Toshihide IBARAKI

Mikio KUBO

Tomoyasu MASUDA

Takeaki UNO

Mutsunori YAGIURA

Effective Local Search Algorithms for the Vehicle Routing Problem with General Time Window Constraints

ProblemInput:

Output: minimum cost vehicle routes

Constraints: capacity and time window constraints

1q2q

3q4q

5q

6q7q

8q9q

10q11q

12q

)(,},,1{},,,1,0{ ijdDmMnV ik qQ , )( ijtT

General Time WindowsEach customer indicates the time to be serviced

ttime windows of a customer i

)(tpi can be non-convex and discontinuousas long as it is a piecewise linear function

pena

lty penalty function )(tpi

Objective function

)()()()( QTDstco

the total distance

the total time penalty

the total capacity excess

)(D)(T)(Q

a vehicle schedule

time penalty and capacity constraints

soft constraints

Problem structureWe have to determine:

・（ a ） and （ b ）simultaneously done by the localsearch procedure

（ a） the assignment of customers to the vehicles（ b） the visiting order of customers for each vehicle（ c） the optimal start times of services of each vehicle

・ (c) determined by using dynamic programming

)( 0σN

Local search（ LS）LS repeats replacing with a better solution

In its neighborhood σ

a locally optimal

1σ2σ

3σ4σ

)( 4σN

)( 3σN

)( 2σN)( 1σN

)(σN

0σ

an initial solution

Neighborhoods

• the CROSS exchange neighborhood

• the ２ -opt* exchange neighborhood

• the Intra-Route exchange neighborhood

• the cyclic exchange neighborhood

The cross exchange neighborhood

1cross lL

))(( 22cross nLO

cross2 Ll

The ２ -opt* exchange neighborhood

)( 2nO

The intra-route neighborhood

intrapathpath Ll

insintrains lL

)( intrapath

intrapath nLLO

The cyclic exchange neighborhood

a set of solutions obtained by exchanging paths of length at most among several routes of at mostcyclicL Ψ

The neighborhood size grows exponentially with and Ψn

cyclicL以下

cyclicL以下

Effective search via an improvement graph

• an arcbelong to different routes

The improvement graphAn improvement graph is defined with respect to the current solution σ

))(),(()( σEσVσG

corresponds to a path)(σVvil

][ 1ikσ

11liv22liv

)()'( ][][ 22 ikik σostcσostc 11lip

22lip

customer1i 2i

1l 2l

][ 2ikσ

ilp• a node

)(),(2211

σEvv lili exists if paths and11lip22lip

customer

The improvement graph・ a cycle C　 is subset-disjoint　：　 all paths corresponding to nodes in C belongs to different routes

・ valid cycle　：　 subset-disjoint cycle with a negative cost

Identifying a valid cycle is NP-hard

Effective heuristic is proposed

a valid cycle a corresponding operationis cost-decreasing

Find the optimal start times

Dynamic Programming Approach

Problem

Input: the customer order of the vehicle (which is denoted by )Output: the start times of services that minimize the total time penalty of the vehicle

k

k

)(),(,),1(),0( 1kkkkkk nn σσσσ

)()()()( QTDstco objective function

DP Algorithm)(tf k

h : the minimum penalty value if customers of the vehicle are serviced before time t

11)),'()'((min)(

),[,0

),(,)(

11'

0

0

0

khh-k

htt

kh

k

nhtptftf

et

ettf

τ

k

(h)σ,),(σ),(σ kkk 10

)(tpkh

0eht

)(hkσ: a time penalty function for customer

: traveling time from the (h-1) st to the h th customer

: the departure time of the vehicles from the depot

))'()'((min)( 11'

tptftf hh-k

htt

kh

τ

t)(1 tf k

h

)( 11 hk

h tf τ

)()( 11 tptf khh

kh τ

)(tpkh

)(tf kh

pena

lty

Time complexity of DP

0 1 2 3 1knkn

)( kknO time

kδ : the total pieces of piecewise linear functions for customers in route k

kn : the number of customers in route k

)(tf knk

)( kO

…

)( kO time

)(0 tf k

)( kO

)(1 tf k

)( kO

)(2 tf k

)( kO

)(3 tf k

)( kO

)(1 tf knk

)( kO

Optimal penalty obtained

Iterated Local Search（ ILS）•The operation that repeats LS more than once.•Initial solutions are generated using the information of the previous search.• Final output is the best solution of the entire search.

Adaptive Multi-start Local Search （ AMLS）

• LS is repeatedly applied.• a set of locally optimal solutions obtainedin the previous search is maintained. • an initial solution for LS is generated bycombining • Final output is the best solution of the entire search.

