A novel binary artificial bee colony algorithm based on genetic operators

Accepted Manuscript

A novel binary artificial bee colony algorithm based on genetic operators

Celal Ozturk, Emrah Hancer, Dervis Karaboga

PII: S0020-0255(14)01053-6

DOI: http://dx.doi.org/10.1016/j.ins.2014.10.060

Reference: INS 11228

To appear in: Information Sciences

Received Date: 25 November 2013

Revised Date: 20 October 2014

Accepted Date: 27 October 2014

Please cite this article as: C. Ozturk, E. Hancer, D. Karaboga, A novel binary artificial bee colony algorithm based

on genetic operators, Information Sciences (2014), doi: http://dx.doi.org/10.1016/j.ins.2014.10.060

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http://dx.doi.org/10.1016/j.ins.2014.10.060

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* Corresponding Author: Celal OZTURK Tel: +90 352 207 66 66 #32581

A Novel Binary Artificial Bee Colony Algorithm BasedOn Genetic Operators

Celal OZTURK*, Emrah HANCER, and Dervis KARABOGAErciyes University, Engineering Faculty, Computer Engineering Department, Kayseri, Turkey

{celal,emrahhancer,karaboga}@erciyes.edu.tr

Abstract—This study proposes a novel binary version of the artificial bee colony algorithm

based on genetic operators (GB-ABC) such as crossover and swap to solve binary optimization

problems. Integrated to the neighborhood searching mechanism of the basic ABC algorithm,

the modification comprises four stages: 1) In neighbourhood of a (current) food source,

randomly select two food sources from population and generate a solution including zeros

(Zero) outside the population; 2) apply two-point crossover operator between the current, two

neighborhood, global best and Zero food sources to create children food sources; 3) apply swap

operator to the children food sources to generate grandchildren food sources; and 4) select the

best food source as a neighbourhood food source of the current solution among the children and

grandchildren food sources. In this way, the global-local search ability of the basic ABC

algorithm is improved in binary domain. The effectiveness of the proposed algorithm GB-ABC

is tested on two well-known binary optimization problems: dynamic image clustering and 0-1

knapsack problems. The obtained results clearly indicate that GB-ABC is the most suitable

algorithm in binary optimization when compared with the other well-known existing binary

optimization algorithms. In addition, the achievement of the proposed algorithm is supported by

applying it to the CEC2005 benchmark numerical problems.

Keywords; Binary optimization, Dynamic clustering, Knapsack problem, Artificial bee

colony, Genetic algorithm.

1 Introduction

Concentrating on local interactions of the swarm individuals (including bird

flocks, fish schools, ants, bees and etc.) with each other and with their environments

[5], Swarm Intelligence (SI) has become a significant research area among computer

scientists, engineers, economists, bioinformaticians and several other disciplines on

account of the fact that the remarkable ability of natural intelligent swarms on solving

their related problems (finding food, building nests etc.) can be simulated to deal with

real world problems [32]. It is known that an intelligent swarm should have the

following properties [47]: 1) the ability of proximity, 2) the ability of receiving and

responding to quality factors, 3) the ability of protecting behaviours against

fluctuations and 4) the ability of adapting to diverse situations. In the 1990s, two

important SI based algorithms which are ant colony optimization (ACO) [18] and


particle swarm optimization (PSO) [20] were developed and both of them have

attracted researchers’ attention.

After realizing the above four mentioned properties of SI in honey bee swarms,

researchers, especially from the beginning of the 2000s, started to concentrate on

modelling various intelligent behaviours of these swarms; for instance, the behaviours

of dance and communication, collective decision, task allocation, nest site selection,

mating, foraging, marriage, floral and pheromone laying [29]. Some well-known bee

swarm intelligence based algorithms are the virtual bees algorithm (VBA) [70], the

bees algorithm (BA) [57], BeeAdHoc [64], honey bee mating optimization (HMBO)

[22], the BeeHive [65], bee system (BS) [43], bee colony optimization (BCO) [61]

and the artificial bee colony (ABC) algorithm [27, 28]. According to [32], ABC is the

most widely used algorithm among bee swarm based algorithms presented in the

literature. Some successful applications of the ABC algorithm are optimization of

numerical problems [31], data clustering [35], neural networks training [34], wireless

sensor network deployment [54] and routing [33], and image analysis [24, 25, 44, 52,

53]. It is clearly seen that there exist various appreciated studies of the ABC algorithm

on continuous space [1, 6, 7, 13, 14, 30, 60, 66, 68, 72]. However, that situation is not

the same for the studies of ABC on binary problems. Through this gained

information, the basic aim of this study is to propose an improved binary model of the

ABC algorithm to solve binary optimization problems.

1.1 Related Works

It is known that the standard ABC algorithm can optimize continuous problems. In

other words, the structure of the basic ABC cannot be directly adapted to the binary

optimization problems. Thus, some modifications are required on the ABC algorithm

for binary optimization applications. Pampapa and Engelbrecht [56] considered

binary solution as four dimensional real vector through angle modulation schema.

Kashan et al. [37] introduced a binary version of the ABC algorithm (DisABC) which

uses the dissimilarity measure of binary structures instead of the arithmetic

subtraction operator. To measure the similarity of binary structures, Jaccard’s

coefficient of similarity is employed and to generate a new solution, two selections

are applied in a probabilistic manner; random selection depending only on

neighbourhood and greedy selection depending on both neighbourhood and the best

solution. The performance of the algorithm was verified with the implementation of

the uncapacited facility location problem (UFLP). The same methodology was also

applied to the differential evolution algorithm to solve UFLP problem [36]. Kıran and

Gunduz [39] proposed a binary version of the ABC algorithm based on XOR logic

operator (XOR-ABC) and its performance was compared with the DisABC algorithm

in terms of UFLP.


