Two Swarm Intelligence Algorithms for the Set Covering Problem

Broderick Crawford

1,2

, Ricardo Soto

1,3

, Rodrigo Cuesta

, Miguel Olivares-Su

arez

Franklin Johnson

and Eduardo Olgu

ın

Pontiﬁcia Universidad Cat

olica de Valpara

ıso, Av. Brasil 2950, Valpara

ıso, Chile

Universidad Finis Terrae, Av. Pedro de Valdivia 1509 Providencia, Santiago, Chile

Universidad Aut

onoma de Chile, Av. Pedro de Valdivia 641 Providencia, Santiago, Chile

Universidad de Playa Ancha, Avda. Gonz

alez de Hontaneda 855, Valpara

ıso, Chile

Universidad San Sebasti

an, Bellavista 7 Recoleta, Santiago, Chile

Keywords:

Weighted Set Covering Problem, Metaheuristics, Fireﬂy Algorithm, Artiﬁcial Bee Colony Algorithm, Swarm

Intelligence.

Abstract:

The Weighted Set Covering problem is a formal model for many industrial optimization problems. In the

Weighted Set Covering Problem the goal is to choose a subset of columns of minimal cost in order to cover

every row. Here, we present its resolution with two novel metaheuristics: Fireﬂy Algorithm and Artiﬁcial

Bee Colony Algorithm. The Fireﬂy Algorithm is inspired by the ﬂashing behaviour of ﬁreﬂies. The main

purpose of ﬂashing is to act as a signal to attract other ﬁreﬂies. The ﬂashing light can be formulated in such

a way that it is associated with the objective function to be optimized. The Artiﬁcial Bee Colony Algorithm

mimics the food foraging behaviour of honey bee colonies. In its basic version the algorithm performs a kind

of neighbourhood search combined with random search. Experimental results show that both are competitive

in terms of solution quality with other recent metaheuristic approaches.

1 INTRODUCTION

The Set Covering Problem (SCP) is a class of re-

presentative combinatorial optimization problem that

has been applied to many real world problems, such

as crew scheduling in airlines (Housos and Elmroth,

1997), facility location problem (Vasko and Wilson,

1984), and production planning in industry (Vasko

et al., 1987).

The SCP is a well-known NP-hard in the strong

sense (Garey and Johnson, 1990). Many algorithms

have been developed to solve it and has been reported

to literature. Exact algorithms are mostly based

on branch-and-bound and branch-and-cut (Balas and

Carrera, 1996; Fisher and Kedia, 1990). However,

these algorithms are rather time consuming and can

only solve instances of very limited size. For this

reason, many research efforts have been focused on

the development of heuristics to ﬁnd good or near-

optimal solutions within a reasonable period of time.

Classical greedy algorithms are very simple, fast,

and easy to code in practice, but they rarely produce

high quality solutions for their myopic and determi-

nistic nature (Chvatal, 1979). Compared with classi-

cal greedy algorithms, heuristics based on Lagrangian

relaxation with subgradient optimization are much

more effective. The most efﬁcient ones are those pro-

posed in (Ceria et al., 1998; Caprara et al., 1999).

As top-level general search strategies, metaheuris-

tics such as genetic algorithms (Beasley and Chu,

1996), simulated annealing (Brusco et al., 1999a),

tabu search (Caserta, 2007), evolutionary algorithms

(Crawford et al., 2007), ant colony optimization

(ACO) (Ren et al., 2010; Crawford et al., 2013b),

electromagnetism (unicost SCP) (Naji-Azimi et al.,

2010), gravitational emulation search (Balachandar

and Kannan, 2010) and cultural algorithms (Crawford

et al., 2013a) have been also successfully applied to

solve the SCP.

In this paper, we propose to solve the SCP with

two recent metaheuristics: Fireﬂy Algorithm and Ar-

tiﬁcial Bee Colony Algorithm. The Fireﬂy Algorithm

(FA) is a recently developed, population-based meta-

heuristic (Yang, 2010; Yang, 2009) where the objec-

tive function of a given optimization problem is based

on differences of light intensity. Thus, ﬁreﬂies are

Crawford B., Soto R., Cuesta R., Olivares-Suárez M., Johnson F. and Olguín E..

Two Swarm Intelligence Algorithms for the Set Covering Problem.

