Paper:

# Interactive Decision Making for Fuzzy Random Multiobjective Linear Programming Problems with Variance Covariance Matrices Through Fractile Optimization

## Hitoshi Yano

School of Humanities and Social Sciences, Nagoya City University, 1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya 467-8501, Japan

In this paper, we focus on fuzzy random multiobjective linear programming problems with variance covariance matrices through fractile optimization, and propose an interactive decision making method to obtain a satisfactory solution. In the proposed method, it is assumed that the decision maker has fuzzy goals for not only permissible probability levels but also the corresponding objective functions. Such fuzzy goals are quantified by eliciting the corresponding membership functions. Using the fuzzy decision, such two kinds of membership functions are integrated, and *D _{f}* -Pareto optimal solution concept is defined in the integrated membership space. By using the bisection method and the convex programming technique, a satisfactory solution is obtained from among a

*D*-Pareto optimal solution set through the interaction with the decision maker.

_{f}*J. Adv. Comput. Intell. Intell. Inform.*, Vol.18, No.3, pp. 383-390, 2014.

- [1] H. Kwakernaak, “Fuzzy Random Variable-1,” Information Sciences, Vol.15, pp. 1-29, 1978.
- [2] M. L. Puri and D. A. Ralescu, “Fuzzy Random Variables,” J. of Mathematical Analysis and Applications, Vol.14, pp. 409-422, 1986.
- [3] H. Katagiri, H. Ishii, and T. Itoh, “Fuzzy Random Linear Programming Problem,” In: Proc. of Second European Workshop on Fuzzy Decision Analysis and Neural Networks for Management, Planning and Optimization, pp. 107-115, 1997.
- [4] M. K. Luhandjula and M. M. Gupta, “On Fuzzy Stochastic Optimization,” Fuzzy Sets and Systems, Vol.81, pp. 47-55, 1996.
- [5] G.-Y.Wang and Z. Qiao, “Linear Programming with Fuzzy Random Variable Coefficients,” Fuzzy Sets and Systems, Vol.57, pp. 295-311, 1993.
- [6] J. R. Birge and F. Louveaux, “Introduction to Stochastic Programming,” Springer, 1997.
- [7] A. Charnes and W.W.Cooper, “Chance Constrained Programming,” Management Science, Vol.6, pp. 73-79, 1959.
- [8] G. B. Danzig, “Linear Programming under Uncertainty,” Management Science, Vol.1, pp. 197-206, 1955.
- [9] P. Kall and J.Mayer, “Stochastic Linear Programming Models, Theory, and Computation,” Springer, 2005.
- [10] V. J. Lai and C. L. Hwang, “Fuzzy Mathematical Programming,” Springer, Berlin, 1992.
- [11] H. Rommelfanger, “Fuzzy Linear Programming and Applications,” European J. of Operational Research, Vol.92, pp. 512-527, 1997.
- [12] H.-J. Zimmermann, “Fuzzy Sets, Decision-Making and Expert Systems,” Kluwer Academic Publishers, Boston, 1987.
- [13] H. Katagiri, M. Sakawa, K. Kato, and I. Nishizaki, “Interactive Multiobjective Fuzzy Random Linear Programming: Maximization of Possibility and Probability,” European J. of Operational Research, Vol.188, pp. 530-539, 2008.
- [14] M. Sakawa, I. Nishizaki, and H. Katagiri, “Fuzzy Stochastic Multiobjective Programming,” Springer, 2011.
- [15] H. Yano, “Interactive Decision Making for Fuzzy Random Multiobjective Linear Programming Problems with Variance-CovarianceMatrices Through Probability Maximization,” Proc. of The 6th Int. Conf. on Soft Computing and Intelligent Systems and 13th Int. Symposium on Advanced Intelligent Systems, pp. 965-970, 2012.
- [16] M. Sakawa, “Fuzzy Sets and Interactive Multiobjective Optimization,” Plenum Press, 1993.
- [17] J. J. Glen, “Mathematical models in farm planning: a survey,” Operations Research, Vol.35, pp. 641-666, 1987.
- [18] K. Hayashi, “Multicriteria analysis for agricultural resource management: A critical survey and future perspectives,” European J. of Operational Research, Vol.122, pp. 486-500, 2000.
- [19] T. Itoh, H. Ishii, and T. Nanseki, “A model of crop planning under uncertainty in agricultural management,” Int. J. of Production Economics, Vol.81-82, pp. 555-558, 2003.
- [20] T. Toyonaga, T. Itoh, and H. Ishii, “A Crop Planning with Fuzzy Random Profit Coefficients,” Fuzzy Optimization and Decision Making, Vol.4, pp. 51-69, 2005.
- [21] H.-J. Zimmermann, “Fuzzy programming and linear programming with several objective functions,” Fuzzy Sets and Systems, Vol.1, pp. 45-55, 1978.
- [22] D. Dubois and H. Prade, “Fuzzy Sets and Systems,” Academic Press, 1980.