This study explores Cooperative Continuous Static Games (CCSGs) characterized by fuzzy cost functions that are piecewise quadratic. It introduces an effective approximation technique, specifically the close interval approximation for Piecewise Quadratic Fu zzy Numbers (PQFNs). The paper employs the weighted Tchebycheff method to derive an optimal compromise solution and establishes the corresponding stability set of the first kind. Additionally, a numerical example is provided to illustrate the effectiveness of the proposed method in computational terms.

Cooperative continuous static games, Close interval approximation, Piecewise quadratic fuzzy numbers, ? - efficient solution, Weighted Tchebycheff method, Optimal compromise solution, Stability .

The possibility of competition among the system controllers is called '' P layers ,'' the optimization problem under consideration is therefore termed a '' G ame''. Each player in the game controls a specified subset of the system parameters ( Called his/her control vector) and seeks to minimize his/her own scaler cost criterion subject to specified constraints. Game theory applications may be found in economics, engineering, biology, etc.

Three major classes of games are: Matrix games Differential games Corresponding Author : s.a.edalatpanah@aihe.ac.ir https://doi.org/10.22105/scfa.v1i1.19 Licensee System Analytics . This article is an open - access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0). The three basic solution concepts for the games are [1] : Nash equilibrium solutions Min - max solutions Pareto minimal solutions In many scientific areas, such as system analysis and operator research, a model has to be setup - using data, which is only approximately known. Fuzzy sets theory, introduced by Zadeh [2], makes this possible. Fuzzy numerical data can be represented by empl oying fuzzy subsets of the real line, known as fuzzy numbers.

Dubois and Prade [3] extended the use of algebraic operations on real numbers to fuzzy numbers by use of a fuzzification principle. Despite vast decision - making experience, the decision - maker ca nnot articulate the goals precisely. Decision - making in a fuzzy environment, developed by [4], improved and is a great help in managing decision problems. Zimmermann [5] proposed the fuzzy set theory and its applications.

Orlovski [6] studied Multi - Objective Nonlinear Programming (MONLP) problems with fuzzy parameters. Osman and El - Banna [7] introduced the stability of fuzzy MONLP. Many pieces of research have been introduced in ( CCSGs ) [8] ² [11]. In his earlier work, Osman [12], [13] analyzed the notions of the solvability set, stability set of the first kind, and stability set of the second kind for parametric convex nonlinear programming.

In this paper, Fuzzy Cooperative Continuous Static Games (F - CCSG) are introduced. The weighted Tchebycheff method is applied to obtain the optimal compromise solution. The remainder of the paper is organized as follows: Section 2 introduces some preliminaries needed in this paper. Section 3 formulated the mathematical model for the continuous cooperative static games. Section 4 proposed a solution approach for obtaining an optimal compromise solution. Section 5 gives a numerical example for illustration. Finally, some concluding remarks are reported in section 6.

To discuss the problem easily, it recalls basic rules and findings related to fuzzy numbers, Piecewise Quadratic Fuzzy Numbers (PQFNs), close interval approximation, and its arithmetic operations. Definition 1 ( [2] ). Fuzzy number: A fuzzy number ? is a fuzzy set with a membership function defined as ? ? ? ? š ? ? ? ? ? ? ? ? ? , and satisfies: I. ? is fuzzy convex, i.e., ? ? ? ? ? š ? ?

? ? ? ? › ? ? ‹ ? ? ? ? ? š ? ? ? ? ? ? › ? ? ? ˆ‘” ƒŽŽ š ? › ? ? ? ? ? ? ? ? . II. ? is normal, i.e., ? š ? ? ? for which ? ? ? ? š ? ? ? ? . III. —’’ ? ? ? ? ? š ? ? ? ? ? ? ? š ? ? ? ? is the support of ? . IV. ? ? ? ? š ? is an upper semi - continuous (i. e., for each ? ? ? ? ? ? ? WKH? - cut set ? ? ? ? š ? ? ? ? ? ? ? ? ? is closed. Definition 2 ( [14] ). A PQFN is denoted by ?

?? ? ? ƒ ? ? ƒ ? ? ƒ ? ? ƒ ? ? ƒ ? ? , where ƒ ? ? ƒ ? ? ƒ ? ? ƒ ? ? ƒ ? are real numbers and is defined by if its membership function ? ? ? is given by ( Fig. 1 ). ?? E dalatpanah et al. | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 19 - 2 7 ? ? š ? ƒ ? ? ?

