MIN-MAX SOLUTIONS FOR PARAMETRIC CONTINUOUS STATIC GAME UNDER ROUGHNESS (PARAMETERS IN THE COST FUNCTION AND FEASIBLE REGION IS A ROUGH SET)

Yousria A. Aboelnaga     (Higher Technological Institute, Tenth of Ramadan City, 44629, Egypt)
Mai Zidan     (Faculty of Engineering, Tanta University, Al-Geish St., Tanta, 31512, Egypt)

Abstract


Any simple perturbation in a part of the game whether in the cost function and/or conditions is a big problem because it will require a game re-solution to obtain the perturbed optimal solution. This is a waste of time because there are methods required several steps to obtain the optimal solution, then at the end we may find that there is no solution. Therefore, it was necessary to find a method to ensure that the game optimal solution exists in the case of a change in the game data. This is the aim of this paper. We first provided a continuous static game rough treatment with Min-Max solutions, then a parametric study for the processing game and called a parametric rough continuous static game (PRCSG). In a Parametric study, a solution approach is provided based on the parameter existence in the cost function that reflects the perturbation that may occur to it to determine the parameter range in which the optimal solution point keeps in the surely region that is called the stability set of the \(1^{st}\) kind. Also the sets of possible upper and lower stability to which the optimal solution belongs are characterized.
Finally, numerical examples are given to clarify the solution algorithm.


Keywords


Continuous static game, Rough programming, Non-linear programming, Rough set theory, Parametric linear programming, Parametric non-linear programming

Full Text:

PDF

References


Bank B., Guddat J., Klatte D., Kummer B., Tammer K. Non-Linear Parametric Optimization. Basel: Birkhäuser, 1982. 228 p. DOI: 10.1007/978-3-0348-6328-5

Bazaraa M.S., Sherali H.D., Shetty C.M. Nonlinear Programming: Theory and Algorithms. 3rd Ed. Verlag: J. Wiley & Sons Inc., 2013. 872 p.

Bertsekas P.D. Nonlinear Programming. 2rd Ed. Belmont, Massachusetts: Athena Scientific, 1999. 791 p.

Budhiraja A., Dupuis P. Representations for functional of Hilbert space valued diffusions. In: Stochastic Analysis, Control, Optimization and Applications. McEneaney W.M., Yin G.G., Zhang Q. (eds.) Ser. Systems Control Found. Appl. Boston, MA: Birkhäuser, 1999. P. 1–20. DOI: 10.1007/978-1-4612-1784-8_1

Elsisy M.A., Eid M.H., Osman M.S.A. Qualitative analysis of basic notions in parametric rough convex programming (parameters in the objective function and feasible region is a rough set). OPSEARCH, 2017. Vol. 54. P. 724–734. DOI: 10.1007/s12597-017-0300-2

Jongen H.Th., Jonker P., Twilt F. Nonlinear Optimization in Finite Dimensions. Boston, MA: Springer, 2000. 513 p. DOI: 10.1007/978-1-4615-0017-9

Kalaiselvi R., Kousalya K. Statistical modelling and parametric optimization in document fragmentation. Neural Comput. Applic., 2020. Vol. 32. P. 5909–5918. DOI: 10.1007/s00521-019-04068-1

Lijun X., Yijia Z., Bo Y. Robust Optimization Model with Shared Uncertain Parameters in Multi-Stage Logistics Production and Inventory Process. Mathematics, 2020. Vol. 8, No. 2. Art. no. 211. P. 1–12. DOI: 10.3390/math8020211

Matsumoto A., Szidarovszky F. Continuous Static Games. In: Game Theory and Its Applications. Tokyo: Springer, 2016. P. 21–47. DOI: 10.1007/978-4-431-54786-0_3

Miettinen K. Nonlinear Multiobjective Optimization. Ser. Internat. Ser. Oper. Res. Management Sci., vol. 12. NY: Springer, 1998. 298 p. DOI: 10.1007/978-1-4615-5563-6

Nguyen V., Gupta S., Rana S. et al. Filtering Bayesian optimization approach in weakly specified search space. Knowl. Inf. Syst., 2019. Vol. 60. P. 385–413. DOI: 10.1007/s10115-018-1238-2

Osman M.S.A. Qualitative analysis of basic notions in parametric convex programming. I. Parameters in the constraints. Aplikace Matematiky, 1977. Vol. 22., No. 5. P. 318–332. DOI: 10.21136/AM.1977.103710

Osman M., Lashein E.F., Youness E.A., Elsayed T. Mathematical programming in rough environment. Optimization, 2011. Vol. 60, No. 5. P. 603–611. DOI: 10.1080/02331930903536393

Patil A., Desai A. D. Parametric optimization of engine performance and emission for various n-butanol blends at different operating parameter condition. Alexandria Eng. J., 2020. Vol. 59, No. 2. P. 851–864. DOI: 10.1016/j.aej.2020.02.006

Sawaragi Y., Nakayama H., Tanino T. Theory of Multiobjective Optimization. Math. Sci. Eng., vol. 176. Academic Press, 1985. 322 p.

Schneider J.J., Kirkpatrick S. Stochastic Optimization. Berlin Heidelberg: Springer-Verlag, 2006. 568 p. 2 DOI: 10.1007/978-3-540-34560-2

Sun W., Yuan Y.-X. Optimization Theory and Methods: Nonlinear Programming. Springer Optim. Appl., vol. 1. US: Springer-Verlag, 2006. 688 p. DOI: 10.1007/b106451

Tuy H. Minimax: existence and stability. In: Pareto Optimality, Game Theory and Equilibria. A. Chinchuluun, P.M. Pardalos, A. Migdalas, L. Pitsoulis (eds.). Springer Optim. Appl., vol 17. NY: Springer. P. 3–21. DOI: 10.1007/978-0-387-77247-9_1

Youness E. Characterizing solutions of rough programming problems. European J. Oper. Res., 2006. Vol. 168, No. 3. P. 1019–1029. DOI: 10.1016/j.ejor.2004.05.019

Zhang J., Liu N., Wang S. A parametric approach for performance optimization of residential building design in Beijing. Build. Simul., 2019. Vol. 13. P. 223–235. DOI: 10.1007/s12273-019-0571-z




DOI: http://dx.doi.org/10.15826/umj.2020.2.001

Article Metrics

Metrics Loading ...

Refbacks