COMPARYING A TABU SEARCH PROCESS

Using and Not Using and Intensification Strategy to Solve

the Vehicle Routing Problem

Etiene Pozzobom Lazzeris Simas

Universidade do Vale do Rio dos Sinos. Av. Unisinos, 950, São Leopoldo, Brasil

Arthur Tórgo Gómez

Universidade do Vale do Rio dos Sinos. Av. Unisinos, 950, São Leopoldo, Brasil

Keywords: Vehicle Routing Problem; Tabu

Search, Intensification Strategy.

Abstract: In this paper we propose a Tabu Search algorithm to solve the Vehicle Routing Problem. The Vehicle

Routing Problem are usually defined as the problem that concerns in creation of least cost routs to serve a

set of clients by a fleet of vehicles. We develop an intensifications strategy to diversify the neighbours

generated and to increase the neighbourhood size. We had done experiments using and not using the

intensification strategy to compare the performance of the search. The experiments we had done showed

that an intensification strategy allows an increase on the solutions quality.

1 INTRODUCTION

The Vehicle Routing Problem (VRP) is a NP-Hard

problem (Lenstra and Rinooy Kan, 1981) that is

usually dealt within the logistic context (Ho and

Haugland, 2004; Xu and Kelly, 1996). It can be

described as a set of customers that have to be

served by a fleet of vehicles, satisfying some

constraints (Laporte, 1992; Xu and Kelly, 1996).

The Transport is one of the most costly activities in

logistic, typically varying in one or two thirds of the

total costs (Ballou, 2001). Therefore, the necessity

of improving the efficiency of this activity has great

importance. A small percentage saved with this

activity could result in a substantial saving total

(Bodin, Golden and Assad, 1983). There many

variants and constraints that can be considered, i.e. it

can be considered that the fleet may be

heterogeneous the vehicles must execute collections

and deliveries, there may exist more than one depot,

etc. In this paper we are dealing with the classic

version of this problem, were just the vehicle

capacity constraint are considered.

2 THE VEHICLE ROUTING

PROBLEM

A classical definition is presented in Barbarasolgu

and Ozgur (Barbarasolgu and Ozgur, 1999). The

VRP is defined in a complete, undirected graph

G=(V.A) where a fleet of Nv vehicle of

homogeneous capacity is located. All remaining

vertices are customers to be served. A non-negative

matrix C=(c

) is defined on A representing the

distance between the vertices. The costs are the same

in both directions. A non-negative demand di, is

associated with each vertex representing the

customer demand at vi. The routes must start and

finish at the depot. The clients must be visited just

once, by only one vehicle and the total demand of

the route can’t exceed the capacity Qv of the

vehicle. In some cases, there is a limitation on the

total route duration. In this case, t

is defined to

represent travel time for each (vi,vj), ti represents

the service time at any vertex vi and is required that

the total time duration of any route should not

exceed Tv. A typical formulation based on

Barbarasoglu and Ozgur ones (Barbarasolgu and

zgur, 1999) are used in this paper:

Pozzobom Lazzeris Simas E. and Tórgo Gómez A. (2007).

COMPARYING A TABU SEARCH PROCESS - Using and Not Using and Intensiﬁcation Strategy to Solve the Vehicle Routing Problem.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 77-84

DOI: 10.5220/0001638700770084

 SciTePress

Minimize

∑∑∑

(1)

Subject:

∑∑

for all j

(2)

∑∑

for all i

(3)

0=−

∑∑

for all p,v

(4)

d <=

⎟

⎠

⎞

⎜

⎝

⎛

∑∑

for all v

(5)

∑

for all v

(6)

∑

for all v

(7)

∈

for all i,j and v

(8)

Where are binary variables indicating if

arc(vi,vj) is traversed by vehicle v. The objective

function of distance/cost/time is expressed by eq.

(1). Constraints in eqs (2) and (3) together state that

each demand vertex is served by exactly one vehicle.

The eq. (4) guarantees that a vehicle leaves the

demand vertex as soon as it has served the vertex.

