Evaluating solutions and solution sets under multiple objectives

In this study we address evaluating solutions and solution sets that are defined by multiple objectives based on a function. Although any function can be used, we focus on mostly weighted Tchebycheff functions that can be used for a variety of purposes when multiple objectives are considered. One such use is to approximate a decision maker’s preferences with a Tchebycheff utility function. Different solutions can be evaluated in terms of expected utility conditional on weight values. Another possible use is to evaluate a set of solutions that approximate a Pareto set. It is not straightforward to find the Pareto set, especially for large-size multi-objective combinatorial optimization problems. To measure the representation quality of approximate Pareto sets and to compare such sets with each other, there are some performance indicators such as the hypervolume measure, the ε indicator, and the integrated preference functional (IPF) measure. A Tchebycheff function based IPF measure can be used to estimate how well a set of solutions represents the Pareto set. We develop the necessary theory to practically evaluate solutions and solution sets. We develop a general algorithm and demonstrate it for two, three, and four objectives.