Triangular type-2 fuzzy goal programming approach for bimatrix games

Document Type : Original paper


1 Birjand University of Technology

2 Department of Mathematics, University of Birjand; Birjand; Iran

3 Researcher in Army command and staff university

4 Department of Mathematics; Kosar University of Bojnord, Bojnord; Iran


This paper addresses a bimatrix game in which the satisfactory degrees of the players are vague. Type-2 fuzzy goal programming technique is used to describe the game. Then, the notion of equilibrium points is introduced and an optimization problem is given to calculate them. Moreover, the special case that the type-2 fuzzy goals are triangular is investigated. Finally, an applicable example is presented to illustrate the results.


Main Subjects

[1] A. Aggarwal, A. Mehra, and S. Chandra, Application of linear programming with i-fuzzy sets to matrix games with i-fuzzy goals, Fuzzy Optim. Decis. Making 11 (2012), no. 4, 465–480.
[2] S.Z. Alparslan Gök and I. Özcan, On big boss fuzzy interval games, Eur. J. Oper. Res. 306 (2023), no. 3, 1040–1046.
[3] Z. Bashir, J. Wątróbski, T. Rashid, W. Sałabun, and J. Ali, Intuitionistic-fuzzy goals in zero-sum multi criteria matrix games, Symmetry 9 (2017), no. 8, Article ID:158
[4] C. Bector, S. Chandra, and V. Vidyottama, Matrix games with fuzzy goals and fuzzy linear programming duality, Fuzzy Optim. Decis. Making 3 (2004), no. 3, 255–269.
[5] H.P. Benson, Existence of efficient solutions for vector maximization problems, J. Optim. Theory Appl. 26 (1978), no. 4, 569–580.
[6] H. Bigdeli and H. Hassanpour, A satisfactory strategy of multiobjective two person matrix games with fuzzy payoffs, Iran. J. Fuzzy Syst. 13 (2016), no. 4, 17–33.
[7] H. Bigdeli and H. Hassanpour, An approach to solve multi-objective linear production planning games with fuzzy parameters, Yugosl. J. Oper. Res. 28 (2018), no. 2, 237–248.
[8] H. Bigdeli, H. Hassanpour, and J. Tayyebi, Multiobjective security game with fuzzy payoffs, Iran. J. Fuzzy Syst. 16 (2019), no. 1, 89–101.
[9] M. Bisht and R. Dangwal, Fuzzy ranking approach to bi-matrix games with interval payoffs in marketing-management problem, Int. Game Theory Rev. 25 (2023), no. 1, Article ID: 2250016.
[10] A. Chakeri, J. Habibi, and Y. Heshmat, Fuzzy type-2 nash equilibrium, Computational Intelligence for Modelling Control & Automation, 2008 International Conference on IEEE, 2008, pp. 398–402.
[11] A. Charnes and W.W. Cooper, Goal programming and multiple objective optimizations: Part 1, European J. Oper. Res. 1 (1977), no. 1, 39–54.
[12] J.C. Figueroa-García, E.J. Medina-Pinzón, and J.D. Rubio-Espinosa, Non-cooperative Games Involving Type-2 Fuzzy Uncertainty: An Approach, Computer Information Systems and Industrial Management (Berlin, Heidelberg) (K. Saeed
and V. Sn´aˇsel, eds.), Springer Berlin Heidelberg, 2014, pp. 387–396. 36
[13] J.C. Figueroa-García, A. Mehra, and S. Chandra, Optimal solutions for group matrix games involving interval-valued fuzzy numbers, Fuzzy Sets and Systems 362 (2019), 55–70.
[14] A. González and M. A. Vila, A discrete method for studying indifference and order relations between fuzzy numbers, Inform. Sci. 56 (1991), no. 1-3, 245–258.
[15] A. González and M. A. Vila, Dominance relations on fuzzy numbers, Inform. Sci. 64 (1992), no. 1-2, 1–16.
[16] E. L. Hannan, On fuzzy goal programming, Decis. Sci. 12 (1981), no. 3, 522–531.
[17] E. Hosseinzadeh, H. Hassanpour, and M. Arefi, A weighted goal programming approach to fuzzy linear regression with crisp inputs and type-2 fuzzy outputs, Soft Comput. 19 (2015), no. 5, 1143–1151.
[18] D. Hunwisai and P. Kumam, Linear programming model for solution of matrix game with payoffs trapezoidal intuitionistic fuzzy number, Bull. Comput. Appl. Math. 5 (2017), no. 1, 9–32.
[19] J. Jana and S. K. Roy, Dual hesitant fuzzy matrix games: based on new similarity measure, Soft Comput. 23 (2019), no. 18, 8873–8886.
[20] S. Karmakar, M. R. Seikh, and O. Castillo, Type-2 intuitionistic fuzzy matrix games based on a new distance measure: Application to biogas-plant implementation problem, Appl. Soft Comput. 106 (2021), Article ID: 107357.
[21] S. Kumar, Max-min solution approach for multi-objective matrix game with fuzzy goals, Yugosl. J. Oper. Res. 26 (2016), no. 1, 51–60.