DOI: 10.5220/0005093500600069

In Proceedings of the 9th International Conference on Software Engineering and Applications (ICSOFT-EA-2014), pages 60-69

ISBN: 978-989-758-036-9

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

characterized by their light intensity which helps ﬁre-

ﬂies to change their position iteratively towards more

attractive locations in order to obtain optimal solu-

tions. The canonical FA algorithm is developed to

tackle continuous optimization problems (Fister et al.,

2013; Yang and He, 2013). However, the effective-

ness of the FA algorithm to solve discrete NP-hard

problems such as image compression and processing

(Horng, 2012), shape and size optimization (Miguel

and Fadel Miguel, 2012) and manufacturing cell pro-

blems (Sayadi et al., 2013) encourage researchers to

design novel FAs for discrete optimization problems.

The approach developed in this paper focus on trans-

fer functions which force ﬁreﬂies to move in binary

space. To the best of our knowledge, this is the ﬁrst

work proposing a binary coded FA to solve the SCP.

Artiﬁcial Bee Colony Algorithm (ABC) is one of the

most recent algorithms in the domain of the collective

intelligence. Created by Dervis Karaboga in 2005,

who was motivated by the intelligent behavior ob-

served in the domestic bees to take the process of for-

aging (Karaboga and Basturk, 2007). ABC mimics

the foraging strategy of honey bees to look for the best

solution to an optimization problem. Each candidate

solution is thought of as a food source and a colony

of bees is used to search in the solution space.

The rest of this paper is organized as follows. In

Section 2, we give a formal deﬁnition of the SCP. The

Section 3 describes FA and the Section 4 describes

ABC. In Section 5, we present experimental results

obtained when applying the algorithm for solving the

65 instances of SCP contained in the OR-Library. Fi-

nally, in Section 6 we conclude and highlight future

directions of research.

2 PROBLEM DESCRIPTION

The Set Covering Problem (SCP) can be formally

deﬁned as follows. Let A = (a

i j

) be an m-row, n-

column, zero-one matrix. We say that a column j

covers a row i if a

i j

= 1. Each column j is associa-

ted with a nonnegative real cost c

. Let I = {1,...,m}

and J = {1,...,n} be the row set and column set, res-

pectively. The SCP calls for a minimum cost subset

S ⊆ J, such that each row i ∈ I is covered by at least

one column j ∈S. A mathematical model for the SCP

Minimize f (x) =

∑

j=1

(1)

subject to

∑

j=1

i j

≥ 1, ∀i ∈ I (2)

∈ {0,1}, ∀j ∈J (3)

The goal is to minimize the sum of the costs of the

selected columns, where x

= 1 if the column j is in

the solution, 0 otherwise. The restrictions ensure that

each row i is covered by at least one column.

3 THE FIREFLY ALGORITHM

Nature-inspired methodologies are among the most

powerful algorithms for optimization problems. The

Fireﬂy Algorithm is a novel nature-inspired algorithm

inspired by the social behavior of ﬁreﬂies. By ideali-

zing some of the ﬂashing characteristics of ﬁreﬂies,

a ﬁreﬂy-inspired algorithm was presented in (Yang,

2010; Yang, 2009).

The canonical FA was developed using the follo-

wing three idealized rules:

• All ﬁreﬂies are unisex and are attracted to other

ﬁreﬂies regardless of their sex.

• The degree of the attractiveness of a ﬁreﬂy is pro-

portional to its brightness, and thus for any two

ﬂashing ﬁreﬂies, the one that is less bright will

move towards to the brighter one. More bright-

ness means less distance between two ﬁreﬂies.

However, if any two ﬂashing ﬁreﬂies have the

same brightness, then they move randomly.

• The brightness of a ﬁreﬂy is determined by the

value of the objective function. For a maximiza-

tion problem, the brightness of each ﬁreﬂy is

proportional to the value of the objective function.

As the attractiveness of a ﬁreﬂy is proportional to

the light intensity seen by adjacent ﬁreﬂies, the attrac-

tiveness β of a ﬁreﬂy is deﬁned as follows:

β(r) = β

−γr

, m ≥1 (4)

where r is the distance between two ﬁreﬂies, β

is the

attractiveness at r = 0 and γ is a ﬁxed light absorption

coefﬁcient. The distance r

i j

between two ﬁreﬂies i

and j at positions x

and x

is determined by

i j

= ||x

−x

|| =

∑

k=1

−x

)

