# On Stackelberg Leader with Min-Max Followers to Solve Fuzzy Continuous Static Games

Authors: Hamiden Abd El Wahed Khalifa; Dragan Pamucar; Alhanouf Alburaikan; W. A. Afifi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This study attempts to introduce continuous static games with fuzzy parameters. Piecewise quadratic fuzzy numbers (PQFNs) are used to represent the fuzzy parameters. The close interval approximation of the PQFNs, which is one of the most accurate approximations, is applied. To overcome the problem, Stackelberg leaders with min-max follower’s solutions are examined. The given solution is accompanied by a parametric analysis. An algorithm for identifying the first-kind stability set is presented. Finally, to demonstrate and clarify the method’s steps, a numerical example is given


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## Body

## 1. Introduction

Game theory plays a vital role in economics, engineering, biology, and other computational cum mathematical sciences with wide range of applications in real-world problems. Matrix games, continuous static games, and differential games are the three main types of games. Matrix games derive their name from a discrete relationship between a finite/countable number of possible decisions and the corresponding costs. The three basic solution concepts for these games (Vincent and Grantham [1]) are:
(1)
Nash equilibrium solution(2)
Min-max solutions(3)
Pareto minimal solutionsSeveral researchers worked on fuzzy set theory in the beginning (Zadeh [2], Zadeh and Bellman [3], Prade and Dubois [4], and Gupta and Kaufmann [5]). Osman [6] has formulated different types of parametric problems of continuous static games. For solving continuous static games with fuzzy parameters, Osman et al. [7] offered a Stackelberg leader with min-max follower solution, as well as a parametric analysis for the solution. For large-scale, continuous static games with all cost functions and constraints parameterized, Osman et al. [8] proposed the Nash equilibrium solution, where in this type of games, each participant strives to decrease his or her cost functions on their own, without collaborating with any of the other players. In addition, the cost functions and limitations are the only pieces of information available to each player. Elshafei [9] demonstrated an interactive method for solving Nash Cooperative Continuous Static Games, as well as the first-of-its-kind stability set that corresponds to the reached compromised solution. Khalifa and ZeinEldein [10] suggested an interactive technique for solving cooperative continuous static games with fuzzy parameter coefficients in the objective functions. Kacher and Larbain [11] developed the concept of equilibrium for a noncooperative game with fuzzy goals and fuzzy parameters. Instead of using a payout function, the notion of ordinal games was given by Cruz and Simaan [12], in which players can compare their decision-making choices to those of other players. To solve the challenge of multiresponse optimization, a new game theoretic-based strategy was proposed by Navidi et al. [13]. Corley [14] defined a mixed dual to the Nash equilibrium for n-person in strategic form, where in a Nash equilibrium, given the tactics of the other n-1 players, the mixed approach of each player optimizes his or her own predicted reward. Furthermore, the dual and related to the mixed Nash equilibrium is compared, as well as topological and algebraic criteria. Farooqui and Niazi [15] published a comprehensive multidisciplinary assessment and classification of game theory models of complicated agent interactions. Based on compromise programming and the compromise weights method, an interactive method for solving Nash continuous cooperative static games was proposed by Sasikala and Kumaraghuru [16]. (NCCSTGs). Khalifa [17] devised an interactive method for solving multiobjective nonlinear programming problems, he then used this information to play cooperative continuous static games. Awaya and Krishna [18] compared the set of equilibria under two regimes and looked at the function of communication in repeated games with private monitoring. A review of noncooperative game theory’s use in inventory management was offered by Silbermayr [19]. Shuler [20] looked into cooperative games in which poor agents do not get anything from cooperating with wealthier agents. Khalifa and Kumar [21] investigated cooperative continuous static games in a crisp environment, defining and determining the first-kind stability set corresponding to the solution without differentiability. Under uncertainty environment, some recent researchers for cooperative games have been introduced (Mallozzi and Messalli [22], Bigdeli and Hassanpour [23], Elshafei [24], Zaichenko [25], Zidan et al. [26], Donahue et al. [27], Ganzfried [28], Zhou et al. [29], Krishankumar et al. [30, 31], Tukhtasinov et al. [32], Sivagami et al. [33], Zhao et al. [34], Megahed [35], Romanuke [36], and Khalifa et al. [37]).In this paper, Stackelberg leader min-max followers approach to solve the continuous static games under fuzzy environment is developed. Parametric analysis for the problem is introduced. An algorithm for identifying the first-kind stability set is proposed.Having been inspired by the material listed above, in this study, piecewise quadratic fuzzy continuous static game (PQF-CCSG), it is imagined that there aren players, each of whom has piecewise quadratic fuzzy numbers in both the cost functions and the right-hand side of the limitations. Moreover, the first and second types of nondifferentiable stability sets are investigated and determined.


