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Resolution of the Location Routing Problem

Resolution of the Location Routing Problem. C. Duhamel , P. Lacomme C. Prins, C. Prodhon Université de Clermont-Ferrand II, LIMOS, France Université de Technologie de Troyes, ISTIT, France EU/MEeting October 23-24, 2008, Troyes. Outline. LRP presentation A memetic algorithm

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Resolution of the Location Routing Problem

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  1. Resolution of the Location Routing Problem C. Duhamel, P. Lacomme C. Prins, C. Prodhon Université de Clermont-Ferrand II, LIMOS, France Université de Technologie de Troyes, ISTIT, France EU/MEeting October 23-24, 2008, Troyes

  2. Outline • LRP presentation • A memetic algorithm • chromosome definition • SPLIT procedure • local search • Computational experiments • Concluding remarks

  3. Problem definition • set of depots • = setup cost of depot i • = capacity of depot i • set of customers • = demand of customer j • set of homogeneous vehicles • = vehicle capacity • = fixed cost of a vehicle • set of nodes • = traveling cost on arc

  4. Problem definition • Objectives • select the depots to use • assign each customer to a depot • solve a VRP for each open depot • Integration: two decision levels • hub location (tactical level) • vehicle routing (operational level)

  5. depot customer Example: the data

  6. Example: a LRP solution for depot node 26 trip 1 : 26, 25, 24, 14, 10, 11, 15, 16, 26 trip 2 : 26, 27, 28, 36, 35, 43, 50, 49, 42, 34, 35, 26 trip 3 : 26, 16, 4, 19, 29, 37, 36, 28, 27, 26

  7. initial Graph G SP-Graph H MA sequence LS sequence trips auxiliary graph H’ Split The memetic algorithm (MA)

  8. Chromosome ordered set of customers fitness = total cost of the solution no information on open depot and assignments Population set of chromosomes crossover and mutation initialization: heuristics + random solutions Mutation local search based on trips Population management based on opening depot nodes MA key features SPLIT population management

  9. Evaluation: SPLIT procedure • SPLIT for the CARP • (Lacomme et al., 2001) • outperformed CARPET • encompass extensions (prohibited turns, etc.) • SPLIT for the VRP • (Prins, 2004) • best published method for the VRP at that time  proved to be efficient for routing problems

  10. SPLIT method (1/4) • Parameters • permutation on the customers • (local) auxiliary graph • Initial label at node 0 • pth label at node i label cost father label nb available vehicles remaining capacity at each depot

  11. OR OR SPLIT method (2/4) • Dominance rules • label • label • (is dominated by) if (4;8,10;1245;*,*) < (4;10,10;1245;*,*)

  12. SPLIT method (3/4) • Label propagation • node i: label • node j: label • new values • add the trip • number of vehicles: • depots capacity: • label cost:

  13. SPLIT method (4/4) • At each node i • set of non dominated labels • ways to split the customers into trip blocks assigned to depots • At node n • sets of feasible solutions given

  14. Split example (1/4) • Shortest paths and demands • Depots • 1: node 7, capacity 10, opening cost 20 • 2: node 8, capacity 15, opening cost 10 • 3: node 9, capacity 8, opening cost 50

  15. Split example (2/4)

  16. Split example (3/4)

  17. Split example (4/4) dominance rule

  18. Mutation: local search (1/2) • Parameters • trips computed by Split • graph H of the shortest paths • Modifications • Swap (1/1 clients) within the trip • Swap (1/1 clients), trips of the same depot • Swap (1/1 clients), trips of different depots • FA strategy, VND-like exploration, it. limit

  19. Mutation: local search (2/2) • Combination Split - LS • mutation: sequence → sequence • Split: sequence → trips • LS: trips → trips • compact: trips → sequence • Purpose • two different search spaces • combination allow a wider exploration • similar to Variable Search Space

  20. Population management Neighborhood:depots used in the best solution + randomly chosen depot initial subset of open depots (heuristic) restart: new subset of open depots value iterations

  21. Numerical experiments • Prodhon’s instances • 4 instances with 20 customers • 8 instances with 50 customers • 12 instances with 100 customers • 6 instances with 200 customers  from 5 to 10 depots • Tuzun & Burke’s instances • 12 instances with 100 customers • 12 instances with 150 customers • 12 instances with 200 customers  from 10 to 20 depots • Barreto’s instances • From 27 to 100 customers • From 5 to 10 depots no depot capacity not a true LRP

  22. Numerical experiments • Protocol • best of 4 runs • 150.000 iterations • population of 40 chromosomes • restart • triggered after 1000 iterations • each time +200 iterations • maximum = 10.000 iterations

  23. Prodhon’s instances (1/3) 20-50 nodes

  24. Prodhon’s instances (2/3) 100 nodes

  25. Prodhon’s instances (3/3) 200 nodes

  26. Tuzun & Burke’s instances (1/3) 100 nodes

  27. Tuzun & Burke’s instances (2/3) 150 nodes

  28. Tuzun & Burke’s instances (3/3) 200 nodes

  29. Barreto’s instances (1/1)

  30. Concluding remarks • Found some new best solutions • Time consuming → reduction strategies • Could handle extensions: • heterogeneous fleet of vehicles • time-windows (customers and depots) • stochastic demands for customers • bin-packing constraints in vehicles load

  31. Thanks !

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