(5)

where x

is the current value of the k

dimension of

the i

ﬁreﬂy and d is the number of dimensions. The

movement of a ﬁreﬂy i is attracted to another more

attractive (brighter) ﬁreﬂy j is determined by

(t + 1) = x

(t) + β

−γr

i j

(t) −x

(t)) + α(rand −

)

(6)

TwoSwarmIntelligenceAlgorithmsfortheSetCoveringProblem

where the ﬁrst term x

(t) is the current value (cu-

rrent position) of the k

dimension of the ﬁreﬂy i

at iteration t. The second term denotes the ﬁreﬂy

attractiveness where γ characterizes the variation of

the attractiveness typically varying from 0.1 to 10

(Yang, 2010), and the last term introduces randomiza-

tion, with α ∈ [0, 1] being the randomization parame-

ter and rand is a random number generator uniformly

distributed between 0 and 1.

3.1 Description of the Fireﬂy Approach

In this section, a discrete FA is proposed to solve the

SCP.

Step 1. Initialization of ﬁreﬂy parameters (γ, β

, size

for the ﬁreﬂy population and the maximum num-

ber of generations for the termination process).

Step 2. Initialization of ﬁreﬂy position. Initialize

randomly M = [X

;...;X

] of m solutions or ﬁreﬂy

positions in the multi-dimensional search space,

where m represents the size of the ﬁreﬂy popula-

tion. Each solution of X is represented by a d-

dimensional binary vector.

Step 3. Evaluation of ﬁtness of the population. For

this case the function of ﬁtness is equal to the ob-

jective function of the SCP model (Eq. 1).

Step 4. Modiﬁcation of ﬁreﬂy position. A ﬁreﬂy

produces a modiﬁcation in its position based on

the brightness w.r.t other ﬁreﬂies. Using Eq. 6

the new position is determined by modifying the

value of each dimension of a ﬁreﬂy. To move from

a continuous search space to a discrete one we

work with the following update rule:

(t +1) =



∗

if rand < T(x

(t +1))

0 otherwise

(7)

where rand is a uniform random number between

0 and 1, x

∗

is the best ﬁreﬂy so far, and T (x) is

the binary transfer function (Mirjalili and Lewis,

2013). The transfer function forces the values of

the dimensions of ﬁreﬂies to move in a binary

space. In this work we use:

T (x) =



arctan(



(8)

Step 5. Evaluation. The new solution is evaluated

and if it is not a feasible solution then it is re-

paired. In order to make feasible solutions we de-

termine which rows have not yet been covered and

choose the columns needed for coverage. The cri-

teria used to choose these columns is based in the

cost of a column/number of rows not covered that

cover the column j. Once the solution has become

feasible we apply an optimization step in order to

eliminate those redundant columns. A column is

redundant when it is removed and the solution re-

mains feasible.

Step 6. Memorization of the best solution achieved

so far and increment the counter of generations.

Step 7. Stop the process and display the result if the

termination criteria is satisﬁed. Termination crite-

ria used in this work is the maximum number of

generations. Otherwise, go to step 3.

The following algorithm shows the pseudo code

of the steps proposed.

Algorithm: FA pseudo-code.

1 Begin

2 Initialize parameters

3 Evaluate the light intensities

4 while t < MaxGeneration do

5 for i = 1 : m (m ﬁreﬂies) do

6 for j = 1 : m (m ﬁreﬂies) do

7 if (I

< I

) then

8 movement = calculates value according

to Eq. 6

9 if (rand() < T(movement)) then

10 f ire f lies[i][ j] = bestFire f ly[ j]

11 else

12 f ire f lies[i][ j] = 0

13 end if

14 end if

15 Repair solutions

16 Update attractiveness

17 Update light intensity

18 end for j

19 end for i

20 t = t + 1

21 end while

22 Output the results

23 End

4 THE ARTIFICIAL BEE

COLONY ALGORITHM

ABC is one of the most recent algorithms in the do-

main of the collective intelligence. Created by Dervis

Karaboga in 2005, who was motivated by the in-

telligent behavior observed in the domestic bees to

take the process of foraging (Karaboga and Basturk,

2007).

ABC is an algorithm of combinatorial optimiza-

tion based on populations, in which the solutions of

ICSOFT-EA2014-9thInternationalConferenceonSoftwareEngineeringandApplications

the problem of optimization, the sources of food, are

modiﬁed by the artiﬁcial bees, that fungen as opera-

tors of variation. The aim of these bees is to discover

the food sources with major nectar.

