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Resolution of the Location Routing Problem

This paper outlines the application of a memetic algorithm for solving the Location Routing Problem (LRP), focusing on chromosome definition, the SPLIT procedure, local search, and computational experiments. The LRP involves selecting depots, assigning customers, and solving the Vehicle Routing Problem for each open depot.

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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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