, u¯1=-b2¯, u¯2=-b2¯}∪{(w,c,d)∈ℝ9:w∉{0, 1}, u¯1=b1¯-b2¯+1-ww+4w1-w, u¯2=2b1¯-b2¯+2(1-w)w+4w1-w}.$

### 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.

### 7. Conclusions and Future Works

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.

### Data Availability

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

### Fig 1. Figure 1.

Graphical representation of a piecewise quadratic fuzzy number (PQFN).

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 365-374https://doi.org/10.5391/IJFIS.2023.23.3.365

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