1.2 Motivations

For decades, researchers have attempted to develop algorithms in optimization

satisfying superior performance with respect to the other algorithms in literature [49].

To design a suitable algorithm, it is crucial to establish a connection between the

algorithm and a given optimization problem i.e. analysing optimization problem is the

starting point. There exist two mostly preferred approaches by the researchers to

improve an existing algorithm, known as hybridization and modification. The former

is the process of mixing of at least two heterogeneous particles through conscious

manipulation or as a natural progressive manipulation [71]. The latter is the process of

modifying some particles of the mechanism through internal or external forces. As

observed in nature, memes, which are the basic unit of cultural transmission and

imitation [17], can encounter modifications and combine with each other to generate

stronger memes. Language life cycle can be given as an example to this concept.

While new words are welcomed by languages on account of the needs and interests of

society or community, some words lose their popularity and hence become

disappeared from languages. From the algorithmic perspective, it is known that two or

more properly combined distinct algorithms and modified algorithms by the operators

of other algorithms can improve the ability of problem-solving mechanism. As the

importance of cultural information transmission has been realized by researchers,

memetic algorithms, “which are population-based metaheuristics composed of an

evolutionary framework and a set of local search algorithms activated within the

generation cycle of the external framework” [49], has gained popularity. That

motivates us to develop a genetically modified artificial bee colony algorithm for

optimizing binary search space which resembles to the memetic algorithms using

genetic operators.

1.3 Contribution

In this paper, a novel binary artificial bee colony algorithm (GB-ABC) based on

genetic operators is introduced. To our knowledge, this study is the first to use genetic

operators in a binary model of the ABC algorithm. Furthermore, the GB-ABC

algorithm is not intended to solve only a specific binary problem, it is also proposed

to overcome general binary optimization problems. This is demonstrated by applying

GB-ABC to well-known binary optimization problems; the dynamic image clustering

problem known as the process of automatically finding the optimum number of

clusters and the 0-1 knapsack problem known as maximizing the cost of a knapsack

with maximum weight capacity. For the dynamic clustering problem, the performance

of the GB-ABC algorithm is analysed by comparing it with the results of the binary

particle swarm optimization for dynamic clustering (known as DCPSO) [51], the


quantum inspired binary particle swarm optimization (QBPSO) [10], the genetic

algorithm for dynamic clustering, the discrete binary artificial bee colony (DisABC)

[37], the self-organizing map (SOM), the unsupervised fuzzy clustering approach

(UFA) and Snob algorithms. As for the knapsack problem, the performance of the

GB-ABC is evaluated by comparing with the results of the BPSO, QBPSO, GA, and

DisABC algorithms. It should be noted that the applied GA in binary problems is one

of the basic type of GA in which the population is represented in bit string form, the

crossover type is scattered, the mutation type is Gaussian, the reproduced population

consists of 2 elitist members, and selection type is stochastic uniform. Besides binary

problems, the suitability of the GB-ABC algorithm to the numerical problems

(CEC2005 benchmarks) is also considered and its results are compared with the same

existing algorithms as in the knapsack problem.

1.4 Organization

The rest of the paper is organized as follows: Section 2 summarizes the ABC

algorithm; Section 3 describes the GB-ABC algorithm; Section 4 explains the

considered problems concerning the perspective of an optimization problem; Section

5 discusses the results; and finally Section 6 concludes the paper.

2 ABC Algorithm

The artificial bee colony (ABC) algorithm is the simulation of the minimalistic

foraging model of honey bee in search process for solving real-parameter, non-

convex, and non-smooth optimization problems [30]. In this algorithm, a colony

consists of three types of artificial bees: employed bees, onlooker bees and scout bees;

Employed bee: Each food source is associated with an employed bee and each

employed bee tries to detect a new food source in the neighbourhood of its current

food source. The detected food source is memorized when the nectar amount of the

detected food source is higher than the nectar amount of current food source. After

completion of the search process, employed bees share their information concerning

the nectar amount of food sources with onlooker bees via waggle dance in the dance

area.

Onlooker bee: An onlooker bee evaluates the information gained from the

employed bees and tries to find a new food source in the neighbourhood of the

selected food based on this evaluated information. Thus, the tendency of onlooker

bees is to search around the food sources with high nectar amount; in this way, more

qualified food sources can be chosen for exploitation.


Scout Bee: The number of scout bees is not predefined in the colony. A scout bee

is produced according to the situation of a food source whether it is abandoned or not.

When an abandoned (fully exploited) food source is detected, its employed becomes a

scout bee, and then randomly searches a new food source in space. After that, the

scout bee returns to its earlier type i.e. employed bee again. The number of scout bees

is controlled by the parameter “limit” representing the number of trials before

defining a food source as “abandoned”. If a food source cannot be improved during

the predetermined number of unsuccessful trials being equal to limit parameter, that

food source is leaved and a new food source is randomly generated. Exploration and

exploitation procedures are carried out together for robust search process in the ABC

module [31].

In the ABC model, a food source denotes a possible solution of an optimization

problem and the nectar amount of a food source represents the quality of a solution.

As mentioned before, each food source is associated with an employed bee in the

colony. Thus, the number of food sources is equal to the number of employed bees.

Also, the number of employed bees is equal to the number of onlooker bees.

Therefore, the number of food sources is half of the population number.

The main steps of the ABC algorithm are as follows:

1. Initialize food sources ( ) by Eq. (1);

= + (0,1) − (1)

where = 1, … , (SN is the number of food sources); = 1, … , (D is the number

of parameters); is the minimum and is the maximum values of

parameter j.

2. Move employed bees to the food sources and determine neighbourhood food

sources by Eq. (2);

İ = +ϕ (x − x ) (2)

where and k are randomly chosen parameters and neighbourhoods, respectively

and ϕ is a random number within [-1,1].

