Improved Heuristic Search Algorithm for Capacitated Vehicle Routing

Problem

Chunyu REN1, a 1 School of Information science and technology, Heilongjiang University, Harbin, China

arency2004@163.com

Keywords: Capacitated vehicle routing problem; 1-exchange; 2-opt*; Improved heuristic search algorithm

Abstract. The paper is focused on the capacitated vehicle routing problem. And solve this type of

problems utilizing improved Heuristic Search Algorithm from the overall situation. According to the

features of the problem, the essay centered the construct initial solution to construct neighborhood

structure. For the operation, 1-exchange and 2-opt* were applied, it can also fasten the speed of

convergence, and boost the search efficiency. Finally, the good performance of this algorithm can be

proved by experiment calculation and concrete examples.

Introduction

The vehicle routing problem of logistics distribution is indispensability contents in logistics

distribution optimization. Optimization of vehicle scheduling can improve logistics economic benefit

and realize to scientific development of logistics. Main research methods of CVRP include accuracy

algorithm, heuristic algorithm and intelligent optimization algorithm. Zhao Yanwei used

double-population genetic algorithm to solve CVRP problem [4]. Chen designed the improved ant

colony algorithm to solve CVRP [2]. Chen designed the hybrid ant colony algorithm that is to solve

VRP problems with ant colony algorithm as well as the simulated annealing algorithm alternately. It

introduced local search algorithm into ant colony algorithm and applied 2-Opt and SWAP operator

into the neighboring structure of the simulated annealing algorithm [3]. Lysgaard present a new

branch-and-cut algorithm for the capacitated vehicle routing problem [4]. Aiming at Capacitated

Vehicle Routing Problem (CVRP), ZHAO Yan-wei constructed a QEA with quantum rotation gate

and cataclysm in her paper [5]. Wu B proposed a novel real number encoding method of particle

swarm optimization for capacitated Vehicle Routing Problem [6].

Therefore, this article established model of CVRP and Improved Heuristic Search Algorithm to

solve it. Computational experiments show that this algorithm is an efficient method for CVRP

through a set of standard test problems.

Model

∑∑ ∑∈ ∈ ∈

=Si Sj Vkl

ij

l

ijkdXMin,

(1)

Restraint condition,

HjXVkl Si

l

ijk ∈=∑ ∑∈ ∈

　,1,

(2)

VklwXql

k

l

ijkHi Sj

i ∈≤∑∑∈ ∈

,,　　 (3)

VklSjYX l

ik

Si

l

ijk ∈∈=∑∈

,,, 　　 (4)

Applied Mechanics and Materials Vols. 361-363 (2013) pp 2079-2082Online available since 2013/Aug/08 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.361-363.2079

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 130.126.162.126, University of Illinois, Urbana, USA-06/10/14,13:51:49)

VklSiYXl

ikSj

l

ijk ∈∈=∑∈

,,, 　　 (5)

HjDdXVkl Si

l

kij

l

ijk ∈≤∑ ∑∈ ∈

　,,

(6)

In the formula, }{ RrgG r ,...,2,1= is the distribution centre muster of a series of R.

}{ NRRihH i ++= ,...1 is the customer muster of a series of N. { } }{HGS ∪ is the summation of all

distribution centre and customers. }{ KkLlvV lk ,...2,1,...2,1 == 　 is l model of the muster of transportation

vehicle k. iq is the demand amount of customer i ( )Hii ∈ . l

kW is l type of the load capacity of

transportation vehicle k. ijd is the beeline distance from customer i to customer j. l

kD is the maximum

running distance of l model of transportation vehicle k.

Application in CVRP of Improved Heuristic Search Algorithm

Sequence of Real Numbers Code. Therefore, consider to select vehicles when coding. Adopt

sequence of real numbers to code.

Initial Solution Forming. Supposed hk is the total number of all customer points for vehicle k.

