HYBRID POPULATION-BASED INCREMENTAL LEARNING

TO ASSIGN TERMINALS TO CONCENTRATORS

Eugénia Moreira Bernardino, Anabela Moreira Bernardino

Research Center for Informatics and Communications, Dep. 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, Terminal assignment problem, Optimisation algorithms, Population-based

incremental learning.

Abstract: In the last decade, we have seen a significant growth in communication networks. In centralised

communication networks, a central computer serves several terminals or workstations. In large networks,

some concentrators are used to increase the network efficiency. A collection of terminals is connected to a

concentrator and each concentrator is connected to the central computer. In this paper we propose a Hybrid

Population-based Incremental Learning (HPBIL) to assign terminals to concentrators. We use this algorithm

to determine the minimum cost to form a network by connecting a given collection of terminals to a given

collection of concentrators. We show that HPBIL is able to achieve good solutions, improving the results

obtained by previous approaches.

1 INTRODUCTION

In last years, we have observed tremendous research

activities in optimisation methods for

communication networks. This is mainly due to the

dramatic growth in the use of the Internet (Salcedo-

Sanz and Yao, 2004; Yao et al. 2005). The

assignment of terminals to concentrators is an

important issue in communication networks'

optimisation. The number of concentrators and

terminals and their locations are known. Each

concentrator is limited in the amount of traffic that it

can accommodate. For that reason, each terminal

must be assigned to one node of the set of

concentrators in a way that no concentrator

oversteps its capacity (Khuri and Chiu, 1997;

Salcedo-Sanz and Yao, 2004; Xu et al. 2004). This

problem is known as Terminal Assignment Problem

(TAP). Our purpose is to minimise the cost to form a

network between a specified set of terminals and

concentrators (Khuri and Chiu, 1997). The objective

is to assign terminals to concentrators under three

constraints (Bernardino et al. 2009b): (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 TAP is a NP-Hard combinatorial

optimisation problem (Salcedo-Sanz and Yao,

2004). This means that we cannot guarantee to find

the best solution in a reasonable amount of time. The

intractability of this problem is a motivation for the

pursuits of an algorithm to produce approximate

solutions.

Estimation of distribution algorithms (EDAs) are

a class of Evolutionary Algorithms (EAs). EDAs use

sampling with probabilities instead of traditional

crossover and mutation operators.

The Population-Based Incremental Learning

(PBIL) algorithm is an EDA, proposed by Baluja

(1994). The PBIL uses a stochastic guide search

process to obtain new solutions based on the

directional information from the previous best

solution. The PBIL maintains statistics about the

search space (learning probabilities) and uses them

182

Moreira Bernardino E., Moreira Bernardino A., Sánchez-Pérez J., Gómez-Pulido J. and Vega-Rodríguez M..

HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS .

DOI: 10.5220/0003076301820189

In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 182-189

ISBN: 978-989-8425-31-7

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

to direct its exploration (Baluja, 1994, 1996, 1997).

The algorithm produces new solutions according

to the learning probabilities. Based on analysing the

principle of PBIL algorithm, in this paper we present

an improvement version of the PBIL algorithm. We

extend the standard PBIL algorithm to work with an

integer representation (terminal-based

representation). The standard PBIL algorithm uses

learning probabilities to build complete solutions. In

our Hybrid PBIL (HPBIL) algorithm, we use the

learning probabilities to perform modifications on

TAP solutions. We incorporate the HPBIL with an

intensification mechanism to allow returning to

previous best solutions. The HPBIL also uses a

diversification mechanism that periodically

reinitialises all the learning probabilities.

We compare the performance of HPBIL with

five algorithms: Local Search Genetic Algorithm

(LSGA), Tabu Search (TS), Hybrid Ant Colony

Optimisation (HACO), Hybrid Differential

Evolution with a Multiple strategy (MHDE), and

Hybrid Scatter Search (HSS), used in literature.

The paper is structured as follows. In Section 2

we describe the TAP; in Section 3 we describe the

implemented HPBIL algorithm; in Section 4 we

discuss the computational results obtained and,

finally, in Section 5 we report about the conclusions.

2 TAP

In the TAP, a communication network will connect

N terminals and each with T

demand (weight) via M

concentrators and each with C

capacity. No

terminal’s demand exceeds the capacity of any

concentrator. A terminal site has a fixed and known

location CT

(x,y). A concentrator site has also a

fixed and known location CP

(x,y).