### 1.1. Research Gap and Motivation

(a)
The phrase “piecewise quadratic fuzzy number” is actually meant for dispensing the fuzzy value to each attribute/subattribute in the domain of single argument/multiargument approximate function(b)
Many authors discussed the fuzzy set-like structures under fuzzy set environment with fuzzy set-like settings(c)
Along these lines, another structure asks for a place in writing to deal with such a problem; therefore, a fuzzy set is imagined to deal with itThe following is a summary of the rest of the paper (Figure1).Figure 1
Layout of remaining paper.


## 1.1. Research Gap and Motivation

(a)
The phrase “piecewise quadratic fuzzy number” is actually meant for dispensing the fuzzy value to each attribute/subattribute in the domain of single argument/multiargument approximate function(b)
Many authors discussed the fuzzy set-like structures under fuzzy set environment with fuzzy set-like settings(c)
Along these lines, another structure asks for a place in writing to deal with such a problem; therefore, a fuzzy set is imagined to deal with itThe following is a summary of the rest of the paper (Figure1).Figure 1
Layout of remaining paper.


## 2. Preliminaries

For appropriate understanding of the proposed study, several fundamental concepts and terms from fuzzy-like literature are recalled in this section.Definition 1 (Jain [38]).
A piecewise quadratic fuzzy number (PQFN) is denoted byA~PQ=a1,a2,a3,a4,a5, where a1≤a2≤a3≤a4≤a5 are real numbers, and whose membership function μA~PQ is given as below and illustrated as in Figure 2.
(1)μA~PQ=0,x<a1,121a2−a12x−a12,a1≤x≤a2,121a3−a22x−a32+1,a2≤x≤a3,121a4−a32x−a32+1,a3≤x≤a4,121a5−a42x−a52,a4≤x≤a5,0,x>a5.
For the PQFN, the confidence interval at levelα is specified as
(2)A~PQα=a1+2a2−a1α,a5−2a5−a4α;∀α∈0,1.Definition 2 (Jain [38]).
An intervalA~PQ=aα−,aα+ of a PQFN. A~PQ is called closed interval approximation if:
(3)aα−=infx∈R:μA~PQ≥0.5,andaα+=supx∈R:μA~PQ≥0.5.Definition 3 (Jain [38]).
Suppose thatA~PQ=a1,a2,a3,a4,a5 and B~PQ=b1,b2,b3,b4,b5 be two PQFNs. Then,
(1)
Addition:A~PQ+B~PQ=a1+b1,a2+a2,a3+b3,a4+b4,a5+b5(2)
Subtraction:A~PQ−B~PQ=a1−b5,a2−b4,a3−b3,a4−b2,a5−b1.(3)
Scalar multiplication:k.A~PQ=ka1,ka2,ka3,ka4,ka5,k>0,ka5,ka4,ka3,ka2,ka1,k<0.Definition 4 (Jain [38]).
LetA~PQ=aα−,aα+ and B~PQ=bα−,bα+ be two inexact interval of PQFN, then the arithmetic operations are:
(1)
Addition:A~PQ+B~PQ=aα−+bα−,aα++bα+(2)
Subtraction:A~PQ−B~PQ=aα−−bα+,aα+−bα−(3)
Scalar multiplication:k.A~PQ=kaα−,kaα+,k>0,kaα+,kaα−,k<0.(4)
Multiplication:A~PQ×B~PQ=aα+bα−+aα−bα+/2,aα−bα−+aα+bα+/2.(5)
Division:A~PQ/B~PQ=2aα−/bα−+bα+,2aα+/bα−+bα+,B~PQ>0andbα−+bα+≠0,2aα+/bα−+bα+,2aα−/bα−+bα+,B~PQ<0andbα−+bα+≠0.Definition 5.
The order relations=−,+,≤−,+,,≥−,+ between A~PQ and B~PQ are defined as
(i)
A~PQ=−,+B~PQ iff aα−=bα− and aα+=bα+.(ii)
A~PQ≤−,+B~PQ iff aα−≤−,+bα− and aα+≤−,+bα+ or(4)aα−+aα+≤−,+bα−+bα+.(iii)
A~PQ≥−,+B~PQ iff aα−≥LUbα+ and aα+≥LUbα− or(5)aα−+aα+≥−,+bα++bα−.Figure 2
Graphical representation of PQFN [38].


