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The Vehicle Routing Problem with Time Windows and Scheduled Departure .

The Vehicle Routing Problem with Time Windows and Scheduled Departure. Ho-Chi-Minh City, March-2010. Cristian Oliva San Martín. Universidad Católica de la Santísima Concepción. Chile. Outline. INTRODUCTION. Algorithms. Remarks. VRPTW. VRP. VRPTWSL. Objective. Methods of Solution.

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The Vehicle Routing Problem with Time Windows and Scheduled Departure .

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  1. The Vehicle Routing Problem with Time Windows and Scheduled Departure. Ho-Chi-Minh City, March-2010 Cristian Oliva San Martín. Universidad Católica de la Santísima Concepción. Chile.

  2. Outline INTRODUCTION Algorithms Remarks

  3. VRPTW VRP VRPTWSL

  4. Objective

  5. Methods of Solution Exact Methods Approximation Methods Heuristics Metaheuristics • Construction. • Local search. • Simulated Annealing • Genetic algorithms • Ant Colony Optimization • Tabu-Search

  6. Data of solomon (C101-25).

  7. The vehicle is loaded in the depot, This time is calculated as: load time = load factor * service time.

  8. Arrival Time: Load time + Travel time + departure time. • Departure time: Arrival time + Service time.

  9. 24 2 25 5 20 3 10 18 36 18 17 55 5 16 16 5 3 5 19 7 15 5

  10. 24 2 25 5 20 3 [0,27] 10 18 36 18 17 55 5 16 16 5 3 5 19 7 15 5

  11. 24 2 25 5 [37,127] 20 3 [0,27] 10 18 36 [27,45] 18 17 55 5 16 16 5 3 5 19 7 15 5

  12. [132,222] 24 2 25 5 [37,127] 20 3 [0,27] 10 18 36 [45,90] [27,45] 18 17 55 [62,152] 5 16 16 5 3 5 19 7 15 5

  13. [132,222] 24 2 [224,314] 25 5 [37,127] [108,198] 20 3 [0,27] 10 18 36 [45,90] [27,45] 18 17 55 [62,152] 5 16 16 5 3 5 19 7 15 5 [170,260]

  14. [132,222] Route 2: [276,1236] 24 Route 1: [331,1236] 2 [224,314] 25 5 [37,127] [108,198] 20 3 [0,27] 10 18 36 [45,90] [234,324] [27,45] 18 17 55 [62,152] 5 16 16 5 3 5 19 7 15 5

  15. [132,222] Route 2: [276,1236] 24 Route 1: [331,1236] 2 [224,314] 25 5 [37,127] [108,198] 20 3 [0,27] 10 18 36 [45,90] [234,324] [27,45] 18 17 55 [62,152] 5 16 16 5 3 5 19 7 15 5 [329,419] [170,260]

  16. [132,222] Route 2: [276,1236] 24 Route 1: [331,1236] 2 [224,314] 25 5 [37,127] [108,198] 20 3 [0,27] 10 18 36 [45,90] [234,324] [27,45] 18 17 55 [62,152] 5 16 16 5 3 5 19 7 15 5 [329,419] [170,260] [424,514]

  17. [132,222] Route 2: [276,1236] 24 Route 1: [331,1236] 2 [224,314] 25 5 [37,127] [108,198] 20 3 [0,27] 10 18 36 [45,90] [234,324] [27,45] 18 17 55 [62,152] [519,609] 5 16 16 5 3 5 19 7 15 5 [329,419] [170,260] [424,514]

  18. [132,222] Route 2: [276,1236] 24 Route 1: [331,1236] 2 [224,314] Route 3: [664,1236] 25 5 [37,127] [108,198] 20 3 [0,27] 10 18 36 [45,90] [234,324] [27,45] 18 17 55 [62,152] [519,609] 5 16 16 5 3 5 19 7 15 5 [329,419] [170,260] [424,514]

  19. Model of VRPTWSL VRPTW SL

  20. Simple Heuristics for VRPTWSL 1. Nearest Neighbor (NN) [for TSP] (By Rosenkrantz, Sterns, Lewis, SIAM J. Computing 6(1977) 563-581) Step 1. Start with node 0 as the beginning of the TSP tour Step 2. - Find the node closest to the last node added to the tour satisfying all the constraints. - Add this node to the tour Step 3.- Repeat Step 2 until all nodes are added to the tour - Then join the first and the last nodes