In the ABC algorithm, an artiﬁcial bee moves in

a multidimensional search space choosing sources of

nectar depending on its past experience and its com-

panions of beehive or ﬁtting his position. Some bees

(exploratory) ﬂy and choose food sources randomly

without using experience. When they ﬁnd a source of

major nectar, they memorize their positions and for-

get the previous ones. Thus, ABC combines methods

of local search and global search, trying to balance

the process of the exploration and exploitation of the

search space.

Although, the performance of different optimiza-

tion algorithm is dependent on applications some re-

cent works demonstrate that the Artiﬁcial Bee Colony

is faster than either Genetic Algorithm or Particle

Swarm Optimization solving certain problems (Zhang

and Wu, 2012; Zhang and Wu, 2011; Zhang et al.,

2011b; Zhang et al., 2011a; Zhang et al., 2011c). Ad-

ditionally, ABC has demonstrated an ability to attack

problems with a lot of variables (high-dimensional

problems) (Akay and Karaboga, 2009).

The pseudocode of Artiﬁcial Bee Colony is as

follows.

Algorithm: ABC pseudo-code.

1 Begin

2 InitPopulation()

3 while remain interations do

4 Select sites for the local search

5 Recruit bees for the selected sites and to

evaluate ﬁtness

6 Select the bee with the best ﬁtness

7 Assign the remaining bees to looking

for randomly

8 Evaluate the ﬁtnes of remaining bees

9 UpdateOptimum()

10 end while

11 return BestSolution

12 End

The procedure for determining a food source in

the neighborhood of a particular food source which

depends on the nature of the problem. Karaboga

(Karaboga, 2005) developed the ﬁrst ABC algorithm

for continuous optimization. The method for deter-

mining a food source in the neighborhood of a par-

ticular food source is based on changing the value of

one randomly chosen solution variable while keeping

other variables unchanged. This is done by adding to

the current value of the chosen variable the product of

a uniform variable in [-1, 1] and the difference in va-

lues of this variable -current food source - and some

other randomly chosen food source. This approach

can not be used for discrete optimization problems for

which it generates at best a random effect.

Singh (Singh, 2009) subsequently proposed a

method, which is appropriate for subset selection pro-

blems. In his model, to generate a neighboring solu-

tion, an object is randomly dropped from the solution

and in its place another object, which is not already

present in the solution, it is added. The object to be

added is selected from another randomly chosen solu-

tion. If there are more than one candidate objects for

addition then ties are broken arbitrarily.

This approach is based on the idea that if an ob-

ject is present in one good solution then it is highly

likely that this object is present in many good solu-

tions. This method provides another advantage, con-

sisting in that if the method fails to ﬁnd an object di-

fferent from the others objects in the original solution

it means that the two solutions are equal. Then, the

employed bee associated with the original solution is

converted in a scout bee eliminating duplication.

4.1 Description of the Bee Approach

In this section, an ABC is proposed to solve the SCP.

Step 1. Initialization.

To initialize the parameters of ABC as size of the

colony, number of workers and curious (onlookers

or “in wait”) bees, limit of attempts and maximum

number of cycles.

Step 2. Generation of initial population.

To generate the initial population by every row

(or SCP constraint) a column (or SCP variable) is

selected at random from the set of columns with

covering possibilities. After we run a duplicates

drop process, we check that there are no columns

duplicated. A solution is represented by means

of an entire vector like appears in Figure 1 stay-

ing the columns considered in the solution (a “-1”

means that the column was removed due to dupli-

cation). Then, we use an integer encoding as the

encoding rule.

333 10 5 300 −1 ... 657 99

Figure 1: Representation of a solution.

Step 3. Evaluation of the ﬁtness of the population.

The ﬁtness function is equal to the objective func-

tion of the SCP.

Step 4. Modiﬁcation of position and selection of

sites for worker bees.

TwoSwarmIntelligenceAlgorithmsfortheSetCoveringProblem

A hard-working bee modiﬁes its position by

means of the creation of a new solution based on a

different food source selected randomly. It sees if

at least it has a different column, in case of having

not even a different column, the hard-working bee

is transformed in an explorer in order to eliminate

duplicated solutions. In opposite case, it proceeds

to add a certain random number of columns bet-

ween 0 and the maximum numbers columns to

add.

After this, it proceeds to eliminate a certain ran-

dom number of columns between 0 and the max-

imum numbers columns to eliminate. In case that

new solution does not meet constraints, it is re-

paired. The ﬁtness of the solution is evaluated,

if the ﬁtness (cost) is minor that the solution had

in a beginning, the solution is replaced. In op-

posite case, it increases the number of attempts

for improving this solution (parameter limit of

attempts).