? ? ? ? ? š ? ƒ ? ? ? ƒ ? ? š ? ƒ ? ? ? ? ? ƒ ? ? ƒ ? ? ? ? ? ? ? ? ? š ? ƒ ? ? ? ? ? ƒ ? ? ? š ? ƒ ? ? ? ? ƒ ? ? ƒ ? ? ? ? ? ? ?? ? ? ? ? ? š ? ƒ ? ? ? ? ? ƒ ? ? ? ? š ? ƒ ? ? ? ? ƒ ? ? ƒ ? ? ? ? ? ? ? ? š ? ƒ ? ? ? ƒ ? ? š ? ƒ ? ? ? ? ? ? ? ƒ ? ? ƒ ? ? ? ? š ? ƒ ? ? Fig. 1. Graphical representation of a piecewise quadratic fuzzy number .

Definition 3 ( [14] ). Let ? ?? ? ? ƒ ? ? ƒ ? ? ƒ ? ? ƒ ? ? ƒ ? ? and ? ?? ? ? „ ? ? „ ? ? „ ? ? „ ? ? „ ? ? be two PQFNs. The arithmetic operations on ? ?? and ? ?? are: I. Addition: ? ?? ? ? ? ? ?? ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ? II. Subtraction: ? ?? ? ? ? ? ?? ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ? ?  ƒ ?

?  ƒ ? ?  ƒ ? ?  ƒ ? ?  ƒ ? ? ?  ? ? ? III. Scalar multiplication:  ? ?? ? ? ?  ƒ ? ?  ƒ ? ?  ƒ ? ?  ƒ ? ?  ƒ ? ? ?  ? ? ? ? ? Definition 4 ( [14] ). An interval approximation ? ? ? ? ƒ ? ? ? ƒ ? ? of a PQFN is called closed interval ? ? approximation if: ƒ ? ? ‹ˆ ? š ? ? ? ? ? ? ? ? ? ? ? ? ƒ† ƒ ? ? •—’ ? š ? ? ? ? ? ? ? ? ? ? ? ? ? ? Definition 5 ( [14] ).

Associated ordinary numbers [14] . If ? ? ? ? ƒ ? ? ƒ ? ? is the close interval approximation ? ? ? ? ? ? ? of PQFN, the Associated ordinary number of ? ? is defined as ? ? . ? ? ? ? ? Definition 6 ( [14] ). Let ? ? ? ? ƒ ? ? ƒ ? ? , and ? ? ? ? „ ? ? „ ? ? be two interval approximations of PQFN. Then the arithmetic operations are : ? ? ? ? I. Addition: ? ? ? ? ? ? ? ? ? ƒ ? ? „ ?

? ƒ ? ? „ ? ? ? ? ? ? ? II. Subtraction: ? ? ? ? ? ? ? ? ? ƒ ? ? „ ? ? ƒ ? ? „ ? ? ? ? ? ? ? ƒ ? ? ? ƒ ? ? ? ? ? ? ? III. Scalar multiplication: ? ? ? ? ? ? ? ? ? ƒ ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? IV. Multiplication: ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ? ? „ ? „ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? V. Division: ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? „ ? ? „ ? ? ? ? ? ? ? ? ? ? ? ? ? ? VI. The order relations: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? if ? ? ? ? ? and ? ? ? ? ? or ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? is preferred to ? ? ? if and only if ? ? ? ? ? ? ? ? ? ? ? ? It is noted that ? ? ? ? ? ?

? t, where ? ? ? , and ? ? ? are the sets of all PQFNs and close in interval approximation of PQFN, respectively.

Consider the following F - CCSG s with n - players having piecewise quadratic fuzzy parameters in the cost functions of the players. These players respectively have the costs. ? ? „ ? ? ? ? ƒ ? ? ? ? ? ? „ ? ? ? ƒ ? ? ? ? ? ? ? ? „ ? ? ? ƒ ? ? ? ? (1) s.t. ‰ ? ? „ ? ? ? ? ? ? Œ ? ? ?  ? ( 2 ) ? ? ? ? ? ? ? ? ? ? Š ? „ ? ? ? ? ? ? Ž ? ? ? ” ? ? ( 3 ) ? ? ? where ? ? „ ? ? ? ƒ ? ? ? ? Œ‹ ? ? ?  are convex functions on ?

