NEW PROPOSAL FOR A MULTI-OBJECTIVE TECHNIQUE

USING TRIBES AND TABU SEARCH

Nadia Smairi, Sadok Bouamama

National School of Computer Sciences, University of Manouba, Manouba 2010, Tunisia

Khaled Ghedira, Patrick Siarry

High Institute of Management, University of Tunis, Tunisia

University of Paris 12 (LiSSi, E.A. 3956),France

Keywords: Particle Swarm Optimization, Tribes, Tabu Search, Multi-objective Optimization.

Abstract: The aim of this paper is to present a new multi-objective technique which consists on a hybridization

between a particle swarm optimization approach (Tribes) and tabu search technique. The main idea of the

approach is to combine the high convergence rate of Tribes with a local search technique based on Tabu

Search. Besides, in our study, we proposed different places to apply local search: the archive, the best

particle among each tribe and each particle of the swarm. As a result of our study, we present three versions

of our hybridized algorithm. The mechanisms proposed are validated using twelve different functions from

specialized literature of multi-objective optimization. The obtained results show that using this kind of

hybridization is justified as it is able to improve the quality of the solutions in the majority of cases.

1 INTRODUCTION

One of many drawbacks of evolutionary algorithms

is that each one of them has many parameters to be

tuned each time we want to solve a different

problem. Tribes, an adaptative Particle Swarm

Optimization (PSO) technique, has the advantage to

be designed as a black box; the user defining only

the search space, the function to minimize, the

required accuracy and a maximum number of

evaluations. At the beginning, it was designed to

treat mono-objective problems. The aim of this work

is to design a competitive multi-objective algorithm

free from parameters based on Tribes. However, it

has become evident that the concentration on a sole

metaheuristic is restrictive. A skilled combination of

Tribes with other optimization techniques can

provide a more efficient behaviour and higher

flexibility when dealing with the real-world

problems. Therefore, in this paper, we propose a new

multi-objective technique based on Tribes and Tabu

Search (TS). In fact, TS is used to cover widely the

solution space and to avoid the risk of trapping in

non Pareto solutions and Tribes is used to accelerate

the convergence. In our study, we use twelve well-

known multi-objective test functions in order to find

the best one from the proposed techniques and to

justify the use of the local search.

In section 2 of this paper we present the existing

multi-objective PSO techniques. In section 3, we

consider Tribes. In addition, in section 4, we present

our proposed approach. Then comparative results are

described in section 5, from which conclusions are

drawn in section 6.

2 STATE OF ART

In the last few years, several PSO algorithms have

been proposed to tackle the multi-objective

optimization problem. Here we briefly review the

most relevant of them.

Parsopoulos and Vrahatis (2002) propose three

different types of aggregation: a classic linear

aggregation, for which the weights are fixed, a

dynamic aggregation where the weights are

gradually modified during the treatment and an

86

Smairi N., Bouamama S., Ghedira K. and Siarry P. (2010).

NEW PROPOSAL FOR A MULTI-OBJECTIVE TECHNIQUE USING TRIBES AND TABU SEARCH.

In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 86-91

Copyright

c

SciTePress

aggregation the weights of which are brutally

modified during the treatment.

Hu, Eberhart and Shi (2003) propose an

algorithm optimizing each time one single objective

using a lexicographical order.

The VEPSO strategy was introduced by

Parsopoulos, Tasoulis and Vrahatis (2004). It

presents an adaptation of VEGA to the particle

swarm optimization.

Moore and Chapman (1999) propose an

algorithm based on the Pareto dominance and a PSO

algorithm with a circular topology of the

neighbourhood. In this approach, the choice of the

personal guide, for every particle, is arbitrarily made

from a list containing the not dominated positions

that are found.

Ray and Liew (2002) propose a PSO algorithm

using the Pareto dominance. They combine

evolutionary techniques with those of the OEP. They

also use an operator of density on the neighbourhood

to promote the density in the swarm.

This approach, proposed by Coello and Lechuga

(2002), is based on having an external archive to

store the not dominated positions. Furthermore, the

updates of the archive are executed considering a

geographical system which decomposes the space of

the objectives to a set of hypercubes. The archive is

also used to identify a leader which will drive the

search.

The authors propose a multi-objective PSO

algorithm, called DOPS in which several techniques

are integrated for the selection of the leaders and the

update of archive (Bartz-Beielstein, Limbourg,

Parsopoulos, Vrahatis, Mehnen and Shmitt, 2003).

