Models and Solutions of Resource Allocation Problems based on Integer Linear and Nonlinear ...

108 Pages·2016·1.52 MB·English

by  Dimitri Thomopulos

Alma Mater Studiorum - Universita` di Bologna
DOTTORATO DI RICERCA IN
Automatica e Ricerca Operativa
Ciclo XXVIII
Settoreconcorsualediafferenza: 01/A6-RICERCAOPERATIVA
Settorescientificodisciplinare: MAT/09-RICERCAOPERATIVA
Models and Solutions of Resource
Allocation Problems based on Integer
Linear and Nonlinear Programming
Presentata da: Dimitri Thomopulos
Coordinatore Dottorato Relatore
Prof. Daniele Vigo Prof. Enrico Malaguti
Correlatore
Prof. Andrea Lodi
Esame finale anno 2016

Contents
1 Introduction 1
1.1 On Resource Allocation Problems . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Applied Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis Methodological Outline . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Two-Dimensional Guillotine Cutting Problem . . . . . . . . . . . 6
1.4.2 Mid-Term Hydro Scheduling Problem . . . . . . . . . . . . . . . 7
2 Two-Dimensional Guillotine Cutting Problem 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Families of cutting problems. . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Structure of general guillotine cuts. . . . . . . . . . . . . . . . . . 13
2.1.3 Literature review. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.4 Contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 MIP modeling of guillotine cuts . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Definition of the cut position set . . . . . . . . . . . . . . . . . . 20
2.2.2 Model extensions: Cutting Stock problem and Strip Packing
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 An effective solution procedure for the PP-G2KP Model . . . . . . . . . 27
2.3.1 Variable pricing procedures . . . . . . . . . . . . . . . . . . . . . 27
2.4 Computational Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Lower bound (feasible solution) computation . . . . . . . . . . . 32
2.4.2 Iterative Variable Pricing . . . . . . . . . . . . . . . . . . . . . . 34
2.4.3 Models size and reductions . . . . . . . . . . . . . . . . . . . . . 36
2.4.4 Overall solution procedure . . . . . . . . . . . . . . . . . . . . . . 41
2.4.5 Comparison with state-of-the-art approaches . . . . . . . . . . . 46
2.4.6 Relevance of guillotine cuts . . . . . . . . . . . . . . . . . . . . . 48
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Mid-Term Hydro Scheduling Problem 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Mid-term hydro scheduling . . . . . . . . . . . . . . . . . . . . . 54
3.1.2 Choice of the objective function . . . . . . . . . . . . . . . . . . . 55
3.2 Decomposition algorithm for nonlinear (CCP) . . . . . . . . . . . . . . . 56
3.2.1 Overview of the approach . . . . . . . . . . . . . . . . . . . . . . 58
3.2.2 Separation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.3 Termination of the Branch-and-Cut algorithm . . . . . . . . . . . 62
3.2.4 Comparison with generalized Benders cuts . . . . . . . . . . . . . 64
3.3 (CCP) for mid-term hydro scheduling . . . . . . . . . . . . . . . . . . . 66
3.3.1 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1.1 Electricity generation function . . . . . . . . . . . . . . 68
3.3.1.2 Demand and price function . . . . . . . . . . . . . . . . 69
3.3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.2 Computational performance . . . . . . . . . . . . . . . . . . . . . 73
3.4.3 Quadratic electricity generation function . . . . . . . . . . . . . . 80
3.4.4 The effect of α on the profit . . . . . . . . . . . . . . . . . . . . . 84
3.4.5 Other solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography 91
Keywords
• Two-dimensional Cutting Problems
• Guillotine Knapsack Problems
• Chance Constrained Programming
• Exact Algorithms
• Integer Linear Programming
• Nonlinear Programming
• Pricing
• Stochastic Programming
• Computational Experiments

Ai miei Genitori, mio Fratello, i miei Amici, i miei Nonni e in
particolar modo a mio Nonno Dimitris

Acknowledgements
I want to express my deeply-felt thanks to my thesis advisors, Professors Enrico
Malaguti and Andrea Lodi, for their warm encouragement, thoughtful guidance.
I also want to express my gratitude to Dr.Valentina Cacchiani for her continuous
support and sound advice.
Lastly, I wish to thank Professor Daniele Vigo and Professor Giacomo Nannicini.