? ? , Š ? ? „ ? ? ? ? Ž ? ? ? ” are concave functions on ? ? ? ? ? ? ? , ‰ ? ? „ ? ? ? ? Œ ? ? ?  are convex functions on ? ? ? Assume that there exists a function „ ? ˆ ? ? ? ? If the ? ? ? ? ? ? ? ? function ‰ ? ? „ ? ? ? ? ? ? Œ ? ? ?  are differentiable than the Jacobian ? ? ? ? ? Œ ? “ ? ? ?  in the neighborhood ? ? ? of a solution point ? „ ? ? ? to Eq. (2) ? „ ? ˆ ? ? ? is the solution to Eq. (2)

generated by ? ? ? ? differentiability assumptions are not needed here for all the functions ? ? „ ? ? ? ƒ ? ? ? ? ‹ ? ? ?  and Š ? ? „ ? ? ? , ? is a regular and compact set. ƒ ? ? ? ‹ ? ? ?  , represents a vector of PQFNs [14] . Let ƒ ? ? ? ƒ ? ? ? ? ? ƒ ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? be the PQFNs in the F - CCSG problem with convex membership functions, respectively.

For a certain degree of ? , the F - CCSG problem can be rewritten in the following fuzzy form [15], [16] : ? - ? ? „ ? ? ? ƒ ? ? ? ? ? „ ? ? ? ƒ ? ? ? ? ? ? ? „ ? ? ? ƒ ? ? ? ( 4 ) s.t. ‰ ? ? „ ? ? ? ? ? ? Œ ? ? ? ? ? ? ?  ? ( 5 ) ? ? ? ? ? ? ? ? Š ? ? „ ? ? ? ? ? ? Ž ? ? ? ” ? , ( 6 ) ƒ ? ? ƒ ? ? ? ‹ ? ? ?  ? ( 7 ) ? ? ? ? Definition 7 ( [1] ). Let „ ? ˆ ? ? ? be the solution to Eq. (5) generated by ?

? ? . A point ? ? ? LVFDOOHGDQ? - 3DUHWRRSWLPDOVROXWLRQWRWKH? - CCSG problem, if and only if there does not exist ? ? ? ƒ ? ? ? ? ? ? ƒ ? ? ? such that: ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ƒ ? ? ? ? ? ˆ ? ? ? ? ? ? ƒ ? ? ? ˆ‘” ƒŽŽ ‹ ? ? ?  and ? ? ˆ ? ? ? ? ? ? ƒ ? ? ? ? ? ˆ ? ? ? ? ? ? ƒ ? ? for some ‹ ? ? ? ? ? ? ? ?  ? , ? where ƒ ? DUHFDOOHG? - level minimal parameters. E dalatpanah et al.

| Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 19 - 2 7 ? )URPWKH? - 3DUHWRRSWLPDOVROXWLRQWRWKH? - CCSG problem concept, one can show that a point ? ? ? is an ? ? - 3DUHWRPLQLPDOVROXWLRQWRWKH? - CCSG problem if and only if ? LVDQ? - Pareto minimal solution to the IROORZLQJ? - multi - objective optimization problem : ? - MOP ? ‹ ? ? ? ? ? ƒ ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ƒ ? ?

? ( 8 ) ? ? s.t. ? ? ? ? ? ? ? ? Š ? ? „ ? ? ? ? ? ? Ž ? ? ? ” ? ? ƒ ? ? ? ? ƒ ? ? ? ? ‹ ? ? ?  ? ? ? ? where ? ? ? ? ƒ ? ? ? ‹ ? ? ?  are convex functions on ? ? ? and Š ? ? ? ? ? Ž ? ? ? ? ? ? ? ” are concave functions on ? and ? ? ? ? ƒ ? ? ? ? ? ˆ ? ? ? ? ? ? ƒ ? ? ? Š ? ? ? ? ? Š ? ? ˆ ? ? ? ? ? ? ? $VVXPHWKDWWKH? - MOP is to be stable [17] . Problem Eq. (8) will be solved by the weighting Tchebycheff problem ?

? ‹ ƒš ? ™ ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ƒ ? ? ? ? ƒ ? ? ? ? ƒ ? ? ? ? ‹ ? ? ?  ? ? (9) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? or ? ? ‹ ? ? ? ™ ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ƒ ? ? ? ? ‹ ? ? ?  ? ? ( 10 ) ? ? where, ™ ? ? ? ‹ ? ? ?  , and ? ? ? ? ƒ ? ? ? ‹ ? ? ?  are the ideal targets. It is noted that the stability of ? - MOP implies the stability of the Eq. (10)

.

The solution method is based on determining the ? - best compromise solution within the close interval ? ? approximation of PQFNs and the minimum deviation from the ? ? ? ? ƒ ? ? , where ? ? ? ? ? ? ƒ ? ? ? ‹ ? ? ? ? ƒ ? ? ? ‹ ? ? ?  ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ??? Step 1. Calculate ? , and ? ( i.e., individual minimum and maximum) at ? ? ? and ? ? ? , respectively. Step 2.

