Multiobjective Combinatorial Optimization: Foundations, Theory, and Methods
Abstract
This tutorial presents the foundations, theory, and methods of multiobjective integer programming, with a particular focus on multiobjective combinatorial optimization (MOCO). Many real-world optimization problems are inherently combinatorial in nature and involve multiple, often conflicting, objectives. MOCO has been widely used to model these complex decision problems across various domains such as transportation, logistics, finance, energy, and healthcare. Although the feasible set is typically finite in MOCO problems, these solutions are not always explicitly available to the decision makers. Adding to this complexity, there is rarely a single solution that optimizes all objectives simultaneously in multiobjective optimization problems. This tutorial provides a structured framework for students, researchers, and practitioners to approach and solve these complex decision problems. We first define key terminology, describe the main characteristics of MOCO problems, and discuss their scalarization. We then explore the advanced methods and exact algorithms to generate all nondominated points, for which an improvement in one objective cannot be made without sacrificing performance in another. Because these algorithms become intractable in real-world problem settings with the increase in the number of nondominated points, we also discuss the methods that generate a representative set of solutions with a prespecified level of quality or find preferred solutions. The tutorial provides a broad and accessible overview of existing methods, supported by illustrative examples, discussions, figures, and comprehensive references that clarify their main ideas, strengths, and limitations.
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