Generating representative sets for multiobjective discrete optimization problems with specified coverage errors

Abstract

We present a new approach to generate representations with a coverage error quality guarantee for multiobjective discrete optimization problems with any number of objectives. Our method is based on an earlier exact algorithm that finds the entire nondominated set using epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-constraint scalarizations. The representation adaptation requires the search to be conducted over a p-dimensional parameter space instead of the (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)$$\end{document}-dimensional one of the exact version. The algorithm uses rectangles as search elements and for each rectangle, two-stage mathematical programs are solved to obtain efficient solutions. The representation algorithm implements a modified search procedure and is designed to eliminate a rectangle if it can be verified that it is not of interest given a particular coverage error requirement. Since computing the coverage error is a computationally demanding task, we propose a method to compute an upper bound on this quantity in polynomial time. The algorithm is tested on multiobjective knapsack and assignment problem instances with different error tolerance levels. We observe that our representation algorithm provides significant savings in computational effort even with relatively low levels of coverage error tolerance values for problems with three objective functions. Moreover, computational effort decreases almost linearly when coverage error tolerance increases. This makes it possible to obtain good quality representations for larger problem instances. An analysis of anytime performance on two selected problem instances demonstrates that the algorithm puts together a diverse representation starting from the early iterations.