Received July 17, 2019, accepted August 8, 2019, date of publication August 22, 2019, date of current version October 10, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2937021

An Enhanced Version of Black Hole Algorithmvia Levy Flight for Optimization and DataClustering ProblemsHANEEN A. ABDULWAHAB1, A. NORAZIAH2, ABDULRAHMAN A. ALSEWARI 2,AND SINAN Q. SALIH 3,41Faculty of Computer Systems and Software Engineering, University Malaysia Pahang, Gambang 26300, Malaysia2Faculty of Computing, College of Computing and Applied Sciences, Universiti Malaysia Pahang, Gambang 26300, Malaysia3Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam4Computer Science Department, College of Computer Science and Information Technology, University of Anbar, Ramadi, Iraq

Corresponding author: Sinan Q. Salih (sinanq.salih@duytan.edu.vn)

This work was supported by the Novel Hybrid Harmony Search Algorithm With Nomadic People Optimizer Algorithm for GlobalOptimization and Feature Selection under Grant RDU190334.

ABSTRACT The processes of retrieving useful information from a dataset are an important data miningtechnique that is commonly applied, known as Data Clustering. Recently, nature-inspired algorithms havebeen proposed and utilized for solving the optimization problems in general, and data clustering problemin particular. Black Hole (BH) optimization algorithm has been underlined as a solution for data clusteringproblems, in which it is a population-based metaheuristic that emulates the phenomenon of the black holesin the universe. In this instance, every solution in motion within the search space represents an individualstar. The original BH has shown a superior performance when applied on a benchmark dataset, but it lacksexploration capabilities in some datasets. Addressing the exploration issue, this paper introduces the levyflight into BH algorithm to result in a novel data clustering method ‘‘Levy Flight Black Hole (LBH)’’, whichwas then presented accordingly. In LBH, themovement of each star dependsmainly on the step size generatedby the Levy distribution. Therefore, the star explores an area far from the current black hole when the valuestep size is big, and vice versa. The performance of LBH in terms of finding the best solutions, preventgetting stuck in local optimum, and the convergence rate has been evaluated based on several unimodaland multimodal numerical optimization problems. Additionally, LBH is then tested using six real datasetsavailable fromUCImachine learning laboratory. The experimental outcomes obtained indicated the designedalgorithm’s suitability for data clustering, displaying effectiveness and robustness.

INDEX TERMS Optimization, data clustering, black hole, levy flight, metaheuristic, computational intelli-gence.

I. INTRODUCTIONData clustering is a method that consists of placing similarobjects together, where like items are placed in one and differ-ent items are grouped in different ones. It is an unsupervisedlearning technique characterized by the grouping of objectsin unspecified predetermined clusters. The conceptualizationcontrasts with classification, which is a form of supervisedlearning that involves objects being allocated to predeter-mined classes (clusters) [1]. Data clustering is widely usedin many areas including data mining, statistical data anal-

The associate editor coordinating the review of this article and approvingit for publication was Zhenliang Zhang.

ysis, machine learning, pattern recognition, image analysis,information retrieval, and more. This is due to clusteringmethods that can be categorized into various methods, suchas partitional, hierarchical, density-based, grid-based, andmodel-based methods, accordingly [2].

Per the above methods, partitional clustering methods arethe type that is commonly used, in which the K-means algo-rithm is an example of partitional and center-based cluster-ing algorithms. Due to cluster centers being initialized, thek-means clustering algorithm is limited to the localoptima [3]. Regardless, the past few decades have witnessedthe development of many nature-inspired evolutionary algo-rithms in order to resolve engineering design optimization

VOLUME 7, 2019 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ 142085

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

problems. They are known to emulate the behaviors of livingthings within nature, rendering them to be also described asSwarm Intelligence (SI) algorithms. SI algorithms typicallysearch for global optima while being associated with speedyconvergence [4].

Meanwhile, metaheuristic searching optimization isrecently heavily discussed on in literature over wide-rangingengineering applications, such as power optimization con-trol [5], robotic [6], communications and networking [7],engineering [8]–[12], information security [13], [14], andmachine learning [15], [16] Even though the approaches ofthe knowledge branch are characterizable by different con-cepts and inspirations, one fundamental attribute underlinestheir goal. All of the approaches make use of a selectivesearching process that is inspired by heuristic knowledge inthe solution space to obtain a solution. The solution shouldoptimize a given objective function or a set of objectivefunctions in case of multi-optimization, provided that theset of constraints is maintained. These algorithms are highlyattractive to researchers nowadays due to the fast enhance-ment of hardware speed and improved feasibility in solvingmany engineering problems. This is done by adhering to theheuristic searching conceptualization, with a simple designof objective function and constraints.

Various natural phenomena have led to the formulation ofnatural-inspired searching optimization algorithms [17], [18]such as hunting behavior of grey wolves [19]; krill herds [20];black holes [21]; egg-laying behavior of cuckoos [22];hunting behavior of bats [23]; food-searching behavior ofbees [24]; and improvisation process of jazz musicians [25].