Quintero, Santiago and Coello (2008) suggest a

hybridization of a PSO algorithm with local search

techniques such as scatter search and rough sets

theory.

The proposed algorithm (Sierra and Coello,

2005) is based on the dominance of Pareto: every

not dominated position presents a potential

candidate to be selected as a leader. A crowd

function is also used to filter all the leaders. This

approach (Sierra and Coello, 2007) also integrates

the concept of the ε-dominance to fix the size of the

archive.

The author has developed a multi-objective

version of Tribes. In fact, Mo-Tribes use an

approach based on the Pareto dominance. The not

dominated particles are stored in an external archive

which size and updates are automatically defined.

Furthermore, the variety is maintained thanks to a

criterion of maximization of the crowd distance and

also thanks to the multiple restarts of the swarm. The

results of Mo-Tribes are very encouraging

(Cooren, 2008).

3 TRIBES

Tribes is a PSO algorithm that works in an

autonomous way. Indeed, it is enough to describe

the problem to be resolved and the way of making it

at the beginning of the execution. Then, it is the role

of the program to choose the strategies to be adopted

(Clerc, 2006).

At the beginning, we start with a single particle

forming a tribe. After the first iteration, the first

adaptation takes place and we generate a new

particle which is going to form a new tribe, while

keeping in touch with the generative tribe. In the

following iteration, if the situation of both particles

does not improve, then every tribe creates two new

particles: we form a new tribe containing four

particles. In this way, if the situation deteriorates,

then the size of the swarm grows (creation of new

particles). However, if we are close to an optimal

solution, the process is reversed and we begin to

eliminate particles, even tribes. In fact, the removal

or the generation of a particle are not arbitrary. The

removal of a particle consists in eliminating a

particle without risking the missing of the optimal

solution. For that purpose, only the good tribes are

capable of eliminating their worst elements. The

creation of a particle is made for bad tribes as they

need new information to improve their situations.

4 OUR APPROACH

4.1 Preliminary Study

The adaptation of Tribes to the multi-objective

optimization consists in using the Pareto dominance

to respect the completeness of every objective and to

add an external archive to save the found not

dominated solutions. Furthermore, as the PSO

algorithm, Tribes can be considered neither a global

optimization algorithm nor a local optimization one

(Bergh, 2002). Therefore, the hybridization between

Tribes and a local search algorithm can be

considered as a competitive approach to handle

difficult problems of multi-objective optimization.

In order to improve the capacity of exploitation of

Tribes, we apply a local search technique: TS. In

fact, the local search is not going to be inevitably

applied in a canonical way that is on all the particles

of the swarm: we also propose two other manners,

the first one consists in applying the local search

only among the best particle of every tribe. The

second one consists in applying it among the

particles of the archive. We shall have then three

versions of the algorithm.

NEW PROPOSAL FOR A MULTI-OBJECTIVE TECHNIQUE USING TRIBES AND TABU SEARCH

87

The first one consists in applying the TS only to

the particles of the archive which are situated in the

least crowded zones. Let us note that, in this case,

the local search is not applied unless the archive is

full so that some time is allowed to the information

to propagate in the swarm.

Figure 1: TS-TribesV1 pseudo-code.

The second version of the algorithm consists in

applying the TS only to the best particles of the

tribes. In fact, we consider that those particles are

situated in promising zones and probably they need

further intensification to find out other solutions.

The third version consists in applying the TS to

all the particles of the swarm. It is made at the

moment of the swarm adaptation.

The detailed description of TS-TribesV2 and TS-

TribesV3 was omitted due to space restrictions.

4.2 Updating the External Archive

The update of the archive consists in adding all the

not dominated particles to the archive and deleting

the already present dominated ones. If the number of

particles in the archives exceeds a fixed number, we

apply a crowd function to reduce the size of the

archive and to maintain its variety. Indeed, Crowd

divides the objective space into a set of hypercube.

4.3 Choosing the Particle Informer

The choice of the particle informer or guide is

similar to the case of mono-objective Tribes. Indeed,

if we take a particle which is not the best of its tribe,

his guide is then the best particle of the tribe. If we

consider, on the other hand, the best particle of a

given tribe, the informer is then some random

particle from the archive.

