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Abstract

This study proposes a novel methodology for solving constrained bi-matrix games with payoffs represented by fuzzy rough numbers, addressing uncertainties common in real-world decision-making. By integrating fuzzy rough set theory with α-cut techniques, the method establishes the existence of an fuzzy rough equilibrium value. Five linear programming models are developed to compute the mean equilibrium and its lower-lower, lower-upper, upper-lower, and upper-upper bounds, based on 0-cut and 1-cut representations. For any confidence level α, corresponding equilibrium bounds are determined through α-cut-based optimization. The methodology is validated through a case study on corporate environmental behavior, demonstrating its effectiveness and practical relevance.

Keywords

constrained bi-matrix Fuzzy rough α-cut Matrix game Linear programming

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References

Achemine

Larbani

(2023). Nash equilibrium in Fuzzy random bi-matrix games. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 31(6), 1005–1031. https://doi.org/10.1142/S0218488523500459

Ahuja

Kumar

(2024). Mehar approach to solve hesitant fuzzy linear programming Problems. The Journal of Analysis, 32(1), 335–371. https://doi.org/10.1007/s41478-023-00629-9

Ammar

E. S.

Brikaa

M. G.

(2017). A study on fuzzy rough continuous differential games. Journal of Fuzzy Set Valued Analysis, 2017(3), 102–119. https://doi.org/10.5899/2017/jfsva-00361

Ammar

E. S.

Brikaa

M. G.

(2019). On solution of constraint matrix games under rough interval approach. Granular Computing, 4(3), 601–614. https://doi.org/10.1007/s41066-018-0123-4

Ammar

Muamer

(2016). On solving fuzzy rough linear fractional programming problem. Int. Res. J. Eng. Technol, 30(4), 2099–2120.

J. J.

D. F.

(2019). A linear programming approach to solve constrained bi-matrix games with intuitionistic fuzzy payoffs. International Journal of Fuzzy Systems, 21(3), 908–915. https://doi.org/10.1007/s40815-018-0573-5

Bhaumik

Roy

S. K.

(2021). Intuitionistic interval-valued hesitant fuzzy matrix games with a new aggregation operator for solving management problem. Granular Computing, 6(2), 359–375. https://doi.org/10.1007/s41066-019-00191-5

Bhaumik

Roy

S. K.

(2021a). (α,β,γ) -Cut set based ranking approach to solving bi-matrix games in neutrosophic environment. Soft Computing, 25(4), 2729–2739. https://doi.org/10.1007/s00500-020-05332-6

Bhaumik

Roy

S. K.

Weber

G. W.

(2020). Hesitant interval-valued intuitionistic fuzzy-linguistic term set approach in Prisoners’ dilemma game theory using TOPSIS: A Case Study on human-trafficking. Central European Journal of Operations Research, 28(2), 797–816. https://doi.org/10.1007/s10100-019-00638-9

10.

Bhaumik

Roy

S. K.

Weber

G. W.

(2021b). Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management. Journal of Dynamics & Games, 8(2), 101–118. https://doi.org/10.3934/jdg.2020031

11.

Bigdeli

Hassanpour

Tayyebi

(2018). Constrained bimatrix games with fuzzy goals and its application in nuclear negotiations. Iranian Journal of Numerical Analysis and Optimization, 8(1), 81–110. https://doi.org/10.22067/ijnao.v8i1.55385

12.

Bigdeli

Mousazadeh

(2023). Analytical hierarchy process in modeling and solving matrix games in neutrosophic environment and its application in military problems. Military Science and Tactics, 19(64), 5–33. https://doi.org/10.22034/qjmst.2023.544038.1627

13.

Bisht

Beg

Rawat

(2024). A new method to solve matrix game with interval payoffs and its MATLAB code. International Game Theory Review, 26(3), 2450001. https://doi.org/10.1142/S0219198924500014

14.

Bisht

Dangwal

(2023). Fuzzy ranking approach to bi-matrix games with interval payoffs in marketing-management problem. International Game Theory Review, 25(1), 2250016. https://doi.org/10.1142/S0219198922500165

15.

