Optimization Modelling Using R

299 Pages·2022·9.079 MB·English

by  Timothy R. Anderson

Optimization
Modeling Using R
Chapman & Hall/CRC Series in Operations
Research
Series Editors:
Malgorzata Sterna, Bo Chen, Michel Gendreau, and Edmund Burke
Rational Queueing
Refael Hassin
Introduction to Theory of Optimization in Euclidean Space
Samia Challal
Handbook of The Shapley Value
Encarnación Algaba, Vito Fragnelli and Joaquín Sánchez-Soriano
Advanced Studies in Multi-Criteria Decision Making
Sarah Ben Amor, João Luís de Miranda, Emel Aktas, and Adiel Teixeira
de Almeida
Handbook of Military and Defense Operations Research
Natalie Scala, and James P. Howard II
Understanding Analytic Hierarchy Process
Konrad Kulakowski
Introduction to Optimization-Based Decision Making
João Luís de Miranda
Applied Mathematics with Open-source Software
Operational Research Problems with Python and R
Vince Knight, Geraint Palmer
Optimization Modeling Using R
Timothy R. Anderson
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Chapman--HallCRC-Series-in-Operations-Research/book-series/CRCOPSRES
Optimization
Modeling Using R
Timothy R. Anderson
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ISBN: 978-0-367-50789-3 (hbk)
ISBN: 978-1-032-29076-8 (pbk)
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DOI: 10.1201/9781003051251
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Contents
List of Figures xiii
List of Tables xv
Preface xix
Author xxiii
1 Introduction 1
1.1 What is Operations Research . . . . . . . . . . . . . . . . . . . 1
1.2 Purpose of this Book . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Range of Operations Research Techniques . . . . . . . . . . . . 2
1.4 Relationship between Operations Research and Analytics . . . 3
1.5 Importance of Optimization . . . . . . . . . . . . . . . . . . . . 3
1.6 Why R? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 Conventions Used in this Book . . . . . . . . . . . . . . . . . . 6
2 Introduction to Linear Programming 7
2.1 What Is Linear Programming . . . . . . . . . . . . . . . . . . . 7
2.2 Two Variable Base Case . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Graphically Solving a Linear Program . . . . . . . . . . . . . . 9
2.4 Implementing and Solving with ompr . . . . . . . . . . . . . . . 14
2.4.1 Preparing to Implement the Linear Program . . . . . . 14
2.4.2 Implementing the Base Case with Piping . . . . . . . . 20
2.5 Adding a Third Product (Variable) . . . . . . . . . . . . . . . . 22
2.5.1 Three Variable Base Case Formulation . . . . . . . . . . 22
2.5.2 Three Variable Base Case Implementation . . . . . . . . 23
2.5.3 Three Variable Case Results and Interpretation . . . . . 24
2.6 Linear Programming Special Cases . . . . . . . . . . . . . . . . 24
2.6.1 Case 1: No Feasible Solution . . . . . . . . . . . . . . . 25
2.6.2 Case 2: Multiple Optima. . . . . . . . . . . . . . . . . . 26
2.6.3 Case 3: Redundant Constraint . . . . . . . . . . . . . . 30
2.6.4 Case 4: Unbounded Solution . . . . . . . . . . . . . . . 31
2.7 Abstracting the Production Planning Model . . . . . . . . . . . 33
2.8 Methods of Solving Linear Programs . . . . . . . . . . . . . . . 34
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vii
viii Contents
3 More Linear Programming Models 39
3.1 Types of LP Models . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 The Algebraic Model . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Tips and Conventions for Algebraic Models . . . . . . . 40
3.2.2 Building the Generalized Model in R . . . . . . . . . . . 41
3.2.3 Examining the Results . . . . . . . . . . . . . . . . . . . 44
3.2.4 Changing the Model . . . . . . . . . . . . . . . . . . . . 46
3.3 Common Linear Programming Applications . . . . . . . . . . . 47
3.3.1 Blending Problems . . . . . . . . . . . . . . . . . . . . . 47
3.4 Allocation Models . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.1 Covering Models . . . . . . . . . . . . . . . . . . . . . . 51
3.4.2 Transportation Models . . . . . . . . . . . . . . . . . . . 53
3.4.3 Transshipment Models . . . . . . . . . . . . . . . . . . . 57
3.4.4 Production and Inventory Planning . . . . . . . . . . . 59
3.4.5 Standard Form . . . . . . . . . . . . . . . . . . . . . . . 60
3.5 Vector and Matrix Forms of LPs . . . . . . . . . . . . . . . . . 63
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Sensitivity Analysis 71
