یک کلاس از مدل برنامه ریزی خطی چندهدفه با ضرایب تصادفی فازی

The aim of this paper is to deal with a multiobjective linear programming problem with fuzzy random coefficients. Some crisp equivalent models are presented and a traditional algorithm based on an interactive fuzzy satisfying method is proposed to obtain the decision maker’s satisfying solution. In addition, the technique of fuzzy random simulation is adopted to handle general fuzzy random objective functions and fuzzy random constraints which are usually hard to be converted into their crisp equivalents. Furthermore, combined with the techniques of fuzzy random simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy random multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.

Among types of uncertainty surrounding real life problems, randomness (stochastic variation) and fuzziness (vagueness) play a pivotal role. Accordingly, stochastic programming and fuzzy programming have been proposed to make decisions under an uncertainty environment. Different types of stochastic programming and fuzzy programming models have been developed to suit the different purposes of management, such as the expectation model [1], chance constrained programming [2] and [3], the minimum risk problem [4], the modality approach and the fractile approach [5] etc. In these models, randomness and fuzziness are considered as separate aspects. However, in a decision-making process, we may face a hybrid uncertain environment where fuzziness and randomness coexist. In such cases, the concept of fuzzy random variable introduced by Kwakernaak [6] is a useful tool dealing with the two types of uncertainty simultaneously. Recently, several researchers have considered the issue of combining fuzziness and randomness in an optimization framework such as Wang and Qiao [7] and [8], Luhandjula [9] and [10], Katagiri et al. [11], and Liu [3], [12], [1] and [4]. In [7] and [8], Wang and Qiao discussed the distribution problems for linear programming with fuzzy random coefficients. In [9], Luhandjula employed a semi-infinite approach in order to convert the original fuzzy random linear programming problem into a stochastic programming one so that the techniques of stochastic optimization can apply. In [10], Luhandjula proposed a unifying methodological approach to transform the constraints with fuzzy random coefficients into crisp constraints. In [11], Katagiri et al. proposed an interactive satisfying method to solve the fuzzy random multiobjective 0-1 programming problem. Besides, in [3] and [12] Liu presented fuzzy random chance-constrained programming, fuzzy random dependent-chance programming, and designed some hybrid intelligent algorithms in order to solve them effectively. In [1] and [4], Liu and Liu presented an expected value model and minimum risk problem for the fuzzy random multiobjective programming problem and designed some hybrid intelligent algorithms. The purpose of this paper is to present two approaches of solving multiobjective linear programming with fuzzy random coefficients. Our research is based on the chance measure of fuzzy random events [3]. This paper is organized as follows. Section 2 recalls some definitions and results about fuzzy random variables. Section 3 studies the prob-pos constrained multiobjective programming model. A crisp equivalent model is proposed for a special type of fuzzy random variables, and an interactive fuzzy satisfying method is adopted to obtain the decision maker’s satisfactory solution. Section 4 considers the prob-nec constrained multiobjective programming model. Fuzzy random simulation and fuzzy random simulation-based genetic algorithm using compromise approach are presented in Sections 5 and 6, respectively. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms. The results show that the fuzzy random simulation-based genetic algorithm is effective.

In this paper, we have considered the multiobjective linear programming with fuzzy random coefficients. For a special type of fuzzy random variables, some crisp equivalent models are proposed for prob-pos and prob-nec constrained multiobjective programming models. Two approaches have been presented for solving this kind of problem. One is the interactive fuzzy satisfying method which is used to solve a special type of fuzzy random multiobjective programming problem, and the other is the fuzzy random simulation-based genetic algorithm using compromise approach which is effective to solve the general fuzzy random multiobjective programming problem. Though the fuzzy random simulation-based genetic algorithm proposed in this paper usually spends more CPU time than traditional algorithms, it is a viable and efficient way to deal with complex optimization problems involving randomness and fuzziness. Furthermore, the genetic algorithm proposed in this paper can be extended to solve the prob-pos (prob-nec) constrained multiobjective nonlinear programming problems. In the future, fuzzy random simulation-based multiobjective genetic algorithm is another field that we will consider.