Document Type : Research Paper

Author

Department Operations Research, Institute of Statistical Studies and Research, Cairo University, Giza, Egypt.

Abstract

In this paper, an interactive approach for solving multi- objective nonlinear programming (MONLP) problem is introduced. This approach is combined with the Reference Direction (RD) introduced by Narula et al. (1993) and the Attainable Reference Point (ARP) method introduced by Wang et al. (2001). In the interactive approach, we still starting with a weak efficient solution as the first step and use the corresponding objective values to improve the weighting coefficients of the augmented Lexicographic Weighted Tchebycheff problem and hence modify the reference point in the case of an unsatisfactory solution for the decision maker (DM) he (she) wishes. Cooperative continuous static game is introduced as an application and its solution is obtained under the proposed interactive approach. Finally, a numerical example is given to the utility of our proposed method.

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Main Subjects

[1]   Cruz, J. B., & Simaan, M. A. (2000). Ordinal games and generalized Nash and Stackelberg solutions. Journal of optimization theory and applications107(2), 205-222.
[2]   Chankong, V., & Haimes, Y. Y. (2008). Multi-objective decision making: theory and methodology. Courier Dover Publications.
[3]   Elshafei, M. M. K. (2007). An interactive approach for solving Nash cooperative continuous static games. International journal of contemporary mathematical sciences, 2(24), 1147- 1162.
[4]   Gardiner, L. R., & Steuer, R. E. (1994). Unified interactive multiple objective programming. European journal of operational research74(3), 391-406.
[5]   Hwang, C. L., & Masud, A. S. M. (2012). Multiple objective decision making—methods and applications: a state-of-the-art survey (Vol. 164). Springer Science & Business Media.
[6]   Khalifa, H. A., & Zeineldin, R. A. (2015). An interactive approach for solving fuzzy cooperative continuous static games. International journal of computer applications113(1).
[7]   Khalifa, H. A. (2016). An interactive approach for solving fuzzy multi-objective non-linear programming problems. The journal of fuzzy mathematics, 3(24), 535- 545.
[8]   Molina, C., & Earn, D. J. (2017). Evolutionary stability in continuous nonlinear public goods games. Journal of mathematical biology74(1-2), 499-529.
[9]   Muhammed, D., Anisi, M., Zareei, M., Vargas-Rosales, C., & Khan, A. (2018). Game theory-based cooperation for underwater acoustic sensor networks: Taxonomy, review, research challenges and directions. Sensors18(2), 425.
[10]Narula, S. C., Kirilov, L., & Vassilev, V. (1994). An interactive algorithm for solving multiple objective nonlinear programming problems. In G. H. Tzeng H. F. Wang U. P. Wen & P. L. Yu (Eds.). Multiple criteria decision making (pp. 119-127). Springer, New York, NY.
[11]Navidi, H., Amiri, A., & Kamranrad, R. (2014). Multi-responses optimization through game theory approach. International journal of industrial engineering and production research, 25(3), 215- 224.
[12]Osman, M. S. A. (1977). Qualitative analysis of basic notions in parametric convex programming. I. Parameters in the constraints. Aplikace matematiky22(5), 318-332.
[13]Osman, M., & Dauer, J. (1983). Characterization of basic notions in multi-objective convex programming problems. Retrieved from Lincoln, University of Nebraska database.
[14]Osman, M. S., El-Kholy, N. A., & Soliman, E. I. (2015). A recent approach to continuous time open loop stackelberg dynamic game with min- max cooperative and non-cooperative followers. European scientific journal, 11(3), 94-109.
[15]Sadrabadi, M. R., & Sadjadi, S. J. (2009). A new interactive method to solve multiobjective linear programming problems. JSEA2(4), 237-247.
[16]Shin, W. S., & Ravindran, A. (1992). A comparative study of interactive tradeoff cutting plane methods for MOMP. European journal of operational research56(3), 380-393.
[17]Wang, X. M., Qin, Z. L., & Hu, Y. D. (2001). An interactive algorithm for multicriteria decision making: the attainable reference point method. IEEE transactions on systems, man, and cybernetics-part A: systems and humans31(3), 194-198.
[18]Wendell, R. E., & Lee, D.  N. (1977). Efficiency in multi-objective optimization problems. Mathematical programming, 12(1), 406- 414.