Vehicle capacity is expressed by (5) where Q

f b

is the

capacity. Constraints (6) and (7) express that vehicle

availability can’t be exceeded. The sub tour

elimination constraints are given in eq.(8) where Z

can be de ined y:

(

)

{

1: −<==

∑∑

∈∈

BjBi

for

}

2};0/{ >=⊆ BVB

(8)

3 RESOLUTIONS METHODS

Since VRP is Np-Hard to obtain good solutions in an

acceptable time, heuristics are used and this is the

reason why the majority of researchers and scientists

direct their efforts in heuristics development

(Thangiah and Petrovik, 1997; Nelson et al, 1985;

Xu and Kelly, 1996). Osman and Laporte (Osman

and Laporte, 1996) define heuristic as a technique,

which seeks good solutions at a reasonable

computational cost without being able to guarantee

the optimality. Laporte et al (Laporte et al, 2000)

define two main groups of heuristics: classical

heuristics, developed mostly between 1960 and

1990, and metaheuristics. The classical heuristics

are divided in three groups: constructor methods,

two-phase methods and improvement methods.

Since 1990, the metaheuristics have been applied to

the VRP problem. To Osman and Laporte (Osman

and Laporte, 1996) a metaheuristic is formally

defined as an iterative generation process which

guides a subordinate heuristic by combining

intelligently different concepts for exploring and

exploiting the search space in order to find

efficiently near-optimal solutions. Several

metaheuristics have been proposed to solve the VRP

problem. Among these ones, Tabu Search are

considered the best metaheuristic for VRP. To

review some works with Tabu Search and others

metaheuristics some readings are suggested

(Cordeau et al, 2002; Tarantilis et al, 2005).

3.1 Tabu Search

It was proposed by Glover (Glover, 1989) and had

its concepts detailed in Glover and Laguna (Glover

and Laguna, 1997). It’s a technique to solve

optimization combinatorial problems (Glover, 1989)

that consists in an iterative routine to construct

neighborhoods emphasizing the prohibition of

stopping in an optimum local. The process that Tabu

Search searches for the best solution is through an

aggressive exploration (Glover and Laguna, 1997),

choosing the best movement for each iteration, not

depending on if this movement improves or not the

value of the actual solution. In Tabu Search

development, intensification and diversification

strategies are alternated through the tabu attributes

analysis. Diversification strategies direct the search

to new regions, aiming to reach whole search space

while the intensification strategies reinforce the

search in the neighborhood of a solution historically

good (Glover and Laguna, 1997). The stop criterion

makes it possible to stop the search. It can be

defined as the interaction where the best results were

found or as the maximum number of iteration

without an improvement in the value of the objective

function. The tabu list is a structure that keeps some

solution’s attributes that are considered tabu. The

objective of this list is to forbid the use of some

solutions during some defined time.

4 VRP APPLICATION

The application developed are divided into three

modules:

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

a) Net Generation module: This module generates

the nets that will be used in the application using

vertices coordinates and demands given.

b) Initial Solution module: This module generates

the initial solutions of the nets. The initial solutions

are created thought the use of an algorithm

implementing the Nearest Insertion heuristic (Tyagi,

1968; Cook et al, 1998).

c) Tabu Search module: This module performs the

tabu search algorithm. The Tabu Search elements

that were used are now detailed. The stop criterion

adopted is the maximum number of iterations

without any improvement in the value of the

objective value. The tabu list keeps all the routes and

the cost of the solution, forbidden these routes to be

used together during the tabu tenure defined. And

elite solution list is used to keep the best results that

were found during the search. It was proposed an

intensification strategy to be used every time when

the search executes 15 iterations without an

improvement in the objective function value. In this

strategy we visit every solution that is in elite list

generating a big neighbourhood for each one. There

were defined two movements to neighbourhood

generation. V1, that makes the exchange of vertices

and V2, that makes the relocation of vertices. In V1,

one route r1 is selected and than one vertex of this

route is chosen. We try to exchange this vertex with

every vertex of all the other routes. The exchange is

done if the addition of the two new demands doesn’t

exceed the vehicle’s capacity of both routes. This

procedure is done for every vertex of the route r1.