Compute the weight from the relation . ??? ??? ? ? ? ™ ? ? ? ??? ??? ? ? ? ? ? ? ? ? ? ? Step 3. Formulate and solve the following problem: ‹ ? ? s.t. ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ‹ ? ? ?  ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ‹ ? ? ?  ? ? ? ? where ? ? ? ? ‹ ? ? ?  ? ? ? ? ? ™ ? ? ? ? ? ? ƒ ?? ? ? ? ? ƒ ?? ? ? ? ? ? ? ƒ ? ? ? ? ‹ ? ? ?  ? ? ? Let ?

? ? ƒ ? ? EHWKH? - optimal compromise solution. ? ? Step 4. Determine the stability set of the first kind ? ? ? ƒ ? ? . ?? ? ? Let † ? ? † ? ? † ? ? ? ? ? where † ? ? ? † ?? ? ? ? † ?? ? ? † ? ? ? † ?? ? ? ? † ?? ? . Suppose that Eq. (11) is solvable for ? ? ?? ? ? ? ? ? ™ ? † ? ? ? ZLWKDFRUUHVSRQGLQJ? - Pareto optimal solution ? ? ? ƒ ? ? be given. ? ? ? ƒ ? ? is determined by applying the following conditions: ?

? ? ? ? ƒ ? ? † ?? ? ? ? ? ‹ ? ? ?  ? ? ? ? ? ? † ?? ? ƒ ? ? ? ? ? ‹ ? ? ?  ? ? ? ? ? ? ? ? ? ? ? ? ? † ?? ? † ?? ? ? ? ? ? ƒ ?? ? ? ? ? ƒ ?? ? ? ? ? ? ? ƒ ? ? ? ? ‹ ? ? ?  ?

Consider the following two - player game with ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? , wher e player 1 controls ? ? ? ? ? and player 2 controls ? ? ? ? with ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Let ƒ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? and ƒ ? ? ? ? ? ? ? ? ? ? ? ? ?? ? with the close interval approximation are ? ? ƒ ?

? ? ? ? ? ? ? ? ? ? and ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? . Step 1. Solve ? ? ‹ ? ? ? ? ? ? ? ? ? ? ? ? ? ? s.t. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ??? Let ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? with ? ? ? . Solve ? ? ‹ ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? s.t. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ƒ ? ?

? ? ? ? ? ??? Let ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ? with ? ? ? . Solve ? ? ƒš ? ? ? ? ? ? ? ? ? ? ? ? ? ? s.t. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? E dalatpanah et al. | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 19 - 2 7 ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? Solve ? ? ƒš ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? s.t. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ƒ ? ? ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ??? Let ? ? ? ? ? ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ? with ? ? ?? ? ??? ??? ??? ??? ? ? ? ? ? ? ? ? ? ? Step 2. ™ ? ? ??? ??? ??? ??? ? ? ? ??? and ™ ? ? ??? ??? ??? ??? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Step 3. Solve ‹ ? ? s.t. ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

? ? ? ? ? ? ? ? ? ? ? ? ? ? ƒ† ? ? ƒ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Yields, ? ? ? ? ? ?????? ? ? ? ? ? ? ƒ ? ? ? ? ƒ ? ? ? and ? ? ? ? ??????? ? Step 4. Determine ? ? ? ?????? ? ? ? ? ? ? ? by applying the following conditions: ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? … ?? ? … ?? ? ? ? ? ƒ ? ? ? ? ‹ ? ? ? ? ? ? ? We have ?? ? ?? ? ? ? ? ? ? , for ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? For ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? then , ? ? † ? ? † ? ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? ? ?????? ? ? ? ? ? ? ? ? † ?? ? ? ? ? ?? ?? ? ? † ?? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? For ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . For ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

then , ? ? † ? ? † ? ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? ? ?????? ? ? ? ? ? ? ? ? † ?? ? ? ? ? ?? ?? ? ? † ? ? ? ? ?? ? ? † ?? ? ? ? ? ? ? ? ? For ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . For ?? ? ? ? ? ? ? ? ? ? ? ? ? ? then , ? ? † ? ? † ? ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? ? ?????? ? ? ? ? ? ? ? ? † ? ? ? ? ?? ?? ?? ? ? † ? ?? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? For ?? ? ? ? ? ?

? ? ? ? ? ? ? . For ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? then , ? ? † ? ? † ? ? ? ? ? ? ? ? † ?? ? ? ? ? ? ? ? ? ? ? ?????? ? ? ? ? ? ? ? ? † ? ? ? ? ?? ?? ?? ? ? † ?? ? ? ? ? ? ? † ?? ? ? ? ? Hence, ? ? ? ? ?????? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?????? ? ? ? ? ? ? ? ? ?? ?? ? ? ?