3. Determine the new target food source as and remove from the memory, if the

nectar amount (fitness value) of is better than .

4. Move onlooker bees probabilistically to the food sources depending on the roulette

wheel using Eqs. (3) and (4) and determine new neighbourhoods as in Steps 2 and

3;

=∑

(3)


=1

1 + , ≥ 0

1 + ( ), < 0 (4)

where is the cost value (value of objective function) of the food source and

fitnessi the nectar amount (fitness value) of the food source .

5. Randomly initialize food source by a scout bee using Eq. (1), if a better food

source cannot be found in the neighborhood of present one after a number of

trials, called as “limit”.

6. Repeat steps (2) to (5) until Cycle=MCN

A basic pseudo code of the ABC algorithm can be seen in Fig. 1.

Fig. 1 The pseudo code of the ABC algorithm

3 The Proposed Algorithm

As mentioned in Introduction section, the structure of basic ABC is not suitable

for binary optimization. Thus, Eq. (1) and (2) need to be modified. In the proposed

genetic binary artificial bee colony algorithm (GB-ABC), the initial solutions are

generated by Eq. (5) instead of Eq. (1);

Xi: i=1...SN = 0 (0,1) ≤ 0.51 (0,1) > 0.5 (5)

where (0,1) is a uniformly generated value.

The genetic operators such as two-point crossover and swap operators are

integrated into ABC as a modification (referred as “genetic neighbourhood generator

(GNG)”) in place of Equation (2). For two-point crossover operator [46], two


positions on binary vectors are randomly chosen and then the parts of the vectors

between these positions are exchanged. For the swap operator, two positions are

randomly chosen with the values of 1 and 0 on the binary vector. Then, the position

with the value 1 is varied to 0, and the other with the value 0 is changed as 1.

The genetic neighbourhood generator (GNG) is as follows:

1. For a binary current food source = ( , , … . , ), randomly select two

neighborhoods = ( , , … . , ) and = ( , , … . , ) where is

the dimension of food source.

2. Determine a dimensional zero food source = 0 .

3. Define , , , and as parent food sources, where is the best

food source of the population.

4. Randomly match the parent food sources to each other based on one-to-one and

onto relation as seen in Fig. 2, and then apply a two-point crossover on the

matched parent food sources as seen in Fig. 3.

Fig. 2 An example of one-to-one and onto matching between parent food sources for crossover application.

Fig. 3 An illustrative example of exchanging positions between food sources through two-point crossover.


5. If any food source with zeros exists around children food sources, change ℎ

number of positions to 1 on the zero food source where ℎ is a predetermined

threshold value.

6. Apply swap operator to the children food sources as seen in Fig. 4 and define the

generated food sources as “grandchildren”.

Fig. 4 An illustrative example of applying swap operator on children food sources.

7. Select the best food source among the children and grandchildren food sources as

.

8. is determined as current food source instead of if the fitness value of is

better than the fitness value of , as in basic ABC.

In GB-ABC, the food source is determined to improve the global searching

mechanism since the food source satisfies diversity in children generation

through crossover. For crossover, one-to-one and onto functional relation is

determined between the parent food sources, an example of which can be found in

Fig. 2. In this way, the chance of getting the optimal solution is improved. As for the

determination of Th parameter, it is chosen as 3 for all the applications. The main aim

of using Th is to avoid possible errors resulting from empty solutions during the

objective function evaluation of children and grandchildren food sources. Any

number between 1 and D can be defined as Th parameter on account of the fact that

the proposed algorithm has an efficient search mechanism and its performance is not

adversely affected from this situation.

The pseudo code of the GB-ABC algorithm is presented in Fig. 5 and an

illustrative example of GNG implementation is given as follows:

Let = (1001010101) is the current solution; = (0010101011) and

= (1011101101) are randomly selected food sources; = (1101001100)

and = (0000000000). Assign these food sources as parents and make a


randomly one-to-one and onto relation them as seen in Fig. 2 and apply two-point

crossover to generate children between the matched food sources as in Fig. 3. After

the generation of children food sources, apply swap operator to create grandchildren

shown in Fig. 4. Finally, call the objective function to evaluate the quality of children

and grandchildren and assign the best food source as the food source.

Fig. 5 The pseudo code of the GB-ABC algorithm

4 Implementation of GB-ABC Algorithm

Three different problems such as dynamic image clustering, knapsack and

numerical optimization are chosen to show the effectiveness of the proposed

algorithm. The definition and implementation of problems through the GB-ABC

algorithm are given as follows;


4.1 Image Clustering

Image clustering is the process of identifying similar image primitives such as

pixels, regions, border elements etc. and collecting them together [59]. The act of

grouping is based on some similar measures like distance and intervals within

multidimensional data [26]. The groups satisfy “a concise summarization and

visualization of the image content that can be used for different tasks related to image

database management” [21]. In recent decades, image clustering has obtained a

significant role in biomedical imaging, remote sensing, geology, computer vision and

etc. all of which aim to achieve accurately partitioned and meaningful clustered

images.

Image clustering algorithms can be categorized into two groups: supervised and

unsupervised. Supervised classification is based on prior knowledge of the group or

members i.e. training examples [48]. A supervised learning clustering algorithm

analyzes the training data and produces an inferred output according to the test data

set (prior knowledge). Gaussian maximum likelihood, minimum-distance-to-mean,

artificial neural networks (ANN) and the naïve Bayes classifier are some of the most

widely used supervised algorithms. The main drawback of supervised algorithms is

the requirement of prior knowledge which may not be available in large unlabeled

datasets.