## 3. Problem Statement

Based on the control notion, consider the followingr+1 player game with piecewise quadratic fuzzy parameters in constraints and the cost function as
(6)PQF−CSGminG0x,u,v,a~PQ0,(7)minGx,u,v,b~PQ,Subject to
(8)hx,u,v,c~PQ=0,(9)Ψ=u,v∈Rm×Rs:gx,u,v,d~PQ≥0,where the leader players 0 controls u∈Rm; players 1,2,⋯,r controls Vi∈RsiV=V1,V2,⋯,Vr∈Rs under the given u, based on some of the optimality concept, where G.=G10,⋯,Gr0T is r vector. The state x=su,V is a solution to (4)
(10)b~PQ=b~PQ1,b~PQ2,⋯,b~PQr,c~PQ=c~PQ1,c~PQ2,⋯,c~PQr,d~PQ=d~PQ1,d~PQ2,⋯,d~PQq,a~PQ0=a~PQ01,a~PQ02,⋯,a~PQ0r,b~PQi=b~PQi1,b~PQi2,⋯,b~PQir,c~PQj=c~PQj1,c~PQj2,⋯,c~PQjn.d~PQk=d~PQk1,d~PQk2,⋯,d~PQkq represents a vector of piecewise quadratic fuzzy parameters in all of the cost functions and constraints, respectively. Let us assume that: ∂hx,u,v,c~PQ/∂x≠0 is a ball about a solution x,u,v. The function G0.:Rn×Rm×Rs⟶R is of class C1 in u. The functions G.:Rn×Rm×Rs⟶Rr,h.:Rn×Rm×Rs⟶Rn and g.:Rn×Rm×Rs⟶Rp are of C2 of x and v. Piecewise quadratic fuzzy numbers are supposed to represent these fuzzy parameters (Jain [38]).Definition 6 (According to Dubois and Prade [4]).
The set of fuzzy numbers at theα−level a~PQ0,b~PQi,c~PQj,d~PQk is defined as the ordinary set Lαa~PQ0,b~PQi,c~PQj,d~PQk which exceeds the level α in terms of the degree of their membership functions:
(11)Lαa~PQ0,b~PQi,c~PQj,d~PQk=a0,bi,cj,dk:μa~PQ0a0≥α,μb~PQibi≥α,i=1,r¯,μc~PQjcj≥α,j=1,n¯;μd~PQkdk≥α,k=1,p¯.
Based on the concept ofα−cut (Definition 6) and using introduced by Sakawa and Yano [39, 40], the PQF-CSG in a nonfuzzy version, problems 2–4 can be expressed as follows: (α-CSG) minG0x,u,v,a0(12)minGx,u,v,b,
Subject to(13)hx,u,v,c=0,(14)Ψ=u,v∈Rm×Rs:gx,u,v,d≥0,(15)a0,b,c,d∈Lαa~PQ0,b~PQi,c~PQj,d~PQk,i=1,r¯;j=1,n¯;k=1,p¯.
Alternatively, theα-CSG problem can be phrased as follows in the near interval approximation of PQFNs: (CIA-CSG) minG0x,u,v,a0(16)minGx,u,v,b,Subject to
(17)hx,u,v,c=0,(18)Ψ=u,v∈Rm×Rs:gx,u,v,d≥0,(19)a0α−≤a0≤a0α+,biα−≤bi≤biα+,i=1,r¯cjα−≤cj≤cjα+,j=1,n¯,(20)dkα−≤dk≤dkα+,j=1,p¯.


