Logistics: The Vehicle Routing Problem

Abstract

1 An Overview of Vehicle Routing Problems / P. Toth, D. Vigo 1 -- 1.2 Problem Definition and Basic Notation 5 -- 1.2.1 Capacitated and Distance-Constrained VRP 5 -- 1.2.2 VRP with Time Windows 8 -- 1.2.3 VRP with Backhauls 9 -- 1.2.4 VRP with Pickup and Delivery 10 -- 1.3 Basic Models for the VRP 11 -- 1.3.1 Vehicle Flow Models 11 -- 1.3.2 Extensions of Vehicle Flow Models 17 -- 1.3.3 Commodity Flow Models 19 -- 1.3.4 Set-Partitioning Models 21 -- 1.4 Test Instances for the CVRP and Other VRPs 22 -- I Capacitated Vehicle Routing Problem 27 -- 2 Branch-and-Bound Algorithms for the Capacitated VRP / P. Toth, D. Vigo 29 -- 2.2 Basic Relaxations 30 -- 2.2.1 Bounds Based on Assignment and Matching 30 -- 2.2.2 Bounds Based on Arborescences and Trees 32 -- 2.2.3 Comparison of the Basic Relaxations 33 -- 2.3 Better Relaxations 35 -- 2.3.1 Additive Bounds for ACVRP 35 -- 2.3.2 Further Lower Bounds for ACVRP 39 -- 2.3.3 Lagrangian Lower Bounds for SCVRP 40 -- 2.3.4 Lower Bounds from a Set-Partitiong Formulation 41 -- 2.3.5 Comparison of the Improved Lower Bounds 42 -- 2.4 Structure of the Branch-and-Bound Algorithms for CVRP 44 -- 2.4.1 Branching Schemes and Search Strategies 44 -- 2.4.2 Reduction, Dominance Rules, and Other Features 46 -- 2.4.3 Performance of the Branch-and-Bound Algorithms 47 -- 2.5 Distance-Constrained VRP 48 -- 3 Branch-and-Cut Algorithms for the Capacitated VRP / D. Naddef, G. Rinaldi 53 -- 3.1 Introduction and Two-Index Flow Model 53 -- 3.2 Branch-and-Cut 55 -- 3.3 Polyhedral Studies 58 -- 3.3.1 Capacity Constraints 59 -- 3.3.2 Generalized Capacity Constraints 61 -- 3.3.3 Framed Capacity Constraints 62 -- 3.3.4 Valid Inequalities from STSP 62 -- 3.3.5 Valid Inequalities Combining Bin Packing and STSP 67 -- 3.3.6 Valid Inequalities from the Stable Set Problem 69 -- 3.4 Separation Procedures 71 -- 3.4.1 Exact Separation of Capacity Constraints 71 -- 3.4.2 Heuristics for Capacity and Related Constraints 72 -- 3.4.3 STSP Constraints 75 -- 3.5 Branching Strategies 75 -- 3.6 Computational Results 78 -- 4 Set-Covering-Based Algorithms for the Capacitated VRP / J. Bramel, D. Simchi-Levi 85 -- 4.2 Solving the Linear Programming Relaxation of P 87 -- 4.3 Set-Covering-Based Solution Methods 89 -- 4.3.1 Branch-and-Bound Algorithm for Problem CG 89 -- 4.3.2 Polyhedral Branch-and-Bound Algorithm 91 -- 4.3.3 Pseudo-Polynomial Lower Bound on cmin 92 -- 4.3.4 Solving P[subscript D] via Dual-Ascent and Branch-and-Bound 94 -- 4.4 Solving the Set-Covering Integer Program 96 -- 4.4.1 A Cutting Plane Method 97 -- 4.4.2 Branch-and-Price 99 -- 4.4.3 Additional Comments on Computational Approaches 100 -- 4.5 Computational Results 100 -- 4.6 Effectiveness of the Set-Covering Formulation 102 -- 4.6.1 Worst-Case Analysis 103 -- 4.6.2 Average-Case Analysis 103 -- 5 Classical Heuristics for the Capacitated VRP / G. Laporte, F. Semet 109 -- 5.2 Constructive Methods 110 -- 5.2.1 Clarke and Wright Savings Algorithm 110 -- 5.2.2 Enhancements of the Clarke and Wright Algorithm 111 -- 5.2.3 Matching-Based Savings Algorithms 113 -- 5.2.4 Sequential Insertion Heuristics 114 -- 5.3 Two-Phase Methods 116 -- 5.3.1 Elementary Clustering Methods 116 -- 5.3.2 Truncated Branch-and-Bound 118 -- 5.3.3 Petal Algorithms 120 -- 5.3.4 Route-First, Cluster-Second Methods 120 -- 5.4 Improvement Heuristics 121 -- 5.4.1 Single-Route Improvements 121 -- 5.4.2 Multiroute Improvements 122 -- 6 Metaheuristics for the Capacitated VRP / M. Gendreau, G. Laporte, J.