4.4 Hybridizing Tribes with TS

The TS is introduced by Glover. It consists in the

examination of a neighbourhood of a current

solution x and retains the best neighbour x

0

even if x

0

is worse than x. However, this strategy can pull

cycles. To prevent this kind of situation from

appearing, we store the k last visited configurations

in a short-term memory and we forbid to hold any

other configuration which is already a part of it.

However, TS is essentially intended for the

resolution of the combinatorial problems. Few works

considered its adaptation for the continuous

optimization. Among whom we can mention the

approach of Chelouah and Siarry (2000). In that case,

this method is similar to the classic TS. The

difference lies essentially in the generation of the

neighbourhood. It is necessary to define first of all a

way to discretize the search space. In fact, the

neighbourhood is defined by using the concept of

“ball”. A ball B(x, r) centered on x (current solution)

with radius r. To obtain a homogeneous exploration

of the space, we consider a set of balls centered on

the current solution x with radius r

0

, r

1

, r

2

,…r

n

.

Hence the space is partitioned into concentric

crowns. The n neighbours of x are obtained by

random selection of a point which does not belong to

the tabu list inside each crown C

i

, for i varying from

1 to n. Finally, we select the best neighbour x ' even

if it is worse than x and we insert it in the tabu list.

5 EXPERIMENTATIONS

AND RESULTS

5.1 Test Functions

In order to compare the proposed techniques, we

perform a study using twelve well-known test

functions taken from the specialized literature on

evolutionary algorithms. The detailed description of

these functions was omitted due to space

restrictions. However, all of them are unconstrained,

minimization and have between 3 and 30 decision

Begin

Swarm initialization

Swarm evaluation

Archive initialization

While f<fmax

For each tribe

For each particle i

Determination of the state of the particle

Choice of the strategy of movement

Choice of the informer

Update of the position

Evaluation

Update of pi (best position visited by i)

Update the best particle of the tribe

Update the archive

EndFor

EndFor

If criterion of adaptation verified

Determination of the quality of the tribe

Adaptation of the swarm

Update of the adaptation criterion

EndIf

For each particle of the archive situated in

the least crowded zones

TS (stopping criterion)

EndFor

EndWhile

End

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

88

variables. Indeed, we fix the maximal size of the

archive to 100 for the two-objective functions and to

150 to the three-objective ones. We also varied the

size of the neighbourhood for the TS algorithm: 5,

10 and 20. Moreover, we fix the maximal number of

evaluations in the experimentations to 5e+4.

Table 1: Properties of the test functions.

Test

functions

Objective

Modality

Geometry

Oka2

f

1

f

2

Uni-modal

Multi-modal

Concave

Sympart

f

1:2

Multi-modal

Concave

S_ZDT1

f

1:2

Uni-modal

Convex

S_ZDT2

f

1:2

Uni-modal

Concave

S_ZDT4

f

1

f

2

Uni-modal

Multi-modal

Convex

R_ZDT4

f

1:2

Multi-modal

Convex

S_ZDT6

f

1:2

Multi-modal

Concave

S_DTLZ2

f

1:3

Uni-modal

Concave

S_DTLZ3

f

1:3

Multi-modal

Concave

WFG1

f

1:3

Uni-modal

Convex

WFG8

f

1:3

Uni-modal

Concave

WFG9

f

1:3

Multi-modal

Concave

5.2 Metrics of Comparison

For assessing the performance of the algorithms,

there are many existent unary and binary indicators

measuring quality, diversity and convergence. In the

literature, there are many proposed combination in

order to perform a convenient study and comparison.

We choose the combination of two binary indicators

that was proposed in (Knowles, Thiele and Zitler,

2006): R indicator and hypervolume indicator.

5.2.1 R indicator (I

R2

)

It computes the difference between the maximum

value of the augmented Tchebycheff utility function

of the reference set and the obtained solutions from

the procedure.

5.2.2 Hypervolume Indicator (𝑰

𝑯

)

The hypervolume indicator measures the

hypervolume of that portion of the objective space

that is weakly dominated by an approximation set A,

and is to be maximized. Here we consider the

hypervolume difference to a reference set R; where

smaller values correspond to higher quality.

5.2.3 Results

The binary indicators used to make the comparison

measure both convergence and diversity. The results

regarding the R indicator are given in tables 2, 3 and

4 (R can take values between -1 and 1 where smaller

values correspond to better results). The

hypervolume difference is given for all test functions

in table 5, 6 and 7. Again, smaller values mean

better quality of the results because the difference to

a reference set is measured.

