Universal Method for Solving Optimization Problems Under the Conditions of Uncertainty in the Initial Data

Abstract

This paper proposes a method to solve a mathematical programming problem under the conditions of uncertainty in the original data.

The structural basis of the proposed method for solving optimization problems under the conditions of uncertainty is the function of criterion value distribution, which depends on the type of uncertainty and the values of the problem’s uncertain variables. In the case where independent variables are random values, this function then is the conventional theoretical-probabilistic density of the distribution of the random criterion value; if the variables are fuzzy numbers, it is then a membership function of the fuzzy criterion value.

The proposed method, for the case where uncertainty is described in the terms of a fuzzy set theory, is implemented using the following two-step procedure. In the first stage, using the membership functions of the fuzzy values of criterion parameters, the values for these parameters are set to be equal to the modal, which are fitted in the analytical expression for the objective function. The resulting deterministic problem is solved. The second stage implies solving the problem by minimizing the comprehensive criterion, which is built as follows. By using an analytical expression for the objective function, as well as the membership function of the problem’s fuzzy parameters, applying the rules for operations over fuzzy numbers, one finds a membership function of the criterion’s fuzzy value. Next, one calculates a measure of the compactness of the resulting membership function of the fuzzy value of the problem’s objective function whose numerical value defines the first component of the integrated criterion. The second component is the rate of deviation of the desired solution to the problem from the previously received modal one.

Absolutely similarly designed is the computational procedure for the case where uncertainty is described in the terms of a probability theory. Thus, the proposed method for solving optimization problems is universal in relation to the nature of the uncertainty in the original data. An important advantage of the proposed method is the ability to use it when solving any problem of mathematical programming under the conditions of fuzzily assigned original data, regardless of its nature, structure, and type.

Keywords: mathematical programming problem, uncertainty in the original data, universal solution method