To every exchange that is made, one neighbour is

generated. In V2, we select one route and choose

one vertex and then we try to reallocate it into all

others routes, if it doesn’t exceed the vehicle

capacity of the route. When a vertex can be insert

into a route, we try to insert it into all possible

positions inside this route. To every position that a

vertex is inserted, one neighbour is generated.

When these movements are used in the search with

intensification, they are called V1’ and V2’ because

with intensification, not only one route is selected

like in V1 and V2, but also all routes of the solution

are chosen. Aiming increase the neighbourhood size

and the diversification between the solutions, we

proposed to use the movements alone and together.

5 COMPUTACIONAL

EXPERIENCE

The computational experiments were conducted on

problems 1, 2, 3, 4 and 5 of Christofides Mingozzi

and Toth (Christofides Mingozzi and Toth, 1979).

These problems contain 50, 75, 100, 150 and 199

vertices and one depot respectively and they are

frequently used in papers for tests purposes. The

objective of the experiments was to compare the

search process using and not using intensification

strategy using the different movements proposed.

There were proposed 9 values to Nbmax {100, 250,

500, 750, 1000, 1250, 1500, 1750, and 2000} and 6

values to Tabu List size {10, 25, 50, 75, 100, 200}.

For every problem, the experiments were divided

into 6 groups according with the used movements.

Table 1 shows these groups.

Table 1: Groups of experiments divided by movements.

Search mode Used movement

Using Intensification V1

Using Intensification V2

Using Intensification V1,V2

Not Using Intensification V1 + V1’

Not Using Intensification V2 + V2’

Not Using Intensification V1V2 + V1’ V2’

There were generated 54 experiments for each group

combining all values proposed to Nbmax with all

Tabu List size. So, for each problem there were

generated 162 experiments using intensification and

162 experiments not using it. Two types of analyses

were done. In one type it was evaluated the best

result obtained for a fixed value of Nbmax used

with all Tabu List size and in other type it was

evaluated the best results obtained for a fixed size of

Tabu List used with all values proposed to Nbmax.

Analyses had also been made comparing the best

result found for each group of experiment, in this

case comparing the quality of the different

movements.

5.1 Analysing the Nbmax Variation for

Each Tabu List Size

By analysing the results in this perspective it will be

evaluate the variation of the Nbmax for each Tabu

List size. The objective is verified if big values of

Nbmax can improved the quality of Tabu Search

process. We create a “lower average” for the average

from results obtained with Nbmax = 100 and Nbmax

= 250 and an “upper average” for the average from

results obtained with Nbmax = 1750 and 2000. For

all problems, analysing each one of the 6 groups of

experiments done, the “upper average” were always

lesser than the “lower average”, indicating that big

values of Nbmax can improve the search quality.

Figure 1 shows an example of the graphics

COMPARYING A TABU SEARCH PROCESS - Using and Not Using and Intensification Strategy to Solve the Vehicle

Routing Problem

generated with the results of the search process. It’s

clear to see that an increase in Nbmax value can

improve the search quality, by decreasing the results

costs of the solutions.

Results obtained for a Nbmax variation with fixed Tabu list size for a

search process not using intensification with V1,V2

1300

1320

1340

1360

1380

1400

1420

1440

1460

250

500

1250

1750

Nbmax value

Cost

Solution cost

w ith Tabu List

size = 10

Solution cost

w ith Tabu List

size = 25

Solution cost

w ith Tabu List

size = 50

Solution cost

w ith Tabu List

size = 75

Solution cost

w ith Tabu List

size = 100

Solution cost

w ith Tabu List

size = 200

Figure 1: Costs obtained from Nbmax variation with

different Tabu List size for problem 5 using V1 and V2

without intensification.

Tables 2 to 6 show the number of best results that were

found in each Nbmax value:

Table 2: Quantity and localization of the best Results

found for problem 1.

