Paper:

# Multiobjective Random Fuzzy Linear Programming Problems Based on the Possibility Maximization Model

## Takashi Hasuike^{*}, Hideki Katagiri^{**}, and Hiroaki Ishii^{*}

^{*}Graduate School of Information Science and Technology, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan

^{**}Graduate School of Engineering, Hiroshima University, 1-4-1, Kagamiyama Higashi Hiroshima 739-8527, Japan

Two multiobjective random fuzzy programming problems considered based on the possibility maximization model using possibilistic and stochastic programming are not initially well defined due to the random variables and fuzzy numbers included. To solve them analytically, probability criteria are set for objective functions and chance constraints are introduced. Taking into account the decision maker’s subjectivity and the original plan’s flexibility, a fuzzy goal is introduced for each objective function. The original problems are then changed into deterministic equivalent problems to make the possibility fractile optimization problem equivalent to a linear programming problem. The possibility maximization problem for probability is changed into a nonlinear programming problem, and an analytical solution is constructed extending previous solution approaches.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.13, No.4, pp. 373-379, 2009.

- [1] E. M. L. Beale, “On optimizing a convex function subject to linear inequalities,” Journal of the Royal Statistical Society, Vol.17, pp. 173-184, 1955.
- [2] A. Charnes and W. W. Cooper, “Deterministic equivalents for optimizing and satisfying under chance constraints,” Operations Research, Vol.11, pp. 18-39, 1963.
- [3] G. B. Dantzig, “Linear programming under uncertainty,” Management Science, Vol.1, pp. 197-206, 1955.
- [4] D. Dubois and H. Prade, “Fuzzy Sets and Systems,” Academic Press, New York, 1980.
- [5] M. Inuiguchi and T. Tanino, “Portfolio selection under independent possibilistic information,” Fuzzy Sets and Systems, Vol.115, pp. 83-92, 2000.
- [6] R. Kruse and K. D. Meyer, “Statistics with Vague Data,” D. Riedel Publishing Company, 1987.
- [7] H. Kwakernaak, “Fuzzy random variable-1, ”Information Sciences, Vol.15, pp. 1-29, 1978.
- [8] M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” Journal of Mathematical Analysis and Applications, Vol.14, pp. 409-422, 1986.
- [9] B. Liu, “Theory and Practice of Uncertain Programming,” Physica Verlag, 2002.
- [10] B. Liu, “Uncertainty theory,” Physica Verlag, 2004.
- [11] H. Katagiri, H. Ishii, and M. Sakawa, “On fuzzy random linear knapsack problems, Central European Journal of Operations Research,” Vol.12 No.1, pp. 59-70, 2004.
- [12] H. Katagiri, M. Sakawa, and H. Ishii, “A study on fuzzy random portfolio selection problems using possibility and necessity measures,” Scientiae Mathematicae Japonocae, Vol.65, No.2, pp. 361-369, 2005.
- [13] T. Hasuike, H. Katagiri, and H. Ishii, “Portfolio selection problems with random fuzzy variable returns,” Proc. of 2007 IEEE Int. Conf. on Fuzzy Systems, pp. 416-421, 2007.
- [14] X. Hung, “Two new models for portfolio selection with stochastic returns taking fuzzy information,” European Journal of Operational Research, Vol.180, pp. 396-405, 2007.
- [15] X. Huang, “Optimal project selection with random fuzzy parameters,” Int. Journal of Production Economics, Vol.106, pp. 513-522, 2007.
- [16] T. Hasuike, H. Katagiri, and H. Ishii, “Probability Maximization Model of 0-1 Knapsack Problem with Random Fuzzy Variables,” Proc. of 2008 IEEE Int. Conf. on Fuzzy Systems, pp. 548-554, 2007.
- [17] H. Katagiri, M. Sakawa, K. Kato, and I. Nishizaki, “Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability,” European Journal of Operational Research, Vol.188, No. 2, pp. 530-539, 2008.
- [18] S. Kataoka, “A stochastic programming model,” Econometrica, Vol.31, pp. 181-196, 1963.
- [19] A. M. Geoffrion, “Stochastic programming with aspiration or fractile criteria,” Management Science, Vol.13, pp. 672-679, 1967.
- [20] L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, Vol.1, pp. 3-28, 1978.