Recently, a meta-heuristic optimization called a ‘‘blackhole’’ (BH) that mimics the black hole behavior of pulling insurrounding stars has been invented by [21]. BH optimizationis particularly inspired by the nature or physics of BH, as wellas its interaction with the surrounding stars. With the assump-tion that in a given iteration, a set of star is representativeof the total number of solutions and each star is subjectedto a pulling force towards the best solution representing BH.Then, a new set of solutions in the next iteration is generatedby moving the stars toward the black hole, whereupon the starbeing within the predetermined distance to BH will render itswallowed and for alternative stars to be arbitrarily generated.This allows the algorithm to initiate an exploration in thesearching space, rather than consuming the optimization timewith an area fully discovered with solutions. In case of itsimplementation to solve a data clustering issue, it remainsrelevant despite performance evaluation showing that it issuperior compared to other similar processes. Similarly, fur-ther enhancement for the approach will allow the discov-ery of powerful phenomenon in the solution space, whilealso making space for effectual clustering processing. In thisperspective, the work of [21] can be developed from theobjective function which does not assure the best possibleaccuracy, even when the cost is at the global optimum theoriginal black hole algorithm suffers from weaknesses inexploration. Therefore, it requires too many reiterations to

attain an optimum resolution. In recent years, the black holealgorithm and its modified versions have been used to solveengineering and optimization problems [26]–[37].

In this study, enhancing BH global search and resolvingthe issue of entrapment in the local minima have been under-taking by combining BH with levy flight. A Levy flight canbe described as a type of arbitrary walk, namely general-ized Brownian motion inclusive of non-Gaussian arbitrar-ily distributed step sizes for the distance moved. Differentnatural and man-made facts are explainable using Levyflight, which include fluid dynamics, earthquake analysis,fluorescent molecule diffusion, cooling behavior, noise, andmore [38], [39]. Pereyra and Hadj have also opted for itin case of Ultrasound in Skin Tissue [40], while Al-teemyutilized it in Ladar Scanning [41]. Its role is also momen-tous in various computer science fields [42], with it beingemployed by Terdik and Gyres in designing Internet TrafficModels [41], Chen’s Delay and Disruption Tolerant Network,Sutantyo et al.’s Multi-Robot Searching procedure. [42], andRhee’s human mobility utilization [43]. Meanwhile, Tasge-tiren [44] and Yang and Deb [45] opted for Levy flightdistribution to generate a novel cuckoo in Cuckoo Search,alongside Yang’s introduction of an updated model of FireflyAlgorithm-FA. The Levy-flight Firefly algorithm (LFA) [46]incorporates Firefly to unite Levy-flight with the search strat-egy so as to attain improved FA randomization. Lee and Yao’sEvolution Algorithm also developed four dissimilar states ofparameters of Levy flight and 4 prospective solutions; thestate offering the best results would be used for mutation pro-cedure. Additionally, it was also utilized as a diversificationtool in optimizing an ant colony.

In this paper, the long jumps have been undertaken viaLevy distribution in order to ensure effectual use of thesearch space in comparison with BH. Previously investi-gated works have aimed to improve BH, whereby the currentproposal calls for BH to perform random walks and globalsearch. Thus, a Levy flight-based method combined with BHalgorithm is proposed to resolve global optimization prob-lems and data clustering problem. Levy flight, in particular,improves the global search capacity for the BH algorithm,preventing one to be stuck in local minima. Additionally,the proposed method enhances the global search ability ofBH algorithm as per the new equation of star movementsunderlined. As BH algorithm is incapable of attaining theoptimum results in a specific number of iterations, an efficientLevy-flight selection is imperative to avoid being stuck inlocal optimum as it results in improved global and localsearch capability concomitantly.

The remaining sections for this work will be arrangedin the following manner: Section 2 will discuss someof the previously proposed research on data clustering.Then, the BH algorithm and proposed modified levy blackhole algorithm is presented in Sections 3 and 4, respec-tively, whereas Section 5 outlines the experimental out-comes obtained. Finally, Section 6 will conclude the worksuccinctly.

142086 VOLUME 7, 2019

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

II. OVERVIEWA. THE PROBLEM OF DATA CLUSTERINGClustering can be described as an essential unsupervised clas-sification approach characterized by the placement of a set ofpatterns or vectors (e.g. observations, data items, or featurevectors) into a multi-dimensional space in clusters or groups.This is achieved by utilizing similarity metrics between dataobjects, whereby the similarity and dissimilarity of objects inthe database are looked into using distancemeasurement [47].The action is propelled by the idea of classifying a datasetprovided using a specific number of clusters via distanceminimization between objects of each cluster itself. Termedas cluster analysis, it is defined as the rearrangement of a bodyof patterns typically presented in two ways: 1) a vector ofmeasurements, or 2) a point in a multi-dimensional space.This is done to obtain clusters that are characterized by theattribute of similarity [48], [49].