Problem Instance:

 Terminals - a set N of n distinct terminals;

 Weights - a vector T, with the capacity required

for each terminal;

 Terminals' Location - a vector CT, with the

location (x,y) of each terminal;

 Concentrators - a set M of m distinct

concentrators;

 Capacities - a vector C, with the capacity

required for each concentrator;

 Concentrators' Location - a vector CP, with the

concentrators' location (x,y).

Each terminal must be assigned to one node of

the set of concentrators, in a way that no

concentrator oversteps its capacity. To minimise the

cost, the distances between concentrators and

terminals assigned to them must be minimised.

Other 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 and terminals and also their capacities.

Figure 1: TAP - Example.

In this work, the solutions are represented using

integer vectors. We use the terminal-based

representation (Figure 1). 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.

3 PROPOSED HPBIL

PBIL is a population-based optimisation method to

solve hard combinatorial optimisation problems.

PBIL combines characteristics from EAs and

reinforcement learning (Baluja, 1994).

The PBIL creates a real-valued probability vector

P, which is used to generate new solutions. The

coding method of Baluja’s PBIL uses binary

representation (Baluja, 1994; He et al. 1999).

Initially, the values of the probability vector are

initialised to 0.5. As the search proceeds, the values

in the probability vector gradually shift to reflect the

search experience of the best solutions found in

previous generations.

The probability vector is used to create a number

of ns solutions. The probability vector is updated,

HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS

183

taking the best solution B of the previous generation.

After the probability vector is updated, the vector is

mutated and a new set of solutions is created by

sampling from the updated probability vector. The

algorithm continues until the stop criterion is

reached (He et al. 1994).

The main steps of the standard PBIL are the

following:

Initialise parameters

Initialise the probability vector P

Update the probability vector: P

=0.5(1<=i<=n)

While stop criterion is not reached:

Generate ns sample solutions according to

the probability vector P.

Evaluate ns solutions.

Find best solution B.

Update probability vector towards best

solution B:

*(1.0-lr)+B

*lr(1<=i<=n)

Mutate probability vector:

if (random(0,1)<mp)

*(1.0-mutShift)+

random(0,1)*(mutShift)

(1<=i<=n)

ns – number of solutions in the population

n - length of encoded solution

lr - learning rate

mp - mutation probability

mutShift - amount for mutation to affect the

probability vector

The probability vector in PBIL serves as

distributed, numerical information which the

algorithm uses to probabilistically construct

solutions to the problem being solved and which the

algorithm adapts during the algorithm execution, to

reflect the search experience. The probabilities

induce a probability distribution over the search

space and determine which parts of the search space

are effectively sampled.

To extend the binary PBIL algorithm for an

integer representation, the probabilities need to be

maintaned in a matrix. For the TAP, the set of

probabilities 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 standard PBIL algorithm uses the probability

vector to construct complete solutions. Our HPBIL

algorithm uses the probabilities to perform

modifications on TAP solutions.

In HPBIL, we incorporate a parameter q to

control the exploration and exploitation processes.

We use these two processes to modify the solutions.

The management of the probabilities is the most

important component of PBIL. Exploration is a

stochastic process, in which the choice of the

component used to modify a solution to the problem

is made in a probabilistic way. Exploitation chooses

the component that maximises a blend of probability

values and partial objective function evaluations.

In this paper, we also explore one of the most

successful emerging ideas combining Local Search

(LS) with a population-based search algorithm.

HPBIL uses a modified PBIL to explore several

regions of the search space and simultaneously

integrates a LS algorithm to intensify the search

around some selected regions.

For the best solution B in some generation, the

corresponding learning formula to update the

probabilities is: T

iBi

= T

iBi

+ lr.

HPBIL uses an intensification mechanism. This

mechanism allows returning to previous best

solutions. The algorithm also uses a diversification

mechanism after a predefined number of nid

iterations, without improving the best solution found

so far. When combined with appropriate choices for

the probabilities update, the diversification

mechanism can be very useful to refocus the search

on a different search space region and to avoid the

early convergence of the algorithm.

The main steps of HPBIL are the following:

Initialise parameters

iteration = 0

Generate initial population S

for s=1 to ns do

Apply Local Search to S

Evaluate Solutions in S

Find best global solution G in S

Initialise the probability matrix T

=1/M, (1<=i<=N), (1<=j<=M)

intensification=true

While stop criterion is not reached:

iteration = iteration+1

for s=1 to ns do

Modify s solution according to the

probability matrix T:

S’

=ModifyProcess(S

)

Apply Local Search to S’

cond=true

if (intensification=true)

if(fitness(S’

fitness(S

))

S’

else

cond=false

if (cond=true) //no solution improved

ICEC 2010 - International Conference on Evolutionary Computation

184

intensification=false

Find best solution B in S’

if(fitness(B)<fitness(G))

G=B

Intensification=true

Update probability matrix towards best

solution B:

iBi

= T

iBi

+ lr

Mutate probability matrix:

if (random(0,1)<mp)

*(1.0-mutShift)+

random(0 or 1)*(mutShift)

(1<=i<=N), (1<=j<=M)

if (iteration % nid =0)

Apply Diversification Mechanism

S=S’

S - population of solutions

lr - learning constant

The next subsections describe each step of the

algorithm in detail.

