Introduction to optimization-based decision making

1. First Notes on Optimization for Decision Support. 1.1. Introduction. 1.2. First Steps. 1.3. Introducing Proportionality. 1.4. A Non-Proportional Instance. 1.5. An Enlarged and Non-Proportional Instance. 1.6. Concluding Remarks. 2. Linear Algebra. 2.1. Introduction. 2.2. Gauss Elimination on the Linear System. 2.3. Gauss Elimination with the Augmented Matrix. 2.4. Gauss-Jordan and the Inverse Matrix. 2.5. Cramer’s Rule and Determinants. 2.6. Concluding Remarks. 3. Linear Programming Basics. 3.1. Introduction. 3.2. Graphical Approach. 3.3. Algebraic Form. 3.4. Tableau Form. 3.5. Matrix Form. 3.6. Updating the Inverse Matrix. 3.7. Concluding Remarks. 4. Duality. 4.1. Introduction. 4.2. Primal-Dual Transformations. 4.3. Dual Simplex Method. 4.4. Duality Properties. 4.5. Duality and Economic Interpretation. 4.6. A First Approach to Optimality Analysis. 4.7. Concluding Remarks. 5. Calculus Optimization. 5.1. Introduction. 5.2. Constrained Optimization with Lagrange Multipliers. 5.3. Generalization of the Constrained Optimization Case. 5.4. Lagrange Multipliers for the Furniture Factory Problem. 5.5. Concluding Remarks. 6. Optimality Analysis. 6.1. Introduction. 6.2. Revising LP Simplex. 6.3. Sensitivity Analysis. 6.4. Parametric Analysis. 6.5. Concluding Remarks. 7. Integer Linear Programming. 7.1. Introduction. 7.2. Solving Integer Linear Programming Problems. 7.3. Modeling with Binary Variables. 7.4. Solving Binary Integer Programming Problems. 7.5. Concluding Remarks. 8. Game Theory. 8.1. Introduction. 8.2. Constant-Sum Game. 8.3. Zero-Sum Game. 8.4. Mixed Strategies - LP Approach. 8.5. Dominant Strategies. 8.6. Concluding Remarks. 9. Decision Making Under Uncertainty. 9.1. Introduction. 9.2. Multiple Criteria and Decision Maker Values. 9.3. Capacity Expansion for the Furniture Factory. 9.4. A Comparison Analysis. 9.5. Concluding Remarks. 10. Robust Optimization. 10.1. Introduction. 10.2. Notes on Stochastic Programming. 10.3. Robustness Promotion on Models and Solutions. 10.4. Models Generalization onto Robust Optimization. 10.5. Concluding Remarks. Selected References

The large and complex challenges the world is facing, the growing prevalence of huge data sets, and the new and developing ways for addressing them (artificial intelligence, data science, machine learning, etc.), means it is increasingly vital that academics and professionals from across disciplines have a basic understanding of the mathematical underpinnings of effective, optimized decision-making. Without it, decision makers risk being overtaken by those who better understand the models and methods, that can best inform strategic and tactical decisions.

Introduction to Optimization-Based Decision-Making provides an elementary and self-contained introduction to the basic concepts involved in making decisions in an optimization-based environment. The mathematical level of the text is directed to the post-secondary reader, or university students in the initial years. The prerequisites are therefore minimal, and necessary mathematical tools are provided as needed. This lean approach is complemented with a problem-based orientation and a methodology of generalization/reduction. In this way, the book can be useful for students from STEM fields, economics and enterprise sciences, social sciences and humanities, as well as for the general reader interested in multi/trans-disciplinary approaches.