Clusters are oftentimes utilized for various applications,such as image processing, data statistical analysis, and med-ical imaging analysis, as well as other research fields of thescience and engineering branch. Moreover, it is synonymouswith statistical data analysis and known as a primary taskfor exploratory data mining in a multitude of fields, such asmachine learning, pattern recognition, image analysis, infor-mation retrieval, and bioinformatics. Figure 1 displays thedifference between clusters that may be due to their shapes,sizes, and densities.

However, noise present in the data may render clusterdetection challenging, in which the ideal cluster is generallydescribed as a compact and solitary set of points. Despitehuman beings having known to be proficient in cluster seek-ing in two and probably three dimensions, high-dimensionaldata calls for automatic algorithms. This fact, coupled withthe unspecified number of clusters yet for data set provided,has continuously generated thousands of clustering algo-rithms in publication. In the context of pattern recognition,the data analysis section is particularly correlated with pre-dictive modeling, in which training data is provided and theunknown test data’s behavior is predicted. Such task is termedas learning.

An evaluation of the similarity of data objects requires theuse of distance measurement. The problem may be framed asfollows: given N records of data, each record is allocated toone of K the clusters. Performing clustering has been carriedout using different criteria that serve as an objective func-tion for the process of optimization. One of the commonestattribute is minimizing the sum of squared Euclidean distancebetween each record and the center of the correspondingcluster as defined in [50]. This is displayed per equation (1)below.

F (O � Z ) =∑N

i=1

∑K

j=1Wij

∥∥Oi − Zj∥∥2, (1)

where N and K are the numbers of data records and thenumbers of clusters, respectively. While

∥∥Oi − Zj∥∥ is theEuclidean distance between a data record Oi and the cluster

FIGURE 1. The difference between clusters a) input data b) fit desiredClustering.

center Zj which is calculated as follows:

Zj =1|Nj|

∑N

i=1WijOi (2)

where Nj is the number of patterns in the ith cluster, Wij theassociation weight of pattern Oi with cluster j. Wij is 1 whenOi is allocated to cluster j, otherwise it is 0.

B. RELATED WORKSThe utilization of metaheuristic algorithms for the pur-pose of clustering problems has been discussed in vari-ous studies. This section is specifically driven to reviewmetaheuristic-based clustering algorithms that are restrictedto techniques that are linked to the proposed algorithm.

Xiao et al. [51] had first proposed the data clusteringapproach using two means. The first is particle swarmoptimization (PSO), whereby optimal centroids are foundand utilized as a seed in the K-means algorithm. Mean-while, the second approach entails the PSO usage in refining

VOLUME 7, 2019 142087

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

K-means formed clusters. Both have been tested and indi-cated their extensive potential.

Next, the Ant Colony Optimization (ACO) method hasbeen discussed by Shelokar et al. [52]. It is characterizedby the use of distributed agents mimicking the manner inwhich ants locate the shortest distance to a food source fromtheir nest and return. The resulting observation indicates thatit may be viable as an effectual heuristic for near-optimalcluster representation.

Senthilnath et al. [53] comparatively studied threenature-inspired algorithms, namely GA, PSO, and CuckooSearch (CS) on clustering problem. During the analysis CSwas used with levy flight and the heavy-tail property of levyflight was exploited. The performance of these algorithmswas evaluated on three standard datasets and one real-timemulti-spectral satellite dataset while the results were analysedusing various analytical techniques. The authors concludedthat based on the given set of parameters, CS works better formost of the dataset due to the important role played by levyflight.

Singh and Sood [54] proposed a hybrid approach toshow the swarm behaviour of clusters. They used a Krillherd algorithm to simulate the herding behaviour of eachkrill. The clusters were discovered using a density-basedapproach; it was also used to show the regions with suf-ficiently high-density krill clusters. The minimum distancefrom each krill to the food source and from high-density ofherds were considered as the objective function of the krillmovement. The movement of each krill is determined by therandom diffusion and foraging movement.

An approach based on the combination of Levy flight withmodified Bat algorithm to improve the clustering result hasbeen proposed [55]. The proposed approach was tested onten datasets and the experimental results showed that theproposed algorithm clusters the data objects efficiently. It alsoillustrates that it escapes from local optima and explores thesearch space effectively.

