Optimization Models by Fabio Schoen

Optimization Models by Fabio Schoen is an e-book on modeling techniques for Operations Research. A new book on the art of modeling optimization problems. This text was used in the “Optimization and Data Science for Management” course, Master degree in Management Engineering at the University of Florence.

• Free version available on-line.

• Printed Book available on Amazon.

Abstract

Optimization is the tool for good decision making or, to be more precise, for good decision support. In order to be able to suggest rational and effective decisions the Operations Research scientific community have offered courses, developed methods, analyzed theoretical aspects of optimization. However I feel there is still a wide gap which separates academic research from day to day adoption. And, in my opinion, part of this is due to the lack of sufficient modeling skills. If we cannot model a problem, no matter how sophisticated our optimization algorithm are, we will not be able to solve it nor suggest any decision. We need to communicate our problem to a solver. And to do this we need a deep knowledge of models.

Modeling is a creative skill, which is best learned by doing; however, especially for more advanced models involving integer variables or graph theory, modeling practice should be paired with a deep theoretical knowledge, if we wish to be able to solve large scale problems.

This book has been written with this in mind. It was born from many years of teaching modeling mainly to Management and Computer Science Engineers. The book has an on line web based version, which is freely available and will be the subject of future modifications and additions.

Table of Contents

• 1. Preface

• 2. Structured linear optimization models

• 3. Product Mix Models

  • 3.1. Dual of the product mix problem

• 4. Blending Models

  • 4.1. Dual of blending problems

  • 4.2. Dual of the diet model

• 5. Multi Period Models

• 6. Network flow models

  • 6.1. Minimum cost flow

  • 6.2. Applications of minimum cost flow models

• 7. Shortest (Minimum Cost) Path Problems

  • 7.1. The problem of arcs with negative cost

  • 7.2. Dual of the minimum cost path model

  • 7.3. Applications of the minimum cost path model

• 8. Maximum Flow

  • 8.1. Dual of the maximum flow problem

• 9. The transportation model

• 10. Assignment or bi-partite matching

• 11. Dynamic flows

• 12. Objectives in optimization models

  • 12.1. Feasibility problems (no objective function)

  • 12.2. Multiple Objectives

• 13. Some nonlinear problems that can be transformed into linear ones

  • 13.1. Minimax problems

  • 13.2. Absolute values in the objective

  • 13.3. Regression (or best approximation) models

    • 13.3.1. Minimax regression

    • 13.3.2. Absolute regression

    • 13.3.3. Quadratic (least squares) regression

  • 13.4. Rational Objectives

    • 13.4.1. Comparative efficiency evaluation

      • 13.4.1.1. Dual of the efficiency evaluation model

• 14. Constraints in linear optimization models

  • 14.1. Soft constraints

  • 14.2. Piece-wise linear convex cost functions

• 15. Uncertainty modeling: Stochastic optimization models

  • 15.1. Probabilistic constraints

  • 15.2. Stochastic linear optimization models

• 16. Mixed Integer Linear Models

  • 16.1. Introduction

  • 16.2. Modeling with integer variables

    • 16.2.1. Using binary variables in logical constraints

    • 16.2.2. Using binary variables to impose constraints

    • 16.2.3. Propositional logic and binary variables

      • 16.2.3.1. Normal forms

• 17. The fixed charge problem

• 18. Lot sizing

  • 18.1. Branch and cut - dynamical simplex method

  • 18.2. Advanced formulations for the lot sizing model

  • 18.3. Variants of production planning models

  • 18.4. Advanced lot sizing models

    • 18.4.1. Master Production Scheduling models

    • 18.4.2. Material Requirement Planning

• 19. Knapsack, Bin Packing and Plant Location problems

  • 19.1. Introduction

    • 19.1.1. Advanced modeling: cover inequalities

  • 19.2. Multiple knapsacks, Cutting Stock

  • 19.3. Bin packing - container optimization

    • 19.3.1. Comparison between the formulations

  • 19.4. Plant location

• 20. Set covering, packing, partition

• 21. Sequencing problems: the Traveling Salesperson Problem

  • 21.1. Sub-tour elimination constraints

• 22. Vehicle Routing

• 23. Arc routing

• 24. Quadratic Optimization Models

  • 24.1. Solving linear electrical networks

  • 24.2. Elementary portfolio optimization

  • 24.3. Support Vector Machines

  • 24.4. Quadratic Assignment

• 25. Appendix

  • 25.1. Basic optimization notions

  • 25.2. Graphs: basic definitions

• 26. Bibliographic References

• 27. Version history

• 28. Book Errata Corrige

Authors

• Fabio Schoen, University of Florence

Citation

If you wish to cite this work, please use

@book{schoen2024optimization,
  title     = {Optimization Models},
  author    = {Schoen, Fabio},
  year      = {2024},
  publisher = {Self-published},
  isbn      = {979-1221052633},
  note      = {417 pages}
}