Contrary to the supervised clustering algorithms, unsupervised clustering

algorithms do not need prior information that makes these algorithms more acceptable

in the literature. Unsupervised clustering algorithms are classified into two groups:

hierarchical and partitional. Partitional clustering algorithms are more widely used in

image analysis and pattern recognition which lead us to use these algorithms in this

study. Partitional clustering algorithms can be accepted as optimization problems

since they attempt to partition the data into a set of disjoint clusters using some

similarity measures (eg. square error function) minimized by assigning clusters to

peaks in the probability density function, or the global structure. The most well-

known partitional clustering algorithm K-means [45] is applied by assigning data

members to the closest centroids via Euclidean distance measure. Various

modifications on K-means such as Fuzzy C-Means (FCM)[19], possibilistic C-means

(PCM) [41], robust fuzzy C-means (RFCM) [58] etc. are introduced to improve the

clustering performance. However, the number of clusters needs to be predefined in

these algorithms.

Clustering can be carried out in two different forms: crisp (hard) and soft (fuzzy).

In crisp clustering, all patterns should be assigned to only one cluster i.e. the clusters

are disjoint and have at least one pattern. As distinct from crisp clustering, a pattern


may be assigned to multiple clusters with fuzzy membership property in soft

clustering. For instance, K-means is a hard clustering algorithm and Fuzzy C-means

is a soft clustering algorithm.

Today’s technology allows us to store a large scale data set. Thus, clustering

algorithms requiring a predefined number have recently become disabled and the

demand for dynamic clustering has intensified. SNOB [63], the computer program

developed by Chris Wallace, applies the Minimum [Message|Description] Length

[Encoding] (MML|MDL) principle to estimate the number of classes, the relative

abundance of each class, the distribution parameters for each variate within each

class, and the class to which each thing in the sample most probably belongs.

Kohonen’s “Self Organizing Maps” (SOM) [40] is a single-layered unsupervised

neural network model and uses competitive learning to train the weight vectors and

provides a way for corresponding multidimensional data in lower dimensional spaces,

which is a data compression method known as vector quantization. Thus, SOM is

applied for automatically determining the number of clusters. However, SOM suffers

from the following drawbacks [26]: 1) it depends on initial conditions, 2) its

performance depends on the learning rate parameter and the neighborhood function,

3) it only works well with hyper-spherical clusters and 4) it uses a fixed number of

outputs. Lorette et al. [2] proposed a new fuzzy objective function in order to find the

optimum number of clusters (referred as UFA). UFA starts with a large number of

clusters and modifies the membership values and centroids until a user predefined

value where the performance of the algorithm is very sensitive.

Evolutionary computation based algorithms have also been applied to overcome

the dynamic clustering problem. The GA has been applied to the dynamic clustering

problem by using its standard bit-string structure [69]. Omran et al. [51] proposed a

binary PSO algorithm for dynamic image clustering in which the activation of

centroids is based on the selection of 1 and 0. Das and Konar [11] introduced a

modified differential evolution algorithm (ACDE) based on adaptive crossover rate

for dynamic clustering, which determines active clusters through the activation

threshold. In addition, The fuzzy version of ACDE was suggested to overcome

dynamic image fuzzy clustering [15]. Kuo et al. [42] applied a hybrid algorithm of

PSO and GA for dynamic clustering in order to prevent the local minima problem. In

this algorithm, PSO and GA correspondingly try to improve the solutions and then

elitist selection is performed between the solutions of PSO and GA. Although a great

research effort has been put forward to obtain an ideal, accurately constructed and

visually meaningful image clustering performance by researchers, it still remains a

challenge.


4.1.1 Dynamic Clustering Methodology

Clustering validity indexes are defined as a class of statistical-mathematical

functions depending on the similarity or dissimilarity between the data points [12].

Clustering validity indexes are used for two main purposes. The first is to measure the

quality of clustering relative to others created by other methods or by the same

methods using different parameter values. The second is to determine the number of

clusters. The traditional approach for determining the number of clusters is to

repeatedly run the algorithm as input with a different number of classes and then to

choose the clustered results with the best validity measure [23]. A validity index

needs to consider two important aspects [11]: patterns in the same cluster should be as

similar to each other as possible, known as cohesion (compactness), and clusters

should be well separated (separation).

Many various validity indexes have been proposed. For hard clustering, Dunn’s

index (DI) [19], the Calinski-Harabasz index [8], the DB index [16], the CS measure

[9], the VI index [62] etc. and for fuzzy clustering, Partition coefficient [4], Partition

Entropy [3] and the Xie-Beni [67] index are the most known indexes. According to

[12], the DB, PC, CE and XB indexes cannot obtain the optimal cluster number in

some images and according to [50, 62] the performance of the VI index is superior to

DB and Dunn’s index which leads us to use the VI index as an objective function.

The VI index consists of two issues:

a) Intra-Cluster Separation: Intra-cluster separation satisfies well collected groups

known as compactness by calculating the average of all distances between the

patterns and their related centroids. It should be minimized and defined by

Equation (6);

=1

−∈

(6)

where = , , … , , … , is whole dataset, is a pattern with

dimensional feature space and is the number of patterns in .

b) Inter-Cluster Separation: Inter-cluster separation, which is the minimum

Euclidean distance between any pairs of cluster centroids, provides well

separated clusters and is used to measure the cluster separation. It should be

maximized and defined by Eq. (7);

= min{‖ − ‖ } ℎ ≠ (7)

where is the cluster centroid of and is the cluster centroid of .


In terms of intra and inter-cluster separation, the VI index is defined by Equation (8);

= ( × (μ, ) + 1) × (8)

where c is a user specified constant (generally chosen as 20, 25 or 30) and N(μ, ) is a

Gaussian distribution with mean and standard deviation, defined by Eq. (9):

(μ, ) =1

√2[ ( ) ](9)

where K is the number of clusters.

