HYBRIDIZING ANT COLONY SYSTEMS AND TABU SEARCH FOR

A VEHICLE ROUTING PROBLEM WITH TIME WINDOWS

Juan Carlos Figueroa D., M. Ang

elica Pinninghoff J. and Ricardo Contreras A.

Department of Computer Science, University of Concepci

on, Concepci

on, Chile

Keywords:

Ant colony system, VRPTW, Tabu search.

Abstract:

This paper describes a new approach for solving the vehicle routing problem that considers time windows

(VRPTW). The proposal presents a hybrid approach that takes into account an ant colony system ACS and the

meta-heuristic Tabu Search. Hybridizing meta-heuristics is one of the alternatives used for solving VRPTWs.

Authors believe that a hybrid approach, with ACS providing good initial solutions for the Tabu Search heuristic

can help to get acceptable ﬁnal solutions. Tabu Search plays the role of keeping diversity in the population

considered while searching a solution. The proposal was implemented and tested, and results obtained are

discussed in the ﬁnal part of this presentation.

1 INTRODUCTION

The key idea concerning VRP (Toth and Vigo, 2001)

involves a routing problem with vehicles that have a

speciﬁed capacity (CVRP); this is known as a prob-

lem having multiple variables to optimize, and there-

fore, is a classical combinatorial problem. The most

popular version associated to this problem is the ve-

hicle routing problem with time windows (VRPTW),

due to his wide applicability to real problems.

All problems labeled as VRP may be described

by using a graph G deﬁned as G = (V, A), where V =

{

0. . . n

}

is the set of nodes belonging to the graph,

and A is the set of edges that connect a speciﬁc pair

of nodes. Meta-heuristics are used to generate a set of

circuits that include the V nodes exactly once, except

one speciﬁc node that represents the starting and ﬁnal

node, known as node 0 (the depot in this work).

There have been proposed different mechanisms

to deal with this problem; genetic algorithms (GA) as

in (Zhu, 2000), where authors test different crossover

operators hybridizing with hill climbing, or in (Om-

buki et al., 2006) for optimizing distances and the

number of vehicles. In (Jaszkiewicz and Kominek,

2003) the CVRP problem is presented under a genetic

algorithm approach. Finally (Berger and Barkaoui,

2004) develops a parallel GA, where a population

evolves focusing in minimizing distances.

Ant Colony System (ACO) is also used to solve

this problem. In (Tao et al., 2009) the VRPTW is

solved by combining ACO with local search algo-

rithms. Vehicles after returning to the depot can be

re-used to visit different customers if time windows

allow it. (Yu et al., 2011) deal with VRPTW com-

bining ACO and Tabu Search. We consider, when

selecting the next node in the ACS algorithm, not

only distance and pheromone, we consider the un-

used capacity in the vehicles and the waiting intervals

in time windows. Besides that, we use 2-opt search

trying to ﬁnd improvements between two paths, this

is one of the differences we introduce. The objec-

tive of this proposal is to verify if hybridizing ACS

with Tabu Search allows to improve results obtained

by Solomon (Solomon, 2006).

This article is structured as follows; the ﬁrst sec-

tion is made up of the present introduction; the second

section describes the problem; the third section is de-

voted to introduce the proposal, while in section four

we present some results obtained. In section ﬁve we

end with the conclusions.

2 PROBLEM DESCRIPTION

The practical problem we are interested in, that takes

as a starting point the routing vehicle problem, deals

with routing garbage trucks, a problem that presents

various objectives to optimize, such as the number of

trucks to be used, which in turn is associated to the

number of paths and the distance involved in each

469

Carlos Figueroa D. J., Angélica Pinninghoff J. M. and Contreras A. R..

HYBRIDIZING ANT COLONY SYSTEMS AND TABU SEARCH FOR A VEHICLE ROUTING PROBLEM WITH TIME WINDOWS.

DOI: 10.5220/0003701104690472

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 469-472

ISBN: 978-989-8425-95-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

path. Not less important, it is necessary to consider

the time involved in each path and constraints such

as speciﬁc (and limited) time period in which the

garbage collection can be done. The last constraint

shows that we are solving a problem that can be clas-

siﬁed as VRPTW.