500 750 1000 1250 1500 1750 2000

Int. 1 1 3 3 3 4 3

Int.

0 0 1 2 4 6 5

Table 3: Quantity and localization of the best Results

found for problem 2.

500 750 1000 1250 1500 1750 2000

Int. 1 0 1 3 4 2 7

Int.

0 0 1 2 3 6 6

Table 4: Quantity and localization of the best Results

found for problem 3.

500 750 1000 1250 1500 1750 2000

Int. 0 1 0 2 1 4 10

Int.

0 1 0 1 2 1 12

Table 5: Quantity and localization of the best Results

found for problem 4.

500 750 1000 1250 1500 1750 2000

Int. 0 0 1 2 3 1 11

Int.

0 0 1 4 1 2 10

Table 6: Quantity and localization of the best Results

found for problem 5.

500 750 1000 1250 1500 1750 2000

Int. 0 0 2 3 0 1 12

Int.

0 0 0 1 1 5 11

These tables shown that most best results were

found when the search used big values of Nbmax.

5.2 Analysing the Tabu List Size

Variation for Each Nbmax Value

By analysing the results in this perspective it will be

evaluate the variation of the tabu list size for each

Nbmax value. The objective is verified if big Tabu

list size can improved the quality of Tabu Search

process. For problem 1, not using intensification,

66,66% of the results were found with Tabu list size

>=75. Using intensification it was 48,14%. For

problem 2 the percentage were 55,55% and 59,25%,

not using and using intensification. For problem 3

the percentage were 77,77% and 96,29% not using

and using intensification. For problem 4 these

percentage were 74,05% and 77,77% and for

problem 5 they were 63,88% and 92,59%. So by

analysing these results we can see that big tabu list

size can improve the quality of the search process.

5.3 Comparing the Search Process

using and not using Intensification

This analysis intend to compare the search process

using and not using the intensification strategy to see

if it can improve the results generated.

Figure 2 shows an example of the graphics done

with the results obtained in both search process to

compare the quality of the different search process.

Results from search process using V2

1250

1300

1350

1400

1450

1500

75/10

75/50

000

5/125

75/1500

5/175

5/2000

Tabu List/Nbmax

Cost

Not Using Intensification Using intensification

Figure 2: Costs obtained from both search process for

Tabu List size = 75 and using V2 for problem 5.

This figure shows that an intensification strategy

increase all results of the search process using V2

for problem 5. A comparison with the results

generated by the search process using and not using

intensification was done. Figures 3 to 7 shows the

percentage of results that were improved with the

intensification strategy. Figure 3 shows that for

problem 1, from 162 results that were generated 97

were improved with intensification.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

Quantity of best results obtained in search

process using and not using intensification for

problem 1

100

120

not using

intensification

using

intensification

Search type

Results quantity

Best results quantity

Figure 3: Improve caused by the intensification search for

problem 1.

For problem 2, from 162 results, 121 were improved

using intensification strategy.

Quantity of best results obtained in search

process using and not using intensification for

problem 2

100

120

140

not using

intensification

using

intensification

Search type

Resuls quantyti

Best results quantyti

Figure 4: Improve caused by the intensification search for

problem 1.

For problem 3, from 162 results 102 were improved

using intensification strategy.

Quantity of best results obtained in search

process using and not using intensification for

problem 3

100

120

not using

intensification

using intensification

Search type

Results quantyti

Best results quantyti

Figure 5: Improve caused by the intensification search for

problem 3.

For problem 4, from 162 results 117 were improved

using intensification strategy.

Quantity of best results obtained in search

process using and not using intensification for

problem 4

100

120

140

not using

intensification

using

intensification

Search type

Results quantyti

Best results quantyti

Figure 6: Improve caused by the intensification search for

problem 4.

For problem 5, from 162 results 135 were improved

using intensification strategy.

Quantity of best results obtained in search

process using and not using intensification for

problem 5

100

120

140

160

not using

intensification

using

intensification

Search Process

Results quantyti

Best results quantyti

Figure 7: Improve caused by the intensification search for

problem 5.

