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Multicriteria Optimization and Decision Making: Principles, Algorithms and Case Studies

Michael Emmerich, André Deutz

TL;DR

The work provides a comprehensive, foundation-first treatment of Multicriteria Optimization and Decision Analysis (MODA), linking order theory, Pareto dominance, and landscape analysis to practical solution methods. It starts from formal problem definitions in mathematical programming, traverses problem difficulty, and builds toward both scalarization techniques and the geometry of Pareto fronts. It then reviews algorithmic strategies for Pareto optimization, including deterministic and exact methods, as well as state-of-the-art evolutionary approaches, and closes with methods for evaluating and reasoning about Pareto sets in high-dimensional and many-objective contexts. The combined emphasis on theoretical underpinnings, algorithmic frameworks, and practical decision-support tools aims to equip researchers and practitioners to model, compute, and reason about trade-offs in complex, real-world systems. The material highlights the trade-offs between achieving Pareto-optimal fronts and the computational considerations inherent in solving large-scale, multiobjective problems across engineering, economics, and beyond.

Abstract

Real-world decision and optimization problems, often involve constraints and conflicting criteria. For example, choosing a travel method must balance speed, cost, environmental footprint, and convenience. Similarly, designing an industrial process must consider safety, environmental impact, and cost efficiency. Ideal solutions where all objectives are optimally met are rare; instead, we seek good compromises and aim to avoid lose-lose scenarios. Multicriteria optimization offers computational techniques to compute Pareto optimal solutions, aiding decision analysis and decision making. This reader offers an introduction to this topic and has been developed on the basis of the revised edition of the reader for the MSc computer science course "Multicriteria Optimization and Decision Analysis" at the Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands. This course was taught annually by the first author from 2007 to 2023 as a single semester course with lectures and practicals. Our aim was to make the material accessible to MSc students who do not study mathematics as their core discipline by introducing basic numerical analysis concepts when necessary and providing numerical examples for interesting cases. The introduction is organized in a unique didactic manner developed by the authors, starting from more simple concepts such as linear programming and single-point methods, and advancing from these to more difficult concepts such as optimality conditions for nonlinear optimization and set-oriented solution algorithms. Besides, we focus on the mathematical modeling and foundations rather than on specific algorithms, though not excluding the discussion of some representative examples of solution algorithms.

Multicriteria Optimization and Decision Making: Principles, Algorithms and Case Studies

TL;DR

The work provides a comprehensive, foundation-first treatment of Multicriteria Optimization and Decision Analysis (MODA), linking order theory, Pareto dominance, and landscape analysis to practical solution methods. It starts from formal problem definitions in mathematical programming, traverses problem difficulty, and builds toward both scalarization techniques and the geometry of Pareto fronts. It then reviews algorithmic strategies for Pareto optimization, including deterministic and exact methods, as well as state-of-the-art evolutionary approaches, and closes with methods for evaluating and reasoning about Pareto sets in high-dimensional and many-objective contexts. The combined emphasis on theoretical underpinnings, algorithmic frameworks, and practical decision-support tools aims to equip researchers and practitioners to model, compute, and reason about trade-offs in complex, real-world systems. The material highlights the trade-offs between achieving Pareto-optimal fronts and the computational considerations inherent in solving large-scale, multiobjective problems across engineering, economics, and beyond.

Abstract

Real-world decision and optimization problems, often involve constraints and conflicting criteria. For example, choosing a travel method must balance speed, cost, environmental footprint, and convenience. Similarly, designing an industrial process must consider safety, environmental impact, and cost efficiency. Ideal solutions where all objectives are optimally met are rare; instead, we seek good compromises and aim to avoid lose-lose scenarios. Multicriteria optimization offers computational techniques to compute Pareto optimal solutions, aiding decision analysis and decision making. This reader offers an introduction to this topic and has been developed on the basis of the revised edition of the reader for the MSc computer science course "Multicriteria Optimization and Decision Analysis" at the Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands. This course was taught annually by the first author from 2007 to 2023 as a single semester course with lectures and practicals. Our aim was to make the material accessible to MSc students who do not study mathematics as their core discipline by introducing basic numerical analysis concepts when necessary and providing numerical examples for interesting cases. The introduction is organized in a unique didactic manner developed by the authors, starting from more simple concepts such as linear programming and single-point methods, and advancing from these to more difficult concepts such as optimality conditions for nonlinear optimization and set-oriented solution algorithms. Besides, we focus on the mathematical modeling and foundations rather than on specific algorithms, though not excluding the discussion of some representative examples of solution algorithms.
Paper Structure (84 sections, 24 theorems, 91 equations, 42 figures, 2 tables, 6 algorithms)

This paper contains 84 sections, 24 theorems, 91 equations, 42 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Viewing mulicriteria optimization as a task in system design and analysis
  3. Formal Problem Definitions in Mathema-tical Programming
  4. Other Problem Classes in Optimization
  5. Linear Programming
  6. Geometrical Aspects of Linear Programming
  7. Graphical Solution of LP
  8. Linearization Techniques
  9. Min-max problems.
  10. Products of binary variables.
  11. Disjunctive constraints.
  12. No-subtour constraints.
  13. Multiobjective Optimization
  14. Problem Difficulty in Optimization
  15. Problem Difficulty in Continuous Optimization
  16. ...and 69 more sections

Key Result

Lemma 1

The feasible subset in linear programming is a convex subset of $\mathbb{R}^n$.

Figures (42)

  • Figure 1: Different tasks in systems analysis.
  • Figure 2: Graphical solution of an LP. The feasible region is indicated in yellow. The isoheightlines (levels) of the objective funtion are indicated in red dashed lines.
  • Figure 3: Examples of continuous optimization problems.
  • Figure 4: The Hasse Diagram for the set of all non-empty subsets partially ordered by means of $\subseteq$.
  • Figure 5: Different orders over the set $X=\{a,b,c,d\}$
  • ...and 37 more figures

Theorems & Definitions (89)

  • Lemma 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6: NP hard
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 79 more