For both indicators, we present the summary of

the results obtained. In each case, we present the

average of I

R2

and hypervolume measures over 10

independent runs. These values are given for the

different sizes of neighbourhood. According to these

tables, we notice that:

The found fronts for test functions S_ZDT1,

S_ZDT2 and S_DTLZ2 are very close to the

reference set (for all the versions). Moreover,

the found fronts for test functions OKA2,

WFG8 and WFG9 are better than the

proposed reference fronts (for all the

versions).

Bad performance behaviour is noticed for

S_ZDT4 and R_ZDT4 for all the versions

except TS-TribesV3. We note that bad

convergence behaviour is detected also with

another PSO algorithm for ZDT4 in (Hu,

Eberhart and Shi, 2003).

TS-TribesV1 outperforms generally the other

versions except for test functions S_ZDT4

and R_ZDT4 where TS-TribesV3 gives the

best results.

The neighbourhood size has no big effect on

the performances of the considered

algorithms. In fact, they keep the same

tendency with the neighbourhood size

variation.

Finally, we recapitulate that TS-Tribes is very

competitive as it supports both intensification and

diversification. In fact, the choice of particle’s

informer is done in order to accelerate the swarm’s

convergence towards the search space zones where

are situated the archive’s particles. This can be

considered as an intensification process. Moreover,

the archive’s updating is done thanks to the Crowd

function that maintains the archive’s diversity. This

can be considered as a diversification process.

Indeed, TS supports both intensification and

diversification. The good neighbourhood exploration

intensifies the search towards specific zones in the

search space. Besides, the TS mechanisms such as

tabu list allow avoiding the risk of trapping in non

Pareto solutions.

NEW PROPOSAL FOR A MULTI-OBJECTIVE TECHNIQUE USING TRIBES AND TABU SEARCH

89

Table 2: Results for R indicator (neighbourhood size = 5).

Test

Functions

TS-TribesV1

TS-TribesV2

TS-

TribesV3

OKA2

-1.23e-3

-1.22e-3

-1.21e-3

Sympart

6.74e-5

2.91e-5

8.38e-5

S_ZDT1

7.21e-4

1.26e-3

1.05e-3

S_ZDT2

4.01e-5

1.48e-3

3.27e-5

S_ZDT4

2.84e-3

4.84e-3

4.10e-3

R_ZDT4

8.21e-3

2.24e-3

1.46e-2

S_ZDT6

4.50e-3

7.78e-3

2.19e-3

S_DTLZ2

2.52e-4

2.19e-4

2.70e-4

S_DTLZ3

4.24e-4

2.99e-4

7.68e-4

WFG1

2.44e-2

3.93e-2

4.94e-2

WFG8

-2.01e-2

-1.18e-2

-2.25e-3

WFG9

-6.73e-3

-6.10e-3

-2.63e-3

Table 3: Results for R indicator (neighbourhood

size = 10).

Test

Functions

TS-TribesV1

TS-TribesV2

TS-TribesV3

OKA2

-1.15e-3

-1.03e-3

-1.02e-3

Sympart

2.99e-5

3.20e-5

4.68e-5

S_ZDT1

5.17e-4

1.19e-3

1.21e-3

S_ZDT2

3.72e-5

1.02e-3

1.23e-4

S_ZDT4

2.82e-3

8.78e-3

1.68e-4

R_ZDT4

4.24e-3

3.35e-3

2.38e-3

S_ZDT6

3.05e-3

8.79e-3

2.42e-3

S_DTLZ2

1.69e-4

2.32e-4

2.13e-4

S_DTLZ3

2.08e-4

3.37e-4

4.72e-4

WFG1

2.49e-2

4.39e-2

4.89e-2

WFG8

-1.69e-2

-1.22e-2

-2.26e-3

WFG9

-9.21e-3

-4.93e-3

-8.44e-3

Table 4: Results for R indicator (neighbourhood

size = 20).

