Resolution%20of%20the%20Location%20Routing%20Problem

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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

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Outline

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

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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

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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)

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Example the data
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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
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The memetic algorithm (MA)
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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

SPLIT
population management
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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

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SPLIT method (1/4)

• Parameters
• permutation on the customers
• (local) auxiliary graph
• Initial label at node 0
• pth label at node i

nb available vehicles
remaining capacity at each depot
label cost
father label
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SPLIT method (2/4)

• Dominance rules
• label
• label
• (is dominated by) if
• ? (48,101245,) lt (410,101245,)

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SPLIT method (3/4)

• Label propagation
• node i label
• node j label
• new values
• add the trip
• number of vehicles
• depots capacity
• label cost

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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

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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

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Split example (2/4)
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Split example (3/4)
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Split example (4/4)
dominance rule
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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

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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

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Population management
Neighborhooddepots used in the best solution
randomly chosen depot
initial subset of open depots (heuristic)
restart new subset of open depots
value
iterations
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Numerical experiments

• Prodhons 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 Burkes instances
• 12 instances with 100 customers
• 12 instances with 150 customers
• 12 instances with 200 customers
• ? from 10 to 20 depots
• Barretos instances
• From 27 to 100 customers
• From 5 to 10 depots

no depot capacity not a true LRP
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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

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Prodhons instances (1/3)
MA GRASP MAPM LRGTS
instance LB sol sol sol sol
20-5-1 54793 54793 55021 54793 55131
20-5-1b 39104 39104 39104 39104 39104
20-5-2 48908 48908 48908 48908 48908
20-5-2b 37542 37542 37542 37542 37542
50-5-1 84750,6 90111 90632 90160 90160
50-5-1b 59574,9 63242 64761 63242 63256
50-5-2 82057,1 88643 88786 88298 88715
50-5-2b 63841,4 67340 68042 67893 67698
50-5-2bis 82356,6 84055 84055 84055 84181
50-5-2bbis 51085,3 51902 52059 51822 51992
50-5-3 82703,8 86203 87380 86203 86203
50-5-3b 59473,8 61830 61890 61830 61830
gap/LB 3,15 3,71 3,18 3,29
20-50 nodes
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Prodhons instances (2/3)
MA GRASP MAPM LRGTS
instance LB sol sol sol sol
100-5-1 272082 280370 279437 281944 277935
100-5-1b 207037 216813 216159 216656 214885
100-5-2 186917 196086 199520 195568 196545
100-5-2b 153827 157989 159550 157325 157792
100-5-3 194202 201836 203999 201749 201952
100-5-3b 149986 154447 154596 153322 154709
100-10-1 258243 327467 323171 316575 291887
100-10-1b 218826 272267 271477 270251 235532
100-10-2 226905 246615 254087 245123 246708
100-10-2b 194628 206142 206555 205052 204435
100-10-3 222353 256054 270826 253669 258656
100-10-3b 189308 205554 216173 204815 205883
gap/LB 9,32 10,75 8,59 6,69
100 nodes
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Prodhons instances (3/3)
MA GRASP MAPM LRGTS
instance BKS sol sol sol sol
200-10-1 479425 492602 490820 483497 481676
200-10-1b 378773 404131 416753 380044 380613
200-10-2 450468 477048 512679 451840 453353
200-10-2b 374435 392157 379980 375019 377351
200-10-3 472898 484911 496694 478132 476684
200-10-3b 364178 368963 389016 364834 365250
gap/BKS 3,99 6,59 0,49 0,58
200 nodes
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Tuzun Burkes instances (1/3)
MA GRASP MAPM LRGTS
instance sol sol sol sol
P111112 1492,11 1525,25 1493,92 1490,82
P111122 1463,42 1526,90 1471,36 1471,76
P111212 1429,81 1423,54 1418,83 1412,04
P111222 1436,13 1482,29 1492,46 1443,06
P112112 1180,91 1200,24 1173,22 1187,63
P112122 1103,63 1123,64 1115,37 1115,95
P112212 804,06 814,00 793,97 813,28
P112222 731,05 787,84 730,51 742,96
P113112 1288,24 1273,10 1262,32 1267,93
P113122 1250,05 1272,94 1251,32 1256,12
P113212 905,66 912,19 903,82 913,06
P113222 1026,25 1022,51 1022,93 1025,51
gap/BKS 0,50 2,40 0,53 0,81
100 nodes
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Tuzun Burkes instances (2/3)
MA GRASP MAPM LRGTS
instance sol sol sol sol
P131112 1985,63 2006,7 1959,39 1946,01
P131122 1934,36 1888,9 1881,67 1875,79
P131212 2038,33 2033,93 1984,25 2010,53
P131222 1913,12 1856,07 1855,25 1819,89
P132112 1462,53 1508,33 1448,27 1448,65
P132122 1481,15 1456,82 1459,83 1492,86
P132212 1219,52 1240,4 1207,41 1211,07
P132222 947,40 940,8 934,79 936,93
P133112 1762,32 1736,9 1720,3 1729,31
P133122 1420,97 1425,74 1429,34 1424,59
P133212 1227,52 1223,7 1203,44 1216,32
P133222 1163,60 1231,33 1158,54 1162,16
gap/BKS 1,91 2,09 0,31 0,54
150 nodes
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Tuzun Burkes instances (3/3)
MA GRASP MAPM LRGTS
instance sol sol sol sol
P121112 2367,56 2384,01 2293,99 2296,52
P121122 2356,01 2288,09 2277,39 2207,5
P121212 2350,47 2273,19 2274,57 2260,87
P121222 2352,34 2345,1 2376,25 2259,52
P122112 2195,39 2137,08 2106,26 2120,76
P122122 1834,96 1807,29 1771,53 1737,81
P122212 1480,79 1496,75 1467,54 1488,55
P122222 1133,80 1095,92 1088 1090,59
P123112 2021,04 2044,66 1973,28 1984,06
P123122 2057,22 2090,95 1979,05 1986,49
P123212 1821,20 1788,7 1782,23 1786,79
P123222 1477,22 1408,63 1396,24 1401,16
gap/BKS 3,94 2,55 0,91 0,33
200 nodes
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Barretos instances (1/1)
MA GRASP MAPM LRGTS
instance LB sol sol sol sol
Christofides69-50x5 551,1 586,3 599,1 565,6 586,3
Christofides69-75x10 791,4 855,3 861,6 866,1 863,5
Christofides69-100x10 818,1 867,1 861,6 850,1 842,9
Daskin95-88x8 347,0 355,8 356,9 355,8 368,7
Daskin95-150x10 39470,5,0 45656,2 44625,2 44011,7 44386,3
Gaskell67-21x5 424,9 424,9 429,6 424,9 424,9
Gaskell67-22x5 585,1 585,1 585,1 611,8 587,4
Gaskell67-29x5 512,1 512,1 515,1 512,1 512,1
Gaskell67-32x5 562,2 562,2 571,9 571,9 584,6
Gaskell67-32x5 504,3 504,3 504,3 534,7 504,8
Gaskell67-36x5 460,4 463,9 460,4 485,4 476,5
Min92-27x5 3062,0 3062,0 3062,0 3062,0 3065,2
Min92-27x5 5423,0 5927,4 5965,1 5950,1 5809,0
gap/LB 3,75 4,02 4,42 4,03
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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

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• Thanks !