-Y. Potvin 129 -- 6.2 Simulated Annealing 130 -- 6.2.1 Two Early Simulated Annealing Algorithms 130 -- 6.2.2 Osman's Simulated Annealing Algorithms 131 -- 6.2.3 Van Breedam's Experiments 133 -- 6.3 Deterministic Annealing 133 -- 6.4 Tabu Search 134 -- 6.4.1 Two Early Tabu Search Algorithms 134 -- 6.4.2 Osman's Tabu Search Algorithm 134 -- 6.4.3 Taburoute 135 -- 6.4.4 Taillard's Algorithm 137 -- 6.4.5 Xu and Kelly's Algorithm 137 -- 6.4.6 Rego and Roucairol's Algorithms 137 -- 6.4.7 Barbarosoglu and Ozgur's Algorithm 138 -- 6.4.8 Adaptive Memory Procedure of Rochat and Taillard 138 -- 6.4.9 Granular Tabu Search of Toth and Vigo 138 -- 6.4.10 Computational Comparison 140 -- 6.5 Genetic Algorithms 140 -- 6.5.1 Simple Genetic Algorithm 140 -- 6.5.2 Application to Sequencing Problems 141 -- 6.5.3 Application to the VRP 142 -- 6.6 Ant Algorithms 144 -- 6.7 Neural Networks 146 -- II Important Variants of the Vehicle Routing Problem 155 -- 7 VRP with Time Windows / J.-F. Cordeau, G. Desaulniers, J. Desrosiers, M.M. Solomon, F. Soumis 157 -- 7.2 Problem Formulation 158 -- 7.2.1 Formulation 158 -- 7.2.2 Network Lower Bound 159 -- 7.2.3 Linear Programming Lower Bound 159 -- 7.2.4 Algorithms 160 -- 7.3 Upper Bounds: Heuristic Approaches 160 -- 7.3.1 Route Construction 160 -- 7.3.2 Route Improvement 161 -- 7.3.3 Composite Heuristics 161 -- 7.3.4 Metaheuristics 162 -- 7.3.5 Parallel Implementations 165 -- 7.3.6 Optimization-Based Heuristics 165 -- 7.3.7 Asymptotically Optimal Heuristics 165 -- 7.4 Lower Bounds from Decomposition Approaches 166 -- 7.4.1 Lagrangian Relaxation 166 -- 7.4.2 Capacity and Time-Constrained Shortest-Path Problem 167 -- 7.4.3 Variable Splitting 168 -- 7.4.4 Column Generation 169 -- 7.4.5 Set-Partitioning Formulation 169 -- 7.4.6 Lower Bound 170 -- 7.4.7 Commodity Aggregation 171 -- 7.4.8 Hybrid Approach 172 -- 7.5 Integer Solutions 173 -- 7.5.1 Binary Decisions on Arc Flow Variables 173 -- 7.5.2 Integer Decisions on Arc Flow Variables 173 -- 7.5.3 Binary Decisions on Path Flow Variables 174 -- 7.5.4 Subtour Elimination and 2-Path Cuts 175 -- 7.5.5 k-Path Cuts and Parallelism 176 -- 7.5.6 Integer Decisions on (Fractional and Integer) Time Variables 176 -- 7.6 Special Cases 177 -- 7.6.1 Multiple TSP with Time Windows 177 -- 7.6.2 VRP with Backhauls and Time Windows 177 -- 7.7 Extensions 178 -- 7.7.1 Heterogeneous Fleet, Multiple-Depot, and Initial Conditions 178 -- 7.7.2 Fleet Size 179 -- 7.7.3 Multiple Time Windows 179 -- 7.7.4 Soft Time Windows 179 -- 7.7.5 Time- and Load-Dependent Costs 180 -- 7.7.6 Driver Considerations 180 -- 7.8 Computational Results for VRPTW 181 -- 8 VRP with Backhauls / P. Toth, D. Vigo 195 -- 8.1.1 