Test

Functions

TS-TribesV1

TS-TribesV2

TS-TribesV3

OKA2

-1.01e-3

-1.01e-3

-1.03e-3

Sympart

4.03e-5

4.84e-5

5.40e-5

S_ZDT1

6.26e-4

1.26e-3

1.26e-3

S_ZDT2

3.93e-5

1.35e-3

3.95e-5

S_ZDT4

2.31e-3

9.67e-3

2.53e-6

R_ZDT4

8.30e-3

2.78e-3

1.08e-4

S_ZDT6

3.37e-3

6.02e-3

4.32e-3

S_DTLZ2

1.52e-4

1.71e-4

2.41e-4

S_DTLZ3

1.43e-4

2.96e-4

7.36e-4

WFG1

2.88e-2

4.33e-2

3.02e-2

WFG8

-1.96e-2

-1.32e-2

-8.68e-3

WFG9

-1.18e-2

-7.59e-3

-8.26e-4

Table 5: Results for I

H

(neighbourhood size = 5).

Test

Functions

TS-TribesV1

TS-TribesV2

TS-TribesV3

OKA2

-1.23e-3

-1.22e-3

-1.21e-3

Sympart

2.01e-4

8.80e-5

2.49e-4

S_ZDT1

5.81e-4

5.13e-3

4.59e-3

S_ZDT2

3.40e-4

3.87e-3

3.08e-4

S_ZDT4

7.89e-3

1.38e-2

1.15e-2

R_ZDT4

1.47e-2

6.85e-3

4.30e-2

S_ZDT6

6.51e-3

1.65e-2

4.67e-3

S_DTLZ2

1.67e-3

8.78e-4

1.81e-3

S_DTLZ3

5.62e-3

8.30e-4

2.12e-2

WFG1

1.65e-1

2.08e-1

2.58e-1

WFG8

-1.25e-1

-7.21e-2

-1.42e-2

WFG9

-4.06e-2

-3.23e-2

-3.86e-3

Table 6: Results for I

H

(neighbourhood size = 10).

Test

Functions

TS-TribesV1

TS-TribesV2

TS-TribesV3

OKA2

-1.20e-3

-1.20e-3

-1.20e-3

Sympart

8.95e-5

9.47e-5

1.41e-4

S_ZDT1

2.45e-3

5.16e-3

5.11e-3

S_ZDT2

3.51e-4

2.74e-3

5.28e-4

S_ZDT4

7.84e-3

2.52e-2

4.57e-3

R_ZDT4

1.52e-2

7.07e-3

1.04e-3

S_ZDT6

6.38e-3

1.93e-2

5.21e-3

S_DTLZ2

8.09e-4

8.78e-4

1.81e-3

S_DTLZ3

6.10e-4

4.88e-3

1.07e-2

WFG1

1.70e-1

2.56e-1

2.55e-1

WFG8

-1.09e-1

-7.03e-2

-1.30e-2

WFG9

-2.29e-2

-3.01e-2

-5.43e-3

Table 7: Results for 𝐼

𝐻

(neighbourhood size = 20).

Test

Functions

TS-TribesV1

TS-TribesV2

TS-TribesV3

OKA2

-1.21e-3

-1.18e-3

-1.20e-3

Sympart

1.20e-4

1.44e-4

1.61e-4

S_ZDT1

1.50e-3

1.70e-3

5.24e-3

S_ZDT2

3.29e-4

8.65e-4

5.14e-4

S_ZDT4

6.52e-3

2.78e-2

1.52e-5

R_ZDT4

2.46e-2

8.55e-3

3.22e-4

S_ZDT6

9.59e-3

2.19e-2

2.92e-2

S_DTLZ2

1.30e-4

5.93e-4

1.94e-3

S_DTLZ3

2.98e-4

3.40e-3

1.74e-2

WFG1

1.63e-1

2.17e-1

1.70e-1

WFG8

-1.28e-1

-8.96e-2

-5.74e-2

WFG9

-7.22e-2

-2.49e-2

-1.05e-2

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

90

6 CONCLUSIONS

We have introduced a new hybrid multi-objective

evolutionary algorithm based on Tribes and TS. This

hybrid aims to combine the high convergence rate of

Tribes with the good neighbourhood exploration

performed by the TS algorithm. Therefore, we have

studied the impact of the place where we apply TS

technique on the performance of the algorithm. The

proposed version TS-TribesV1 gave the best results

almost for all the test functions except for S-ZDT4

and R-ZDT4 for which the TS-TribesV3 gave the

best results.

The results showed that the hybridization is a

very promising approach to multi-objective

optimization. As part of our ongoing work we are

going to compare the proposed algorithms with other

techniques that are representative of the state of art

of the multi-objective optimization. Moreover, we

are going to study other hybridization between

Tribes and other local search techniques.

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