Fig. 6 The way to activate centroids

1. Choose randomly Nc number of cluster centroids from image data set;

2. for t=1:TC (Termination criteria)

Initialize the initial solutions by Eq. (5);

for cycle=1:MCN(Maximum Cycle Number)

for each solution Xi

· Activate the centroids as seen in Fig. 6;

· Find the Euclidean distance of all patterns to the activated centroidsby Eq. (10);

, = ( , − , ) = − (10)

· Assign patterns to the closest clusters;

· Calculate VI index by Eq. (8);

· Improve the solution using GNG;

b. Apply K-means to the remaining centroids in order to decrease theadverse effects of initial conditions;

c. Generate new cluster centroids as in step 1;

d. Merge new cluster set with remaining centroids;

Fig. 7 The pseudo code of the GB-ABC dynamic clustering algorithm

The main steps of the GB-ABC dynamic clustering algorithm introduced in this study

are presented in Fig. 7. As seen in Fig. 7, a set of centroids (S) is first randomly

generated or chosen from dataset. After that, each food source is uniformly initialized

by {0,1}. Thirdly, each food source is employed to activate centroids for evaluating


the value of VI index as seen in Fig. 6 and then each food source is updated through

GNG. When the stopping criterion (MCN) is satisfied, K-means is applied to the

activated centroid subset (Sbest) of the global best food source to decrease the adverse

effects of initial conditions. After that, the corresponding positions of S cluster set are

updated with the centroids of Sbest. This procedure is repeated again with the updated

S until the termination criteria (TC) is satisfied.

4.2 Knapsack Problem

Searching for the best configuration or set of parameters has become a significant

point in order to accomplish some objective criteria in the fields of industry,

transportation, logistic, computer science etc. Thus, the knapsack problem plays

important role in these applications. The knapsack problem is defined as selecting the

most valuable items without extrapolating the weight capacity. Some types of

knapsack problem are given below [38]:

· Single knapsack problem: For the items, there is only one knapsack.

· Multidimensional knapsack problem: More than one knapsack.

· Multiple-choice knapsack problem: The items are subdivided into pre-defined

classes and for each class, exactly one item must be selected.

· Bounded knapsack problem: Only a limited number of items can be selected.

The basic knapsack problem can be defined by Eqs. 10 and 11;

max ( ) = (10)

− ≤ 0

∈ {0,1} ℎ = (0,1 … , )

(11)

where and are the ℎ item weight and cost and W is the capacity of the

knapsack.

In order to describe the knapsack problem as a binary optimization problem, is

the ℎ binary solution such that = [ , , … … … , ]. The number of n is the

dimension of the whole population and is the ℎ location of ℎ solution. If the

value of is 1, the ℎ corresponding item is loaded in the knapsack. The objective

function can be defined as;

= + 0, − (12)

where is a predefined penalty factor and Eq. (12) is a maximization problem.


4.3 Numerical Problems

To apply a binary optimization algorithm to the numerical problems, bit strings

are needed to be converted to numerical values within the given boundaries. The bit

string is represented in real form by Eq. (13);

= +( )

2 − 1 ∗ ( − )(13)

where lb and ub are lower and upper boundaries of reel number representations, b

is the length of bitstring and bittodec represents the decimal form of bitstring.

To evaluate the performance of the proposed algorithm, 25 special benchmark

functions, comprising unimodal, multimodal, shifted, rotated and hybrid composite

functions provided by CEC2005 [55] are employed. The number of variables (n) for

the search space is considered as 10 and the length of bit string for each variable is

chosen as 15. Thus, the length of a solution string is 10x15 (150).

5 Experimental Results

The experimental results are considered in three subsections: performance analysis

of the proposed algorithm on dynamic image clustering, on knapsack problem and on

numerical test problems. The results are presented over 30 simulations in terms of

mean values and standard deviations in the tables. The best results are denoted in

bold.

5.1 Dynamic Image Clustering

To compare the GB-ABC algorithm with other algorithms, including BPSO,

QBPSO, GA and DisABC in terms of dynamic image clustering on six benchmark

images such as Tahoe, Brain MRI, Jet, Lena, Mandril and Pepper, two simple criteria

are considered: 1) the quality of clustering determined by the VI index and 2) the

ability of obtaining optimal number of cluster. In addition to the evolutionary

computation based algorithms, the results of SOM and UFA from [50] and the results

of Snob algorithms from [62] are put forward to enhance the comparative analysis.

The optimal number of clusters for each image, Tahoe, Brain MRI, Jet, Lena, Mandril

and Pepper is demonstrated by a research group [62] in the range of [3,7], [4,8], [5,7],

[5,10], [5,10] and [6,10], respectively. However, that makes impossible to analyse the

clustering algorithms in terms of optimal number of clusters. In this work, the optimal

number of clusters is defined by Equation (14);

= + 1(14)


Therefore, the optimal number for Tahoe, Brain MRI and Pepper is determined as 4, 5

and 7, respectively; for the others, Jet, Lena and Mandril it is defined as 6 by Eq. (14).

It should be notified that the optimal number for the pepper image was also selected

as 7 in [11] and the number of clusters for each clustered image shown in Fig. 8 and 9

can be accepted as sufficient i.e. no more clusters need to be specified.

In the experiments of dynamic image clustering, the population size is chosen as

50 and the maximum number of evaluations is restricted to 20000. The other

parameters of the evolutionary computation based algorithms are as follows: the limit

values of DisABC and GB-ABC are chosen as 100 and 20, respectively; the

parameters of BPSO are selected as c = 2, c = 2, w = 0.9, w = 0.4 and

V = 6; the initeria and end weight parameters of QBPSO are selected as

w =0.5π and w =0.0001π; the values of crossover rate and mutation rate in

GA are selected as 0.8 and 0.2, respectively.

