شرایط بهینگی برای مسائل برنامه ریزی خطی با ضرایب فازی

The optimality conditions for linear programming problems with fuzzy coefficients are derived in this paper. Two solution concepts are proposed by considering the orderings on the set of all fuzzy numbers. The solution concepts proposed in this paper will follow from the similar solution concept, called the nondominated solution, in the multiobjective programming problem. Under these settings, the optimality conditions will be naturally elicited.

The occurrence of fuzziness in the real world is inevitable owing to some unexpected situations. Therefore, imposing fuzziness upon conventional optimization problems becomes an interesting research topic. The collection of papers on fuzzy optimization edited by Słowiński [1] and Delgado et al. [2] gives the main stream of this topic. Lai and Hwang [3] and [4] also give an insightful survey. On the other hand, the book edited by Słowiński and Teghem [5] provides comparisons between fuzzy optimization and stochastic optimization for multiobjective programming problems. Bellman and Zadeh [6] inspired the development of fuzzy optimization by providing the aggregation operators, which combined the fuzzy goals and fuzzy decision space. After this motivation and inspiration, there appeared a lot of articles dealing with fuzzy optimization problems. Some interesting articles are Buckley [7] and [8], Julien [9] and Luhandjula et al. [10] using possibility distribution, Herrera et al. [11] and Zimmermann [12] and [13] using fuzzified constraints and objective functions, Inuiguchi et al. [14] and [15] using modality measures, Tanaka and Asai [16] using fuzzy parameters, and Lee and Li [17], [18] and [19] considering the de Novo programming problem. The duality of the fuzzy linear programming problem was firstly studied by Rodder and Zimmermann [20] considering the economic interpretation of the dual variables. After that, many interesting results regarding the duality of the fuzzy linear programming problem was investigated by Bector et al. [21], [22] and [23], Liu et al. [24], Ramík [25], Verdegay [26] and Wu [27]. In this paper, we investigate the optimality conditions for linear programming problems with fuzzy coefficients. In Section 2, we introduce some basic properties and arithmetics of fuzzy numbers. In Section 3, we formulate two linear programming problems with fuzzy coefficients. One considers crisp (conventional) linear constraints, and the other considers fuzzy linear constraints. Two solution concepts are proposed for these two problems. In Section 4, we derive the optimality conditions for these two problems by introducing the multipliers. Finally, in Section 5, three examples are provided to illustrate the discussions in linear programming problems with fuzzy coefficients.