4.1 Base Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Shadow Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Extraction and Interpretation . . . . . . . . . . . . . . . 72
4.2.2 Example of Adding an Hour to Assembly . . . . . . . . 74
4.2.3 Shadow Prices of Underutilized Resources . . . . . . . . 75
4.3 Reduced Costs of Variables . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Reduced Cost of Ants . . . . . . . . . . . . . . . . . . . 77
4.3.2 Reduced Price of Bats . . . . . . . . . . . . . . . . . . . 79
4.4 Using Sensitivity Analysis to Evaluate a New Product . . . . . 81
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Data Envelopment Analysis 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Creating the Data . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Graphical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 The Linear Programs for DEA . . . . . . . . . . . . . . . . . . 90
5.4.1 An Explicit Linear Program for DEA . . . . . . . . . . 90
5.4.2 A Generalized Linear Program for DEA . . . . . . . . . 91
5.5 Creating the LP – The Algebraic Approach . . . . . . . . . . . 93
5.6 Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.7 Multiple Inputs and Multiple Outputs . . . . . . . . . . . . . . 105
5.8 Extracting Multiplier Weights from Sensitivity Analysis . . . . 113
5.9 Slack Maximization. . . . . . . . . . . . . . . . . . . . . . . . . 116
5.10 DEA Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.11 DEA Model Building . . . . . . . . . . . . . . . . . . . . . . . . 119
5.11.1 Selection of Inputs and Outputs . . . . . . . . . . . . . 120
Contents ix
5.11.2 Model Choices . . . . . . . . . . . . . . . . . . . . . . . 121
5.11.3 Application Area Expertise . . . . . . . . . . . . . . . . 121
5.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 Mixed Integer Optimization 125
6.1 Example of Minor Integrality Impact . . . . . . . . . . . . . . . 125
6.2 Example of Major Integality Impact . . . . . . . . . . . . . . . 128
6.3 The Branch and Bound Algorithm . . . . . . . . . . . . . . . . 130
6.3.1 The LP Relaxation . . . . . . . . . . . . . . . . . . . . . 131
6.3.2 Subproblem I . . . . . . . . . . . . . . . . . . . . . . . . 132
6.3.3 Subproblem III . . . . . . . . . . . . . . . . . . . . . . . 133
6.3.4 Subproblem IV . . . . . . . . . . . . . . . . . . . . . . . 135
6.3.5 Subproblem V . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.6 Subproblem VI . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.7 Subproblem VII . . . . . . . . . . . . . . . . . . . . . . 138
6.3.8 Subproblem VIII . . . . . . . . . . . . . . . . . . . . . . 139
6.3.9 Subproblem II . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Computational Complexity . . . . . . . . . . . . . . . . . . . . 141
6.4.1 Full Enumeration . . . . . . . . . . . . . . . . . . . . . . 142
6.5 Binary Variables and Logical Relations . . . . . . . . . . . . . . 143
6.6 Fixed Charge Models . . . . . . . . . . . . . . . . . . . . . . . . 146
6.6.1 Fixed Charge Example-Introduction . . . . . . . . . . . 147
6.6.2 Linking Constraints with “Big M” . . . . . . . . . . . . 151
6.6.3 Fixed Charge Implementation . . . . . . . . . . . . . . . 154
6.7 Model Results and Interpretation . . . . . . . . . . . . . . . . . 155
7 More Integer Programming Models 157
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Revisiting the Warehouse Location Problem . . . . . . . . . . . 157
7.2.1 Implementing the Warehouse Model . . . . . . . . . . . 159
7.2.2 Solving the Warehouse Location Problem . . . . . . . . 165
7.2.3 Warehouse Discussion . . . . . . . . . . . . . . . . . . . 172
7.3 Solving MIPs with Different Solvers . . . . . . . . . . . . . . . 173
7.3.1 Performance of glpk . . . . . . . . . . . . . . . . . . . . 173
7.3.2 Performance of symphony . . . . . . . . . . . . . . . . . . 174
7.3.3 Performance of lpsolve . . . . . . . . . . . . . . . . . . 177
7.3.4 Performance of gurobi . . . . . . . . . . . . . . . . . . . 178
7.3.5 Comparing Results across Solvers . . . . . . . . . . . . . 179
7.3.6 Popularity of LP Solvers . . . . . . . . . . . . . . . . . . 181
7.4 Solving Sudoku Puzzles using Optimization . . . . . . . . . . . 184
7.4.1 Introduction to Sudoku and Optimization . . . . . . . . 184
7.4.2 Formulating the Sudoku Problem . . . . . . . . . . . . . 185
7.4.3 Implementing Sudoku in ompr . . . . . . . . . . . . . . 187
7.4.4 Sudoku Discussion . . . . . . . . . . . . . . . . . . . . . 192