In the experiments, the following parameters are chosen for the VI index; c is

chosen 25; for Tahoe, µ is chosen 1, for the images brain MRI, jet, Lena and mandril,

µ is chosen as 2 and for pepper, µ is chose as 3 since it requires more than 6 clusters.

In other words, µ is set to 0 or 1 for fewer clusters (3 or 4), µ is set to 2 for 5 or 6

clusters and to 3 for more than 6 clusters. For all cases, σ is set to 1. Termination

criterion (TC) is set to 2 i.e. algorithms run two times to get optimal solution for each

simulation.

Tables 1 and 2 demonstrate the results of the VI index and the obtained number of

clusters. Table 1 reveals that the GB-ABC algorithm outperforms the others in terms

of minimizing the objective function except Mandril and as a result it can be extracted

that it dominates the other algorithms in satisfying clustering quality. In fact, the GB-

ABC algorithm can be accepted as the most robust algorithm by satisfying the

minimum standard deviation values almost in all cases in the values obtained from the

VI index. Table 2 indicates that the GB-ABC algorithm generally performs better than

the others in terms of finding the optimal number of clusters. Meanwhile, it is clearly

seen from Table 2 that the GA, SOM, UFA and Snob algorithms generally have a

much poorer performance in terms of finding the optimal cluster number. For

instance, the mean value of cluster number for jet is 6.733, 14, 20 and 22 for GA,

SOM, UFA and Snob, respectively; however, the optimal number of clusters is only 6

for the jet. For the images Jet and Pepper where the GB-ABC only cannot get the best

mean values of the obtained number of clusters, BPSO and QBPSO gets the better

results, but the obtained standard deviation values of BPSO and QBPSO are higher

than GB-ABC. It can be concluded that the GB-ABC algorithm performs well in

terms of minimizing the VI index and finding the optimal number of clusters.


Fig. 10 shows the convergence graphics of the algorithms for the selected images

Jet, Lena and Pepper where TC is set to 2 and MCN is set to 200 i.e. the number of

iterations becomes 400. It should be noted that the number of iterations are taken

similar in all algorithms only for the case study of convergence graphs since it is not

possible to establish a comparative study of algorithm convergence through the

number of evaluations. As seen in Fig. 10, the algorithms may have fluctuations after

200 iterations since TC is set to 2 i.e. the algorithms carry out their processes in two

stages: during the first stage from iteration 1 to 200, the algorithms search for the best

solution on randomly generated cluster centroid set and during the second stage from

iteration 201 to 400, the algorithms begin the search process again with the new

randomly generated population on the modified cluster centroid set obtained from the

first stage (see Fig. 7). It can be clearly inferred from the Fig. 10 that GB-ABC has

less fluctuations than the others, which shows its robustness.Table 1. The Obtained Mean VI Index Values for Image Clustering

Images GB-ABC DisABC QBPSO BPSO GATahoe 0.0569

(0.0067)0.0574

(0.0062)0.0604

(0.0073)0.0644

(0.0178)0.0884

(0.0377)MRI 0.0485

(0.0068)0.0499

(0.0069)0.0521

(0.0061)0.1339

(0.1181)0.1778

(0.2129)Jet 0.0945

(0.0145)0.0959

(0.0305)0.1238

(0.1000)0.1366

(0.1183)0.1517

(0.0595)Lena 0.0982

(0,0088)0.1032

(0.0141)0.1077

(0.0167)0.1126

(0.0130)0.1395

(0.0259)Mandril 0.1082

(0,0124)0.1045

(0,0111)0.1023

(0.0116)0.1077

(0,0115)0.1419

(0.0324)Pepper 0.1069

(0.0154)0.1201

(0.0150)0.1202

(0.0317)0.1323

(0.0460)0.1662

(0.0595)

Table 2. The Obtained Mean Cluster Numbers for Image Clustering

Images# of

clustersGB-ABC DisABC QBPSO BPSO GA SOM UFA Snob

Tahoe 4 4.066(0.253)

4.133(0.345)

4.566(0.954)

4.2(0.482)

4.791(1.178)

4 20 NA

MRI 5 5.133(0.434)

5.4(0.621)

5.724(1.130)

6.166(1.341)

7.29(2.123)

NA NA NA

Jet 6 5.7(0.836)

5.5(0.861)

6.366(1.299)

6.033(0.964)

6.733(1.680)

14 20 22

Lena 6 5.7(0.595)

5.666(0.547)

6.7(1.417)

6.695(1.063)

7.1(1.516)

20 20 31

Mandril 6 5.933(0.739)

6.166(1.019)

7.066(1.172)

6.9(1.471)

7.4(1.652)

20 20 42

Pepper 7 6.633(0.668)

6.566(0.568)

7.133(1.224)

7.333(1.212)

7.466(1.105)

20 20 39

5.2 Knapsack Problem

Six benchmark datasets, including 250, 500 and 750 items are chosen from [73] to

show the effectiveness of the GB-ABC algorithm over the BPSO, QBPSO, DisABC

and GA algorithms. In the experiments of the knapsack problem, population size is

chosen as 30 and maximum number of evaluations is set to 60000. The limit values of


DisABC and GB-ABC are selected as 100 and 500, respectively and the parameters

of BPSO and GA are selected as in the experiments of dynamic image clustering. It

should be noted that the convergence structure of the DisABC algorithm leads us to

use higher limit values.Table 3. The Obtained Results for Knapsack

ItemSet

Capacity GB-ABC DisABC QBPSO BPSO GA

250_1 6184

Best 9548 9023 9559 7926 9504

Mean 9520(13.913)

8824.67(117.26)

9518(22.368)

7767.6(86.300)

9448.23(31.854)

Worst 9491 8526 9463 7614 9353

p-val 0,9 2,9e-11 N/A 2,9e-11 4,9e-11

250_2 6536

Best 10217 9716 10219 8760 10172

Mean10177.3(23.111)

