Interactive navigation of multiple convex patches

Abstract

Among the approaches to multi-criteria decision making, Pareto navigation is a powerful, interactive tool that has been successfully applied to a variety of real-world problems with continuous decision variables, including chemical process design, drug manufacturing, logistical vehicle routing problems, and radiotherapy treatment planning. However, many real-life problems are formulated using both continuous and binary decision variables. In this work, we introduce patch navigation as an algorithmic concept that extends Pareto navigation to this type of problem where the number of binary variables is relatively small. The underlying idea is the navigation across a finite set of individual, convex fronts each associated with a specific configuration of the binary variables (patches). We show how the user interactions employed in current Pareto front navigation, namely selection and restriction, can be adopted to handle multiple patches. These routines enable the decision maker (DM) to change the solution in small increments while controlling the related trade-offs. We also describe additional, patch-specific routines that enable the DM to consider only an individually chosen subset of patches in the navigation. To illustrate patch navigation, and to demonstrate its usefulness for real-life problems, we present numerical examples of patch navigation along with an application motivated by radiotherapy planning.