THE LINEAR-FRACTIONAL PROGRAMMING PROBLEM UNDER UNCERTAINTY CONDITIONS

Abstract

This paper addresses the problem of linear-fractional programming under uncertainty. The uncertainty here is understood as the ambiguity of the coefficients’ values in the optimized functional. We give two mathematical formulations of the problem. In the first one, the uncertainty refers to the numerator: there are several sets of objective function coefficients, each coefficient can determine the numerator of the problem’s criterion at the stage of its solution implementation. The uncertainty in the second formulation refers to the denominator of the functional. We propose several compromise criteria for evaluating solutions to the problem we consider. We study the following two criterions in detail: 1) finding a compromise solution in which the deviation of the values of the partial functionals from their optimal values is within the specified limits; 2) finding a compromise solution according to the criterion of minimizing the total weighted excess of the values of partial functionals in relation to the specified feasible deviations from their optimal values (the values of concessions). We formulate an auxiliary linear programming problem to find a compromise solution to the linear-fractional programming problems by these two criteria. The constraints of the auxiliary problem depend on the optimization direction in the original problem. We carried out a series of experiments of four types to study the properties of the problem. The purposes of the experiments were: 1) to study how changes in the values of the specified feasible deviations of partial objective functions impact the values of actual deviations and the values of concessions; 2) to study how changes in the expert weights of partial objective functions impact the values of actual deviations and the values of concessions for the compromise solutions we obtain. We propose in this work the schemes of experiments and present their results in graphical form. We have found that the obtained relations depend on the optimization direction in the original problem.
 Keywords: optimization, uncertainty, convolution, linear-fractional programming, linear programming problem, compromise solution