This paper studies CCSG with PQFNs. The weighted Tchebycheff PHWKRGKDVEHHQDSSOLHGWRREWDLQWKH? - optimal compromise solution; hence, the stability set of the first kind corresponding to the obtained solution has been determined. The advantage of the approach is that it enables the decision - maker to have a satis factory solution.

This research received no specific grant from any funding agency in the public, commercial, or not - for - profit sectors.

The authors declare no potential conflicts of interest with respect to the research, authorship, or publication of this work. All co - authors have confirmed the absence of any financial or personal relationships that could inappropriately influence or bias the content of this paper.

[1] Vincent, T. L., Grantham, W. J., & Stadler, W. (1983). Optimality in parametric systems. Journal of applied mechanics , 50 (2), 476. https://doi.org/10.1115/1.3167074 [2] Zadeh, L. A. (1965). Fuzzy sets. Information and control , 8 (3), 338 ? 353. https://doi.org/10.1016/S0019 - 9958(65)90241 - X [3] Dubois, D. J., & Prade, H. (1980). Fuzzy sets and systems: Theory and applications (Vol. 144) . Academic press.

https://B2n.ir/kx4935 [4] Bellman, R. E., & Zadeh, L. A. (1970). Decision - making in a fuzzy environment. Management science , 17 (4), B - 141. https://doi.org/10.1287/mnsc.17.4.B141 [5] Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy sets and systems , 1 (1), 45 ? 55. https://doi.org/10.1016/0165 - 0114(78)90031 - 3 [6] Orlovski, S. A. (1984). Multiple programming problems with fuzzy parameters.

Control and cybernetics , 13 , 175 ? 183. https://cir.nii.ac.jp/crid/1574231874182526592 E dalatpanah et al. | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 19 - 2 7 [7] Osman, M. S., & El - Banna, A. Z. H. (1993). Stability of multiobjective nonlinear programming problems with fuzzy parameters. Mathematics and computers in simulation , 35 (4), 321 ? 326. https://doi.org/10.1016/0378 - 4754(93)90062 - Y [8] Khalifa, H. A., & Zeineldin, R. A. (2015). An interactive approach for solving fuzzy cooperative International journal of computer applications , 113 (1), 16 ?

20. https://www.researchgate.net/profile/Ramadan - Zeineldin/publication/276344678 [9] Khalifa, H. (2019). Study on cooperative continuous static games under fuzzy environment. International journal of computer applications, foundation of computer science , 13 , 20 ? 29. https://www.researchgate.net/profile/Hamiden - Khalifa/publication/331043836 [10] Donahue, K., Hauser, O. P., Nowak, M. A., & Hilbe, C. (2020). Evolving cooperation in multichannel games.

Nature communications , 11 (1), 3885. https://doi.org/10.1038/s41467 - 020 - 17730 - 3 [11] Zhou, J., Tur, A., Petrosian, O., & Gao, H. (2021). Transferable utility cooperative differential games with continuous updating using pontryagin maximum principle. Mathematics , 9 (2), 163. https://doi.org/10.3390/math9020163 [12] Osman, M. S. A. (1977). Qualitative analysis of basic notions in parametric convex programming. II.

Parameters in the objective function. Aplikace matematiky , 22 (5), 333 ? 348. https://dml.cz/handle/10338.dmlcz/103711 [13] Osman, M. S. A. (1977). Qualitative analysis of basic notions in parametric convex programming. I. Parameters in the constraints. Aplikace matematiky , 22 (5), 318 ? 332. https://dml.cz/handle/10338.dmlcz/103710 [14] Jain, S. (2010). Close interval approximation of piecewise quadratic fuzzy numbers for fuzzy fractional program.

Iranian journal of operations research , 2 (1), 77 ? 88. chrome - extension://efaidnbmnnnibpcajpcglclefindmkaj/https://iors.ir/journal/browse.php?a_id=168&slc_lang=fa& sid=1&ftxt=1 [15] Sakawa, M., & Yano, H. (1989). An interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters. Fuzzy sets and systems , 30 (3), 221 ? 238. https://doi.org/10.1016/0165 - 0114(89)90017 - 1 [16] Sakawa, M., & Yano, H. (1989). Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters.

Fuzzy sets and systems , 29 (3), 315 ? 326. https://doi.org/10.1016/0165 - 0114(89)90043 - 2 [17] Rockafellar, R. (1967). Duality and stability in extremum problems involving convex functions. Pacific journal of mathematics , 21 (1), 167 ? 187. http://dx.doi.org/10.2140/pjm.1967.21.167