The figures show that an increase in solution quality

of, at least, 50% happens when intensification

strategy is used

The average results and the standard deviation are

shown for problem 1, 2, 3, 4 and 5 in tables 7 to 11.

Table 7: Average results and standard deviation for

problem 1.

Problem 1

Average

Standard Deviation

Not Using Intensive Using Intensive

V1 657,55 650,95 24,79 28,30

V2 582,08 570,30 21,16 22,21

V1,V2 537,36 542,01 8,56 13,62

Table 8: Average results and standard deviation for

problem 2.

Problem 2

Average

Standard Deviation

Not Using Intensive. Using Intensive

V1 951,07 943,02 21,68 25,84

V2 895,75 883,03 20,67 21,71

V1,V2 867,96 863,06 13,00 14,70

COMPARYING A TABU SEARCH PROCESS - Using and Not Using and Intensification Strategy to Solve the Vehicle

Routing Problem

Table 9: Average results and standard deviation for

problem 3.

Problem 3

Average

Standard Deviation

Not Using Intensive. Using Intensive

V1 954,01 948,12 19,60 16,95

V2 903,63 901,17 14,50 17,84

V1,V2 879,90 870,10 15,05 20,73

Table 10: Average results and standard deviation for

problem 4.

Problem 4

Average

Standard Deviation

Not Using Intensive. Using Intensive

V1 1215,35 1210,79 12,79 10,70

V2 1124,93 1118,83 15,66 12,52

V1,V2 1087,72 1079,54 18,15 16,70

Table 11: Average results and standard deviation for

problem 5.

Problem 5

Average

Standard Deviation

Not Using Intensive. Using Intensive

V1 1569,11 1561,59 10,12 16,22

V2 1421,41 1393,56 34,40 12,75

V1,V2 1387,93 1377,82 26,64 16,37

From the results presented in tables 7 to 11 we can

see that the results generated by the movements

grouped are better than the results obtained using the

movements alone. The reason for this is that when

movements are used together the size of the

neighbourhood generated is bigger than the

neighbourhood generated by V1 or V2 alone. The

movements together also cause an increase of the

diversification of the solutions. And when the search

generates more results, it is doing a deeper search in

the space. Of course, as was shown, the

intensification strategy helps the search to produce

more qualified results.

When comparing the V1 and V2 movements, we can

see that V2 produce results more qualified. If we

analyse the policy behind the movement, we can say

that V2 is more flexible than V1. V1 needs that two

constraints are satisfied to generate one neighbour.

While in V2, just one demand capacity must be

verified (the capacity of the vehicle that serve the

route where the vertex are being allocated) in V1,

both routes must be verified to see if the vehicles

capacities aren’t exceeded.

Next table shows the best results obtained for each

problem. All the best results were obtained during

the search using movements V1 and V2 together and

with the intensification strategy.

Table 12: Best results obtained.

Problem Best Result

1 525,42

2 847,82

3 837,79

4 1061,07

5 1352,74

5.4 Comparisons

Aiming to evaluate the quality of the application

developed, some papers were selected from the

literature to compare the results. There were selected

some classical heuristic and some papers that also

used Tabu Search to solve the VRP. The paper

selected were: {WL} Willard (1989), {PF}Pureza

and França (1991), {OM1} Osman (1991), {OM2}

Osman (1993), {RG} Rego (1998), {GHL}

Gendreau, Hertz and Laporte (1994), {BO}

Barbarasoglu and Ozgur (1999), {XK} Xu and

Kelly (1996), {TV} Toth and Vigo (2003), {CW}

Clarke and Wright (1964), {GM} Gillet and Miller

(1974), {MJ} Mole and Jamenson (1976), {CMT}

Christofides, Mingozzi and Toth (1979).