## 4. Stackelberg Leader-Follower Solutions forCIA-CSG Problem

Based on the Stackelberg solution concept ofCIA-CSG problem, the leader will announce his/her control first, then the followers will select their compost control depending on the their optimality concepts, where several games of this type can be formulated. The choice of the type of Stackelberg leader with min-max followers is examined in this paper.Theorem 7 (Necessary conditions ofα-Stackelberg leader solutions with min-max followers).
Ifu∗,v∗=u∗,v∗1,v∗2,⋯,v∗r∈Ψ is a completely regular local α-Stackelberg leader solutions with min-max followers, and x∗=su∗,v∗ is the solution to hx,u,v,c=0, then for each follower, i=1,r¯, there exist vectors wi∈Rn,δi∈Rp,s0∈R,s¯0∈R,Ni∈Rr,εi∈Rn,ε¯i∈Rn,ϵi∈Rq, and ϵ¯i∈Rq,i=1,r¯, such that:
(21)∂Li∂x=0,∂Li∂v=0,∂Li∂a0=0,∂Li∂b=0,∂Li∂c=0,∂Li∂d=0,(22)hx∗,u∗,v∗,c∗=0,gx∗,u∗,v∗,d∗=0,(23)a0−a0α−≥0,a0α+−a0≥0,bi−biα−≥0,biα+−bi≥0,(24)cj−cjα−≥0,cjα+−cj≥0,dk−dkα−≥0,dkα+−dk≥0,(25)δti,gx∗,u∗,v∗,d∗=0,(26)st0a0−a0α−=0,s¯t0a0α+−a0=0,(27)Ntibi−biα−=0,N¯tibiα+−bi=0,(28)εticj−cjα−=0,ε¯ticjα+−cj=0,(29)ϵtidk−dkα−=0,ϵ¯tidkα+−dk=0,
δi,s0,s¯0,Ni,N¯0εi,ϵi and ϵ¯0≥0, where
(30)Lix,u,v,a0,b,c,d,wi,δi,s0,s¯0,Ni,N¯iεi,ε¯i,ϵi,ϵ¯i=Gix,u,v,b−wtihx,u,v,c−δtigx,u,v,d−st0a0−a0α−−s¯t0a0α+−a0−Ntibi−biα−−N¯tibiα+−bi−εticj−cjα−−ε¯t0cjα+−cj−ϵtidk−dkα−−ϵ¯tidkα+−dk.
For the leader, there exist vectorsw0∈Rn,δ0∈Rp,s0∈R,s¯0∈R,Ni∈Rr,N¯i∈Rrεi∈Rn,ε¯i∈Rn,ϵi∈Rp,ϵ¯i∈Rp,αi∈Rn,βi∈Rsi,Ai∈R,Bi∈Rr,Ci∈Rn,Di∈Rp,ρi∈Rp,Οi∈R,Ο¯i∈R,Εi∈Rr,Ε¯i∈Rr,Γi∈Rr,Γ¯i∈Rr,Ζi∈Rp and Ζ¯i∈Rp,i=1,r¯, such that:
(31)∂L0∂x=0,∂L0∂v=0,∂L0∂a0=0,∂L0∂b=0,∂L0∂c=0,∂L0∂d=0,δt0gx,u,v,d=0,s0a0−a0α−=0,s¯t0a0α+−a0=0,Nt0bi−biα−=0,N¯t0biα+−bi=0,εt0cj−cjα−=0,ε¯t0cjα+−cj=0,ϵt0dk−dkα−=0,ϵ¯t0dkα+−dk=0.
δ0,s0,s¯0,N0,N¯0ε0,ε¯0,ϵ0 and ϵ¯0≥0,i=1,r¯.
(32)∂L0∂wi=0,∂L0∂δi=0,∂L0∂si=0,∂L0∂s¯i=0,∂L0∂Ni=0,(33)∂L0∂N¯i=0,∂L0∂εi=0,∂L0∂ε¯i=0,∂L0∂εi=0,∂L0∂ϵ¯i=0,(34)ρtiδi=0,ρi≥0,Ο¯isi=0,Οi≥0,Ο¯tis¯i=0,O¯i≥0,ΕiNi=0,Εi≥0,Ε¯iN¯i=0,Ε¯i≥0,Γiεi=0,Γi≥0,Γ¯iε¯i=0,Γ¯i≥0,Ζiϵi=0,Ζi≥0,Ζ¯iϵ¯i=0,Ζ¯i≥0,(35)L0x,u,v,a0,b,c,d,w1,δ1,s1,s¯1,N1,N¯1,ε1,ε¯1,ϵ1,ϵ¯1,α1,β1,A1,Bi,C1,D1,ρ1,Ο1,Ο¯1,Ε1,Ε¯1,Γ1,Γ¯1,Ζ1,⋯,wr,⋯,Ζ¯r=G0x,u,v,a0−w00hx,u,v,c−δ¯0gx,u,v,d−st0a0−a0α−−s¯t0a0α+−a0−Nt0bi−biα−−N¯t0biα+−bi−εt0cj−cjα−−ε¯t0cjα+−cj−ϵt0dk−dkα−−ϵ¯t0dkα+−dk−∑i=1r∂L1∂xαi+∂L1∂vtβi+∂L1∂a0Ai+∂L1∂bBi+∂L1∂cCi+∂L1∂dDi+ρtiδi+Οtisi+Ο¯tis¯i+ΕtiNi+,Ε¯tiN¯i+Γtiεi+Γ¯tiε¯i+Ζtiϵi+Ζ¯tiϵ¯i.