3.1 Initialisation of Parameters

The following parameters, must be defined by the

user: (1) ms – number of seconds; (2) ns – number

of solutions in the population; (3) mp – mutation

probability; (4) mutShift – amount for mutation

to affect the probability matrix; (5) nm – number of

modifications; (6) q – exploitation/exploration

probability and (7) nid – number of iterations

without improvement (used for diversification).

3.2 Create Initial Population

The solutions are created using a deterministic form.

The deterministic form is based in the Greedy

algorithm proposed by Abuali et al. (1994). The

Greedy algorithm randomly assigns terminals to the

closest feasible concentrators.

3.3 Local Search Procedure

HPBIL uses the LS algorithm proposed by

Bernardino et al. (2008b). The evaluation process is

the most time-consuming step of the algorithm,

which is usually the case in many real-life problems.

We improve the LS proposed by Bernardino et al.

(2008b). After creating a neighbour, the algorithm

does not perform a full examination to calculate the

new fitness value; it only updates the fitness value

based on the modifications made to create the

neighbour. The running time is considerably

reduced. We observe 80% of improvement in terms

of execution time.

3.4 Evaluation of Solutions

The fitness function is the same used in Bernardino

et al. (2008a, 2008b, 2009a, 2009b, 2010a, 2010b).

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(1) the total number of terminals connected to each

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distribution of terminals among concentrators);

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(2) the distances between concentrators and

terminals assigned to them (the goal is to minimise

the distances);

(3) the penalisation if a solution is not feasible (the

objective is to penalise the solutions when the total

capacity of one or more concentrators is

overloaded).

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The main objective is to minimise the fitness

function.

3.5 Initialisation of Probability Matrix

All the values in the probability matrix are initialised

with the same probability:

=1/M, (1<=i<=N), (1<=j<=M)

For example, if we have 10 terminals and 4

concentrators, all the values are initialised with the

same probability ¼=0.25. All the concentrators

have the same probability of being selected.

(

)

(

)

(

)

HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS

185

3.6 Modification of Solutions

It consists of repeating nm modifications. The

modification is done assigning a terminal t to a

concentrator c. First, a terminal t is randomly

chosen (between 1 and N) and then a concentrator c

is selected. Then, a random number x is generated

between 0 and 1. If x is smaller than q (parameter),

the best feasible concentrator c is chosen in a way

that T

will be maximum. If x is higher than q,

the feasible concentrator c is chosen with a

probability proportional to the values contained in

the pheromone trail. We only consider feasible

concentrators. This means that we only consider the

concentrators that have a free capacity equal or

higher than the demand of terminal t.

3.7 Intensification Mechanism

The intensification mechanism allows a more

complete exploration of the neighbourhood and

allows returning to previous best solutions. The

objective is to return towards attractive regions to

search them thoroughly. If the intensification is

active and the solution S

in the beginning of the

iteration is better, the new solution S’

returns to the

initial solution S

. The intensification is activated

when the best solution found so far has been

improved and remains active while at least one

solution succeeds on improving its solution during

the iteration. In the end of each iteration, if no

solution improves its last solution, the intensification

is deactivated. The objective is to explore other

regions, avoiding the algorithm to become trapped in

a local minimum.

3.8 Probability Matrix Update

The probability matrix is updated by taking into

account only the best solution of the previous

generation. The probability matrix is updated by

setting: T

iBi

= T

iBi

+ lr

Based on preliminary observations, we consider

the value 0.5 for lr (learning constant).

3.9 Probability Matrix Mutation

PBIL does not use a crossover operator and a

selection mechanism like the most EAs. Instead, the

values in T are mutated once per iteration. During

this step, a random number between 0 and 1 is

generated. If this random value is smaller than mp

(mutation probability), the probability is mutated by

setting:

*(1.0-mutShift)+

random(0 or 1)*(mutShift)

3.10 Diversification Mechanism

The diversification mechanism restarts the

probability matrix and creates new solutions. For the

following iteration, we kept the best solution found

so far (G).

3.11 Termination Criterion

The algorithm stops when a maximum number of

seconds (ms) is reached.