[22] D. F. Li and J. C. Liu, A parameterized nonlinear programming approach to solve matrix games with payoffs of i-fuzzy numbers, IEEE Trans. Fuzzy Syst. 23 (2015), no. 4, 885–896.
[23] S. Li and G. Tu, Bi-matrix games with general intuitionistic fuzzy payoffs and application in corporate environmental behavior, Symmetry 14 (2022), no. 4, Article ID: 671.
[24] V. Mazalov, Mathematical game theory and applications, John Wiley & Sons, 2014.
[25] J. M. Mendel, Computing with words, when words can mean different things to different people, Proc. of Third International ICSC Symposium on Fuzzy Logic and Applications, 1999, pp. 158–164.
[26] J. M. Mendel and John R., Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Syst. 10 (2002), no. 2, 117–127.
[27] R. H. Mohamed, The relationship between goal programming and fuzzy programming, Fuzzy sets and systems 89 (1997), no. 2, 215–222.
[28] J. X. Nan and D. F. Li, Linear programming approach to matrix games with intuitionistic fuzzy goals, Int. J. Comput. Intell. Syst. 6 (2013), no. 1, 186–197.
[29] J. X. Nan, D. F. Li, and J. J. An, Solving bi-matrix games with intuitionistic fuzzy goals and intuitionistic fuzzy payoffs, J. Intell. Fuzzy Syst. 33 (2017), no. 6, 3723–3732.
[30] R. Narasimhan, Goal programming in a fuzzy environment, Decis. Sci. 11 (1980), no. 2, 325–336.
[31] P. K. Nayak and M. Pal, Bi-matrix games with intuitionistic fuzzy goals, Iran. J. Fuzzy Syst. 7 (2010), no. 1, 65–79.
[32] P. K. Nayak and M. Pal, Bi-matrix games with intuitionistic fuzzy goals, Iran. J. Fuzzy Syst. 7 (2010), no. 1, 65–79.
[33] I. Nishizaki and M. Sakawa, Equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals, J. Optim. Theory Appl. 86 (1995), no. 2, 433–457.
[34] I. Nishizaki and M. Sakawa, Equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals, Fuzzy Sets and Systems 111 (2000), no. 1, 99–116.
[35] I. Nishizaki and M. Sakawa, Fuzzy and Multiobjective Games for Conflict Resolution, Springer-Verlag, Berlin Heidelberg, 2001.
[36] J. S. Patiño Callejas, K. Y. Espinosa-Ayala, and J. C. Figueroa-García, Type-2 fuzzy uncertainty in goal programming, Proceedings of the 2015 International Conference on Fuzzy Logic in Artificial Intelligence, vol. 1424, 2015, pp. 21–25.
[37] M. Sakawa and I. Nishizaki, Max-min solutions for fuzzy multiobjective matrix games, Fuzzy Sets and Systems 67 (1994), no. 1, 53–69.
[38] M. R. Seikh, S. Karmakar, and 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), no. 5, 155–172.
[39] M. R. Seikh, S. Karmakar, and P. K. Nayak, Matrix games with dense fuzzy payoffs, Int. J. Intell. Syst. 36 (2021), no. 4, 1770–1799.
[40] M. R. Seikh, S. Karmakar, and M. Xia, Solving matrix games with hesitant fuzzy pay-offs, Iran. J. Fuzzy Syst. 17 (2020), no. 4, 25–40.
[41] M. R. Seikh, P. K. Nayak, and M. Pal, Aspiration level approach to solve matrix games with i-fuzzy goals and i-fuzzy pay-offs, Pac. Sci. Rev. A: Natural Science and Engineering 18 (2016), no. 1, 5–13.
[42] R. Verma and A. Aggarwal, On matrix games with 2-tuple intuitionistic fuzzy linguistic payoffs, Iran. J. Fuzzy Syst. 18 (2021), no. 4, 149–167.
[43] R. Verma, N. Singla, and R.R. Yager, Matrix games under a pythagorean fuzzy environment with self-confidence levels: formulation and solution approach, Soft Comput. (In press),
[44] T. Verma and A. Kumar, Ambika methods for solving matrix games with atanassov’s intuitionistic fuzzy payoffs, IEEE Trans. Fuzzy Syst. 26 (2017), no. 1, 270–283.
[45] V. Vidyottama, S. Chandra, and C. R. Bector, Bi-matrix games with fuzzy goals and fuzzy, Fuzzy Optimization and Decision Making 3 (2004), no. 4, 327–344.
[46] V. Vijay, S. Chandra, and C. R. Bector, Matrix games with fuzzy goals and fuzzy payoffs, Omega 33 (2005), no. 5, 425–429.
[47] C. Xu, F. Meng, and Q. Zhang, Pn equilibrium strategy for matrix games with fuzzy payoffs, Int. J. Fuzzy Syst. 32 (2017), no. 3, 2195–2206.
[48] W. Xue, Z. Xu, and X. J. Zeng, Solving matrix games based on ambika method with hesitant fuzzy information and its application in the counter-terrorism issue, Appl. Intell. 51 (2021), no. 3, 1227–1243.
[49] R. R. Yager, A procedure for ordering fuzzy sets of the unit interval, Inf. Sci. 24 (1981), no. 2, 143–161.
[50] H. J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, 1991.