Modeling and Simulation of Logistics Flows 1 Theory and Fundamentals

Foreword xiii About This Book xvii Introduction xxiii Chapter 1. Operational Research 1 1.1. A history 1 1.2. Fields of application, principles and concepts 2 1.2.1. Identification 2 1.2.2. Modeling 3 1.2.3. Solution 5 1.2.4. Validation 6 1.2.5. Implementation 6 1.2.6. Improvement 6 1.3. Basic models 7 1.4. The future of OR 7 Chapter 2. Elements of Graph Theory 9 2.1. Graphs and representations 9 2.2. Undirected graph 10 2.2.1. Multigraph 10 2.2.2. Planar and non-planar graph 11 2.2.3. Connected and unconnected graph 11 2.2.4. Complete graph 11 2.2.5. Bipartite graph 12 2.2.6. Partial graph, subgraph, clique and stable 12 2.2.7. Degree of a vertex and a graph 13 2.2.8. Chain and cycle in a graph 13 2.2.9. Level of connectivity (or beta index) 15 2.2.10. Eulerian graph 15 2.2.11. Hamiltonian graph 16 2.2.12. Planar graph 17 2.2.13. Isthmus 18 2.2.14. Tree and forest 19 2.2.15. Arborescence 20 2.2.16. Ordered arborescence 21 2.3. Directed graph or digraph 22 2.3.1. Path and circuit in a digraph 22 2.3.2. Absence of circuit in a digraph 23 2.3.3. Adjacency matrix 24 2.3.4. Valued graph matrix 24 2.4. Graphs for logistics 25 Chapter 3. Optimal Paths 27 3.1. Basic concepts 27 3.2. Dijkstra's algorithm 28 3.2.1. An example of calculating minimal paths 28 3.2.2. Interpreting the results of the calculations 30 3.3. Floyd--Warshall's algorithm 30 3.3.1. Creating the starting matrices (initialization of the algorithm) 30 3.3.2. Filling the matrices for the following repetitions 31 3.3.3. An example of calculating minimal paths 32 3.3.4. Interpreting the results 34 3.4. Bellman--Ford's algorithm 35 3.4.1. Initialization 36 3.4.2. The next repetitions with relaxation 36 3.4.3. An example of calculation 37 3.4.4. Interpreting the results 39 3.5. Bellman--Ford's algorithm with a negative circuit 40 3.5.1. Example 40 3.6. Exercises 43 3.6.1. Exercise 1: Optimizing journey time 43 3.6.2. Exercise 2: A directed graph with negative cost side 44 3.6.3. Exercise 3: Routing data packets 45 3.6.4. Solutions to exercise 1 45 3.6.5. Solutions to exercise 2 46 3.6.6. Solutions to exercise 3 48 Chapter 4. Dynamic Programming 51 4.1. The principles of dynamic programming 51 4.2. Formulating the problem 52 4.2.1. Example 1: The pyramid of numbers 52 4.2.2. Example 2: The Fibonacci sequence 54 4.2.3. Example 3: The knapsack 56 4.3. Stochastic process 60 4.4. Markov chains 60 4.4.1. Property of Markov chains 61 4.4.2. Classes and states of a chain 62 4.4.3. Matrix and graph 63 4.4.4. Applying Markov chains 64 4.5. Exercises 66 4.5.1. Exercise 1: Levenshtein distance 66 4.5.2. Exercise 2 67 4.5.3. Exercise 3: Ehrenfest model 67 4.5.4. Solutions to exercise 1 68 4.5.5. Solutions to exercise 2 69 4.5.6. Solutions to exercise 3 70 Chapter 5. Scheduling with PERT and MPM 73 5.1. Fundamental concepts 73 5.2. Critical path method 74 5.3. Precedence diagram 74 5.4. Planning a project with PERT-CPM 77 5.4.1. A brief history 77 5.4.2. Methodology 78 5.5. Example of determining a critical path with PERT 86 5.5.1. Using the example to create a precedence table 87 5.5.2. Creating the graph 87 5.5.3. Numbering of vertices 89 5.5.4. Determining earliest dates of each of the tasks 89 5.5.5. Determining the latest dates for each of the tasks 90 5.5.6. Determining the critical paths 92 5.6. Slacks 93 5.6.1. Total slack 94 5.6.2. Free slack 94 5.6.3. Certain slack (or independent slack) 94 5.6.4. Properties 94 5.7. Example of calculating slacks 95 5.8. Determining the critical path with the help of a double-entry table 96 5.8.1. Creating a table using our example 96 5.8.2 Filling out the table 96 5.8.3. ES dates 97 5.8.4. LF dates 99 5.8.5. Critical path 100 5.9. Methodology of planning with MPM 101 5.9.1. A brief history 101 5.9.2. Formalizing the graph 102 5.9.3. Rules of construction 103 5.9.4. Earliest and latest dates 104 5.9.5. Determining the critical path 105 5.10. Example of determining a critical path with MPM 106 