Step 5. Recruitment of curious bees for the selected

sites.

A curious bee evaluates the information of the

nectar through the workers and it chooses a source

of food with the ﬁtness proportionate selection

method or roulette-wheel selection.

Step 6. Modiﬁcation of position for the curious bees.

They work alike to hard-working bees in Step 4.

Step 7. To leave a source exploited by the bees.

If the solution representing a source of food

does not improve for a predetermined number of

attempts (limit), then the source of food is left and

is replaced by a new source of food generated as

in Step 1.

Step 8. Memorization of the best solution and to in-

crease the counter of the cycle.

Step 9. The process stops if the criteria of satisfac-

tion expires, in opposite case to return to Step 3.

5 EXPERIMENTAL RESULTS

In order to test the effectiveness of FA and ABC they

were tested using the 65 SCP test instances from OR-

Library (Beasley, 1990). These instances are divided

into 11 groups and each group contains 5 or 10 instan-

ces. Table 3 shows their detailed information where

“Density” is the percentage of non-zero entries in the

SCP matrix. The algorithms were implemented using

C language and conducted on a 1.8 GHz Intel Core 2

Duo T5670 CPU with 3GB RAM running Windows

In all experiments, the algorithms were executed

30 times over each SCP instance. Parameter va-

lues have a profound inﬂuence on the performance

of ABC. The parameters were empirically adjusted,

we determined their values in an experimental way,

for each parameter, a set of candidate values were

considered. We modiﬁed the value of one parame-

ter while keeping the others ﬁxed. According to the

best results, as parameter values in our experiments,

we used:

In FA the maximum number of generations was

set to 50. We used a population of 25 ﬁreﬂies and the

values of γ, β

are initialized to 1. ABC runs 1000 ite-

rations with a population of 200 bees, where 100 co-

rresponds to hard-working and 100 to curious. Limit

= 50, maximum number of columns to add = 0,5%

of columns in the SCP instance, maximum number

of columns to eliminate = 1, 2% of the SCP instance.

These parameters showed good results but they can-

not be the ideal ones for all the instances.

Tables 1 and 4 show the results obtained over the

65 instances. The quality of a solution is evaluated

using the relative percentage deviation (RPD). The

RPD value quantiﬁes the deviation of the objective

value Z from Z

opt

which in our case is the best known

cost value for each instance (see the second column).

We report the minimum cost reached, maximum, and

average of the obtained solutions. To compute RPD

we use Z = Min. This measure is computed as fo-

llows:

RPD = (Z −Z

opt

)/Z

opt

×100 (9)

The results expressed in terms of the RPD show

the effectiveness of our approach. It provides high

quality near optimal solutions and it has the ability to

generate them for a variety of instances.

5.1 Comparison with Other Works

In comparison with very recent works solving SCP -

- with Cultural algorithms (Crawford et al., 2013a)

and Ant Colony + Constraint Programming tech-

niques (Crawford et al., 2013c) - our ABC proposal

performs better with SCP instances reported in those

works.

In order to bring out the efﬁciency of our proposal

the solutions of the complete set of instances have

been compared with other metaheuristics. We com-

pared our algorithm solving the complete set of 65

standard non-unicost SCP instances from OR Library

with the newest ACO-based algorithm for SCP in the

literature: Ant-Cover + Local Search (ANT+LS) (Ren

et al., 2010), Genetic Algorithm (GA) proposed by

Beasley and Chu (1996) (Beasley and Chu, 1996) and

ICSOFT-EA2014-9thInternationalConferenceonSoftwareEngineeringandApplications

Simulated Annealing (SA) proposed by Brusco et al.

(1999) (Brusco et al., 1999b).

Tables 1 and 4 show the detailed results obtained

by the algorithms. Column 2 reports the optimal or

the best known solution value of each instance. The

third and fourth columns show the best value and the

average obtained by our ABC algorithm in the 30

runs (trials). The ﬁfth and sixth columns show the

best value and the average obtained by our FA in the

30 runs (trials). The next columns show the average

values obtained by GA, SA and ANT+LS respecti-

vely. The last columns show the Relative Percentage

Deviation (RPD) value over the instances tested with

ABC and FA.

In relation with ABC, we can observe in Tables 1

and 4 that:

• ABC is able to ﬁnd the optimal solution consis-

tently - i.e. in every trial- for 43 of 65 problems.

