A HYBRID ANT COLONY OPTIMIZATION ALGORITHM FOR

SOLVING THE TERMINAL ASSIGNMENT PROBLEM

Eugénia Moreira Bernardino, Anabela Moreira Bernardino

Department of Computer Science, School of Technology and Management, Polytechnic Institute of Leiria, Leiria, Portugal

Juan Manuel Sánchez-Pérez, Juan Antonio Gómez-Pulido, Miguel Angel Vega-Rodríguez

Dep. of Technologies of Computers and Communications, Polytechnic School, University of Extremadura, Cáceres, Spain

Keywords: Communication networks, Optimization algorithms, Ant colony optimization algorithm, Terminal

assignment problem.

Abstract: The past two decades have witnessed tremendous research activities in optimization methods for

communication networks. One important problem in communication networks is the Terminal Assignment

Problem. This problem involves determining minimum cost links to form a network by connecting a

collection of terminals to a collection of concentrators. In this paper, we propose a Hybrid Ant Colony

Optimization Algorithm to solve the Terminal Assignment Problem. We compare our results with the

results obtained by the standard Genetic Algorithm, the Tabu Search Algorithm and the Hybrid Differential

Evolution Algorithm, used in literature.

1 INTRODUCTION

In the last decades the literature on

telecommunication network problems has grown

explosively. This is mainly due to the dramatic

growth in the use of the Internet (Salcedo-Sanz and

Yao, 2004; Yao et al. 2005). Terminal assignment

(TA) is an important issue in telecommunication

networks optimization.

The target of the TA problem implies fixing the

minimum cost links to form a network between a

specified set of terminals and concentrators (Khuri

and Chiu, 1997). The objective is to connect

terminals to concentrators under three constraints:

1. each terminal is assigned to one (and only

one) concentrator;

2. the total number of terminals assigned to

any concentrator does not overload that

concentrator, i.e. is within the concentrators

capacity and,

3. balanced distribution of terminals among

concentrators.

Under these constraints, an assignment with the

lowest possible cost is sought.

The TA problem is a NP-complete combinatorial

optimization problem (Salcedo-Sanz and Yao,

2004). This means that the time required to solve the

problem increases very quickly as the size of the

problem grows. The intractability of this problem is

a motivation for the pursuits of a metaheuristic that

produce approximate, rather than exact, solutions. In

(Dorigo, 1991; Dorigo et al. 1991; Dorigo et al.

1996) the use of an Ant Colony Optimization

algorithm as a new metaheuristic was proposed in

order to solve combinatorial optimization problems.

An Ant Colony Optimization algorithm (ACO) is

essentially a system based on agents which simulate

the natural behavior of ants, including mechanisms

of cooperation and adaptation. This new

metaheuristic has been shown to be both robust and

versatile. The ACO algorithm has been successfully

applied to a range of different combinatorial

optimization problems (ACO HomePage).

In this paper we present a Hybrid Ant Colony

Optimization (HACO) algorithm coupled with a

local search, applied to the TA problem. Our

algorithm is based on the HACO algorithm proposed

by Gambardella et al. (1999) for solving the

quadratic assignment problem. The HACO uses

pheromone trail information to perform

modifications on TA solutions, unlike more

traditional ant systems that use pheromone trail

144

Moreira Bernardino E., Moreira Bernardino A., Manuel Sánchez-Pérez J., Antonio Gomez Pulido J. and Ángel Vega-Rodríguez M. (2009).

A HYBRID ANT COLONY OPTIMIZATION ALGORITHM FOR SOLVING THE TERMINAL ASSIGNMENT PROBLEM.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 144-151

DOI: 10.5220/0002322001440151

 SciTePress

information to construct complete solutions. The

HACO uses also a diversification mechanism that

periodically reinitializes all the pheromone trails.

We compare the performance of HACO with

three algorithms: Genetic Algorithm (GA), Tabu

Search (TS) Algorithm, Hybrid Differential

Evolution (HDE) Algorithm, used in literature.

