resolution of the location routing problem

Resolution of the Location Routing

ProblemC. Duhamel, P. Lacomme

C. Prins, C. Prodhon

Université de Clermont-Ferrand II, LIMOS, FranceUniversité 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

1I m

iO

iW

1J n

jd

1K k

Q

F

V I J

ijc ,i j

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

depot

customer

6

Example: a LRP solution

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

7

The memetic algorithm (MA)

initial Graph G

SP-Graph H MA

Splitauxiliary graph H’

LSsequence

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 vehiclesremaining capacity at each depotlabel cost father label

Parameters permutation on the customers (local) auxiliary graph

Initial label at node 0

pth label at node i

1 n ' ; ;H X A Z

1 10 0,1 0,1 0,1; ;0; 1, 1mL K W W

1, , ,; ; ; ,p m p

i i p i p i p iL K W W z k j

11

Dominance rules label label (is dominated by) if

(4;8,10;1245;*,*) < (4;10,10;1245;*,*)

SPLIT method (2/4)

1, , ,; ; ;*,*p m p

i i p i p i p iL K W W z

1, , ,; ; ;*,*q m q

i i q i q i q iL K W W z p qi iL L

OR OR

, ,

, ,,i p i q

r ri p i q

p qi i

K K

r W W

z z

, ,

, ,,i p i q

r ri p i q

p qi i

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, ,' ', ,

',

',

i p i qr ri p i q

r ri p i q

p qi i

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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)

1, , ,; ; ;*,*p m p

i i p i p i p iL K W W z

1, , ,; ; ; ,q m q

j j q j q j q jL K W W z i p

q p r

j i ijz z c

, , 1j q i pK K

, ,' r r

j q i pr r W W ' '

, ,1

k

r r

j q i pk i j

W W d

1, ,i jr r

1 1

1 1i k k j

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ij r r r rk i j

c O y F c c c

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)

1 i

1 n

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)

0 1 2

(3;10;15;8;0)

*;4;* *;3;*

(2;5;15;8;30)

(2;10;10;8;13)

(2;10;15;3;16)

4

7

4

8

4

9

5

25

5

8

15

15

(4;3;2;6;5;1)

16

Split example (3/4)

4

7

5

10 3

158 9

5

2

4

7

5

10

158 9

3

10

10

4

7

5

25

3

15

8 9

15

0 1 2

(3;10;15;8;0)

*;4;* *;3;*

(2;5;15;8;30)

(2;10;10;8;13)

(2;10;15;3;16)

(1;1;15;8;60)

(1;5;11;8;37)

(1;5;15;4;30)

17

(2;1;15;8;30)

0 1 2

(3;10;15;8;0)

*;4;* *;3;*

(2;5;15;8;30)

(2;10;10;8;13)

(2;10;15;3;16)

(1;1;15;8;60)

(1;5;11;8;37)

(1;5;15;4;30)

Split example (4/4)

4

7

510

3

15

8 9

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

valu

e

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 capacitynot a true LRP

Numerical experiments

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)

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

24


MA GRASP MAPM LRGTS


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

25


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

26

Tuzun & Burke’s 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

27


MA GRASP MAPM LRGTS


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

28


MA GRASP MAPM LRGTS


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

29

Barreto’s instances (1/1)

MA GRASP MAPM LRGTS


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

31

Thanks !

resolution of the location routing problem

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