• ABC is able to ﬁnd the best known value in all

instances of Table 4.

• ABC is able to ﬁnd the best known value in all

trials of Table 4.

• ABC has higher success rate compared to genetic

algorithm, simulated annealing and ants in sets

NRE, NRF, NRG and NRH. The RPD of BEE is

0,00%, the RPD of GA is 1,04%, the RPD of SA

is 0,72% and the RPD of ANT+LS is 0,86%.

• ABC can obtain optimal solutions in some instan-

ces where the others metaheuristics failed.

In relation with FA, we can observe in Tables 1

and 4 that:

• A direct implementation of Binary FA shows good

results considering its simplicity. FA is capable of

solving problems which have continuous search

space. However, there are many optimization pro-

blems as SCP, which have discrete binary search

spaces. They need binary algorithms to be solved.

In the original FA, ﬁreﬂies can move around the

search space because of having position vectors

with continuous real domain. Consequently, the

concept of position updating can be easily imple-

mented for ﬁreﬂies based on the brightness w.r.t.

other ﬁreﬂies using 6. However, the meaning of

position updating is different in a discrete binary

space. In binary space, due to dealing with only

two numbers: “0” and “1”, the position updating

process cannot be done using 6. Therefore, we

have to ﬁnd a way to use the movement equation

to change the positions from “0” and “1” or vice

versa. This switching should be done based on

the attractiveness between the ﬁreﬂies. The idea

is to change position of a ﬁreﬂy with the proba-

bility of its movement to another more attractive

(brighter). In order to do this, a transfer function

is needed to map the movement values to proba-

bility values for updating the positions.

• A novel v-shape transfer function with a new rule

(7) was used in our Binary FA. Traditionally, it

is used a s-shape transfer function as S2 in Ta-

ble 2. Transfer functions force ﬁreﬂies to move

in a binary space. The transfer function should be

able to provide a high probability of changing the

position for a large absolute value of the move-

ment. It should also present a small probability of

changing the position for a small absolute value of

the movement. Moreover, the range of a transfer

function should be bounded in the interval [0,1]

and increased with the increasing of movement.

The binary version of FA proposed here uses the

transfer function V4 in Table 2. For us, this func-

tion had presented better experimental results than

others transfer functions showed.

5.2 Computational Cost and

Convergence to the Best Solution

Our Binary FA shows an excellent tradeoff between

the quality of the solutions obtained and the computa-

tional effort required. In all cases, FA converged very

quickly (mainly from the 3th iteration) and its compu-

tation time in the runs was less than 1 second (except

for NRG and NRH instances where the computation

time was less than 3 secs).

Due to a more sophisticated implementation, our

ABC algorithm requires a major computational effort.

In our ABC approach we considered speciﬁc opera-

tors and parameters to try with SCP. Anyway, we be-

lieve it is worth given the high quality of the solu-

tions obtained. In all cases, ABC converged quickly

(mainly from the 10th iteration) and its computation

time in the runs was less than 2 seconds (except for

NRG and NRH instances where the computation time

was less than 30 secs).

6 CONCLUSION

In this paper we have presented two recent swarm

based metaheuristics for optimizing the Weighted Set

Covering Problem. We have performed experiments

throught several ORLIB instances. Our approach

has demostrated to be very effective, providing unat-

tended solving methods, for quickly producing solu-

tions of a good quality. Experiments shown interest-

ing results in terms of robustness, where using the

TwoSwarmIntelligenceAlgorithmsfortheSetCoveringProblem

same parameters for different instances giving good

results.

The promising results of the experiments open up

opportunities for further research. We visualize diffe-

rent directions for future work:

• The fact that the presented algorithm is easy to

implement, clearly implies that ABC could also

be effectively applied to other combinatorial opti-

mization problems.

• An interesting proposal by Teodor Crainic et al. at

(Glover and Kochenberger, 2003) involves para-

llelizing strategies for metaheuristics. The author

sets a basis on the idea that the central goal of pa-

rallel computing is to speed up computation by di-

viding the work load among several threads of si-

multaneous execution, then a type of metaheuris-

tic parallelism could come from the decomposi-

tion of the decision variables into disjoint subsets.

The particular heuristic is applied to each subset

and the variables outside the subset are conside-

red ﬁxed.

• An interesting extension of this work would be

related to hybridization with other metaheuristics

or to apply a hyperheuristic approach (Valenzuela

et al., 2012).

