یک روش پیشنهادی برای شاخص های تسلط امکان و ضرورت در برنامه ریزی خطی فازی تصادفی

This paper presents a suggested approach for solving a stochastic fuzzy linear programming problem. This approach utilizes two possibility and two necessity dominance indices that have been introduced by Dubois and Prade [D. Dubois, H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Information Sciences 30 (1983) 183–224]. The chance-constrained approach and the αα-cut are used to transform the stochastic fuzzy problem to its deterministic-crisp equivalent, according to each of the four dominance indices. A numerical example is given.

Comparison of fuzzy numbers is considered one of the most important topics in fuzzy logic theory. The early and most important work in the field of comparing fuzzy numbers has been presented by Dubois and Prade [1]. A comparison between their work and other attempts that have been made in this area has been given by Bortolan and Degani [2]. On the other hand, the dominance possibility indices, which have been introduced by Dubois and Prade, were utilized in the field of fuzzy mathematical programming [3] and [4] and the field of stochastic fuzzy mathematical programming [5] and [6]. The approach used in these fields was based on formulating a possibility function, whether in the case of trapezoidal fuzzy numbers or the case of triangular fuzzy numbers. In this paper, we are going to utilize Dubois and Prade’s dominance possibility and necessity indices, within a different approach, in the case of stochastic fuzzy linear programming problem. The dominance possibility and necessity, as well as the strict dominance possibility and necessity criteria, are utilized according to the chance-constrained method to transform the suggested problem to its deterministic-crisp equivalent. This approach helps avoiding any approximation that may exist due to comparing the inverse distribution function of fuzzy tolerance measures.

The suggested approach for comparing fuzzy numbers in the case of stochastic fuzzy linear programming problems can be applied for different types of fuzzy numbers, in addition to the trapezoidal and triangular fuzzy numbers that have been used in this paper. Also, any approximation that may exist, due to using another approach, can be avoided. Utilizing the αα-cut technique for the membership functions to derive closed crisp intervals represents the main step in our approach. Thus, for different values of αα, and by comparing the closed crisp intervals, results are generated according to each of the four dominance indices, whereas the most convenient one can be chosen.