VRPTW needs to satisfy a time window deﬁned

for a customer i as [a

, b

], which expect to accom-

plish the job in a speciﬁc time interval. The starting

time cannot be previous to a

and the arrival time w

plus the time devoted to a particular customer i can-

not exceed b

. The central depot, represented by 0 is

also associated to a time window, which is deﬁned as

[E, L], where E represents the time in which the trucks

begin to traverse their routes, and L the time in which

all trucks should be in the depot. If truck k arrives to a

customer i location in a time w

and satisﬁes w

< a

the truck k has to wait until w

≥ a

A good solution optimizes (minimizes) the sum of

costs involved in the K path considered for the vehi-

cles.

The cost associated to a path for a vehicle k is de-

termined by the sum of costs c

i j

corresponding to the

edges (i, j) that belong to the path. This solution must

satisfy the following constraints:

i) A customer is visited by, exactly, one vehicle,

ii) every path has only one edge from the depot, iii)

every node in the path has only one edge arriving to

it, and only one edge starting from it, iv) every path

has only one edge arriving to the depot, v) the sum

involving the customer’s attention time for customer i

and the time involved in going from i to j needs to be

less or equal to the time in which attention for a new

customer j begins, vi) the arrival time to a customer i

location, for one vehicle, needs to be associated to the

time interval deﬁned for customer i, vii) starting time

from the depot, and arrival time to the depot need to

be considered in the interval associated to the depot

and viii) the sum of customer’s demands, for a partic-

ular path cannot exceed the vehicle capacity.

3 THE PROPOSAL

We propose to combine ACS with Tabu Search as fol-

lows: For each iteration in ACS, once the best route is

found by an ant colony, this route is improved through

2-opt. The route obtained in this way is used as input

data for applying the meta-heuristic Tabu Search. If

Tabu Search effectively improves the solution, it is re-

introduced into the colony, just before the process of

global pheromone updating.

Ant Colony System (ACS) was proposed by

Dorigo in (Dorigo and Stuetzle, 2004). As in other

ACO algorithms, ACS is a meta-heuristics inspired

in ants’ behavior when trying to ﬁnd food. In the

collaborative behavior, communication among ants is

accomplished through a pheromone trail, that is se-

creted by the ants when moving from one point to

another, and that is used to inﬂuence the ants behav-

ior that contact this chemical factor. In this particular

work, we use ACS, that is characterized for having

a local and a global mechanism for updating the se-

creted artiﬁcial pheromone, allowing a search process

in which random aspects are more relevant.

Tabu Search is a meta-heuristic algorithm that can

be used for solving combinatorial optimization prob-

lems. Tabu search uses a local or neighborhood search

procedure to iteratively move from a solution x to a

solution x

in the neighborhood of x, until some stop-

ping criterion has been satisﬁed. To explore regions

of the search space that would be left unexplored by

the global search procedure, ACS in this case, Tabu

Search modiﬁes the neighborhood structure of each

solution as the search progresses (Glover and Laguna,

1997).

When building a solution to the VRPTW, each ant

begins their path from the depot and visits different

customers, until the capacity of the vehicle is com-

pleted. Then the ant goes back to the depot forming

a path. This process is repeated until all customers

have been visited. In this way, the solution for one ant

builds as many path as times has started a search from

the depot.

ACS features are: i) Each time an ant choose a

customer, they have the possibility of doing it in a

probabilistic or deterministic way; this possibility is

given by a parameter called q

. ii) Pheromone up-

dating is accomplished through a global process that

considers the best solution that has been reached until

this moment. iii) Each ant diminishes a certain quan-

tity of pheromone each time it goes through an edge,

allowing this way that future ants can explore differ-

ent paths.

In ACS, when an ant that is in node i has to select

a new node, not previously visited, from a set N(i), it

considers the quantity of pheromone and the cost of

traversing an edge (i, j), for each j ∈ N(i). Besides

that, in this proposal two additional variables as pro-

posed in (Bullnheimer et al., 1997) were considered;

the ﬁrst one is the time to arrive to the customer j

location, with respect to their time window (∆v) de-

scribed in equation (1), and the second is the wasted

capacity of a vehicle (∆c) described in equation (2).

In this way equation (3) describes the mechanism an

ant uses for selecting a node j.

