A new approach for solving constrained matrix games with fuzzy constraints and fuzzy payoffs

Document Type : Research Article

Authors

1 Laboratoire de Recherche Operationnelle et de Mathematiques de la Decision, Faculte des sciences,Universite Mouloud Mammeri de Tizi Ouzou, 15000 Tizi-Ouzou, Algeria

2 Laboratoire de Mathematiques Pures et Appliquees, Faculte des sciences, Universite Mouloud Mammeri de Tizi Ouzou, 15000 Tizi-Ouzou, Algeria

Abstract

The main purpose of this study is to construct a new approach for solving a constrained matrix game where the payoffs and the constraints are LR-fuzzy numbers. The method that we propose here is based on chance constraints and on the concept of a comparison of fuzzy numbers. First, we formulate the fuzzy constraints of each player as chance constraints with respect to the possibility measure. According to a ranking function $\mathcal{R}$, a crisp constrained matrix game is obtained. Then, we introduce the concept of $\mathcal{R}$-saddle point equilibrium. Using results on ordering fuzzy numbers, sufficient existence conditions of this concept are provided. The problem of computing this solution is reduced to a  pair of primal-dual linear programs. To illustrate the proposed method, an example of the market competition game is given.

Keywords

Main Subjects

References

[1] F. Abbasi, T. Allahviranloo, Estimation of failure using fault tree analysis based on new operations
on LR-type flat fuzzy numbers, New Math. Nat. Comput. 17 (2021) 153-174.
[2] M.D. Banadaki, H. Navid, A Bernoulli Tau method for numerical solution of feedback Nash differ-
ential games with an error estimation, Comput. Meth. Differ. Equ. 10 (2022) 894–904.
[3] M.D. Banadaki, H. Navidi, A numerical treatment based on Bernoulli Tau method for computing
the open-loop Nash equilibrium in nonlinear differential games, Iran. J. Num. Anal. Optim. 12
(2022) 467–482.
[4] M.D. Banadaki, H. Navidi, Numerical Solution of Open-Loop Nash Differential Games Based on
the Legendre Tau Method, Games 11 3 (2020) 3–28.
[5] C.R. Bector, S. Chandra, Fuzzy Mathematical Programming and Fuzzy Matrix Games, Springer,
Berlin, 2005.
[6] G. Bortolan, R. Degani, A review of some methods for ranking fuzzy subsets, Fuzzy Sets Syst. 15
(1985) 1–19.
[7] S. Chakraborty, A.S. Kumar, A framework of LR fuzzy AHP and fuzzy WASPAS for health care
waste recycling technology, Appl. Soft Comput. 127 (2022) 109–388.
[8] L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets Syst. 32
(1989) 275-289.
[9] A. Charnes, Constrained games and linear programming, Proc. Natl. Acad. Sci., 39 (1953) 639–
641.
[10] G. Dantzig, A proof of the equivalence of the programming problem and the game problem, Act.
Anal. Prod. Alloc. 13 (1951) 330–335.
[11] I. Deli, Matrix games with simplified neutrosophic payoffs, in Fuzzy Multi-criteria Decision-
Making Using Neutrosophic Sets. Cham: Springer International Publishing, (2019) 233–246.
[12] M. Dresher, Games of Strategy Theory and Applications, Prentice-Hall, 1961.
[13] D. Dubois, H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci. 9 (1978) 613–626.
[14] D. Dubois, H. Prade, Fuzzy real algebra: some results, Fuzzy Sets Syst. 2 (1978) 327–348.
[15] M. Ghanbari, R. Nuraei, T. Allahviranloo, W. Pedrycz, A new effective approximate multiplication
operation on LR fuzzy numbers and its application, Soft. Comput. 26 (2022) 4103–4113.
[16] A. Hosseinzadeh, S.A. Edalatpanah, A new approach for solving fully fuzzy linear programming by
using the lexicography method, Adv. Fuzzy Syst. 2016 (2016) 1–6.
[17] S. Jafari, H. Navidi, A game-theoretic approach for modeling competitive diffusion over social
networks , Games 9 (2018) 1–8.
[18] J. Jana, S.K. Roy, Two-person game with hesitant fuzzy payoff: An application in MADM, RAIRO
Oper. Res. 55 (2021) 3087–3105.
[19] S. Karmakar, M.R. Seikh, Bimatrix games under dense fuzzy environment and its application to
natural disaster management, Artif. Intell. Rev. 56 (2022) 2241–2278.
[20] T. Kawaguchi, Y. Maruyama A note on minimax (maximin) programming, Manag. Sci. (1976) 670–
676.
[21] D.F. Li, An effective methodology for solving matrix games with fuzzy payoffs, IEEE Trans. Cyber.
43 (2013) 610-621.