9539.37(90.805)

10187.1(17.961)

8600.53(78.171)

10123.8(30.001)

Worst 10126 9386 10152 8477 10067

p-val N/A 3,0e-11 0,09 2,9e-11 1,9e-08

500_1 13499

Best 20479 18872 20235 16547 20370

Mean 20383.9(33.703)

18531.8(172.92)

20108.6(75.477)

16261(132.60)

20244.1(60.248)

Worst 20331 18224 19951 15994 20102

p-val N/A 3.0e-011 3.0e-011 3.0e-011 1.5e-010

500_2 13743.5

Best 20633 18774 20500 16446 20582

Mean 20563.7(39.893)

18415.7(158.02)

20309.5(80.504)

16301.5(88.076)

20456.8(59.543)

Worst 20467 18086 20126 16129 20349

p-val N/A 3,0e-11 3,3e-11 3,0e-11 6,5e-09

750_1 20351.5

Best 30300 27243 29731 24174 30212

Mean 30097(95.417)

26630.3(297.91)

29475.4(154.61)

23644.7(197.49)

30011(95.943)

Worst 29957 26046 29133 23312 29816

p-val N/A 3,0e-11 3,0e-11 3,0e-11 0.0021

750_2 20450

Best 31641 28275 31008 24918 31660

Mean31430.1(95.149)

27838.6(285.36)

30767.3(114.08)

24528.6(162.61)

31361.4(125.72)

Worst 31197 27039 30584 24280 31077

p-val N/A 3.0e-011 3.0e-011 3.0e-011 0.0021


The maximum capacities and dimensions of knapsack datasets with the obtained

results in terms of worst, mean, best and Wilcoxon rank sum test (p-val) values at 5%

significance level are presented in Table 3. In this table, N/A indicates “non-

applicable” reflecting the corresponding algorithm cannot statistically compare with

itself in test. It can be clearly extracted from Table 3 that GB-ABC remains overall

commendable. In fact, it excels in knapsack problem by achieving higher results in

both worst and best cases compared to its contenders. ‘250_2’ is the only case where

GB-ABC cannot get the best results. It can be also clearly inferred from Table 3 that

the gap between the GB-ABC and the existing algorithms is increased when the size

of the problem becomes larger. Moreover, the difference between GB-ABC and the

others can be statistically demonstrated via p-vals. Consequently, the GB-ABC

algorithm is far superior to the other algorithms in terms of almost all properties such

as maximizing the cost of the knapsack and satisfying the best standard deviation.

5.3 Numerical Problems

In the experiments of numerical optimization, the population size is chosen as 40

and the maximum number of evaluations is restricted to 320000. The limit values of

GB-ABC and DisABC are set to 400 and 1000, respectively and the user-specified

parameters of the BPSO, QBPSO and GA are selected as in the experiments of

dynamic image clustering and knapsack problems.

Concerned with ten dımensional search space, Table 4 presents the obtained results of

the CEC2005 problems in terms of mean, standard deviation, rankings and Wilcoxon

rank sum test (p-val) values at 10% significance. The mean values in Table 4 clearly

reflect that GB-ABC performs well in unimodal functions (F1-F5), almost all cases of

composite functions (F15-F25), except F25. Although GB-ABC cannot get the first

degree in multimodal functions (F6-F14), except F6; it can be clearly seen that the

gap between the GB-ABC and the other algorithms is very narrow i.e. the results of

the algorithms in (F7-F14) are very close to each other. For instance, the mean value

difference between GB-ABC and QBPSO in F8 is only 0.06. Moreover, p-values also

mostly support the achievement of the GB-ABC algorithm except some cases

concerning the QBPSO algorithm. Concerned with the total rankings, it is seen that

GB-ABC gets the first position among all existing algorithms and the other

algorithms such as QBPSO, BPSO, GA and DisABC are ranked as second, third,

fourth and last, respectively.


Table 4. The Obtained Results of CEC2005 Benchmark Functions

GB-ABCMean(Std)

[Rank]p-val

DisABCMean(Std)

[Rank]p-val

QBPSOMean(Std)

[Rank]p-val

BPSOMean(Std)

[Rank]p-val

GAMean(Std)

[Rank]p-val

F1 -336.54(112.75)

[1]N/A

-103.56(142.39)

[5]7.6e-008

-334.94(78.78)

[2]0.0592

-271.06(131.36)

[3]0.0035

-190.99(171.40)

[4]8.1e-004

F2 -280.57(151.46)

[1]N/A

-74.43(170.08)

[5]5.8e-006

-270.45(173.75)

[2]0.1160

-263.83(123.33)

[3]0.0397

-143.43(253.23)

[4]0.0028

F3 -303.56(145.54)

[1]N/A

-44.47(161.15)

[5]5.1e-007

-291.45(153.19)

[2] 0.0991

-257.65(150.35)

[3] 0.0073

-124.02(265.35)

[4]1.9e-004

F4 -341.57(96.855)

[1]N/A

-64.12(154.92)

[5] 1.1e-008

-332.52(122.57)

[2] 0.4517

-290.42(108.15)

[3] 0.0025

-118.42(269.34)

[4]2.1e-005

F5 -136.87(177.21)

[1]N/A

83.19(155.51)

[5]4.4e-006

-115.02(166.90)

[2]0.1302

-112.83(119.63)

[ 3] 0.0107

13.991(254.74)

[4]6.4e-004

F6 533.92(127.21)

[1]N/A

740.04(183.25)

[5]2.4e-006

577.09(138.99)

[3]0.1668

567.83(101.27)

[2]0.1858

719.53(261.13)

[4]9.0e-004

F7 -167.25(9.06)

[3]N/A

-149.06(15.47)