The paper is structured as follows. In Section 2

we describe the TA problem; in Section 3 we

describe the implemented HACO algorithm; in

Section 4 we present the studied examples; in

Section 5 we discuss the computational results

obtained and, finally, in Section 6 we report about

the conclusions.

2 TA PROBLEM

The TA Problem can be described as follows:

1. a set N of n distinct terminals;

2. a set M of m distinct concentrators;

3. a vector C, with the capacity required for

each concentrator (each concentrator is

limited in the amount of traffic that it can

accommodate);

4. a vector T, with the capacity required for

each terminal (the capacity requirement of

each terminal is known and may vary from

one terminal to another). The capacities are

positive integers and Ti is smaller or equal

to min (Ci…Cn);

5. a matrix CP, with the location (x,y) of

each concentrator (the concentrators sites

have fixed and known locations). The M

concentrators are placed on the Euclidean

grid.

6. a matrix CT, with the location (x,y) of

each terminal (the terminals sites have fixed

and known locations). The N terminals are

placed on the Euclidean grid.

The first objective is to assign each terminal to

one node of the set of concentrators, in such a way

that no concentrator oversteps its capacity. The

second objective is to minimize the distances

between concentrators and terminals assigned to

them. Finally, the third objective is to ensure a

balanced distribution of terminals among

concentrators.

Figure 1 illustrates an assignment to a problem with

N = 10 terminal sites and M = 3 concentrator

sites. The figure shows the coordinates for the

concentrators, terminal sites and also their

capacities.

Figure 1: TA Problem - Example.

3 PROPOSED HACO

ACO is a population-based optimization method for

solving hard combinatorial optimization problems.

ACO is based on the indirect communication of a

colony of simple agents, called (artificial) ants,

mediated by (artificial) pheromone trails. In ant

colony natural, ants indirectly communicate with

each other by depositing pheromone trails on the

ground and thereby influencing the decision

processes of other ants. This simple form of

communication between individual ants gives rise to

complex behaviours and capabilities of the colony as

a whole.

The first algorithm which can be classified

within this framework was presented by Dorigo,

Maniezzo and Colorni (1991, 1996), and Dorigo

(1991) and, since then, many diverse variants of the

basic principle have been reported in the literature.

The real ants behaviour is transposed into an

algorithm by making an analogy between:

1. real ants search - set of feasible solutions to

the problem;

2. amount of food in a source - fitness

function;

3. pheromone trail - adaptive memory.

In ant colony natural, while walking from food

sources to the nest or the nest to food sources, each

A HYBRID ANT COLONY OPTIMIZATION ALGORITHM FOR SOLVING THE TERMINAL ASSIGNMENT

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145

ant deposits a pheromone on the ground. All ants

can smell the pheromone while they walks.

Therefore, more pheromone on the path will

increase the probability of all ants to follow. In

short, the best paths will receive a greater deposit of

pheromones.

The pheromone trails in ACO serve as a

distributed, numerical information which the ants

use to probabilistically construct solutions to the

problem being solved and which the ants adapt

during the algorithm execution to reflect their search

experience.

The essential trait of ACO algorithms is the

combination of a priori information about the

structure of a promising solution with a posterior

information about the structure of previously

obtained good solutions.

Any high performing metaheuristic algorithm

has to achieve an appropriate balance between the

exploitation of the search experience gathered so far

and the exploration of unvisited or relatively

unexplored search space regions. In ACO several

ways exist of achieving such a balance, typically

through the management of the pheromone trails. In

fact, the pheromone trails induce a probability

distribution over the search space and determine

which parts of the search space are effectively

sampled. The management of pheromone trails is the

most important component of an ant system.

Exploration is a stochastic process in which the

choice of the component used to construct a solution

to the problem is made in a probabilistic way.

Exploitation chooses the component that maximises

a blend of pheromone trail values and partial

objective function evaluations.

The standard ACO algorithm uses pheromones

trail information to construct complete solutions.