## 5. Stackelberg Leader-Follower PQF-CSG Parametric Analysis

ConsiderPQF−CSG, (2)–(6). Assume that the functions G0.,G.,h., and g. are convex . Let us define the following notion after converting the problem (2)–(6) into problem (11):Definition 8.
Problem (11)’s solvability set is defined as
(36)ℵ=q∈R4r+n+p:thesolutionoftheproblem8exists.


### 5.1. The Stability Set of the First-Kind

IfQ∗∈ℵ has a matching α-Stackelberg leader solution with min-max followers, the first-order stability set associated with (u∗, v∗) is therefore defined asSuppose thatQ∗∈ℵ with a corresponding α-Stackelberg leader solutions with min-max followers, then the first-kind stability set corresponding to (u∗, v∗) is defined as
(37)Su∗,v∗=q∈ℵu∗,v∗:isα‐Stackelbergleadersolutionswithmin‐maxfollowers.


### 5.2. Algorithm

The phases of the algorithm that can be used to identify the first-kind stability set are as follows:Step 1.
Starting withq∗∈ℵ and get α-Stackelbergleadersolutionswithmin‐maxfollowers.Step 2.
To have the set, deduce the Kuhn-Tucker conditions (34); substitute them in
(38)I1=wi,δi,s0,s¯0,Ni,N¯iεi,ε¯i,ϵi,ϵ¯i:∂Li∂x=0,∂Li∂v=0,∂Li∂a0=0,∂Li∂b=0,∂Li∂c=0,∂Li∂d=0,i=1,r¯,where Li is defined in (12), and
(39)I2=q,w1,δ1,s1,s¯1,N1,N¯1,ε1,ε¯1,ϵ1,ϵ¯1,α1,β1,A1,Bi,C1,D1,ρ1,Ο1,Ο¯1,Ε1,Ε¯1,Γ1,Γ¯1,Ζ1,Ζ¯1,⋯,wr,⋯,Ζ¯r:∂L0∂u=0,∂L0∂v=0,∂L0∂a0=0,∂L0∂b=0,∂L0∂c=0,∂L0∂d=0,∂L0∂wi=0,∂L0∂δi=0,∂L0∂si=0,∂L0∂s¯i=0,∂L0∂Ni=0,∂L0∂N¯i=0,∂L0∂εi=0,∂L0∂ε¯i=0,∂L0∂εi=0,∂L0∂ϵ¯i=0,i=1,r¯,where L0 is defined in (13).Step 3.
DetermineSu∗,v∗ as follows:
(40)Su∗,v∗=q∈R4r+n+p:allmultipliersbelongstoI=I1⋃I2.


## 5.1. The Stability Set of the First-Kind

IfQ∗∈ℵ has a matching α-Stackelberg leader solution with min-max followers, the first-order stability set associated with (u∗, v∗) is therefore defined asSuppose thatQ∗∈ℵ with a corresponding α-Stackelberg leader solutions with min-max followers, then the first-kind stability set corresponding to (u∗, v∗) is defined as
(37)Su∗,v∗=q∈ℵu∗,v∗:isα‐Stackelbergleadersolutionswithmin‐maxfollowers.


## 5.2. Algorithm

The phases of the algorithm that can be used to identify the first-kind stability set are as follows:Step 1.
Starting withq∗∈ℵ and get α-Stackelbergleadersolutionswithmin‐maxfollowers.Step 2.
To have the set, deduce the Kuhn-Tucker conditions (34); substitute them in
(38)I1=wi,δi,s0,s¯0,Ni,N¯iεi,ε¯i,ϵi,ϵ¯i:∂Li∂x=0,∂Li∂v=0,∂Li∂a0=0,∂Li∂b=0,∂Li∂c=0,∂Li∂d=0,i=1,r¯,where Li is defined in (12), and
(39)I2=q,w1,δ1,s1,s¯1,N1,N¯1,ε1,ε¯1,ϵ1,ϵ¯1,α1,β1,A1,Bi,C1,D1,ρ1,Ο1,Ο¯1,Ε1,Ε¯1,Γ1,Γ¯1,Ζ1,Ζ¯1,⋯,wr,⋯,Ζ¯r:∂L0∂u=0,∂L0∂v=0,∂L0∂a0=0,∂L0∂b=0,∂L0∂c=0,∂L0∂d=0,∂L0∂wi=0,∂L0∂δi=0,∂L0∂si=0,∂L0∂s¯i=0,∂L0∂Ni=0,∂L0∂N¯i=0,∂L0∂εi=0,∂L0∂ε¯i=0,∂L0∂εi=0,∂L0∂ϵ¯i=0,i=1,r¯,where L0 is defined in (13).Step 3.
DetermineSu∗,v∗ as follows:
(40)Su∗,v∗=q∈R4r+n+p:allmultipliersbelongstoI=I1⋃I2.