Brikaa

M. G.

Zheng

Dagestani

A. A.

Ammar

E. S.

AlNemer

Zakarya

(2022a). Ambika approach for solving matrix games with payoffs of single-valued trapezoidal neutrosophic numbers. Journal of Intelligent & Fuzzy Systems, 42(6), 5139–5153. https://doi.org/10.3233/JIFS-211604

16.

Brikaa

M. G.

Zheng

Ammar

E.-S.

(2019a). Fuzzy multi-objective programming approach for constrained matrix games with payoffs of fuzzy rough numbers. Symmetry, 11(5), 702. https://doi.org/10.3390/sym11050702

17.

Brikaa

M. G.

Zheng

Ammar

E.-S.

(2020). Resolving indeterminacy approach to solve multi-criteria zero-sum matrix games with intuitionistic fuzzy goals. Mathematics, 8(3), 305. https://doi.org/10.3390/math8030305

18.

Brikaa

M. G.

Zheng

Ammar

E.-S.

(2022b). Mehar approach for solving matrix games with triangular dual hesitant fuzzy payoffs. Granul. Comput, 7(3), 731–750.

19.

Chauhan

Gupta

(2021). Matrix games with proportional linguistic payoffs. Soft Computing, 25(24), 15067–15081. https://doi.org/10.1007/s00500-021-06363-3

20.

Chauhan

Gupta

(2023). A methodology for solving bimatrix games under 2-tuple linguistic environment. International Journal of Systems Science: Operations & Logistics, 10(1), 2199368. https://doi.org/10.1080/23302674.2023.2199368

21.

Dengfeng

Chuntian

(2002). Fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10(4), 385–400. https://doi.org/10.1142/S0218488502001545

22.

Devi

R. N.

Sowmiya

(2025). On solving game problem using octagonal neutrosophic fuzzy number. International Journal of Operational Research, 52(1), 116–127. https://doi.org/10.1504/IJOR.2025.143961

23.

Djebara

Achemine

Larbani

(2024). Solving constrained matrix games with fuzzy random linear constraints. The Yugoslav Journal of Operations Research, 35(1), 209–227.

24.

Djebara

Achemine

Zerdani

(2023). A new approach for solving constrained matrix games with fuzzy constraints and fuzzy payoffs. Journal of Mathematical Modeling, 11(3), 425–439.

25.

Dong

Wan

(2024). Type-2 interval-valued intuitionstic fuzzy matrix game and application to energy vehicle industry development. Expert Systems with Applications, 249(1), 123398. https://doi.org/10.1016/j.eswa.2024.123398

26.

Dubois

Prade

(1990). Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems, 17(2), 191–209. https://doi.org/10.1080/03081079008935107

27.

Gaber

Alharbi

M. G.

Dagestani

A. A.

Ammar

E. S.

(2021). Optimal solutions for constrained bimatrix games with payoffs represented by single-valued trapezoidal neutrosophic numbers. J. Math, 1(2021), 5594623. https://doi.org/10.1155/2021/5594623

28.

Hung

I. C.

Hsia

K. H.

Chen

L. W.

(1996). Fuzzy differential game of guarding a movable territory. Inf. Sci, 91(1-2), 113–131. https://doi.org/10.1016/0020-0255(95)00299-5

29.

İzgi

Özkaya

Üre

N. K.

Perc

(2024). Matrix norm based hybrid shapley and iterative methods for the solution of stochastic matrix games. Applied Mathematics and Computation, 473(2024), 128638.

30.

Jana

Roy

S. K.

(2023). Linguistic pythagorean hesitant fuzzy matrix game and its application in multi-criteria decision making. Applied Intelligence, 53(1), 1–22. https://doi.org/10.1007/s10489-022-03442-2

31.

Jangid

Kumar

Sharama

Mishra

V. N.

(2023). A novel approach to solve fuzzy rough matrix game with two players. Applications and Applied Mathematics, 18(2).