ICAART 2012 - International Conference on Agents and Artificial Intelligence

470

∆v =











1, i f w

= a

;

0, i f w

> b

;

−w

, otherwise;

(1)

∆c =











1, i f C =



∑

i∈N

∑

j∈∆

(i)

i jk



;

0, i f C <



∑

i∈N

∑

j∈∆

(i)

i jk



;

|C−



∑

i∈N

∑

j∈∆

(i)

i jk



, otherwise;

(2)

j =











argmax

l∈N

[η

]

, i f q ≤ q

;

[

i j

]

[

i j

]

[∆v]

[∆c]

∑

l∈N

[

i j

]

[

i j

]

[∆v]

[∆c]

, i f j ∈ N

, otherwise;

(3)

Where q is a random number, q

is the param-

eter mentioned above; τ

i j

and η

i j

are the level of

pheromone and the cost of edge (i, j), respectively.

α, β, ζ and θ are the parameters that represent the in-

ﬂuence of each variable.

For Ant Colony System (Dorigo and Gambardella,

1997) it was established the need to get a global

pheromone updating to characterize the distance in-

volved in the solution and the number of vehicles in-

volved in this solution. This is described through the

formula 4.

i j

← (1 − ρ) τ

i j

+ ρ ∗



∆τ

i j

+ ∆v



, (4)

Where ρ is the parameter that indicates the level

of pheromone to be updated; ∆τ

i j

cost−best−solution

and ∆v

vehicles−best−solution

The local updating of pheromone is given by

equation (5)

i j

← (1 − ξ)τ

i j

+ ξτ

, (5)

Where 0 < ξ < 1 is the quantity of pheromone to

evaporate and τ

is the initial level of pheromone ini-

tially allocated to every edge.

Additionally, solutions obtained from ACS have

been improved with 2-opt, that is able to obtain re-

sults rapidly. It is due to his simple approach, intend-

ing to replace the edges under analysis by other edges

having a lower cost, and that it can be done by re-

structuring the nodes associated to those edges.

4 RESULTS

The following table shows the values for parameter

used in testing the hybrid algorithm, that were ob-

tained through experimentation.

In the table, NACS represents the number of ant’s

colonies; H represents the number of ants per colony.

Table 1: Set of parameters for ACS/2-opt.

NACS H α β ζ

10 1000 3 4 2

θ q

ρ ξ τ

1 0, 4 0, 1 0, 1 3.6724741E −6

In graphics shown in ﬁgures 1 and 2, we can see

the hybrid algorithm performance that correspond to

two of the three categories proposed by Solomon.

When analyzing these graphics, it is possible to de-

duce that the proposed algorithm offers a good aver-

age behavior. The worst case corresponds to a differ-

ence of two additional vehicles for only one problem

(R112). There is a difference of (at most) one addi-

tional vehicle in some problems. For all the other in-

stances our proposal offers solutions that are similar

to solutions in Solomon.

Figure 1: Comparison of Best result, ACS/2-opt and Hybrid

approach in problems with preﬁx C (on the number of ve-

hicles).

Figure 2: Comparison of Best result, ACS/2-opt and Hybrid

approach in problems with preﬁx R (on the number of ve-

hicles).

One of the problems this approach presents, is the

distance involved in the vehicles routes, because in

some cases the distance increases. It is important to

notice that this behavior is likely to occur when cus-

tomers are grouped into clusters, having time win-

dows with large intervals.

HYBRIDIZING ANT COLONY SYSTEMS AND TABU SEARCH FOR A VEHICLE ROUTING PROBLEM WITH

TIME WINDOWS

471

Figure 3: Comparison of Best result, ACS/2-opt and Hybrid

approach in problems with preﬁx C (on the distance).

Figure 4: Comparison of Best result, ACS/2-opt and Hybrid

approach in problems with preﬁx R (on the distance).

5 CONCLUSIONS

Hybridizing meta-heuristics appears as a promising

strategy for solving combinatorial problems; but there

is a lot of work still to be done. Results illustrate a

wide variety of cases; but it is necessary to handle

these cases carefully. On one hand, we got good re-

sults in the case of speciﬁc instances, but in most of

other instances results were deﬁnitely poor.

Solomon, on the other hand, presents academic

examples, that not necessarily reﬂect a real city cus-

tomer distribution. The euclidean computation of the

distances should be probably replaced by a Manhattan

distance computation. In the same sense, the random

distribution of customers in a city doesn’t reﬂect, in

most of cases, a real city.

However, the basic idea of combining different

mechanism that provides, from their own features,

promising pre-processed solutions is something that

should be explored more deeply.

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