A new quantum chaotic cuckoo search algorithm (QCCS)was proposed by Boushaki et al. [56] for data clustering. Thesuperiority of CS over the conventional metaheuristics forclustering problems has been confirmed by various studies.However, all the cuckoos have a similar search pattern, andthis may result to the premature convergence of the algorithmto local optima. Similarly, the convergence rate of the CS issensitive to the randomly generated initial centroids seeds.Thus, the authors strived to extend the CS capabilities usingnonhomogeneous update based on the quantum theory in abid to tackle CS clustering problem in terms of the globalsearch ability. They also replaced the randomness at the ini-tialization step with a chaotic map to increase the efficiencyof the search process and improve the convergence speed.An effective strategy was further developed for a proper man-agement of the boundaries. The results of the experiments onsix common real-life datasets show a significant superiorityof the developed QCCS over eight recently developed algo-rithms, including, hybrid cuckoo search, genetic quantum

cuckoo search, differential evolution, hybrid K-means, stan-dard cuckoo search, improved cuckoo search, quantum par-ticle swarm optimization, hybrid K-means chaotic PSO,differential evolution, and GA in terms of external and inter-nal clustering quality.

A new version of Artificial Bee Colony (ABC) algorithmcalled History-driven Artificial Bee Colony (Hd-ABC) wasproposed by Zabihi and Nasiri [57] by applying a memorymechanism to improve the performance of ABC. The pro-posed Hd-ABC uses a binary space partitioning (BSP) tree tomemorize useful information of evaluated solutions. By theapplication of this memory mechanism, the fitness landscapecan be approximated before the actual fitness evaluation.Fitness evaluation is a time and cost inefficient process inclustering problem, but the use of a memory mechanismhas significantly reduced the number of fitness evaluationsand facilitated the optimization process via the estimation ofthe solutions’ fitness value instead of estimating the actualfitness values. The proposed data clustering algorithm wasapplied on 9UCI datasets and 2 artificial datasets and both thestatistical and experimental outcomes showed the proposedalgorithm to perform better than the original ABC, its vari-ants, and the other recent clustering algorithms.

III. METHODOLOGYA. BLACK HOLE (BH) ALGORITHMThe design of the BH algorithm is rooted in the black holeoccurrence and in the fundamental idea of a region of spacehosting an extensive volume of mass concentrated within thatno nearby object is capable of escaping from its gravitationalpull. Upon falling into the phenomenon, one would be elimi-nated from the universe, light included.

The algorithm consists of two components: 1) the starmovement, and 2) the star re-initialization crossing intothe D-dimensional hypersphere around the black hole (i.e.termed as event horizon). It functions as follows: first, theN+1 stars, xi ∈ RD, i = 1, . . . ,N + 1 (where N is populationsize) are arbitrarily initialized in the search space. After theirfitness evaluation, the best value is referred to as the blackhole xBH . Black hole is static; there is no movement until abetter resolution is obtained by other stars. Thus, the numberof individuals looking for the optimum value equals to N .Next, each generation has each star to move towards the blackhole per the equation below:

xi (t + 1) = xi (t)+ rand × (xBH − xi (t))

× i = 1.2. · · ·N , (3)

where rand is a random number within an interval [0, 1].The BH algorithm also indicates that a star that founds

itself too close the black hole beyond the event horizonwill beeliminated. The radius of the event horizon (R) is describedas follows:

R =fBH∑Ni=1 fi

, (4)

142088 VOLUME 7, 2019

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

where fi and fBH are the fitness values of black hole and ithstar. N is the number of stars (candidate solutions).

In case of a distance that is less than R between a candidatesolution and the black hole (best candidate), the particularcandidate collapses and consequently, a new candidate isgenerated and arbitrarily disseminated in the search space.BH is commonly associated with a simple structure and easeof implementation, as well as a parameter-free algorithm.Its convergence to the global optimum occurs in all runs,whereas other heuristic algorithms may encounter entrap-ment in the local optimum solutions [21], [58].

Despite excellent outcomes obtained when BH is utilizedas a clustering technique, it is flawed by its weak balanc-ing between exploration and exploitation capacities. A starmay alter its direction if one of them finds a better solu-tion compared to the solution for the current black hole,thereby transforming into a new black hole. Furthermore,the conceptualization of the event horizon has been madeas the stars may display a relatively speedy convergence forthe search space to be occupied by the black hole, due tothe lack of exploration capabilities. However, it disallows theintensification of exploration or accumulation of knowledgeregarding previously visited solution; it is simply a restartmethod subjected to each star individually [59]. Therefore,this study presents a modified BH algorithm in combinationwith levy flight for efficient data clustering.

B. LEVY FLIGHT BLACK HOLE (LBH) ALGORITHMThe proposed work aims to cluster and group the data objectsin an efficient and effective manner. The method is foundedupon the Levy flight in combination with the black hole (BH)algorithm for the purpose of global optimization and dataclustering problems. Levy flight, in particular, enhances theglobal search capacity of the BH algorithm to prevent beingstuck in local minima. Thus, the method improves the globalsearch ability using a new equation for star movements.As the algorithm is incapable of finding optimum in a certainamount of iterations, Levy flight-based search is more effi-cient as it improves the local and global search concomitantly.

Some examples of Levy flight compared with the Brown-ian walk (random) have been displayed in Figure 2. After thefirst movements around a point, sudden jumps are encoun-tered; it generates the simultaneous local and global search.