Gambardella et al. (1999) in their paper present a

Hybrid Ant Colony System coupled with a local

search (HAS_QAP), applied to the quadratic

assignment problem (QAP). HAS-QAP uses

pheromone trail information to perform

modifications on QAP solutions. Our HACO

algorithm uses also pheromone trail information to

perform modifications on TA solutions, unlike

traditional ant systems that use pheromone trail

information to construct complete solutions.

In this paper we will also explore one of the most

successful emerging ideas combining local search

with a population based search algorithm. HACO

uses a modified ACO to explore several regions of

the search space and simultaneously incorporates a

mechanism (LS algorithm) to intensify the search

around some selected regions.

The first step for the HACO implementation

involves choosing a representation for the problem.

In this work, the solutions are represented using

integer vectors. We use the terminal-based

representation (Figure 2). Each position in the

vector corresponds to a terminal. The value carried

by position i of the chromosome specifies the

concentrator that terminal i is to be assigned to.

Figure 2: Terminal Based Representation.

For the TA, the set of pheromone trails is

maintained in a matrix T of size N*M, where the

entry T

measures the desirability of assigning

terminal i to concentrator j.

The simplest way to exploit the ants search

experience is to make the pheromone update a

function of the solution quality achieved by each

particular ant. In HACO only the best solution found

during the search contributes to pheromone trail

updating (Gambardella et al. 1999). This makes the

search more aggressive and requires less time to

reach good solutions. Moreover, this has been

strengthened by an intensification mechanism. The

intensification mechanics is used to explore

neighbourhood more completely.

The algorithm uses also a diversification

mechanism after a pre-defined number of S

iterations without improving the best solution found

so far. Gambardella et al. (1999) have shown that

pheromone trail reinitialization, when combined

with appropriate choices for the pheromone trail

update can be very useful to refocus the search on a

different search space region and avoid the early

convergence of the algorithm.

HACO is based on the schematic algorithm of

Figure 3.

The main steps of HACO are the following:

 Initialization of solutions – the initial solutions

can be created randomly or in a deterministic

form. The deterministic form is based in the

Greedy Algorithm proposed by Abuali et al.

(1994). This algorithm assigns terminals to the

closest feasible concentrator.

 Evaluation of solutions – the fitness function is

responsible for performing this evaluation and

returning a positive number (fitness value)

that reflects how optimal the solution is. The

fitness function is based on the fitness

function used in (Salcedo-Sanz and Yao,

2004). The fitness function is based on: (1)

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the total number of terminals connected to

each concentrator (the objective is to

guarantee the balanced distribution of

terminals among concentrators); (2) the

distances between the concentrators and the

terminals assigned to them (the objective is to

minimize the distances between concentrators

and terminals assigned to them); (3) the

penalization if a solution is not feasible (the

objective is to penalize the solutions when the

total capacity of one or more concentrators is

overloaded). The objective is to minimize the

fitness function.

{

()

0,9 *

0,1*

,()

20* 1

500

1()

,()

fitness bal

dist

tct

Penalization

if total round

bal

abs round total

if Feasible

Penalization

if c t c

total

dist

tct

−

∑

−

∑

⎧

⎛⎞

⎪

⎜⎟

⎪

⎝⎠

⎨

⎛⎞

⎪

+−

⎜⎟

⎪

⎝⎠

⎩

−

⎧

∑

⎨

⎩

[] []

()

[] []

()

(). .

CP c t x CT t x

CP c t y CT t y

−+

−

 Pheromone trail initialization – all pheromone

trails T

are set to the same value

=1/(Q*f(X*)) (Gambardella et al.

1999). X* is the best solution found so far and

Q a parameter.

 Modification of solutions – it consists in

repeating R modifications. A modification

consists on assigning a terminal t to a

concentrator c. First a terminal t is randomly

chosen (between 1 and N) and after a

concentrator c is chosen. A random number x

is generated between 0 and 1. If x is smaller

than q (parameter), the best concentrator c is

chosen in such a way that Ttc is maximum.

