Abstract

In this paper, a cooperative Continuous static game (F-CCSG) with fuzzy parameters in the cost function of the player is presented. Through the use of the ?-level sets of fuzzy numbers, the F-CCSG is converted to the corresponding ?-CCSG and an extended Pareto optimality concept called the ?-Pareto optimality is introduced. An algorithm for solving the ?-CCSG is suggested. The algorithm is based mainly on the reference attainable point (ARP) method introduced by Wang et al. , [20] and reference direction (RD) method introduced by Narula et al. , [7]. One of the major improvement is the reduction of the number of iterations and hence the computational effort required to obtain the final solution. The stability of the first kind without differentiability corresponding to the final solution is determined. To clarify this approach, a numerical example is given for illustration.

References

• Cruz, J. B. , and Simaan, M. , (2000). Ordinal games and generalized Nash and Stackelberg solutions, Journal of optimization theory and applications, (107): 205 – 222.
• Dudois, D. , and Prade, H. , (1980). Fuzzy Sets and Systems: theory and Applications, Academic Press, New York.
• El–Shafei, K. M. M. , (2007). An interactive approach for solving Nash cooperative continuous static games (NCCSG), International Journal of contemporary Mathematical sciences, (2):1147 – 1162.
• Garagic, D. , and Cruz, B. J. , (2003). An approach to fuzzy Noncooperative Nash games. Journal of Optimization theory and applications, (118): 475 – 491
• Kacher, F. , and Larbani, M. , (2008). Existence of equilibrium solution for a non–cooperative game with fuzzy goals and parameters, Fuzzy Sets and Systems, (159): 164 – 176.
• Kassem, M. A. , and Ammar. E. E. , (1996). A parametric study of multiobjective nonlinear programming problems with fuzzy parameters in the objective functions, Fuzzy Sets and Systems, (8): 187-196.
• Narula, S. C. , Kirilov, L. , and Vassilev, V. , (1993). An Interactive Algorithm for Solving Multiple Objective Nonlinear Programming Problems, Multiple Criteria Decision Making, Proceeding of the 10th International Conference: Expand and Enrich the Domain of Thinking and Application, Springer-Verlag, Berlin.
• Navidi, H. , Ketabchi, S. , and Messi, M. , (2011). An introduction to Game Theory. (In Farsi) Shahed University.
• Navidi, R. H. , Amiri, A. , and Kamranrad, R. , (2014). Multi responses optimization through game theory approach, International Journal of Industrial Engineering and production Research, (25): 215 – 224.
• Osborne, M. J. , (2003). A Course in Game Theory. Oxford University Press.
• Osman, M. , (1984). Different parametric problems in Continuous Static Games, The 1st ORMA conference, M. T. C. , Cairo Egypt.
• Osman, M. , and El-Banna, A. , (1993). Stability of multiobjective nonlinear programming problems with fuzzy parameters, Mathematics and Computers in Simulation, (35): 321-326.
• Osman, M. , El-Banna, A. , and Kamel, M. , (1999). On Fuzzy Continuous Static Games: Stackberg leader with min-max followers, Fuzzy Mathematics, (2): 259-266.
• Osman, M. , El-Banna, A. , and Amer, A. , (1999). Study large scale fuzzy Nash equilibrium solutions, Fuzzy Mathematics, (2): 267-275.
• Tomas, L. , and Waiter, J. , (1981). Optimality in Parametric Systems, John Wiley, Inc. , New York.
• Rockafellar, R. , (1967). Duality and stability in extremum problems involving convex functions. Pacific Journal of Mathematics, (21): 167-181.
• Sakawa, M. , and Yano, H. , (1989). Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters, Fuzzy Sets and Systems, (29):315-326.
• Sakawa, M. , and Nishizaki, I. , (1994). Max–min solution for fuzzy multi objective matrix games, Fuzzy sets and Systems, (67):53 – 69.
• Tomas,L. , and Walter,J. , (1981). Optimality in Parametric Systems, John Wiley, Inc. , New York.
• Wang, X. M. , Qin, Z. L. , and Hu, Y. D. , (2001). An interactive algorithm for solving multi criteria decision making: the attainable reference point method, IEEE Transactions on systems, Man, and cybernetics – part A: systems and Humans, (31): 194 – 198.