Levy flight [60] can be defined as a type of arbitrary pro-cesses that is characterized by a jump size that adheres to thelevy probability distribution function. Its name was derivativeof a French mathematician named Paul Pierre Levy.

As a random walk, the steps in the Levy Flight are definedwith respect to the step lengths. The step lengths have agiven distribution probability and are drawn from a Levydistribution which is represented in Eq (5):

L (s) ∼ |s|−1−β , where β(0 < β ≤ 2) (5)

where β and s represents an index and the step length,respectively.

FIGURE 2. The Levy flight and Brownian (random) walk.

This study utilized a Mantegna algorithm for a symmetricLevy stable distribution to generate the sizes of the randomsteps. The term ‘symmetric’ in this concept implies that thestep size will assume either a positive or negative value. Thestep length s in the Mantegna’s algorithm can be calculatedthus:

s =u

|v|1/β(6)

where u and v are drawn from normal distributions; i.e.,

u ∼ N (0, σ 2u ), v ∼ N (0, σ 2

u ) (7)

where

σu =τ (1+ β) sin πβ2

τ [(1+β2

)β2

β−12

, σv = 1 (8)

The distribution for s follows the anticipated Levy distribu-tion for |s| ≥ |s0|, where s0 represent the least step length andτ (.) represent the Gamma function which is estimated thus:

τ (1+ β) =∫∞

0tβe−1dt (9)

The Levy distribution is used to generate the step sizes in theproposed technique. This is aimed at exploiting the searcharea. The step sizes are calculated thus:

step (t) = 0.01× s (t)× rand(0, 1) (10)

where t represents an iteration counter, s(t) is estimatedas shown in Equation (6) using Levy distribution, whilerand (0, 1) is a random value ranging from [0, 1].The step sizes in the Levy flights are too aggressive; this

implies that they can often generate new solutions whichare off the domain or on the boundary. Since the movementequation represented in the BH algorithm is a stochasticmethod search for new better positions within the searchspace, therefore, 0.01 multiplier is used in Equation (10) toreduce the step sizes when they get large. The positions ofthe stars are updated in the LBH as follows:

xt (t + 1) = xt (t)+ (step (t)× (xBH − xt (t))) (11)

VOLUME 7, 2019 142089

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

FIGURE 3. The pseudocode of LBH algorithm.

where xt is an individual star in iteration t while step (t) is theactual step sizes generated using Equation (10). xBH denotesthe current best solution or the black hole.

Levy flight is characterized by an important parameter ofβ, whereby each star is a solution and an arbitrary numberis produced as β between 0 and 2. Its different values mayresult in dissimilar outcomes. Therefore, larger values of βpose a higher likelihood to result in jumps to unexplored areas(i.e. higher exploration) and avoidance of being trapped inlocal optimums. However, smaller values will provoke thenew positions to be viewed as near the obtained solutions(i.e. higher exploitation). The BH algorithm is particularlywell-perceived for its excellent local search ability [59], butwithin the surround of the optimum point, it is characterizedby a low convergence rate. This is due to higher exploitationrate compared to the exploration rate.

Hence, the suggested algorithm is designed in a mannerthat it allows the BH algorithm’s local search ability, whichwill improve the method’s efficiency in generating the opti-mal resolution and accelerating the convergence rate.

The proposed algorithm is named as Levy Flight BlackHole (LBH) algorithm and utilized to solve optimization anddata clustering problems effectively. The pseudocode of LBHin Figure 3.

IV. EXPERIMENTS AND RESULTSThe assessments were carried out on a personal computer(Core i7, 3.6 GHz, 16 GB of RAM, 64-bit Windows 10 Oper-ating System) using MATLAB 2017a.

A. EVALUATION OF BENCHMARK TEST FUNCTIONSAs stated previously, the main contribution of this paper isto enhance the exploration of BH algorithm via Levy Flight.

FIGURE 4. The 3d plot of sumsqaure (f1). a) The convergence analysis ofLBH and other algorithms. b) The 3D of f1.

In order to further verify that the proposed algorithm has abetter exploration than the standard BH, it has been eval-uated on a set of unimodal and multimodal type of bench-mark test functions in a multi-dimensional space as definedin [61]–[63]. The functions with their main characteristics interms of Name, Dimensions (D), Upper and Lower Bound-aries (UB, LB) and the value of the optimal solution (Opt) arestated in Table 1.

The comparison stage is done by benchmarking againstnine well-knownmetaheuristics comprising of Big Bang–BigCrunch [64], Artificial Bees Colony (ABC) [65], ParticleSwarm Optimization (PSO) [66], and Levy Firefly Algo-rithm [46] (LFFA), Grey Wolf Optimizer (GWO) [19],Gravitational search algorithm (GSA) [67], Bat algorithm(BA) [23], cat swarm algorithm (CSA) [68], and Black hole(BH) [21] respectively. The parameters settings for thesealgorithms are presented in Table 2.