32.

Jangid

Kumar

(2021). Matrix games with single-valued triangular neutrosophic numbers as pay-offs. Neutrosophic Sets and Systems, 45(2021), 196–217.

33.

Jangid

Kumar

(2022a). Hexadecagonal fuzzy numbers: novel ranking and defuzzification techniques for fuzzy matrix game problems. Fuzzy Information and Engineering, 14(1), 84–122. https://doi.org/10.1080/16168658.2021.2019969

34.

Jangid

Kumar

(2022b). A novel technique for solving two-person zero-sum matrix games in a rough fuzzy environment. Yugoslav Journal of Operations Research, 32(2), 251–278. https://doi.org/10.2298/YJOR210617003J

35.

Karmakar

Seikh

M. R.

(2023). Bimatrix games under dense fuzzy environment and its application to natural disaster management. Artificial Intelligence Review, 56(3), 2241–2278. https://doi.org/10.1007/s10462-022-10220-6

36.

Karmakar

Seikh

M. R.

(2024). A nonlinear programming approach to solving interval-valued intuitionistic hesitant noncooperative fuzzy matrix games. Symmetry, 16(5), 573. https://doi.org/10.3390/sym16050573

37.

Karmakar

Seikh

M. R.

Castillo

(2021). Type-2 intuitionistic fuzzy matrix games based on a new distance measure: application to biogas-plant implementation problem. Applied Soft Computing, 106(2021), 107357. https://doi.org/10.1016/j.asoc.2021.107357

38.

Khan

Kumar

(2023). Solving hesitant fuzzy linguistic matrix game with regret theory. Granular Computing, 8(6), 1325–1340. https://doi.org/10.1007/s41066-023-00375-0

39.

Khan

Mehra

(2020). A novel equilibrium solution concept for intuitionistic fuzzy bi-matrix games considering proportion mix of possibility and necessity expectations. Granular Computing, 5(4), 461–483. https://doi.org/10.1007/s41066-019-00170-w

40.

Kirti

Verma

Kumar

(2024). Modified approaches to solve matrix games with payoffs of single-valued trapezoidal neutrosophic numbers. Soft Computing, 28(1), 1–50. https://doi.org/10.1007/s00500-023-09133-5

41.

Kon

(2021). On characterization of equilibrium strategy for matrix games with LR fuzzy payoffs. Journal of the Operations Research Society of Japan, 64(3), 158–174. https://doi.org/10.15807/jorsj.64.158

42.

Kumar

(2021). Piecewise linear programming approach to solve multi-objective matrix games with I-fuzzy goals. Journal of Control and Decision, 8(1), 1–13. https://doi.org/10.1080/23307706.2019.1619491

43.

Kumar

Garg

(2025). A novel two-level fuzzy set theoretic approach to solve multi-objective matrix games and its applications. Ann. Oper. Res, 1–41. https://doi.org/10.1007/s10479-025-06524-9

44.

D. F.

Hong

F. X.

(2012). Solving constrained matrix games with payoffs of triangular fuzzy numbers. Computers & Mathematics with Applications, 64(4), 432–446. https://doi.org/10.1016/j.camwa.2011.12.009

45.

D. F.

Hong

(2013). Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers. Fuzzy Optimization and Decision Making, 12(2), 191–213. https://doi.org/10.1007/s10700-012-9148-3

46.

(2022a). Bi-matrix games with general intuitionistic fuzzy payoffs and application in corporate environmental behavior. Symmetry, 14(4), 671. https://doi.org/10.3390/sym14040671

47.

(2022b). Probabilistic linguistic matrix game based on fuzzy envelope and prospect theory with its application. Mathematics, 10(7), 1070. https://doi.org/10.3390/math10071070

48.

Feng

Zou

(2023). Fuzzy weighted Pareto–Nash equilibria of multi-objective bi-matrix games with fuzzy payoffs and their applications. Mathematics, 11(20), 4266. https://doi.org/10.3390/math11204266

49.

