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

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 Example: the data depotcustomer

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 The memetic algorithm (MA) initial Graph G SP-Graph HMA Splitauxiliary graph H’ LS sequence trips sequence

8 MA key features 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 population management SPLIT

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) nb available vehicles remaining capacity at each depot label costfather label Parameters permutation on the customers (local) auxiliary graph Initial label at node 0 p th label at node i

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

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

13 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 SPLIT method (4/4)

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

21 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 Numerical experiments

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

23 Prodhon’s instances (1/3) MAGRASPMAPMLRGTS instanceLBsol b b , b59574, , b63841, bis82356, bbis51085, , b59473, gap/LB3,153,713,183, nodes

24 Prodhon’s instances (2/3) MAGRASPMAPMLRGTS instanceLBsol b b b b b b gap/LB9,3210,758,596, nodes

25 Prodhon’s instances (3/3) MAGRASPMAPMLRGTS instanceBKSsol b b b gap/BKS3,996,590,490, nodes

26 Tuzun & Burke’s instances (1/3) MAGRASPMAPMLRGTS instancesol P ,111525,251493,921490,82 P ,421526,901471,361471,76 P ,811423,541418,831412,04 P ,131482,291492,461443,06 P ,911200,241173,221187,63 P ,631123,641115,371115,95 P ,06814,00793,97813,28 P ,05787,84730,51742,96 P ,241273,101262,321267,93 P ,051272,941251,321256,12 P ,66912,19903,82913,06 P ,251022,511022,931025,51 gap/BKS0,502,400,530, nodes

27 Tuzun & Burke’s instances (2/3) MAGRASPMAPMLRGTS instancesol P ,632006,71959,391946,01 P ,361888,91881,671875,79 P ,332033,931984,252010,53 P ,121856,071855,251819,89 P ,531508,331448,271448,65 P ,151456,821459,831492,86 P ,521240,41207,411211,07 P ,40940,8934,79936,93 P ,321736,91720,31729,31 P ,971425,741429,341424,59 P ,521223,71203,441216,32 P ,601231,331158,541162,16 gap/BKS1,912,090,310, nodes

28 Tuzun & Burke’s instances (3/3) MAGRASPMAPMLRGTS instancesol P ,562384,012293,992296,52 P ,012288,092277,392207,5 P ,472273,192274,572260,87 P ,342345,12376,252259,52 P ,392137,082106,262120,76 P ,961807,291771,531737,81 P ,791496,751467,541488,55 P ,801095, ,59 P ,042044,661973,281984,06 P ,222090,951979,051986,49 P ,201788,71782,231786,79 P ,221408,631396,241401,16 gap/BKS3,942,550,910, nodes

29 Barreto’s instances (1/1) MAGRASPMAPMLRGTS instanceLBsol Christofides69-50x5551,1586,3599,1565,6586,3 Christofides69-75x10791,4855,3861,6866,1863,5 Christofides69-100x10818,1867,1861,6850,1842,9 Daskin95-88x8347,0355,8356,9355,8368,7 Daskin95-150x ,5,045656,244625,244011,744386,3 Gaskell67-21x5424,9 429,6424,9 Gaskell67-22x5585,1 611,8587,4 Gaskell67-29x5512,1 515,1512,1 Gaskell67-32x5562,2 571,9 584,6 Gaskell67-32x5504,3 534,7504,8 Gaskell67-36x5460,4463,9460,4485,4476,5 Min92-27x53062,0 3065,2 Min92-27x55423,05927,45965,15950,15809,0 gap/LB3,754,024,424,03

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 !