Table 13 shows the comparison done with the results

from the papers. The results were obtained in

Barbarasolgu and Ozgur (Barbarasolgu and Ozgur,

1999) and in Gendreau, Hertz and Laporte

(Gendreau, Hertz and Laporte, 1994). In the first

columns the paper used is shown. Columns 2 and 4

present the best results from the paper and columns

3 and 5 shows the difference in percentage from the

results obtained in this paper to the paper compared.

This difference was called “gap”. The (+) indicate

that our result is that percentage more than the result

from the paper. The (-) indicate that our results is

that percentage minor than the result from the paper.

Table 13: Best Results and gap for problem 1 and 2.

Problem 1 Problem 2

Best %Gap Best %Gap

WL 588 11,91(-) 893 5,33(-)

RG 557,86 6,17(-) 847 0,10(+)

PF 536 2,10(-) 842 0,69(+)

OM1 524,61 0,15(+) 844 0,45(+)

OM2 524,61 0,15(+) 844 0,45(+)

GHL 524,61 0,15(+) 835,77 1,42(+)

BO 524,61 0,15(+) 836,71 1,31(+)

XK 524,61 0,15(+) 835,26 1,48(+)

TV 524,61 0,15(+) 838,60 1,09(+)

CW 578,56 10,11(-) 888,04 4,74(-)

GM 546 3,92(-) 865 2,03(-)

MJ 575 9,44(-) 910 7,33(-)

CMT 534 1,63(-) 871 2,73(-)

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

Table 14: Best Results and gap for problem 3 and 4.

Problem 3 Problem 4

Best %Gap Best %Gap

WL 906 8,14(-) - -

RG 832,04 0,69(+) 1074,21 1,31(+)

PF 851 1,58(-) 1081 1,88(-)

OM1 835 0,33(+) 1052 0,85(+)

OM2 838 0,03(-) 1044,35 1,58(+)

GHL 829,45 1,00(+) 1036,16 2,35(+)

BO 828,72 1,08(+) 1043,89 1,62(+)

XK 826,14 1,39(+) 1029,56 2,97(+)

TV 828,56 1,10(+) 1028,42 3,08(+)

CW 878,70 4,88(-) 1204 13,47(-)

GM 862 2,89(-) 1079 1,69(-)

MJ 882 5,28(-) 1259 18,65(-)

CMT 851 1,58(-) 1093 3,01(-)

Table 15: Best Results and gap for problem 5.

Problem 5

Best %Gap

WL - -

RG 1352,88 0,014(-)

PF - -

OM1 1354 0,09(-)

OM2 1334,55 1,34(+)

GHL 1322,65 2,22(+)

BO 1306,16 3,44(+)

XK 1298,58 4,00(+)

TV 1291,45 4,53(+)

CW 1540 13,84(-)

GM 1389 2,68(-)

MJ 1545 14,21(-)

CMT 1418 4,82(-)

By analysing these tables we can see that our

application produce more qualified results than all

the classical heuristics used in comparison because

our result was better than all of the heuristic results.

Comparing with other tabu search algorithm, we can

say that our algorithm is very competitive. It

dominates at least 2 results from the 9 used for each

problem. Moreover, the results generated were less

than 5% of the other results for all cases. And in 25

cases out of 45 this percentage is minor than 2%.

6 FINAL CONSIDERATIONS

In this paper it was proposed an application using

Tabu Search to solve the vehicle routing problem.

This application was divided into 3 modules: a net

generation module, an initial solution module and

tabu search module. We used two movements based

in relocation of vertices and exchange of vertices to

create the neighbourhood. We use the movements

alone and together, intending to diversify the

solutions. We used an elite list solution to keep the

best results found during the search. We propose an

intensification strategy to use every time the search

executes 15 iterations without improvement in

objective value. We proposed some experiments to

test if the solution quality increase or not with the

increase in Nbmax value and in Tabu List size. We

also compare the search process using and not using

intensifications intending to see if this solution´s

quality is improved with the Intensification strategy.

The experiments showed that big values to Nbmax

and Tabu list size could improve the results. From

the experiments we also can see that an

intensification strategy can improve the quality of

the search.

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