A EVOLUTIONARY APPROACH TO SOLVE SET COVERING

Broderick Crawford, Carolina Lagos

Escuela de Ingenier

ıa Inform

atica, Pontiﬁcia Universidad Cat

olica de Valpara

ıso, Valpara

ıso, Chile

Carlos Castro

Departamento de Inform

atica, Universidad T

ecnica Federico Santa Mar

ıa, Valpara

ıso, Chile

Fernando Paredes

Escuela de Ingenier

ıa Industrial, Universidad Diego Portales, Santiago, Chile

Keywords:

Set Covering Problem, Cultural Algorithm, Genetic and Evolutionary Computation.

Abstract:

In this paper we solve the classical Set Covering Problem comparing two evolutive techniques: Genetic Algo-

rithms and Cultural Algorithms. We solve this problem with a Cultural Evolutionary Architecture maintaining

knowledge of Diversity and Fitness learned over each generation during the search process and we compare

it with a Genetic Algorithm using the same crossover and mutation mechanisms. Our results indicate that the

approach is able to produce very competitive results in compare with other Metaheuristics and Approximation

Algorithms.

1 INTRODUCTION

Cultural algorithms are a technique that incorporates

knowledge obtained during the evolutionary process

trying to make the search process more efﬁcient. Cul-

tural algorithms have been successfully applied to

several types of optimization problems (Coello and

Landa, 2002; Landa and Coello, 2005b). However,

nobody had proposed a cultural algorithm for the Set

Covering Problem.

Set partitioning problem (SPP) and set covering

problem (SCP) are two types of problems that can

model several real life situations (Feo and Resende,

1989; Chu and Beasley, 1998). In this work, we

solve some benchmarks of SCP with a new evolu-

tive approach: cultural algorithms (Reynolds, 1994;

Reynolds, 1999; Reynolds and Peng, 2004).

This paper is organized as follows: In Section 2,

we formally describe SCP using mathematical pro-

gramming models. In section 3 we present the cul-

tural evolutionary architecture. In sections 4 we show

the population space and the belief space considered

in cultural algorithms. In Section 5, we present exper-

imental results obtained when applying the algorithm

for solving some standard benchmarks taken from the

ORLIB (Beasley, 1990). Finally, in Section 6 we con-

clude the paper and give some perspectives for future

research.

2 PROBLEM DESCRIPTION

SPP is the NP-complete problem of partitioning a

given set into mutually independent subsets while

minimizing a cost function deﬁned as the sum of the

costs associated to each of the eligible subsets (Chu

and Beasley, 1998). In the SPP linear programming

formulation we are given a m rows and n columns in-

cidence matrix A = (a

) in which all the matrix ele-

ments are either zero or one. Additionally, each col-

umn is given a non-negative real cost c

. We say that

a column j can cover a row i if a

= 1. Let J denotes

the set of the columns and x

a binary variable which

is one if column j is chosen and zero otherwise. The

SPP can be deﬁned formally as follows:

Minimize f(x) =

∑

j=1

(1)

356

Crawford B., Lagos C., Castro C. and Paredes F. (2007).

A EVOLUTIONARY APPROACH TO SOLVE SET COVERING.

In Proceedings of the Ninth International Conference on Enterpr ise Information Systems - AIDSS, pages 356-360

DOI: 10.5220/0002406703560360

 SciTePress

Subject to

∑

j=1

= 1; ∀i = 1, . . . , m (2)

∈ 0, 1 i = 1, . . . , m (3)

These constraints enforce that each row is covered

by exactly one column. The SPP has been studied ex-

tensively over the years because of its many real world

applications. The SCP is a SPP relaxation (replacing

the equation 2 by 4. The goal in the SCP is to choose

a subset of the columns of minimal weight which cov-

ers every row. The SCP can be deﬁned formally using

constraints to enforce that each row is covered by at

least one column as follows:

∑

j=1

≥ 1; ∀i = 1, . . . , m (4)

3 CULTURAL EVOLUTIONARY

ARCHITECTURE

The cultural algorithms were developed by Robert

G. Reynolds (Reynolds, 1994; Reynolds and Peng,

2004) as a complement to the metaphor used by evo-

lutionary algorithms that are mainly focused on natu-

ral selection and genetic concepts. The cultural algo-

rithms are based on some theories which try to model

cultural as an inheritance process operating at two

levels: a micro-evolutionary level, which consists of

the genetic material that an offspring inherits from its

parent, and a macro-evolutionary level, which is the

knowledge acquired by the individuals through gen-

erations. This knowledge, once encoded and stored,

it serves to guide the behavior of the individuals that

belong to a population.

Considering that evolution can be seen like an op-

timization process, Reynolds developed a computa-

tional model of cultural evolution that can have ap-

plications in optimization (Coello and Landa, 2002;

Landa and Coello, 2005b). He considered the phe-

nomenon of double inheritance with the purpose of

increase the learning or convergence rates of an evo-

lutionary algorithm. In this model each one of the lev-

els is represented by a space. The micro-evolutionary

level is represented by the population space and the

macro-evolutionary level by the belief space.