## 6. Numerical Example

The leader controlsu and seeks to
(41)minG0u,v,a~PQ1=12u−12+12v−22+a~PQ1.A follower controlsv and seeks to
(42)minGu,v=12u−v2.Both the leader and follower constrained by(43)hu,v,c~PQ=2a~PQ2−u2−v2≥0,where a~PQ1=2,3,4,5,6 and a~PQ2=1,2,4,6,8.The membership functionsμa~PQ1a1 and μa~PQ2a2 are shown in Figures 3 and 4 below μa~PQ1a1.Step 1.
The close interval approximation ofa~PQ1=a1=3,5 and a~PQ2=a2=2,6. Then, problem (16)-(21) becomes
(44)minG0u,v,a1=12u−12+12v−22+a1,minGu,v=12u−v2,
Subject to(45)hu,v,a2=2a2−u2−v2≥0,3≤a1≤5,2≤a2≤6.Step 2.
The Stackelberg solution for the leader is(46)u∗=1.5=v∗,a∗1=3,2.3≤a∗2≤6.Step 3.
The first form of stability set corresponds tou∗=1.5=v∗,a∗1=3,
(47)Su∗=1.5=v∗,a∗1=3=q1:a11<a21=10−2.3a11<a31,0<a11≤3,a31=5,a41=5,a1=3.
Foru∗=1.5=v∗,2.3≤a∗2≤6,
(48)Su∗=1.5=v∗,2.3≤a∗2≤6=q2:a12<a22=3.3a12−2.3a2<a13,2.3≤a12≤6,0<a2≤2.3.Figure 3
The function of membershipμa~PQ1a1.Figure 4
The membership functionμa~PQ1a2.


## 7. Comparative Study

To emphasize the benefits of the proposed approach, this section compares the suggested strategy to some current literature as in Table1.Table 1
Comparisons among the proposed and existing techniques.
Author’s nameStackelberg leader-follower solutionsKuhn-Tucker conditionsEfficient solutionParametric studyEnvironmentα-best compromise solutionKhalifa and ZeinEldein [10]NoNoYesNoFuzzyNoKhalifa et al. [20]NoYesYesNoFuzzyNoKhalifa and Kumar [21]NoYesYesNoCrispNoZaichnko et al. [25]NoNoYesNoFuzzyNoDonahue et al. [27]NoNoNoNoCrispNoZhou et al. [29]NoNoYesNoFuzzyNoKhalifa et al. [41]NoNoYesYesFuzzyYesOur investigationYesYesYesYesFuzzyYes


## 8. Conclusion and Future Works

This study investigates a piecewise quadratic fuzzy continuous static game withn players with fuzzy cost functions and right-hand side constraints.The α-pareto optimal solution concept and the closed approximate intervals for the PQF number are specified. Furthermore, the first types of stability sets has defined and stated. This discovery could be extended to other fuzzy-like structures in the future, such as fuzzy sets with interval values, fuzzy sets that are intuitive, fuzzy Pythagorean sets, fuzzy spherical sets, picture set, take a look at some fuzzy sets, and sets that are neutrosophic, with a focus on applications in the real world. The following are the main characteristics of this work:
(i)
The fundamental theory of fuzzy set is developed, and its decision is constructed. A real-world problem is discussed with the support of proposed algorithm and decision support of fuzzy set(ii)
The proposed model and its decision-making-based system are developed. A real-life problem is studied with the help of proposed algorithm, and decision system of fuzzy set is compared professionally via strategy with several existing relevant models that take into account essential evaluation characteristics(iii)
As the proposed model is inadequate with the situation in the domain of multiargument approximate function and is mandatory, as a result, future work may include addressing and determining this constraint.



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