  21. NearestNeighbor. 0

  22. NearestNeighbor. 1 Arrival= 9+18=27 departure= 65+90=155 pi= 9 route1 0 Arrival= 155+18=173

  23. NearestNeighbor. departure= 65+90=155 Arrival = 18+18=36 1 arrival= 155+6=161 pi= 18 3 route1 0 departure= 255+90=345 arrival =345+20= 371

  24. NearestNeighbor. departure= 65+90=155 arrival = 27+18=45 1 arrival= 155+6=161 pi= 27 3 0 departure= 255+90=345 arrival=1002+23=1025 route1 arrival= 345+7=352 4 departure= 912+90=1002

  25. NearestNeighbor. departure= 65+90=155 arrival = 27+18=45 arrival=36+47=83 1 2 arrival= 155+2=157 pi= 27 3 departure=99+90=189 0 departure= 255+90=345 route2 arrival=1002+23=1025 pi= 36 route1 arrival= 345+2=347 arrival=189+57=246 4 departure= 912+90=1002

  26. Results

  27. Results

  28. Results

  29. Local Search Heuristics (LSH) • Simple heuristics: • - generates one solution only • Local search heuristics: • - generates many solutions • - choose the best one Definition: Given a solution S, its neighborhood N(S) is defined as the set of all solutions than can be obtained through a well-defined modification of S

  30. Basic idea of Local Search Heuristics • Start with a solution • (initial solution obtained • by a simple heuristic) • Search its • neighborhood • Select a solution from • the neighborhood • Either reject or • accept the new • solution S1 N(S1) S2 N(S2) S3 N(S3) S4 Three components of a LSH 1. Neighborhood design 2. Search process in the neighborhood 3. Acceptance-rejection criterion

  31. K-Opt heuristic • Lin, Bell Systems Technical J. 44(1965) 2245-2269 • Lin & Kernigham, Operations Research 21(1973) 498-516 Definition:K-exchange is a procedure that replaces K arcs in a given TSP tour by K new arcs so that the resulting tour is still a TSP tour.

  32. K-Opt heuristic Replace (A, B) & (C, D) by (A, C) & (B, D) B B C A A C D “2-exchange” E E D Replace (A, B), (C, D) & (D, E) by (B, D), (A, D) & (C, E) B B C A C A “3-exchange” E D E D

  33. 2-Interchange for VRPTWSL 8 8 4 4 1 7 1 7 6 6 5 5 3 3 2 2 Initial Solution X Total cost: 42 Waiting time: 25 Solution X’ Total cost: 36 Waiting time: 24

  34. Compute thesavings.

  35. 2. Simulated Annealing (SA) Step 1:Find an initial solution using any heuristic, S. Let the incumbent solution S*=S. Select the cooling parameter T > 0. Step 2:Select a candidate solution Scfrom N(S). If F(Sc)<F(S*), update the incumbent, S*=Sc. If F(Sc)<F(S), accept Sc, i.e. let S = Sc. If F(Sc)>F(S), accept Scwith probability exp((F(S)-F(Sc))/T), if Scis accepted, then let S = Sc, o/w, S is not updated.

  36. Step 3:- Update b such that it becomes smaller. - Repeat Step 2 until a certain stopping rule is satisfied(e.g. stop after 1000 iterations). Note: Three things may affect effectiveness of SA - neighborhood design - the way a candidate solution is selected - cooling parameter

  37. 3. Tabu Search (TS) (1 of 3) • Similar to SA with a different acceptance-rejection • criterion • Tabu list of mutations (prohibited moves) • Step 1:- Find an initial solution using any • heuristic, S. • - Let the incumbent solution S*=S. • - Set tabu list length L. Initially the tabu • list is empty.

  38. 3. Tabu Search (TS) (2 of 3) . Step 2:(1) Select a candidate solution Scfrom N(S) such that the move from S to Scis not a mutation on the tabu list. (2) Enter the reverse mutation of the move at the top of the tabu list, push all other entries in the list one position down, & delete the entry at the bottom of list if the list length exceeds L. (3) Accept Sc, i.e. let S = Sc. (4) If F(Sc)<F(S*), update the incumbent, S*=Sc.

  39. 3. Tabu Search (TS) (3 of 3) Step 3:Repeat Step 2 until some stopping rule is satisfied(e.g. stop after 1000 iterations). Three things may affect effectiveness of TS - neighborhood design - the way a candidate solution is selected - Length parameter

  40. END OF THE PRESENTATION.

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