The population space can be adopted by anyone

of the paradigms of evolutionary computation, such

as the genetic algorithms, the evolutionary strategies

or the evolutionary programming. In all of them there

is a set of individuals where each one has a set of in-

dependent characteristics with which it is possible to

determine its ﬁtness. Through time, such individu-

als could be replaced by some of their descendants,

obtained from a set of operators (crossover and muta-

tion, for example) applied to the population.

The belief space is the ”store of the knowledge”

acquired by the individuals along the generations.

The information in this space must be available for

the population of individuals. There is a protocol of

communication established to dictate rules about the

type of information that it is necessary to interchange

between the spaces. This protocol deﬁnes two

functions: acceptance, this function extracts the

information (or experience) from the individuals

of a generation putting it into the belief space; and

Inﬂuence, this function is in charge ”to inﬂuence”

in the selection and the variation operators of the

individuals (as the crossover and mutation in the

case of the genetic algorithms). This means that this

function exerts a type of pressure according to the

information stored in the belief space.

In the belief space of a cultural evolution model there

are different types of knowledge: Situational, Norma-

tive, Topographic, Historical or Temporal, and Do-

main Knowledge. According to Reynolds and Peng

(Reynolds, 1999; Peng, 2005) these types conform

a complete set, that is any other type of knowl-

edge can be generated by means of a combination

of two or more of the previous types of knowledge.

The pseudo-code of a cultural algorithm appears later

(Landa and Coello, 2005a). Most of the steps of a

cultural algorithm correspond with the steps of a tra-

ditional evolutionary algorithm. It can be clearly seen

that the main difference lies in the fact that cultural

algorithm use a belief space. In the main loop of the

algorithm, we have the update of the belief space, at

this point the belief space incorporates the individual

experiences of a select group of members with the ac-

ceptance function, which is applied to the entire pop-

ulation.

Pseudo-code of a Cultural Algorithm

1 Generate the initial population

2 Initialize the space of beliefs

3 Evaluate the initial population

4 Repeat

5 Update the space of beliefs

(with the individuals accepted)

6 Apply the variation operators(under

the influence of the space of beliefs)

7 Evaluate each child

8 Perform selection

9 While the end condition is not satisfied

A EVOLUTIONARY APPROACH TO SOLVE SET COVERING

357

4 POPULATION SPACE AND

BELIEF SPACE

In the design and development of our cultural algo-

rithm solving SCP we considered in the population

space a genetic algorithm with binary representation.

An individual, solution or chromosome is an n-bit

string, where a value 1 in the bit indicates that the

column is considered in the solution and zero in an-

other case (value in j-bit corresponds to value of in

the linear programming model). The initial popula-

tion was generated with n selected individuals ran-

domly with a repair process in order to as-sure the

feasibility of the individuals. For the selection of par-

ents we used binary tournament and the method of the

roulette. For the process of variation we used the op-

erator of basic crossover and the fusion operator pro-

posed by Beasley and Chu (Beasley and Chu, 1996),

for mutation we used interchange and multibit. For

the treatment of unfeasible individuals we applied the

repairing heuristic proposed by Beasley and Chu too.

In the re-placement of individuals we use the strategy

steady state and the heuristic proposed by Lozano et

al. (Lozano et al., 2003), which is based on the level

of diversity contribution of the new offspring. The

genetic diversity was calculated by the Hamming dis-

tance, which is deﬁned as the number of bit differ-

ences between two solutions. The ﬁtness function is

determined for:

∑

j=1

(5)

Where S is the set of columns in the solutions, S

is the value of bit (or column) j in the string corre-

sponding to individual i and c

is the cost of the bit.

The main idea is try to replace a solution with worse

ﬁtness and with lower contribution of diversity than

the one provided by the offspring. In this way, we

are working with two underlying objectives simulta-

neously: to optimize the ﬁtness and to promote useful

diversity.

The main idea is try to replace a solution with

worse ﬁtness and with lower contribution of diversity

than the one provided by the offspring. In this way, we

are working with two underlying objectives simulta-

neously: to optimize the ﬁtness and to promote useful

diversity.

In a cultural algorithm, the shared belief space is

the foundation supporting the efﬁciency of the search

process. In order to ﬁnd better solutions and improve

the con-vergence speed we incorporated information

about the diversity in the belief space. We stored in

the belief space the individual with better ﬁtness of

the current generation and the individual who deliv-

ers major diversity to the population, which will be

considered leaders in the space of beliefs. With this

type of knowledge situational, each of the new indi-

viduals generated try to follow a leader stored in the

space of beliefs.

A situational-ﬁtness knowledge procedure selects

from the initial population the individual with better

ﬁtness, which will be a leader in the situational-ﬁtness

space of beliefs. A situational-diverse knowledge pro-

cedure selects from the initial population the most di-

verse individual of the population, which will be a

leader in the situational-diverse space of beliefs.