The experiments for LBH and the other algorithmswere executed in 30 different runs. The best, mean, errorrate, and standard deviation were recorded and presented

142090 VOLUME 7, 2019

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

TABLE 1. Benchmark test functions.

TABLE 2. Parameter setting.

VOLUME 7, 2019 142091

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

TABLE 3. The results of the standards algorithms and Levy black hole algorithm.

TABLE 4. Main characteristics of the test datasets.

accordingly in Table 3. Additionally, the convergence curveof the searching has been generated for the first benchmarkfunction and compared with other algorithms including theoriginal BH algorithm. LBH has shown faster convergencecurves for the first 100 iterations than the other algorithms.The convergence of BH by Levy flight (LBH) had enhancedthe exploration ability of the algorithm and guided the starstowards better positions rate. Which means that the starsavoid the possibility of trapping in local optima. It can be seenthat GWO and CSA algorithm have attained the second andthe third place respectively, while the original BH attained thefourth place. Figure 4 shows the convergence and the 3D plotof sumsqaure (f1).

B. EVALUATION BASED ON BENCHMARK DATASETSThe performance of the proposed algorithm for data clus-tering was evaluated using six datasets, namely: Iris, Wine,Glass, Cancer, Contraceptive Method Choice (CMC), andVowel. Their respective characteristics are shown in Table 4.

All data sets were sourced from the UCI machine learninglaboratory.• Iris dataset

The dataset consisted of 150 arbitrary samples of flowers hav-ing four features from the iris. They were differentiated into3 groups of 50 instances, whereby each group represented aform of iris plant (Setosa, Versicolor and Virginica).• Wine dataset

The dataset elucidated the quality of wine using the physic-ochemical properties, in which they were grown in theidentical region in Italy but sourced from three cultivars,respectively. Each of the three types of wine was linked to178 instances, with 13 numeric attributes representing thequantities of 13 components elicited in them.• CMC dataset

The dataset was generated by TjenSien Lim, which is a sub-set of Indonesia’s 1987 National Contraceptive PrevalenceSurvey. The sample size consisted of married women whowere either not pregnant or not in the know of their pregnancyduring the interview period. It featured the issue of predictingthe recent contraceptive method choice (i.e. no use, long-termmethod, or short-term methods) according to a woman’sdemographic and socioeconomic attributes.• Cancer dataset

The dataset was a representation of the Wisconsin breastcancer database, consisting of 683 instances having 9 com-ponents. They included: Clump Thickness, Cell Size

142092 VOLUME 7, 2019

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

TABLE 5. The result obtained by LBH and standard algorithms on different data sets.

Uniformity, Cell Shape Uniformity, Marginal Adhesion, Sin-gle Epithelial Cell Size, Bare Nuclei, Bland Chromatin, Nor-mal Nuclei, and Mitoses. Each of the instances was possiblyof one class, either benign or malignant.

• Glass dataset

The dataset consisted of 214 objects with nine features, whichwere: refractive index, sodium, magnesium, aluminum, sil-icon, potassium, calcium, barium, and iron. The data sam-pling was done using six groups of glass, which were: floatprocessed building windows, non-float processed buildingwindows, float-processed vehicle windows, containers, table-ware, and headlamps.

• Vowel dataset

The dataset was comprised of 871 Indian Telugu vowelsounds, inclusive of three attributes that corresponded tothe first, second and third vowel frequencies, as well as sixoverlapping classes.

The algorithm’s performances were assessed and subjectedto a comparison using two features:

• Sum of intra-cluster distances as an internal qualitymeasure: The distance between each data object andthe center of the corresponding cluster was calculatedand totaled up, per equation (1). Generally, a smallersum of intra-cluster distances was linked with a higherclustering quality. The sum of intra-cluster distances wasalso an assessment component for the fitness in thisstudy.

• Error Rate (ER) as an external quality measure: Thepercentage of misplaced data objects as depicted in theequation below:

ER =Number of misplaced objects

total number of objects within dataset100 (12)

The performance showed by the proposed algorithmwas compared against several heuristic methods previ-ously explained in literature, such as K-means [48],PSO [69],ABC [70], BAT [55], GSA [71], BB-BC [72],CS [56], GWO [73] and BH [21].

In contrast, LBH was compared against newer hybrid andmodified meta-heuristics algorithms reported in the litera-ture. They include: improved krill herd algorithm [74] hybridclustering method using artificial bee colony and Mantegnalevy distribution displayed in [75], a new quantum chaoticcuckoo search algorithm [56], Hd-ABC history-driven artifi-cial bee colony [57] (ICAKHM) is regarded as a novelmethodwhich was designed based on a combination of K-harmonicmeans algorithm and a modified version of the imperialistcompetitive algorithm (ICA) presented in [76] and grey wolfoptimizer with levy flight steps presented in [73].