This policy consists in exploiting the

pheromone trail. If x is higher than q the

concentrator c is chosen with a probability

proportional to the values contained in the

pheromone trail. This consists in exploring the

solution space.

 Local Search – the LS algorithm consists on

applying a partial neighbourhood examination.

We generate a neighbour by swapping two

terminals between two concentrators C1 and

C2 (randomly chosen). If isn’t find a better

solution then is created another set of

neighbours. In this case, one neighbour

results of assign one terminal of C1 to C2 or

C2 to C1. The neighbourhood size is

N(C1)*N(C2) or N(C1)*N(C2) +

N(C1)+N(C2). The LS algorithm consists

on the following steps:

C1 = random (number of concentrators)

C2 = random (number of concentrators)

NN = neighbours of ACTUAL-SOL (one

neighbour results of interchange one

terminal of C1 or C2 with one terminal

of C2 or C1)

SOLUTION = FindBest (NN)

If ACTUAL-SOL is best than SOLUTION

NN = neighbours of ACTUAL-SOL (one

neighbour results of assign one

terminal of C1 to C2 or C2 to C1)

SOLUTION = FindBest (NN)

If SOLUTION is best than ACTUAL-SOL

ACTUAL-SOL = SOLUTION

Else

ACTUAL-SOL = SOLUTION

 Intensification – the intensification mechanism

permits to explore the neighbourhood more

completely and permits to return to previous

best solutions. If the intensification is active

and the solution X in the beginning of the

iteration is better, the ant comes back to the

initial solution X. The intensification is

activated when the best solution found so far

has been improved and remains active while at

least one ant succeeds on improving its

solution during the iteration.

 Pheromone trail update – to speed-up the

convergence the pheromone trails are updated

by taking into account only the best solution

found so far (Gambardella et al. 1999). The

pheromone trails are updating by setting:

=(1-x1)*T

, where 0<x1<1 is a

parameter that controls the evaporation of the

pheromone trail

iXi

*=T

iXi

*+x2/f*(X*), where 0<x2<1

is a parameter that controls the influence of

the best solution X* in the pheromone trail.

 Diversification – this mechanism restarts the

pheromone trails and creates new solutions for

each ant. We kept for the following iteration

the best solution found so far.

More information about ACO can be found in

(ACO HomePage).

(

)

(

)

(3)

c(t)= concentrator of terminal t

t = terminal c = concentrator

M = number of concentrators N = number of terminals

A HYBRID ANT COLONY OPTIMIZATION ALGORITHM FOR SOLVING THE TERMINAL ASSIGNMENT

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147

Figure 3: HACO Algorithm.

4 EXAMPLES

In order to test the performance of our approach, we

use a collection of TA instances of different sizes.

We take 9 problems from literature (Bernardino et

al. 2008).

Table 1 presents the 9 problems that were used to

test our algorithm.

Table 1: TA Instances.

Problem N M Total T Total C

1 10 3 35 39

2 20 6 55 81

3 30 10 89 124

4 40 13 147 169

5 50 16 161 207

6 50 16 173 208

7 70 21 220 271

8 100 30 329 517

9 100 30 362 518

5 RESULTS

To compare our results we consider the results

produced with the classical Genetic Algorithm, the

Tabu Search Algorithm and the Hybrid Differential

Evolution Algorithm. The GA was first applied to

TA by Abuali et al. (1994). The GA is widely used

in literature to make comparisons with other

algorithms. The GA adopted uses “one-point”

method for recombination, “change order” method

for mutation and tournament method for selection.

In “change order”, two genes are randomly selected

and exchanged. TS was applied to this problem by

Xu et al. (2004) and Bernardino et al. (2008). We

compare our algorithm with the TS algorithm

proposed by Bernardino et al. (2008). HDE was

applied to this problem by Bernardino et al. ( 2009).

Table 2 presents the best-obtained results with

HACO, GA, TS and HDE. The first column

represents the problem number (Prob) and the

remaining columns show the results obtained

(Fitness, Time – Run Times) by the four algorithms.

