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