Table 5 and Table 6 displayed the sum of intra-clusterdistances and error rate using the standard meta-heuristicsclustering algorithm and the hybrids and modified meta-heuristics algorithms alike to obtain a better comparison ofthe LBH.

VOLUME 7, 2019 142093

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

TABLE 6. The sum of intra-cluster distances and error rate obtained by LBH and modified algorithms on different data sets.

In Table 5, a summary of intra-cluster distance and errorrate is presented. The values for the best, average, worst,standard deviation and the error rate were calculated based onthe simulation of each independent algorithm after 30 inde-pendent implementations. Best obtained values by algorithmsare marked as bold for each dataset. The experimental resultsindicated that LBH better than BH and K-means. Further-more, the suggested algorithm has the smallest standard devi-ation compared to other algorithms, which mean the LBHget to minimum value each time. Other algorithms is a littleworse than LBH.

In Iris dataset, LBH outperforms other algorithms ofintra-cluster distance 96.5403 value and standard deviation0.00014 in comparison to other algorithms. In the case ofthe Wine dataset, the proposed LBH algorithm obtained theoptimum value of 16,291.99 which is remarkably superiorcompared to the other comparative algorithms. Similarly,upon comparison with the CMC dataset, the proposed LBHalgorithm is also far better compared to the other algorithms,with the worst solution achieved at 5532.58940. However,it is still much better than the best solutions found by otheralgorithms. In case of the Cancer dataset, the proposed LBH

algorithm’s performance surpassed the K-means, PSO andGSA algorithms, but the BB–BC algorithm outcomes weresuperior compared to the proposed LBH in terms of standarddeviation.

For the Glass dataset, the suggested LBH algorithmobtained an average of 210.97180, whereas other algorithmsfailed to attain the solution at all. Meanwhile, the Voweldataset was provided the best average solutions and standarddeviation by the suggested LBH algorithm compared to theother algorithms. Therefore, the LBH offered better solutionquality and smaller standard deviation in comparison withthe other algorithms. LBH is capable of locating the optimalsolutions as seen in a majority of the cases, while otheralgorithms may be trapped in local optima.

As per in Table 6, the proposed LBH obtained the best per-formance according to the average intra-cluster distances anderror rate when subjected to a comparison with the remain-ing comparative algorithms. It also displayed better perfor-mance on all six datasets as opposed to the other comparativealgorithms, in which a notable balance between exploitationand exploration enhanced the proposed LBH algorithms’performance.

142094 VOLUME 7, 2019

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

TABLE 7. The results of the statistical analysis tests.

On the Iris dataset, the standard deviation for the suggestedLBH algorithm is 0.00014, which is significantly less than theother comparative algorithms. In contrast, the best solutionis 96.5403 and the Worst is 96.5873, which is far supe-rior compared to other algorithms. Furthermore, the Winedataset indicated that the proposed LBH algorithm obtainedthe optimum value of 16,291.99, which surpassed the otheralgorithms.

The CMC dataset also yielded a proposed LBH algorithmthat was far better compared to other algorithms, in which theworst solution attained is 5532.88940. This remained to befar superior to the best solutions obtained by the other algo-rithms. For the Cancer dataset, the proposed LBH best solu-tions are 2961.95000 and the average solution is 2963.90000,while the standard deviation is 0.00723. This was supe-rior compared to ABCL, QCCS, HD-ABC, ICAKHM andEGWO.

Lastly, the Glass dataset obtained the best 199.86000that was reached by the ICAKHM algorithm. Meanwhile,the Vowel dataset indicated that the suggested LBH algorithmprovided the best average solutions 149,466.52. It passedsufficiently by yielding the best outcomes on almost all ofthe datasets and when compared to the other comparativealgorithms. Thus, it proved that the suggested (LBH) wasexceedingly effectual to resolve complex optimization prob-lems, simply by the addition of new operators.

In addition to the previous presented comparison, the algo-rithms have been compared statistically based on Friedmantest as well as the Iman–Davenport to determine whetherthere are significant differences in the results of the algo-rithms. Table 7 below shows the ranking of the algorithmsbased on them.

V. CONCLUSIONIn this paper, Levy flight was combined with Black Holealgorithm to improve the clustering result. The suggestedapproachwas subjected to testing on six datasets, whereby theexperimental outcomes indicated that the proposed algorithmclustered the data objects efficiently. It also illustrated itsescape from the local optima and exploration into the searchspace effectively. In the future, this workmay be implementedto other applications, such as text document clustering for thepurpose of clustering the set of documents effectively.

REFERENCES[1] M. Sarstedt and E. Mooi, ‘‘Introduction to market research,’’ in A Concise

Guide to Market Research. Berlin, Germany: Springer, 2019, pp. 1–9.[2] S. Arora and I. Chana, ‘‘A survey of clustering techniques for big data

analysis,’’ in Proc. 5th Int. Conf.-Confluence Next Gener. Inf. Technol.Summit, Sep. 2014, pp. 59–65.

