Method of solving fuzzy problems of mathematical programming

Abstract

A brief analysis of traditional methods of solving fuzzy problems of mathematical programming was carried out. The shortcomings of the known approaches, limiting their application for the problems of real dimensionality, were revealed. The solution of the problem is achieved with the use of a two–stage procedure. At the first stage, a usual optimization problem is solved, which is caused by the original problem with the replacement of fuzzy parameters with their modal values. In this case, standard technologies of solving the problems of mathematical programming are used. At the second stage, a distinct solution, which satisfies two special requirements, is searched for. First, this solution must minimally deviate from the modal, obtained at the first stage. Second, membership function of fuzzy value of the optimized function, corresponding to the solution, must have a minimum level of uncertainty. In this connection, a complex criterion, which contains two appropriate components, is formed for solving the problem. A parameter of regularization, which assigns the value of the weight coefficient, determining the value of components, is introduced into the proposed complex criterion. This regulating multiplier provides acceptable level of the ratio between contradictory requirements, corresponding to the components of the criterion. The proposed approach for solving the problem of mathematical programming with not clearly defined parameters has the following benefits. Complex criterion has a distinct meaning and the corresponding computational procedure is simple. The implementation of its first stage is ensured by a traditional set of tools of determined optimization. The problem of the second stage when using standard membership functions, as a rule, comes down to the problem of quadratic programming. The account of theoretical material of the work is accompanied by examples.

Проведено короткий аналіз традиційних методів розв’язання нечітких задач математичного програмування. Виявлені недоліки відомих підходів. Розв’язання задачі досягається з використанням двоетапної процедури. Спочатку розв’язується оптимізаційна задача, яка породжується початковою задачею при заміні нечітких параметрів їх модальними значеннями. Потім відшукується чітке рішення, що забезпечує максимальну компактність нечіткого значення цільової функції і мінімально ухиляється від модального. Виклад теоретичного матеріалу супроводжується прикладами.