• The use of Autonomous Search (AS), AS repre-

sents a new research ﬁeld, and it provides practi-

tioners with systems that are able to autonomously

selftune their performance while effectively sol-

ving problems. Its major strength and origina-

lity consist in the fact that problem solvers can

now perform self-improvement operations based

on analysis of the performances of the solving

process (Crawford et al., 2013d; Monfroy et al.,

2013; Crawford et al., 2012).

• Furthermore, we are considering to use different

preprocessing steps from the OR literature, which

allow to reduce the problem size (Krieken et al.,

2003).

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TwoSwarmIntelligenceAlgorithmsfortheSetCoveringProblem

APPENDIX

Table 1: Experimental results. - Instances with optimal.

Prob. Opt. ABC Min ABC Avg FA Min FA Avg GA Avg SA Avg ANT-LS Avg ABC RPD FA RPD

4.1 429 430 430.5 481 481.03 429.7 - 429 0.35 12.12

4.2 512 512 512 580 580 512 - 512 0 13.28

4.3 516 516 516 619 619.03 516 - 516 0 19.96

4.4 494 494 494 537 537 494.8 - 494 0 8.7

4.5 512 512 512 609 609 512 - 512 0 18.94

4.6 560 561 561.7 653 653 560 - 560 0.30 16.6

4.7 430 430 430 491 491.07 430.2 - 430 0 14.18

4.8 492 493 494 565 565 492.1 - 492 0.41 14.83

4.9 641 643 645.5 749 749.03 643.1 - 641 0.70 14.84

4.10 514 514 514 550 550 514 - 514 0 7

5.1 253 254 255 296 296.03 253 - 253 0.79 16.99

5.2 302 309 310.2 372 372 303.5 - 302 2.72 23.17

5.3 226 228 228.5 250 250 228 - 226 1.11 10.61

5.4 242 242 242 277 277.07 242.1 - 242 0 14.46

5.5 211 211 211 253 253 211 - 211 0 19.9

5.6 213 213 213 264 264.03 213 - 213 0 23.94

5.7 293 296 296 337 337 293 - 293 1.02 15.01

5.8 288 288 288 326 326 288.8 - 288 0 13.19

5.9 279 280 280 350 350 279 - 279 0.36 25.44

5.10 265 266 267 321 321 265 - 265 0.75 21.13

6.1 138 140 140.5 173 173.03 138 - 138 1.81 25.36

6.2 146 146 146 180 180.07 146.2 - 146 0 23.28

6.3 145 145 145 160 160 145 - 145 0 10.34

6.4 131 131 131 161 161 131 - 131 0 22.9

6.5 161 161 161 186 186 161.3 - 161 0 15.52

A.1 253 254 254 285 285 253.2 - 253 0.40 12.64

A.2 252 254 254 285 285.07 253 - 252 0.79 13.09

A.3 232 234 234 272 272 232.5 - 232.8 0.86 17.24

A.4 234 234 234 297 297 234 - 234 1.10 26.92

A.5 236 237 238.6 262 262 236 - 236 0 11.01

B.1 69 69 69 80 80.03 69 - 69 0 15.94

B.2 76 76 76 92 92 76 - 76 0 21.05

B.3 80 80 80 93 93 80 - 80 0 16.25

B.4 79 79 79 98 98.03 79 - 79 0 24.05

B.5 72 72 72 87 87 72 - 72 0 20.83

C.1 227 230 231 279 279 227.2 - 227 1.76 22.9

C.2 219 219 219 272 272 220 - 219 0 24.2

C.3 243 244 244.5 288 288 246.4 - 243 0.62 18.51

C.4 219 220 224 262 262 219.1 - 219 2.28 19.63

C.5 215 215 215 262 262.07 215.1 - 215 0 21.86

D.1 60 60 60 71 71 60 - 60 0 18.33

D.2 66 67 67 75 75 66 - 66 1.52 13.63

D.3 72 73 73 88 88 72.2 - 72 1.39 22.22

D.4 62 63 63 71 71 62 - 62 1.61 14.51