The algorithms have been executed using a

processor Intel Core Duo T2300.

The HDE and GA were applied to populations of

200 individuals. The initial solutions were created

using the Greedy Algorithm.

The run time corresponds to the average time

that the algorithms need to obtain the best feasible

solution.

The values presented have been computed based

on 100 different executions for each test instance.

The four algorithms reach feasible solutions for

all test instances. In comparison, the HACO obtains

better solutions for larger instances. The TS is the

faster algorithm because can find good solutions in a

better running time. In HDE the crossover

probability is applied to each gene, generating

several perturbations by generation, for which the

algorithm slows down. Besides, in HDE is necessary

to carry out a concentrator conversion so that the

concentrator obtained stays always inside of the

defined range.

The better results obtained with HACO use R

between N/20 and N/3, x1>0.4 and x2>0.4

(Figure 4), Q=100, S between N*2 and N*4,

q>0.4 (Figure 4) and Number of ants ={30,40}.

These parameters were experimentally found to be

good and robust for the problems tested.

In our experiments we use a growing number of

ants. The number of ants was set to {10, 20,

30, 40, 50, 60, 70, 80, 90, 100}. We

studied the impact on the execution time, the

average fitness and the number of best solutions

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Table 2: TA Instances.

Prob GA Tabu Search HDE HACO

Fitness Time Fitness Time Fitness Time Fitness Time

1 65,63 <1s 65,63 <1s 65,63 <1s 65,63 <1s

2 134,65 <1s 134,65 <1s 134.65 <1s 134.65 <1s

3 284,07 <1s 270,26 <1s 270,26 <5s 270,26 <1s

4 286,89 <1s 286,89 <1s 286,89 <5s 286,89 <1s

5 335,09 <1s 335,09 <1s 335.09 <5s 335.09 2s

6 371,48 1s 371,12 <1s 371,12 58s 371,12 3s

7 401,45 2s 401,49 1s 401,21 118s 401,21 4s

8 563,75 4s 563,34 1s 563,19 274s 563,19 14s

9 703,78 5s 642,86 2s 642,83 456s 642,83 25s

found. A higher number of ants significantly

increase algorithm execution time (Figure 5).

Figure 4: Influence of parameters – Problem 7.

Figure 5: Number of Ants – Execution Time – Problem 7.

The results show that the best values are 30 and

40. With these values the algorithm can reach in a

reasonable amount of time a higher number of best

solutions (Figure 7). With a higher number of ants

the algorithm can reach a better average fitness

(Figure 6) but it needs much more time.

Figure 6: Number of Ants – Average Fitness – Problem 7.

Figure 7: Number of Ants – Number of Best Solutions –

Problem 7.

We also observe that a small number of ants

allows an initial faster convergence, but a worse

final result, following to an increased amount of

suboptima values (Figure 8). This can be explained

because the quality of the initial best-located

solution previous to the first restart, depends highly

on the population size: they need more population

diversity – it depends on the population size – to

avoid premature stagnation.

For parameter R, the number of swaps executed

using pheromone trail information, R between

[N/20...N/3] has been shown experimentally to

be more efficient (Figure 9). In our experiments R

was set to {0, 1, 2, …, N}.

In case of a high R the resulting permutation

tends to be too close to the best solution used to

perform global pheromone trail updating, which

makes it more difficult to generate new improving

solutions. A high R has also a significant impact on

the execution time (Figure 10). On the contrary, a

small R did not allow the system to escape from

local minima because after the local search, the

resulting solution was in most cases the same as the

starting permutation.

For S<N*2 and S>N*4 phenomena of

stagnation and insufficient intensification have been

observed (Figure 11).

A HYBRID ANT COLONY OPTIMIZATION ALGORITHM FOR SOLVING THE TERMINAL ASSIGNMENT

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149

Figure 8: Number of Ants – Convergence – Three

different initial populations – Problem 7.

Figure 9: Number of modifications – Average Fitness –

Problem 7.