[22] D. Li, C. Cheng, Fuzzy multiobjective programming methods for fuzzy constrained matrix games
with fuzzy payoffs, Int. J. Uncert. Fuzzy Bas. Syst. 10 (2002) 385–400.
[23] D. Li, F. Hong, Alfa-cut based linear programming methodology for constrained matrix games
with payoffs of trapezoidal fuzzy numbers, Fuzzy Optim. Deci. Mak. 12 (2013) 191–213.
[24] D. Li, F. Hong, Solving constrained matrix games with payoffs of triangular fuzzy numbers, Com-
put. Math. Appl. 64 (2012) 432–446.
[25] H.R. Maleki, M. Tata, M. Mashinchi, Linear programming with fuzzy variables, Fuzzy Sets Syst.,
109 (2000) 21–33.
[26] A. Mansoori, M. Eshaghnezhad, S. Effati, Recurrent neural network model: A new strategy to solve
fuzzy matrix games, IEEE Trans. Neur. Netw. Learn. Syst. 30 (2019) 2538–2547.
[27] J. X. Nan, D. Li, Linear programming technique for solving interval-valued constraint matrix
games, J. Ind. Manag. Optim. 10 (2014) 1059–1070.
[28] G. Owen, Game Theory, Academic Press, San Diego, 1995.
[29] G. Owen, Game theory, Second Edition, Academic Press, New York, 1982.
[30] B. P´erez-Ca˜nedo, A. Rosete, J.L. Verdegay, E.R.Concepci´on-Morales, A fuzzy goal programming
approach to fully fuzzy linear regression, Inf. Proc. Manag. Uncer. Know. Bas. Syst. 18 (2020)
677–688.
[31] S. K. Roy, A. Bhaumik, Intelligent water management: a triangular type-2 intuitionistic fuzzy
matrix games approach, Wat. Resour. Manag. 32 (2018) 949-968.
[32] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York,
1993.
[33] M. Sakawa, I. Nishizaki, Max-min solutions for fuzzy multiobjective matrix games, Fuzzy Sets
Syst. 67 (1994) 53–69.
[34] M.R. Seikh, S. Dutta, D.F. Li, Solution of matrix games with rough interval pay-offs and its appli-
cation in the telecom market share problem, Int. J. Intell. Syst. 36 (2021) 6066–6100.
[35] M. R. Seikh, S. Karmakar, Credibility equilibrium strategy for matrix games with payoffs of trian-
gular dense fuzzy lock sets, Sadhana 46 (2021) 158.
[36] M. R. Seikh, S. Karmakar, O. Castillo, A novel defuzzification approach of type-2 fuzzy variable
to solving matrix games: An application to plastic ban problem, Iran. J. Fuzzy Syst 18 (2021)
155–172.
[37] M.R. Seikh, S. Karmakar, P.K. Nayak, Matrix games with dense fuzzy payoffs, Int. J. Intell. Syst.
36 (2021) 1770–1799.
[38] M. R. Seikh, S. Karmakar, M. Xia, Solving matrix games with hesitant fuzzy pay-offs, Iran. J. Fuzzy
Syst. 17 (2020) 25–40.
[39] M. R. Seikh, P.K. Nayak, M. Pal, Application of intuitionistic fuzzy mathematical programming
with exponential membership and quadratic non-membership functions in matrix games, Annal.
Fuzzy Math. Info. 9 (2015) 183–195.
[40] M.R. Seikh, P.K. Nayak, M. Pal, Aspiration level approach to solve matrix games with I-fuzzy goals
and I-fuzzy pay-offs, Pac. Sci. Rev. Nat. Sci. Eng. 18 (2016) 5–16.
[41] M.R. Seikh, P.K. Nayak, M. Pal, Matrix games in intuitionistic fuzzy environment, Inter. J. Math.
Oper. Res. 5 (2013) 693–708.
[42] S. Suneela, S. Chakraverty, New ranking function for fuzzy linear programming problem and system
of linear equations, J. Inf. Opt. Sci. 40 (2019) 141-156.
[43] T. Verma, A novel method for solving constrained matrix games with fuzzy payoffs, J. Intell. Fuzzy
Syst. 40 (2021) 191–204.
[44] T. Verma, A. Kumar, Fuzzy Solution Concepts for Non-cooperative Games: Interval, Fuzzy and
Intuitionistic Fuzzy Payoffs, Springer, New York, 2020.
[45] J. Von Neumann, On the theory of games, Math. Ann. 100 (1928) 295-320.
[46] R.R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Inf. Sci. 24 (1981) 143–161.
[47] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.
[48] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst. 1 (1978) 3–28.
[49] O. Zerdani, F. Achemine, On optimisation over the integer efficient set in fuzzy linear multicriteria
programming, Int. J. Math. Oper. Res. 13 (2018) 281–302.
[50] J. Zhou, F. Yang, K. Wang, Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy pro-
gramming, J. Intell. Fuzzy Syst. 30 (2015) 71–87.
[51] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions,
Fuzzy Sets Syst. 1 (1978) 45–55.
Journal of Mathematical Modeling
Volume 11, Issue 3
October 2023
Pages 425-439

Files

Share

How to cite

Statistics