5.10.1. Creating the graph 106 5.10.2. Determining the earliest dates for each task 107 5.10.3. Determining the latest dates of each task 108 5.10.4. Determining the critical path(s) 109 5.10.5. Slacks 109 5.11. Probabilistic PERT/CPM/MPM 111 5.11.1. Probability of tasks 112 5.11.2. Implementation in an example 113 5.11.3. Calculating average durations and variance 114 5.11.4. Calculating the average duration of the project 114 5.11.5. Calculating the probability of finishing the project in a chosen duration 114 5.11.6. Calculating the duration of the project for a given probability 115 5.12. Gantt diagram 116 5.12.1. Creating the diagram 116 5.12.2. Example 117 5.13. PERT-MPM cost 119 5.13.1. Method 120 5.13.2. Example 121 5.14. Exercises 125 5.14.1. Exercise 1 125 5.14.2. Exercise 2 126 5.14.3. Exercise 3 126 5.14.4. Exercise 4 127 5.14.5. Solutions to exercise 1 129 5.14.6. Solutions to exercise 2 130 5.14.7. Solutions to exercise 3 132 5.14.8. Solutions to exercise 4 133 Chapter 6. Maximum Flow in a Network 137 6.1. Maximum flow 137 6.2. Ford--Fulkerson algorithm 138 6.2.1. Presentation of the algorithm 139 6.2.2. Application of an example 141 6.3. Minimum cut theorem 147 6.3.1. Example of cuts 148 6.4. Dinic algorithm 149 6.4.1. Presenting the algorithm 149 6.4.2. Application in an example 150 6.5. Exercises 154 6.5.1. Exercise 1: Drinking water supply 154 6.5.2. Exercise 2: Maximum flow according to Dinic 155 6.5.3. Solutions to exercise 1 155 6.5.4. Solutions to exercise 2 158 Chapter 7. Trees, Tours and Transport 163 7.1. The basic concepts 163 7.2. Kruskal's algorithm 165 7.2.1. Application to an example 165 7.3. Prim's algorithm 168 7.3.1. Application to an example 170 7.4. Sollin's algorithm 175 7.4.1. Application to an example 176 7.5. Little's algorithm for solving the TSP 182 7.5.1. Application to an example 184 7.6. Exercises 195 7.6.1. Exercise 1: Computer network 195 7.6.2. Exercise 2: Deliveries 196 7.6.3. Solutions to exercise 1 197 7.6.4. Solutions to exercise 2 200 Chapter 8. Linear Programming 205 8.1. Basic concepts 205 8.1.1. Formulation of a linear program 206 8.2. The graphic resolution method 206 8.2.1. Identification 207 8.2.2. Formalization 207 8.2.3. Resolution 212 8.3. Simplex method 215 8.3.1. Steps 215 8.3.2. An example to be addressed 215 8.3.3. Formalization 216 8.3.4. Change into standard form 217 8.3.5. Creation of the table 218 8.3.6. Determination of the pivot 218 8.3.7. Iterations 219 8.3.8. Interpretation 222 8.4. Duality 223 8.4.1. Dual formulation 224 8.4.2. Passage from primal to dual formalization 224 8.4.3. Determination of the pivot 226 8.4.4. Iterations 227 8.4.5. Interpretation 228 8.5. Exercises . 228 8.5.1. Exercise 1: Video and festival 228 8.5.2. Exercise 2: Simplex 228 8.5.3. Exercise 3: Primal and dual 229 8.5.4. Solutions to exercise 1 229 8.5.5. Solutions to exercise 2 232 8.5.6. Solutions to exercise 3 234 Chapter 9. Modeling Road Traffic 237 9.1. A short introduction to road traffic 237 9.2. Scale of models and networks 239 9.3. Models and types 239 9.4. Learning more information about the models 240 9.4.1. Microscopic models 240 9.4.2. Macroscopic models 245 9.4.3. The families of macroscopic models 250 9.4.4. The discretization of models 252 9.4.5. Mesoscopic models 253 9.4.6. Hybrid models 254 9.5. Urban modeling 255 9.6. Intelligent transportation systems 256 9.7. Conclusion 256 Chapter 10. Software Programs 259 10.1. Software programs for OR and logistics 259 10.2. Spreadsheets 260 10.2.1. Existing software programs 262 10.3. Project managers 266 10.3.1. The procedure for creating a project 268 10.3.2. The different software programs available on the market 268 10.4. Flow simulators 271 10.4.1. Generalist software programs 273 10.4.2. Pedestrian simulators 281 10.4.3. Traffic simulators 288 10.4.4. The creation of a simulation process 300 Appendices 303 Appendix 1: Standard Normal Distribution Table 305 Appendix 2: GeoGebra 309 Conclusion 319 Glossary 323 Bibliography 329 Index 337