More information on PBIL can be found in

(Baluja, 1994, 1996, 1997; Baluja and Caruana,

1995; He et al. 1999).

4 RESULTS

In order to test the performance of our approach, we

use a collection of TAP instances of different sizes.

We take 9 problems from literature (Bernardino et

al. 2008a).

To compare our results we consider the results

produced with LSGA, TS, HACO, MHDE and HSS.

We compare our algorithm with the algorithms

proposed by Bernardino et al. (2008a, 2008b, 2009b,

2010a, 2010b), because they (1) used the same test

instances; (2) adopted the same fitness function; (3)

implemented the algorithms using the same

language (C++), and; (4) adopted the same

representation (terminal-based).

Table 1 presents the best-obtained results with

HPBIL, LSGA, TS, MHDE, HACO and HSS. The

first column represents the number of the problem

(Prob) and the remaining columns show the results

obtained (BestF – Best Fitness, Ts – Run Times).

The initial solutions for all algorithms were created

using the Greedy algorithm. The algorithms have

been executed using a processor Intel Core Duo

T2300. The Ts (Run Time) corresponds to the

execution time that each algorithm needs to obtain

the best feasible solution.

The HPBIL algorithm can reach the best-known

solutions for all instances. MHDE, HACO, HSS and

LSGA can also find the best–known solutions, but in

a higher execution time. Since we are not trying to

dynamically assign terminals to concentrators, the

running time is not enough to determine the quality

of the algorithms. The best-known solutions are

ICEC 2010 - International Conference on Evolutionary Computation

186

Table 1: Results.

Prob LSGA TS MHDE HACO HSS HPBIL

BestF Ts BestF Ts BestF Ts BestF Ts BestF Ts BestF Ts

1 65.63 <1s 65.63 <1s 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 134.65 <1s 134.65 <1s

3 270.26 <1s 270.26 <1s 270.26 <1s 270.26 <1s 270.26 <1s 270.26 <1s

4 286.89 <1s 286.89 <1s 286.89 <1s 286.89 <1s 286.89 <1s 286.89 <1s

5 335.09 <1s 335.09 <1s 335.09 <1s 335.09 2s 335.09 <1s 335.09 <1s

6 371.12 1s 371.12 <1s 371.12 <1s 371.12 3s 371.12 1s 371.12 <1s

7 401.21 1s 401.49 1s 401.21 2s 401.21 4s 401.21 1s 401.21

<1s

8 563.19 7s 563.34 1s 563.19 10s 563.19 14s 563.19 4s 563.19

9 642.83 7s 642.86 2s 642.83 15s 642.83 25s 642.83 6s 642.83

Table 2: Results – fitnesses and standard deviations.

Prob LSGA TS MHDE HACO HSS HPBIL

AvgF Std AvgF Std AvgF Std AvgF Std AvgF Std AvgF Std

1 65.63 0.00 65.63 0.00 65.63 0.00 65.63 0.00 65.63 0.00 65.63 0.00

2 134.65 0.00 134.65 0.00 134.65 0.00 134.65 0.00 134.65 0.00 134.65 0.00

3 270.69 0.23 270.76 0.30 270.75 0.15 270.42 0.08 270.35 0.06 270.26 0.01

4 286.99 0.13 287.93 0.75 287.17 0.14 286.96 0.08 286.90 0.01 286.89 0.00

5 335.99 0.60 335.99 0.59 336.55 0.39 335.79 0.34 335.20 0.14 335.09 0.00

6 371.68 0.24 372.44 0.45 373.19 0.42 372.45 0.39 371.70 0.24 371.41 0.13

7 402.41 0.50 403.25 0.73 403.61 0.33 402.28 0.40 401.82 0.34 401.61 0.15

8 564.94 0.52 564.50 0.54 572.04 0.76 565.64 0.84 563.87 0.37 563.76 0.18

9 646.52 0.84 644.18 0.48 648.46 0.48 644.82 0.58 643.94 0.51 643.35 0.19

reached with almost all algorithms (except TS). For

that reason, to establish which is the best algorithm,

we must observe the average quality of the produced

solutions and the standard deviations.

Table 2 presents the average fitnesses and

standard deviations. The first column represents the

number of the problem (Prob) and the remaining

columns show the results obtained (AvgF – Average

Fitness, Std – Standard Deviation). To compute the

results in table 2, we use 0.5 second for instances

1-3, 1 second for instances 4-5, 2 seconds for

instance 6, 5 seconds for instance 7, 10 seconds for

instance 8 and 15 seconds for instance 9.