Muster }0{ kikk hiyR ≤≤= is the customer point of vehicle k. ikY is the transportation tool of vehicle at

i point. kY0 is the distribution centre of starting point for vehicle k. The concrete steps are as

followings.

Step 1: Supposed the initial residual capacity of transportation vehicle as

kk ww =1 , 0=k , 0=kh , Φ=kR .

Step 2: The demand amount of i gene in a chromosome is iq and 1=k .

Step 3: If 1

ki wq ≤ , { }),( 11

kikk wqwMinw −= . Otherwise, it shifts to step 6.

Step 4: If kik wqw ≤−1 and kii DDD ≤+−1 , }{iRR kk ∪= and 1+= kk hh . Otherwise, it shifts to step 6.

Step 5: If Kk > , Kk = . Otherwise, kk = .

Step 6: 1+= kk , shift to step 3.

Step 7: 1+= ii , shift to step 2.

Step 8: Repeat from step 2 to step 7. K memorizes the total amount of all vehicles. kR memorizes a

group of feasible path.

1-exchange Neighborhood Operation. 1-exchange is to delete two clients in two routes,

alternately insert them into their counterpart route, which can effectively boost the local search

capability. Its neighborhood structure is the same as 1-move, but its radius can be larger.

2-opt* Neighborhood Operation. 2-opt* operates on the exchange of two edges in different

routes, in order to realize optimization between routes. That is in the route l , the client points

are )0,,...,2,1,0( n , in the route k , the client points are )0,,...,2,1,0( m , in it, 0 signifies distribution centre.

Step1: Randomly choose n number of client points in the route l , for each client point i , choose

client point j nearby the route k , if exist, exchange chains )1,(),1,( ++ jjii 　 ;

Step2: Conduct opt−2 neighborhood operation in the exchanged routes 1l and 1k , to obtain feasible

solution;

Step3: Calculate the exchanged objective function 1f , if ff >1 , turn to Step4; if not, turn to Step5;

Step4: If the current optimal solution does not exist in the tabu object, update it, input the obtained

optimal solution into the object, simultaneously remove out the ban-lifted elements; otherwise, turn to

Step5;

Step5: 1+= ii , turn to Step1;

Step6: repeat Step1- 5, till the current optimal solution can not update.

2080 Sustainable Cities Development and Environment Protection

Tabu object and length. The study takes the best solution of each iterative as tabu object and puts

them into tabu table. Tabu length is the pivotal parameter of algorithm, and its term will decide how to

select solution. The study sets the tabu length randomly selecting from 5 to 9.

Contempt regular. The study adopts the regular based on fitness value. If all solutions of

candidate muster are tabu solutions, liberate the best solution of candidate muster.

Ending principle. The study adopts iterative times by limited algorithm in advance as the ending

principle, which refers to confirm a big enough positive number so as that the total iterative times

don’t exceed this number. Iterative times in advance can effectively control operation time of

algorithm and is easy to operate.

Experimental Calculation and Analysis

The test question is: in a distribution system, there is a logistics center, 38 customer needs point of

the task, vehicle capacity of 100. The number of vehicles is 6 units. Other data can be found in table 1.

It is required that rationally arrange distribution vehicle so as to the shortest delivery mileage.

Table 1 Known condition of examples

Item 0 1 2 3 4 5 6 7 8 9

x-coordinate 39 79 41 25 63 33 69 57 53 1

y-coordinate 19 19 79 31 93 5 17 73 75 1

amount 0 18 16 22 24 3 19 6 6 6

Item 10 11 12 13 14 15 16 17 18 19

x-coordinate 79 59 1 41 23 37 85 93 85 49

y-coordinate 73 5 37 31 73 27 93 13 45 91

amount 12 18 16 72 7 16 23 4 22 23

Item 20 21 22 23 24 25 26 27 28 29

x-coordinate 55 83 93 87 31 19 41 83 9 13

y-coordinate 43 29 49 23 23 97 9 61 7 13

amount 7 11 11 1 22 16 15 7 5 22

Item 30 31 32 33 34 35 36 37 38

x-coordinate 43 13 71 45 93 5 81 7 7

y-coordinate 37 61 51 93 55 97 11 53 41

amount 9 10 11 9 3 7 15 10 2

Solution of Improved Heuristic Search Algorithm. After many trails, this algorithm adopts the

following parameters as part. The maximum iterative times are itermax_ =500, tabu length is L =8, and

candidate solution amount is 40.

