toshihide ibaraki mikio kubo tomoyasu masuda takeaki uno mutsunori yagiura
Post on 16-Jan-2016
22 Views
Preview:
DESCRIPTION
TRANSCRIPT
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.
top related