Large types of experiments and considerations

have been made to define other parameters.

In general, experiments have shown that the

proposed parameter setting is very robust to small

modifications.

Figure 10: Number of modifications – Execution Time –

Problem 7.

Figure 11: Diversification – Average Fitness – Problem 7.

6 CONCLUSIONS

In this paper we present a new Hybrid Ant Colony

Optimization Algorithm to solve the Terminal

Assignment Problem. The performance of' our

algorithm is compared with three algorithms: a

classical GA, a TS algorithm and a HDE algorithm.

Experimental results demonstrate that the

proposed HACO algorithm is an effective and

competitive approach in composing fairly

satisfactory results with respect to solution quality

and execution time for the Terminal Assignment

Problem.

The HACO presents better results for larger

problems. Our algorithm provides better solutions

with smaller fitness values for larger problems. The

TS is the faster algorithm.

In literature the application of HACO for this

problem is nonexistent, for that reason this article

shows its enforceability in the resolution of this

problem.

The implementation of parallel algorithms will

speed up the optimization process.

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REFERENCES

Abuali, F., Schoenefeld, D., Wainwright, R., 1994.

Terminal assignment in a Communications Network

Using Genetic Algorithms. In Proc. of the 22nd

Annual ACM Computer Science Conference, pp. 74–

81. ACM Press.

Ant Colony Optimization HomePage,

http://iridia.ulb.ac.be/dorigo/ACO/ACO.html

Bernardino, E., Bernardino, A., Sánchez-Pérez, J., Vega-

Rodríguez, M., Gómez-Pulido, J., 2008. Tabu Search

vs Hybrid Genetic Algorithm to solve the terminal

assignment problem. In IADIS International

Conference Applied Computing, pp. 404–409. IADIS

Press.

Bernardino, E., Bernardino, A., Sánchez-Pérez, J., Vega-

Rodríguez, M., Gómez-Pulido, J., 2009. A Hybrid

Differential Evolution Algorithm for solving the

Terminal assignment problem. In International

Symposium on Distributed Computing and Artificial

Intelligence 2009, pp. 178–185. Springer.

Dorigo, M., 1991. Ottimizzazione, apprendimento

automatico, ed algoritmi basati su metafora naturale

(Optimisation, learning and natural algorithms).

Doctoral dissertation, Dipartimento di Elettronica e

Informazione, Politecnico di Milano, Italy.

Dorigo, M., Maniezzo, V., Colorni, A. 1991. Positive

feedback as a search strategy. Technical Report 91-

016, Dipartimento di Elettronica e Informazione,

Politecnico di Milano, Italy.

Dorigo, M., Maniezzo, V., Colorni, A., 1996. The ant

system: Optimization by a colony of cooperating

agents. IEEE Transactions on Systems, Man, and

Cybernetics. 26, 29–41.

Gambardella, L. M., Taillard, E. D., and Dorigo, M., 1999

Ant colonies for the quadratic assignment problem.

Journal of the Operational Research Society, 50(2),

167-176.

Khuri, S., Chiu, T., 1997. Heuristic Algorithms for the

Terminal Assignment Problem. In Proc. of the ACM

Symposium on Applied Computing, pp. 247–251.

ACM Press.

Salcedo-Sanz, S., Yao, X., 2004. A hybrid Hopfield

network-genetic algorithm approach for the terminal

assignment problem. IEEE Transaction On Systems,

Man and Cybernetics, 2343–2353.

Xu, Y., Salcedo-Sanz, S., Yao, X. 2004 Non-standard cost

terminal assignment problems using tabu search

approach. In IEEE Conference in Evolutionary

Computation, vol. 2, pp. 2302–2306.

Yao, X., Wang, F., Padmanabhan, K., Salcedo-Sanz, S.,

2005. Hybrid evolutionary approaches to terminal

assignment in communications networks. In Recent

Advances in Memetic Algorithms and related search

technologies, vol. 166, pp. 129–159. Springer, Berlin.

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