…1σ σ

P0σ

2σ

Computational experiments

Solomon’s benchmark instances•Instance

• only one time window is given.• both capacity and time window constraintsare treated as hard constraints.

•Experiment’s method• ILS and AMLS are run for 15000 seconds.• Compare the costs of the best solutions output by ILS and AMLS with those of the best known solutions.

instance ILS AMLS best instance ILS AMLS bestR101 1650.8 1650.8 1650.8 RC101 1696.95 1696.95 1696.94R102 1487.88 1486.12 1486.12 RC102 1561.66 1641.51 1554.75R103 1293.85 1356.61 1292.85 RC103 1265.24 1261.77 1262.02R104 988.28 986.03 982.01 RC104 1137.03 1138.34 1135.48R105 1377.11 1377.11 1377.11 RC105 1693.96 1643.96 1633.72R106 1261.94 1257.96 1252.03 RC106 1426.6 1448.26 1427.13R107 1124.3 1118.98 1113.69 RC107 1232.26 1232.26 1230.54R108 962.34 963.99 964.38 RC108 1141.76 1146.47 1139.82C101 828.94 828.94 828.94C102 828.94 828.94 828.94C103 828.05 828.05 828.05C104 824.78 824.78 824.78C105 828.94 828.94 828.94C106 828.94 828.94 828.94C107 828.94 828.94 828.94C108 828.94 828.94 828.94

Computational experiments （ type1）

improvedtieinfeasible

3 improved11 tie

instance ILS AMLS best instance ILS AMLS bestR201 1253.23 1253.23 1252.37 RC201 1446.2 1428.1 1406.94R202 1201.24 1201.69 1191.7 RC202 1442.77 1376.03 1389.57R203 953.98 946.2 942.64 RC203 1096.15 1063.68 1060.45R204 853.86 848.59 849.62 RC204 799.16 800.83 799.12R205 1026.25 1006.66 994.42 RC205 1314.4 1300.25 1302.42R206 913.18 914.28 912.97 RC206 1167.28 1152.03 1153.93R207 906.33 908.35 914.39 RC207 1064.05 1086.46 1062.05R208 736.43 726.82 731.23 RC208 838.95 828.14 829.69C201 591.56 591.56 591.56C202 591.56 591.56 591.56C203 591.17 591.17 591.17C204 590.6 590.6 590.6C205 588.88 588.88 588.88C206 588.49 588.49 588.49C207 588.29 588.29 588.29C208 588.32 588.32 588.32

Computational experiments （ type2）

improvedtieinfeasible

6 improved8 tie

Product and Inventory Scheduling

Application of VRPGTW

Collaboration Research with

KOKUYO Co.,Ltd.

ProblemInput : the number of machines,

product demands, setup costs, inventory costs,

Output : minimum cost schedule

M),,1( pi niD

),,1,( pp

ij njic ),,1( p

si nic

An example of a schedule

Machine 1

Machine 2

Machine M

time

Inventory

t

iD

T

T

dt0

(total inventory) (inventory)

accumulated consumption lineof product i

inve

ntor

y

Formulation to VRPGTW (1)is divided by the parameter iD il

icustomers represent the product and each customercorresponds to producing the amount .

il

ii lD

iDii lD

ii lD

ii lD

ii lDii lD

ii lD

ii lD

ii lD

ii lD

Formulation to VRPGTW (2)

｛ ｝

｛ ｝

ijd

setup time ijt

ijd

ijt

setup cost

Formulation to VRPGTW (3)

t

iD

Tdesired produce start time

t

yen

T

jcustomer

Computational experiments

• Compare the costs of the best solutions output by the ILS with those of the current real schedule.• Compare also the costs with different values of .),,( 1 pnlll

Computational experimentsdata cost(yen) CPU(s) data cost(yen) CPU(s)

1236830 755 875371 3171187407 12644 808687 5784

real 2053816 real 1786060

data cost(yen) CPU(s) data cost(yen) CPU(s)1078367 533 1019018 4191033239 9994 964079 8655

real 2196899 real 1865188

data cost(yen) CPU(s) data cost(yen) CPU(s)1079091 310 868245 4091050375 4472 811308 6262

real 1919381 real 1707792

1999.11

1999.12

2000.01

2000.02

2000.03

2000.04

1ll 12ll

2ll 22ll

3ll

32ll

42ll 4ll

52ll 5ll

62ll 6ll

Conclusion• We proposed the local search heuristic for the Vehicle

Routing Problem with General Time Windows Constraints.

• Our general algorithm produced 9 improved solutions and19 tie solutions out of 48 instances.

• The effectiveness of our algorithm was confirmed through the application to the KOKUYO problem.

DP algorithm is incorporated.