Liao

Zeng

X. J.

(2021). The two-person and zero-sum matrix game with probabilistic linguistic information. Information Sciences, 570(1), 487–499. https://doi.org/10.1016/j.ins.2021.05.019

50.

Morsi

N. N.

Yakout

M. M.

(1998). Axiomatics for fuzzy rough sets. Fuzzy Sets and Systems, 100(1), 327–342. https://doi.org/10.1016/S0165-0114(97)00104-8

51.

Namarta

P. K.

Gupta

U. C.

(2021). A new method to solve intuitionistic fuzzy matrix game based on the evaluation of different experts. J. Intell. Fuzzy Syst, 41(2), 2665–2674. https://doi.org/10.3233/JIFS-202232

52.

Nan

J.-X.

D.-F.

(2014). Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial & Management Optimization, 10(4), 1059–1070. https://doi.org/10.3934/jimo.2014.10.1059

53.

Naqvi

Aggarwal

Sachdev

Khan

(2021). Solving I-fuzzy two person zero-sum matrix games: Tanaka and Asai approach. Granular Computing, 6(2), 399–409. https://doi.org/10.1007/s41066-019-00200-7

54.

Naqvi

D. R.

Sachdev

(2023). On uncertain matrix games involving linguistic pythagorean fuzzy sets. In Combinatorial Optimization Under Uncertainty (pp. 95–128). CRC Press.

55.

Naqvi

D. R.

Sachdev

Ahmad

(2023a). Matrix games involving interval-valued hesitant fuzzy linguistic sets and its application to electric vehicles. Journal of Intelligent & Fuzzy Systems, 44(3), 5085–5105. https://doi.org/10.3233/JIFS-222466

56.

Naqvi

D. R.

Verma

Aggarwal

Sachdev

(2023b). Solutions of matrix games involving linguistic interval-valued intuitionistic fuzzy sets. Soft Computing, 27(2), 783–808. https://doi.org/10.1007/s00500-022-07609-4

57.

Pawlak

(1982). Rough sets. International Journal of Computer & Information Sciences, 11(5), 341–356. https://doi.org/10.1007/BF01001956

58.

Qiu

Xiang

(2022). Bi-matrix games with 2-Tuple linguistic information. IAENG International Journal of Applied Mathematics, 52(3).

59.

Roy

S. K.

Maiti

S. K.

(2020). Reduction methods of Type-2 fuzzy variables and their applications to stackelberg game. Applied Intelligence, 50(5), 1398–1415. https://doi.org/10.1007/s10489-019-01578-2

60.

Seikh

M. R.

Dutta

(2021a). A nonlinear programming model to solve matrix games with pay-offs of single-valued neutrosophic numbers. Neutrosophic Sets Syst, 47(2021), 366–383. https://digitalrepository.unm.edu/nss_journal/vol47/iss1/24

61.

Seikh

M. R.

Dutta

(2021b). Solution of interval-valued matrix games using intuitionistic fuzzy optimisation technique: an effective approach. International Journal of Mathematics in Operational Research, 20(3), 297–322. https://doi.org/10.1504/IJMOR.2021.119951

62.

Seikh

M. R.

Dutta

(2022). Solution of matrix games with payoffs of single-valued trapezoidal neutrosophic numbers. Soft Computing, 26(2022), 921–936. https://doi.org/10.1007/s00500-021-06559-7

63.

Seikh

M. R.

Dutta

(2023a). Interval neutrosophic matrix game-based approach to counter cybersecurity issue. Granular Computing, 8(2), 271–292. https://doi.org/10.1007/s41066-022-00327-0

64.

Seikh

M. R.

Dutta

(2023b). Solution of matrix games with pay-offs of single-valued neutrosophic numbers and its application to market share problem. Math. Comput. Sci, 1(2023), 213–230. https://doi.org/10.1002/9781119879831.ch10

65.

