Mathematical Models for Decision Making with Multiple Perspectives

Description

This book brings together, in a single volume, the fields of multicriteria decision making and multiobjective optimization that are traditionally covered separately. Both fields have in common the presence of multiple perspectives of looking at and evaluating decisions to be taken but they differ in the number of available alternatives. Multicriteria approaches deal with decision processes where a finite number of alternatives have to be evaluated while, in multiobjective optimization, this number is infinite and the space of alternatives continuous. This book is written for students of applied mathematics, engineering, and economics and management, with no assumed previous knowledge on the subject, as well as for practitioners in industry looking for techniques to support decision making. The mathematical formalism is very low, so that all materials are accessible to most readers. Nonetheless, a rich bibliography allows interested readers to access more technical literature. The textbook is organized in eleven chapters, each corresponding to a class of about two hours. A comprehensive set of examples is presented, allowing for a didactic approach when presenting the methodologies. Each chapter ends with exercises that are designed to develop problem-solving skills and to promote concepts retention. Cover Title Page Copyright Page Preface Table of Contents 1. Single Criterion Decision Making 1.1 Introduction 1.2 Single Criterion Decision Making Methodologies 1.2.1 Decision Making in an Uncertainty Context 1.2.2 Decision Making in an Risk Context 1.3 Final Remarks 1.4 Proposed Exercises 2. Sequential Decisions and Introduction to Utility Theory 2.1 Introduction 2.2 Sequential Decision Making and Decision Trees 2.2.1 Drawing the Decision Tree 2.2.2 Assessing the Leaves 2.2.3 Assessing the Nodes 2.2.4 Some Comments about Sequential Decisions 2.3 Utility Theory 2.3.1 Probability Equivalence Method 2.3.2 Certainty Equivalence Method 2.3.3 Types of Utility Functions 2.3.4 Final Remarks Concerning Utility Theory 2.4 Proposed Exercises 3. Simple Multi-Attribute Rating Technique – SMART 3.1 Introduction 3.2 The Method 3.2.1 Value Trees 3.2.2 Alternatives’ Assessment 3.2.3 Value Functions 3.2.4 Dominance Analysis 3.2.5 Assigning Weights to the Criteria – Swing Weight 3.2.6 Calculate the Weighted Sum for All Alternatives 3.2.7 Sensitivity Analysis 3.3 Final Remarks 3.4 Proposed Exercises 4. The ELECTRE Methods 4.1 Introduction 4.2 ELECTRE I 4.2.1 Outranking Relation 4.2.2 Concordance Matrix 4.2.3 Discordance and Veto 4.2.4 The Nucleus 4.3 Electre III 4.3.1 Preference and Indifference Relations 4.3.2 Outranking Relation 4.3.3 Concordance Matrix 4.3.4 Discordance Matrices 4.3.5 Credibility Matrix 4.3.6 Alternatives Ranking 4.4 Final Remarks 4.5 Proposed Exercises 5. Analytic Hierarchy Process 5.1 Introduction 5.2 The Method 5.2.1 Pairwise Comparisons, Judgement Matrix and Priority Vector 5.2.2 Consistency of Judgement Matrices. Consistency Index. Random Consistency Index. Consistency Ratio 5.2.3 Concerning the Alternatives 5.2.4 Non-Consistency 5.2.5 Integrating the Problem’s Hierarchic Levels 5.2.6 More Complex Hierarchies 5.3 Final Remarks 5.4 Exercises 6. TOPSIS and Fuzzy TOPSIS 6.1 Introduction 6.2 The Method 6.3 Some of the Method’s Limitations 6.3.1 Normalization Step 6.3.2 Weights 6.3.3 Rank Reversal 6.4 Fuzzy TOPSIS 6.4.1 Fuzzy Sets and Fuzzy Numbers 6.4.2 The Method 6.4.3 Fuzzy TOPSIS for Group Decision Making 6.5 Final Remarks 6.6 Proposed Exercises 7. Multi-Objective Linear Programming 7.1 Introduction 7.2 Efficient Solutions and Objective Functions Space 7.3 Weakly Efficient Solution 7.4 Payoff Table, and Ideal and Nadir Solutions 7.5 Final Remarks 7.6 Exercises 7.6.1 Solved Exercises 7.6.2 Proposed Exercises 8. Lexicographic and ε-Constraint Methods 8.1 Introduction 8.2 The Lexicographic Method 8.2.1 The Algorithm 8.2.2 The Lexicographic Method as a Tool to Identify Multiple Optimal Solutions 8.2.3 Final Remarks 8.3 ε-Constraint Method 8.3.1 Infeasibility 8.3.2 Weak Efficiency 8.3.3 How to Approximate the Pareto Front using ε-Constraint Method? 8.3.4 Final Remarks 8.4 Exercises 8.4.1 Solved Exercises 8.4.2 Proposed Exercises 9. Weighted Sum and Distance Minimization Methods 9.1 Introduction 9.2 Weighted Sum Model 9.2.1 Definition of a Weighted Sum Objective Function 9.2.2 Dealing with Functions in Different Scales 9.2.3 Final Remarks 9.3 Distance Minimization Model 9.3.1 Distance Minimization to the Ideal Solution 9.3.2 L∞-Distance Minimization to the Ideal Solution 9.3.3 L2-Distance Minimization to the Iideal Solution 9.3.4 Final Remarks 9.4 Final Remarks 9.5 Exercises 9.5.1 Solved Exercises 9.5.2 Proposed Eexercises 10. Interactive Methods 10.1 Introduction 10.2 The Step Method (STEM) 10.2.1 Augmented Weighted Tchebychev Distance 10.2.2 Iterative Process 10.2.3 Final Rremarks 10.3 Other Methods 10.3.1 Zionts-Wallenius Method 10.3.2 Pareto Race 10.3.3 NAUTILUS 10.4 Final Remarks 10.5 Exercises 10.5.1 Solved Exercises 10.5.2 Proposed Exercises Appendix A Models in GAMS 11. Goal Programming 11.1 Introduction 11.2 Goals and the Utopian Set 11.2.1 What if the Utopian Set Does Not Intersect the Feasible Region? 11.2.2 What if the Utopian Set is an Empty Set? 11.3 Goal Programming Models 11.3.1 Preemptive GP Model 11.3.2 The Archimedean GP Model 11.3.3 A GP Model Combining Preemptive Priorities and Weighting 11.3.4 The Tchebychev GP Model 11.4 Final Remarks 11.5 Proposed Exercises Appendix B: Definitions A.1 Multi-Criteria Methods A.1.1 An Example A.2 Multi-Objective Methods Index