Approaches to discrete optimization problems with interval objective function

В работе дается обзор подходов к решению задач дискретной оптимизации с интервальной целевой функцией. Эти подходы рассматриваются в общем контексте исследований оптимизационных задач с неопределенностями в постановках. Приводятся варианты концепций оптимальности решений для задач дискретной оптимизации с интервальной целевой функцией - робастные решения, множества решений, оптимальных по Парето, слабые и сильные оптимальные решения, объединенные множества решений и др. Оценивается предпочтительность выбора той или иной концепции оптимальности при решении задач и отмечаются ограничения для применения использующих их подходов Optimization problems with uncertainties in their input data have been investigated by many researchers in different directions. There are a lot of sources of the uncertainties in the input data for applied problems. Inaccurate measurements and variability of the parameters with time are some of such sources. The interval of possible values of uncertain parameter is the natural and the only possible way to represent the uncertainty for a wide share of applied problems. We consider discrete optimization problems with interval uncertainties in their objective functions. The purpose of the paper is to provide an overview of the investigations in this field. The overview is given in the overall context of the researches of optimization problems with uncertainties. We review the interval approaches for the discrete optimization problem with interval objective function. The approaches we consider operate with the interval values and are focused on obtaining possible solutions or certain sets of the solutions that are optimal according to some concepts of optimality that are used by the approaches. We consider the different concepts of optimality: robust solutions, the Pareto sets, weak and strong solutions, the united solution sets, the sets of possible approximate solutions that correspond to possible values of uncertain parameters. All the approaches we consider allow absence of information on probabilistic distribution on intervals of possible values of parameters, though some of them may use the information to evaluate the probabilities of possible solutions, the distribution on the interval of possible objective function values for the solutions, etc. We assess the possibilities and limitations of the considered approaches

Network Models for Multiobjective Discrete Optimization

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework represents these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest-path problem in an auxiliary network. We design techniques for exploiting network models in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions. Summary of Contribution: Multiobjective optimization has a long history of research with applications in several domains. Our paper provides an alternative modeling and solution approach for multiobjective discrete optimization problems by leveraging graphical structures. Specifically, we encode the decision space of a problem as a layered network and propose graph reduction operators to preserve only solutions whose image are part of the Pareto frontier. The nondominated solutions can then be extracted through shortest-path algorithms on such a network. Numerical results comparing our method with state-of-the-art approaches on several problem classes, including the knapsack, set covering, and the traveling salesperson problem (TSP), suggest orders-of-magnitude runtime speed-ups for exactly enumerating the Pareto frontier, especially when the number of objective functions grows.

Computational approach to solving the problem of optimizing the supply of raw materials and components at the enterprise

The results of modelling the problem of supply management of an enterprise using discrete optimization tools are presented. The formulation of the optimization problem of supply management and the found method for its solution are presented. Since there may be cases when the number of variables in the problem is large enough, an algorithm was developed that uses the decomposition of the problem as a solution. A numerical example of the application of the decompositional algorithm for optimizing supplies and comparison of the results using the direct algorithm are given.