Seikh

M. R.

Dutta

(2024). A non-linear mathematical approach for solving matrix games with picture fuzzy payoffs with application to cyberterrorism attacks. Decision Analytics Journal, 10(1), 100394. https://doi.org/10.1016/j.dajour.2023.100394

66.

Seikh

M. R.

Dutta

D. F.

(2021a). Solution of matrix games with rough interval pay-offs and its application in the telecom market share problem. International Journal of Intelligent Systems, 36(10), 6066–6100. https://doi.org/10.1002/int.22542

67.

Seikh

M. R.

Karmakar

(2021). Credibility equilibrium strategy for matrix games with payoffs of triangular dense fuzzy lock sets. Sādhanā, 46(3), 1–17. https://doi.org/10.1007/s12046-021-01666-5

68.

Seikh

M. R.

Karmakar

Castillo

(2021b). A novel defuzzification approach of Type-2 fuzzy variable to solving matrix games: an application to plastic ban problem. International Journal of Applied Mathematics, 18(5), 155–172. https://doi.org/10.22111/ijfs.2021.6262

69.

Seikh

M. R.

Karmakar

Nayak

P. K.

(2021c). Matrix games with dense fuzzy payoffs. International Journal of Intelligent Systems, 36(4), 1770–1799. https://doi.org/10.1002/int.22360

70.

Seikh

M. R.

Karmakar

Xia

(2020). Solving matrix games with hesitant fuzzy pay-offs. Iranian Journal of Fuzzy Systems, 17(4), 25–40. https://doi.org/10.22111/ijfs.2020.5404

71.

Singh

Gupta

Mehra

(2020). Matrix games with 2-Tuple linguistic information. Annals of Operations Research, 287(2), 895–910. https://doi.org/10.1007/s10479-018-2810-6

72.

Singla

Kaur

Gupta

U. C.

(2023). A new approach to solve intuitionistic fuzzy bi-matrix games involving multiple opinions. Iranian Journal of Fuzzy Systems, 20(1), 185–197. https://doi.org/10.22111/ijfs.2023.7354

73.

Verma

(2021). A novel method for solving constrained matrix games with fuzzy payoffs. J. Intell. Fuzzy Syst, 40(1), 191–204. https://doi.org/10.3233/JIFS-191192

74.

Verma

Aggarwal

(2021a). Matrix games with linguistic intuitionistic fuzzy payoffs: basic results and solution methods. Artificial Intelligence Review, 54(7), 5127–5162. https://doi.org/10.1007/s10462-021-10014-2

75.

Verma

Aggarwal

(2021b). On matrix games with 2-Tuple intuitionistic fuzzy linguistic payoffs. Iranian Journal of Fuzzy Systems, 18(4), 149–167. https://doi.org/10.1007/s00500-023-08785-7

76.

Verma

Singla

Yager

R. R.

(2023). Matrix games under a pythagorean fuzzy environment with self-confidence levels: formulation and solution approach. Soft Comput, (2023), 1–23. https://doi.org/10.1007/s00500-023-08785-7

77.

Von Neumann

Morgenstern

(1944). The theory of games in economic bahavior. J. Wiley.

78.

Wan

Qiu

(2024). Pandemic management by using sentiment analysis and trapezoidal Type-2 fuzzy linguistic intuitionistic fuzzy matrix games. J. Intell. Fuzzy Syst, 46(4), 8677–8695. https://doi.org/10.3233/JIFS-237319

79.

Xue

Zeng

X. J.

(2021). Solving matrix games based on Ambika method with hesitant fuzzy information and its application in the counter-terrorism issue. Applied Intelligence, 51(3), 1227–1243. https://doi.org/10.1007/s10489-020-01759-4

80.

Zadeh

L. A.

(1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

81.

Zheng

Brikaa

M. G.

(2022). Solving multi-objective bi-matrix games with intuitionistic fuzzy goals through an aspiration level approach. International Journal of Computing Science and Mathematics, 16(4), 307–326. https://doi.org/10.1504/IJCSM.2022.128650

Optimal Solutions for Constrained Bi-matrix Games with Fuzzy Rough Payoffs

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