Benchmark Instances 197 -- 8.2 Integer Linear Programming Models 198 -- 8.2.1 Formulation of Toth and Vigo 198 -- 8.2.2 Formulation of Mingozzi, Giorgi, and Baldacci 200 -- 8.3 Relaxations 201 -- 8.3.1 Relaxation Obtained by Dropping the CCCs 202 -- 8.3.2 Relaxation Based on Projection 202 -- 8.3.3 Lagrangian Relaxation 203 -- 8.3.4 Overall Additive Lower Bound 204 -- 8.3.5 Relaxation Based on the Set-Partitioning Model 204 -- 8.4 Exact Algorithms 208 -- 8.4.1 Algorithm of Toth and Vigo 208 -- 8.4.2 Algorithm of Mingozzi, Giorgi, and Baldacci 209 -- 8.4.3 Computational Results for the Exact Algorithms 210 -- 8.5 Heuristic Algorithms 214 -- 8.5.1 Algorithm of Deif and Bodin 214 -- 8.5.2 Algorithms of Goetschalckx and Jacobs-Blecha 215 -- 8.5.3 Algorithm of Toth and Vigo 216 -- 8.5.4 Computational Results for the Heuristics 217 -- 9 VRP with Pickup and Delivery / G. Desaulniers, J. Desrosiers, A. Erdmann, M.M. Solomon, F. Soumis 225 -- 9.2 Mathematical Formulation 226 -- 9.2.1 Construction of the Networks 226 -- 9.2.2 Formulation 227 -- 9.2.3 Service Quality 228 -- 9.2.4 Reduction of the Network Size 228 -- 9.3 Heuristics 229 -- 9.3.1 Construction and Improvement 229 -- 9.3.2 Clustering Algorithms 230 -- 9.3.3 Metaheuristics 230 -- 9.3.4 Neural Network Heuristics 231 -- 9.3.5 Theoretical Analysis of Algorithms 231 -- 9.4 Optimization-Based Approaches 232 -- 9.4.1 Single Vehicle Cases 232 -- 9.4.2 Multiple Vehicle Cases 234 -- 9.5 Applications 236 -- 9.6 Computational Results 236 -- III Applications and Case Studies 243 -- 10 Routing Vehicles in the Real World: Applications in the Solid Waste, Beverage, Food, Dairy, and Newspaper Industries / B.L. Golden, A.A. Assad, E.A. Wasil 245 -- 10.2 Computerized Vehicle Routing in the Solid Waste Industry 247 -- 10.2.1 History 247 -- 10.2.3 Commercial Collection 249 -- 10.2.4 Residential Collection 250 -- 10.2.6 Roll-on-Roll-off 252 -- 10.2.7 Further Remarks 254 -- 10.3 Vehicle Routing in the Beverage, Food, and Dairy Industries 254 -- 10.3.2 Beverage Industry 255 -- 10.3.3 Food Industry 259 -- 10.3.4 Dairy Industry 260 -- 10.4 Distribution and Routing in the Newspaper Industry 266 -- 10.4.1 Industry Background 266 -- 10.4.2 Newspaper Distribution Problem 268 -- 10.4.3 Vehicle Routing Algorithms for NDP 271 -- 10.4.4 Three Case Studies 276 -- 11 Capacitated Arc Routing Problem with Vehicle-Site Dependencies: The Philadelphia Experience / J. Sniezek, L. Bodin, L. Levy, M. Ball 287 -- 11.2 Networks, Assumptions, and Goals of the CARP-VSD 288 -- 11.2.1 Travel Network 288 -- 11.2.2 Service Network 289 -- 11.2.3 Vehicle Classes 290 -- 11.2.4 Travel Network and Service Network for a Vehicle Class 291 -- 11.2.5 Vehicle Preference List 291 -- 11.2.6 Other Assumptions 292 -- 11.2.7 Goals and Constraints of the CARP-VSD 292 -- 11.3 Vehicle Decomposition Algorithm (VDA) 293 -- 11.3.1 Step A. Create and Verify Vehicle Class Networks 293 -- 11.3.2 Step B. Estimate Total Work and Determine Initial Fleet Mix 294 -- 11.3.3 Step C. Partition the Service Network 301 -- 11.3.4 Step D. Determine Travel Path and Balance the Partitions 304.