Here, the total distance of best solution is 835.252 and concrete running path can be seen in table 2

and figure 2.

Table 2 Optimal results by IHSA

No. Running route

1 0-13-15-0

2 0-18-22-34-16-10-27-32-20-0

3 0-3-12-38-37-31-35-25-14-0

4 0-6-1-21-23-17-36-11-0

5 0-2-33-19-4-8-7-30-0

6 0-26-5-9-28-29-24-0

Total vehicles 6

Total distances 835.252

Applied Mechanics and Materials Vols. 361-363 2081

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100 Fig. 1 Optimal routes on solving CVRP by IHSA

Comparison with other algorithms. Compared the optimal scheme of reference [5,6],

experiments proved that this algorithm can achieve not only better calculating results, but also better

calculation efficiency and quicker convergence rate. The concrete scheme can be shown in Table 3.

Table3 Algorithm performance comparison

Algorithm Optimal results

Particle Swarm Optimization (PSO) 872

Genetic Algorithm(GA) 866

Quantum Evolutionary Algorithm (QEA) 837

This Algorithm 835.252

Conclusions

This algorithm can also enlarge the search scope of the solution; avoid local optimization so as to

ensure the solution’s diversity.

Acknowledgment

This paper is supported by Philosophy and social science planning project of Heilongjiang Province

(NO. 11D026).

References

[1] ZHAO Yan-wei, WU Bin, JIANG Li, DONG Hong-zhao, WANG Wan-liang. Double populations

genetic algorithm for vehicle routing problem. Computer Integrated Manufacturing Systems, Vol.

10(2004), p.303-306

[2] Chen Ch H, Ting Ch J. An improved ant colony system algorithm for the vehicle routing problem.

Journal of the Chinese Institute of Industrial Engineer, Vol. 23(2006), P. 115-126

[3] C. H. Chen, C. J. Ting, P. C. Chang, Applying a hybrid ant colony system to the vehicle routing

problem, in: Computational Science And Its Applications - ICCSA 2005, Proceedings, IV, Lecture

Notes in Computer Science, Vol. 34(2005),p. 417-426

[4] Lysgaard J, Letchford A N, Eglese R W. A new branch-and-cut algorithm for the capacitated

vehicle routing problem. Mathematical Programming, Vol. 100(2004), P. 423-445

[5] ZHAO Yan-wei, PENG Dian-jun, ZHANG Jing-ling, WU Bin. Quantum evolutionary algorithm

for capacitated vehicle routing problem. Systems Engineering-Theory & Practice, Vol. 29(2009),

P. 159-166

[6] Wu B, Wang W L, Zhao Y W, Xu X L, Yang F Y. A novel real number encoding method of

particle swarm optimization for vehicle routing problem. The 6th World Congress on Intelligent

Control and Automation, VOLS1-12, Conference Processing, (2006), P. 3271-3275

2082 Sustainable Cities Development and Environment Protection

Sustainable Cities Development and Environment Protection 10.4028/www.scientific.net/AMM.361-363 Improved Heuristic Search Algorithm for Capacitated Vehicle Routing Problem 10.4028/www.scientific.net/AMM.361-363.2079

DOI References

[3] C. H. Chen, C. J. Ting, P. C. Chang, Applying a hybrid ant colony system to the vehicle routing problem,

in: Computational Science And Its Applications - ICCSA 2005, Proceedings, IV, Lecture Notes in Computer

Science, Vol. 34(2005), pp.417-426.

http://dx.doi.org/10.1007/11424925_45