In this work, we implemented the inﬂuence

of situational-ﬁtness knowledge in the operator of

crossover. The inﬂuence initially appears at the mo-

ment of the parental selection, the ﬁrst father will be

chosen with the method of binary tournament and the

second father will be the individual with better ﬁtness

stored in the space of beliefs. In relation with the in-

ﬂuence of situational-diverse knowledge in the oper-

ator of cross-over, this procedure works recombining

the individual with better ﬁtness of every generation

with the most diverse stored in the space of beliefs,

with this option we expect to deliver diversity to the

population.

The updating the situational belief space proce-

dure implies that the situational space of beliefs will

be updated in all generations of the evolutionary pro-

cess. The update of the situational space of beliefs

consists in the replacement of the individuals by cur-

rent generation individuals if they are better consider-

ing ﬁtness and diversity.

5 COMPARISON OF RESULTS

The following results present the cost obtained when

applying different operators in our algorithm for solv-

ing SCP41, SCP42, SCP48, SCP51, SCP61, SCP62,

SCP63, SCPa1, SCPb1, SCPc1 benchmarks, these

test problem sets were obtained electronically from

OR-Library (Beasley, 1990). The ﬁrst table shows the

optimal values and the number of rows(m), number of

columns(n) for diverse instances of the Set Covering

Problem. Following the same sequence that the table

1, the table 2 shows the cost from a Genetic Algo-

rithm using the basic proposal described in section 4

not considering diversity. The next column presents

the best cost obtained when applying our Cultural Al-

gorithm. The table 3 shows the results applying Ant

System (AS) and Ant Colony System (ACS) taken

from (Crawford and Castro, 2006) and Round, Dual-

LP, Primal-Dual, Greedy taken from (Gomes et al.,

ICEIS 2007 - International Conference on Enterprise Information Systems

358

Table 1: Test problem details for different instances of the

SCP.

Problem Optimal Rows(m) Columns(n)

SCP41 429 200 1000

SCP42 512 200 1000

SCP48 492 200 1000

SCP51 253 200 2000

SCP61 138 200 1000

SCP62 146 200 1000

SCP63 145 200 1000

SCPa1 253 300 3000

SCPb1 69 300 3000

SCPc1 227 400 4000

Table 2: Obtained costs using Genetic Algorithm (GA) and

Cultural Algorithm (CA) with Basic Crossover and Muta-

tion Interchange.

Problem GA CA

SCP41 519 446

SCP42 627 569

SCP48 642 636

SCP51 303 295

SCP61 179 151

SCP62 183 162

SCP63 198 196

SCPa1 426 300

SCPb1 97 95

SCPc1 238 277

2006). The algorithm has been run with the following

parameters setting: size of the population (n) =100,

size of the tournament (t) =3, number of generations

(g) =30, probability of crossing (pc) =0.6 and prob-

ability of mutation (pm) =0.2. The algorithm was

implemented using ANSI C, GCC 3.3.6, under Mi-

crosoft Windows XP Professional version 2002.

Table 3: Obtained costs using algorithms to solve SCP from

different authors.

Problem AS ACS Round Dual Primal Greedy

-LP -Dual

SCP41 473 463 429 505 521 463

SCP42 594 590 * * * *

SCP48 524 522 * * * *

SCP51 289 280 405 324 334 293

SCP61 157 154 301 210 204 155

SCP62 169 163 347 209 232 170

SCP63 161 157 * * * *

SCPa1 * * 592 331 348 288

SCPb1 * * 196 115 101 75

SCPc1 * * 592 331 348 288

Figure 1: Fitness versus generations are showed for SCP41

instances, using Basic Crossover and Mutation Interchange.

6 CONCLUSIONS

In this paper we have introduced the ﬁrst proposal to

solve SCP with Cultural Algorithms. The main idea

of the Cultural Algorithms is to incorporate knowl-

edge acquired during the search process, integrating

inside of an evolutionary algorithm the so called Be-

lief Space. The objective is to do more robust algo-

rithms with greater rates of convergence. Our compu-

tational results conﬁrm that incorporating information

about the diversity of solutions we can obtain good

results in the majority of the experiments. Our main

conclusion from this work is that we can improve the

performance of Genetic Algorithms considering ad-

ditional information in the evolutionary process. Ge-

netic Algorithms tends to lose diversity very quickly.

In order to deal with this problem, we have shown

that maintaining diversity in the Belief Space we can

improve the computational efﬁciency. Evolutionary

Algorithms can be seen as two phase process. The

ﬁrst phase explores for good solutions, while the sec-

ond phase exploits the better solutions. Both phases

appear to require different diversity. Then, in order

to improve further the performance of our algorithm,

we are now exploring the possibilities for employing

knowledge about diversity from the Belief Space only

in the ﬁrst generations of the Population Space. In this

way, considering that population diversity and selec-

tive pressure (that gives individuals with higher ﬁt-

ness a higher chance of being selected for reproduc-

tion) are inversely related, we also plan to work on

a non-deterministic use of the two functions (accept

and inﬂuence) communication. The main idea behind

this approach is an adaptive solving way that it eval-

A EVOLUTIONARY APPROACH TO SOLVE SET COVERING

359

uates the trade-offs between exploration and explota-

tion during the process.

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