D.5 61 62 62 71 71 61 - 61 1.64 16.39

NRE.1 29 29 29 32 32.03 29 29 29 0 10.34

NRE.2 30 30 30 36 36 30.6 30 30 0 20

NRE.3 27 27 27 35 35 27.7 27 27 0 29.62

NRE.4 28 28 28 34 34 28 28 28 0 21.42

NRE.5 28 28 28 34 34 28 28 28 0 21.42

NRF.1 14 14 14 17 17.03 14 14 14 0 21.42

NRF.2 15 15 15 17 17 15 15 15 0 13.33

NRF.3 14 14 14 21 21 14 14 14 0 50

NRF.4 14 14 14 19 19 14 14 14 0 35.71

NRF.5 13 13 13 16 16 13.7 13.7 13.5 0 23.07

NRG.1 176 176 176 230 230.03 177.7 176.6 176 0 30.68

NRG.2 154 154 154 191 191 156.3 155.3 155.1 0 24.02

NRG.3 166 166 166 198 198 167.9 167.6 167.3 0 19.27

NRG.4 168 168 168 214 214 170.3 170.7 168.9 0 27.38

NRG.5 168 168 168 223 223 169.4 168.4 168.1 0 32.73

NRH.1 63 63 63 85 85.07 64 64 64 0 34.92

NRH.2 63 63 63 81 81.03 64 63.7 67.9 0 28.57

NRH.3 59 59 59 76 76 59.1 59.4 59.4 0 28.81

NRH.4 58 58 58 75 75 58.9 58.9 58.7 0 29.31

NRH.5 55 55 55 68 68 55.1 55 55 0 23.63

ICSOFT-EA2014-9thInternationalConferenceonSoftwareEngineeringandApplications

Table 2: S-shaped and v-shaped transfer functions.

S-shaped family V-shaped family

Name Transfer function Name Transfer function

S1 T (x) =

1+e

−2x

V1 T (x) =



erf



√





√

−t



S2 T (x) =

1+e

−x

V2 T (x) =

tanh(x)

S3 T (x) =

1+e

−x/2

V3 T (x) =



√

1+x



S4 T (x) =

1+e

−x/3

V4 T (x) = |

arctan(

x)|

Table 3: Details of the test instances.

Instance set No. of instances m n Cost range Density (%) Optimal solution

4 10 200 1000 [1, 100] 2 Known

5 10 200 2000 [1, 100] 2 Known

6 5 200 1000 [1, 100] 5 Known

A 5 300 3000 [1, 100] 2 Known

B 5 300 3000 [1, 100] 5 Known

C 5 400 4000 [1, 100] 2 Known

D 5 400 4000 [1, 100] 5 Known

NRE 5 500 5000 [1, 100] 10 Unknown

NRF 5 500 5000 [1, 100] 20 Unknown

NRG 5 1000 10000 [1, 100] 2 Unknown

NRH 5 1000 10000 [1, 100] 5 Unknown

Table 4: Experimental results - Instances with Best Known Solution.

Prob. Opt. ABC Min ABC Avg FA Min FA Avg GA Avg SA Avg ANT-LS Avg ABC RPD FA RPD

NRE.1 29 29 29 32 32.03 29 29 29 0 10.34

NRE.2 30 30 30 36 36 30.6 30 30 0 20

NRE.3 27 27 27 35 35 27.7 27 27 0 29.62

NRE.4 28 28 28 34 34 28 28 28 0 21.42

NRE.5 28 28 28 34 34 28 28 28 0 21.42

NRF.1 14 14 14 17 17.03 14 14 14 0 21.42

NRF.2 15 15 15 17 17 15 15 15 0 13.33

NRF.3 14 14 14 21 21 14 14 14 0 50

NRF.4 14 14 14 19 19 14 14 14 0 35.71

NRF.5 13 13 13 16 16 13.7 13.7 13.5 0 23.07

NRG.1 176 176 176 230 230.03 177.7 176.6 176 0 30.68

NRG.2 154 154 154 191 191 156.3 155.3 155.1 0 24.02

NRG.3 166 166 166 198 198 167.9 167.6 167.3 0 19.27

NRG.4 168 168 168 214 214 170.3 170.7 168.9 0 27.38

NRG.5 168 168 168 223 223 169.4 168.4 168.1 0 32.73

NRH.1 63 63 63 85 85.07 64 64 64 0 34.92

NRH.2 63 63 63 81 81.03 64 63.7 67.9 0 28.57

NRH.3 59 59 59 76 76 59.1 59.4 59.4 0 28.81

NRH.4 58 58 58 75 75 58.9 58.9 58.7 0 29.31

NRH.5 55 55 55 68 68 55.1 55 55 0 23.63

TwoSwarmIntelligenceAlgorithmsfortheSetCoveringProblem