The suggestions from literature helped us to

guide our choice of parameter values for TS

(Bernardino et al. 2008a), LSGA (Bernardino et al.

2008b), MHDE (Bernardino et al. 2010a), HACO

(Bernardino et al. 2009b) and HSS (Bernardino et al.

2010b). For the TS, we consider a number of

elements in the tabu list between 5 and 20. The

parameters of LSGA are set to crossover probability

between 0.3 and 0.4, selection operator=

“tournament”, mutation probability between 0.6

and 0.8, crossover operator=“one-point” and

mutation operator= “multiple”. The parameters of

the MHDE algorithm are set to crossover probability

between 0.3 and 0.4, factor F between 0.9 and

1.6 and strategy=“Best1Exp”. The parameters of

the HACO algorithm are set to the number of

iterations used for diversification between 200

and 400, Q=100, q=0.9, pheromone

influence=0.8, pheromone evaporation=0.8 and

number of modifications between 2 and 10. The

parameters of the HSS algorithm are set to ni=100,

b1=8, b2=8, and nid between N/15 and N/2.

The parameters of HPBIL are set to mp=0.3,

mutShift=0.1, nid<N/20 and q=0.6. The

MHDE and LSGA were applied to populations of

200 individuals, HSS to populations of 100

individuals and HPBIL and HACO to populations of

30 individuals.

The values presented in table 2 have been

computed based on 50 different executions (50 best

executions out of 100 executions) for each test

instance.

As it can be seen in table 2, for larger instances

the standard deviations and the average fitnesses for

the HPBIL algorithm are smaller. It means that the

HPBIL algorithm is slightly more robust than

LSGA, TS, HACO, MHDE and HSS.

The best results obtained with HPBIL use

nm<N/20, mp between 0.2 and 0.7 (Figure 2)

and mutShift>=0.1 (Figure 2), q between 0.3

and 0.6 (Figure 2), number of solutions between

HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS

187

30 and 100 and nid in the range [N*3 and

N*4]. These parameters were experimentally found

to be good and robust for the problems tested.

We perform comparisons between all parameters

(using the 9 instances) in order to establish the

correct parameter setting for the HPBIL algorithm.

We consider the same instance – 7 (a problem with

average difficulty) to show the comparisons between

parameters. To compute the results we use 1000

iterations.

Figure 2: Influence of parameters – Problem 7.

In our experiments we use a growing number of

solutions. The number of solutions (ns) was set to

{10, 20, 30, 40, …, 200}. We studied the

impact on the execution time and the average fitness

(Figures 3 and 4). A high number of solutions

significantly increases the algorithm execution time

(Figure 3). The results show that the best values are

in the range [30 and 100]. With these values, the

algorithm can reach, in a reasonable amount of time,

a reasonable number of good solutions. With a

higher number of solutions, the algorithm can reach

a better average fitness (Figure 4), but it is more

time consuming. We also observe that a small

number of solutions allows an initial faster

convergence, but a worse final result, following to

an increased amount of suboptima values. 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.

Figure 3: ns – Execution Time – Problem 7.

For nid<N*3 and nid>N*4 we observed

phenomena of stagnation and insufficient

intensification.

Figure 4: ns – Average Fitness – Problem 7.

For parameter nm, the number of modifications,

nm<N/20 has been shown experimentally more

efficient (Figure 5). A high nm has a significant

impact on the execution time (Figure 6). A small nm

did not allow the system to escape from local

minima, because after the LS, the resulting solution

was, in most cases, the same as the starting solution.

Figure 5: nm – Average Fitness – Problem 7.

Figure 6: nm – Execution Time – Problem 7.

In general, experiments have shown that the

proposed parameter setting is very robust to small

modifications.

5 CONCLUSIONS

In this paper we present a HPBIL algorithm to solve

the TAP. The performance of' our algorithm is

compared with five algorithms: a LSGA, a TS

algorithm, a HACO algorithm, a MHDE algorithm

ICEC 2010 - International Conference on Evolutionary Computation

188

and a HSS algorithm. All algorithms were applied to

TAP by the same authors.

HPBIL presents better results for TAP. The

experimental results show that the proposed

algorithm is an effective and competitive approach

in composing satisfactory results with respect to

solution quality and execution time for TAP.

Moreover, in terms of standard deviation, the

algorithm also proved to be more stable and robust

than the other algorithms.

For future work we suggest the implementation

of Evolutionary Swarm Intelligence algorithms. The

combination of EAs and SI algorithms can unify the

fast speed of EAs for global solutions and good

precision of SI algorithms for good solutions by the

feedback information.

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