[5] 7.6e-006

-171.42(8.82)

[1]0.0436

-169.07(9.07)

[2]0.4464

-160.71(12.02)

[4] 0.0163

F8 -119.73(0.09)

[2]N/A

-119.66(0.08)

[4] 0.0016

-119.79(0.07)

[1]0.0040

-119.71(0.06)

[3] 0.1031

-119.67(0.12)

[5] 0.0066

F9 -309.67(0.11)

[4]N/A

-309.69(0.07)

[3]0.0056

-309.81(0.06)

[1] 1.7e-006

-309.71(0.06)

[2]0.3555

-309.66(0.14)

[5]0.0877

F10 -309.66(0.12)

[4]N/A

-309.67(0.07)

[3]0.0010

-309.78(0.08)

[1]2.6e-004

-309.70(0.05)

[2]0.2226

-309.66(0.14)

[4] 0.4154

F11 110.32(0.10)

[4]N/A

110.29(0.09)

[3] 1.9e-004

110.19(0.09)

[1]1.0e-005

110.28(0.07)

[2]0.0798

110.36(0.17)

[5] 0.0877

F12 -439.75(0.09)

[2]N/A

-439.67(0.08)

[4]4.1e-008

-439.81(0.07)

[1]0.0184

-439.70(0.06)

[3]0.0224

-439.62(0.12)

[5]6.7e-005

F13 -109.68(0.12)

[3]N/A

-109.68(0.08)

[3]0.0030

-109.81(0.09)

[1]3.3e-004

-109.70(0.05)

[2] 0.2226

-109.64(0.13)

[4]0.0161

F14 -279.64(0.09)

[4]N/A

-279.66(0.08)

[3]0.0055

-279.75(0.08)

[1]1.0e-005

-279.71(0.05)

[2]3.5e-004

-279.64(0.11)

[4] 0.4260

F15 412.57(189.24)

[1]N/A

472.64(48.95)

[5] 0.0169

435.10(174.28)

[3]0.0565

458.75(105.22)

[4]0.0079

419.09(188.33)

[2]0.2489

F16 411.04(164.88)

[1]N/A

472.43(67.26)

[5]0.0070

449.07(160.37)

[4]0.0423

438.23(104.84)

[2]0.0179

444.86(184.35)

[3]0.0250

F17 367.16(165.15)

472.22(59.78)

398.99(183.57)

466.92(111.92)

446.07(171.13)


[1]N/A

[5]6.3e-005

[2]0.1302

[5]7.4e-005

[4]0.0216

F18 284.38(171.02)

[1]N/A

384.49(46.09)

[5]5.9e-004

288.25(201.08)

[2]0.2905

356.14(116.55)

[4]0.0133

332.12(188.89)

[3]0.0682

F19 291.25(152.52)

[1]N/A

372.68(51.72)

[5]0.0033

296.73(189.52)

[2]0.1868

336.29(112.08)

[3]0.0360

342.78(182.99)

[4]0.0019

F20 291.27(204.83)

[1]N/A

369.94(57.39)

[5]0.0066

331.57(173.24)

[3]0.1055

347.48(140.15)

[4]0.0077

314.62(181.76)

[2]0.0807

F21 632.88(178.25)

[1]N/A

696.78(63.13)

[4]0.0018

701.74(167.29)

[5]0.0151

691.12(121.87)

[3]0.0029

672.88(177.95)

[2]0.0743

F22 618.47(176.95)

[1]N/A

726.58(60.58)

[5] 9.3e-005

646.42(171.96)

[2]0.1206

694.21(103.99)

[3]4.2e-004

696.14(185.74)

[4]0.0086

F23 656.22(163.05)

[1]N/A

722.61(64.67)

[5]0.0193

694.85(164.65)

[4]0.0772

665.93(91.22)

[2]0.0345

672.48(184.78)

[3]0.0532

F24 505.23(169.10)

[1]N/A

604.26(52.72)

[5] 1.4e-004

561.22(176.22)

[3]0.0179

553.72(74.82)

[2] 3.7e-004

573.05(199.48)

[4]0.0066

F25 558.69(167.03)

[2]N/A

610.83(57.06)

[4]0.0070

522.70(172.31)

[1]0.0186

631.59(115.15)

[5] 0.0077

587.57(186.89)

[3]0.0842

6 Conclusion

Many engineering problems such as clustering, uncapacited facility location

problem (UFLP) and knapsack problem etc. are accepted as binary optimization

problems. In this paper, a novel binary artificial bee colony algorithm is proposed

based on genetic operators (GB-ABC). The main advantage of the proposed algorithm

is simple to implement. In other words, it does not include any complicated structure.

The other advantage of the proposed algorithm is that it is not only applicable for

specific binary problems and it can be also used for general binary optimization

problems. This is demonstrated by applying it to the different kind of binary

problems: dynamic image clustering and knapsack problems. Besides, its suitability is

tested on numerical optimization problems. From the simulation results, it is clear that

the GB-ABC algorithm outperforms all other algorithms in all handled problems.

Hence, it can be successfully applied to optimize general binary optimization

problems. As future works, it is planned to evaluate the performance of the proposed

algorithm on other binary optimization problems.


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A (Tahoe) C (MRI) E (JET)

B (Tahoe) D (MRI) F (JET)Fig. 8 The original input images and the clustered images by GB-ABC, respectively for tahoe (A,B), MRI (C,D) and jet (E,F) images.

A(Lena) C(mandril) E(pepper)

B(Lena) D(mandril) F(pepper)Fig. 9 The original input images and the clustered images by GB-ABC, respectively for Lena (A,B), mandril (C,D), and pepper (E,F) images.


Fig. 10 The convergence graph of the algorithms for Jet, Lena and Pepper images.

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A novel binary artificial bee colony algorithm based on genetic operators