[3] C. C. Aggarwal, and C. K. Reddy, Data Clustering: Algorithms andApplications. Boca Raton, FL, USA: CRC Press, 2013.

[4] X.-S. Yang, Nature-Inspired Metaheuristic Algorithms. Bristol, U.K.:Luniver Press, 2010.

[5] G. V. N. Kumar, B. V. Rao, D. D. Chowdary, and P. V. S. Sobhan, ‘‘Multi-objective optimal power flow using metaheuristic optimization algorithmswith unified power flow controller to enhance the power system perfor-mance,’’ in Advancements in Applied Metaheuristic Computing. Hershey,PA, USA: IGI Global, 2018, pp. 1–33.

[6] M. N. Janardhanan, Z. Li, G. Bocewicz, Z. Banaszak, and P. Nielsen,‘‘Metaheuristic algorithms for balancing robotic assembly lines withsequence-dependent robot setup times,’’ Appl. Math. Model., vol. 65,pp. 256–270, Jan. 2019.

[7] G. P. Gupta and S. Jha, ‘‘Integrated clustering and routing protocol for wire-less sensor networks using Cuckoo andHarmony Search basedmetaheuris-tic techniques,’’ Eng. Appl. Artif. Intell., vol. 68, pp. 101–109, Feb. 2018.

[8] J. E. Diaz, J. Handl, and D.-L. Xu, ‘‘Integrating meta-heuristics, simulationand exact techniques for production planning of a failure-prone manufac-turing system,’’ Eur. J. Oper. Res., vol. 266, no. 3, pp. 976–989, 2018.

[9] Z. A. Al-Sudani, S. Q. Salih, A. Sharafati, and Z. M. Yaseen, ‘‘Develop-ment of multivariate adaptive regression spline integrated with differentialevolutionmodel for streamflow simulation,’’ J. Hydrol., vol. 573, pp. 1–12,Jun. 2019.

[10] A. A. Al-Musawi, A. A. Alwanas, S. Q. Salih, Z. H. Ali, M. T. Tran,and Z. M. Yaseen, ‘‘Shear strength of SFRCB without stirrups simulation:Implementation of hybrid artificial intelligence model,’’ in EngineeringWith Computers. 2018, pp. 1–11.

[11] H. Tao, L. Diop, A. Bodian, K. Djaman, P. M. Ndiaye, and Z. M. Yaseen,‘‘Reference evapotranspiration prediction using hybridized fuzzy modelwith firefly algorithm: Regional case study in Burkina Faso,’’ Agricult.Water Manage., vol. 208, pp. 140–151, Sep. 2018.

[12] K. N. Sayl, N. S. Muhammad, Z. M. Yaseen, and A. El-shafie, ‘‘Estimationthe physical variables of rainwater harvesting system using integrated GIS-based remote sensing approach,’’ Water Resour. Manage., vol. 30, no. 9,pp. 3299–3313, 2016.

[13] H. A. Ahmed, M. F. Zolkipli, and M. Ahmad, ‘‘A novel efficientsubstitution-box design based on firefly algorithm and discrete chaoticmap,’’ in Neural Computing and Applications. 2018, pp. 1–10.

[14] A. A. Alzaidi, M. Ahmad, H. S. Ahmed, and E. Al Solami, ‘‘Sine-cosine optimization-based bijective substitution-boxes construction usingenhanced dynamics of chaotic map,’’ Complexity, vol. 2018, Dec. 2018,Art. no. 9389065.

[15] A. M. Taha, S.-D. Chen, and A. Mustapha, ‘‘Natural extensions: Batalgorithm with memory,’’ J. Theor. Appl. Inf. Technol., vol. 79, no. 1,pp. 65–75, 2015.

[16] A. M. Taha, S.-D. Chen, and A. Mustapha, ‘‘Bat algorithm basedhybrid filter-wrapper approach,’’ Adv. Oper. Res., vol. 2015, Sep. 2015,Art. no. 961494.

[17] S. Q. Salih, A. A. Alsewari, B. Al-Khateeb, and M. F. Zolkipli, ‘‘Novelmulti-swarm approach for balancing exploration and exploitation in par-ticle swarm optimization,’’ in Proc. Int. Conf. Reliable Inf. Commun.Technol., 2018, pp. 196–206.

[18] S. Q. Salih and A. A. Alsewari, ‘‘Solving large-scale problems usingmulti-swarm particle swarm approach,’’ Int. J. Eng. Technol., vol. 7, no. 3,pp. 1725–1729, 2018.

[19] S. Mirjalili, S. M. Mirjalili, and A. Lewis, ‘‘Grey wolf optimizer,’’ Adv.Eng. Softw., vol. 69, pp. 46–61, Mar. 2014.

[20] A. H. Gandomi and A. H. Alavi, ‘‘Krill herd: A new bio-inspired opti-mization algorithm,’’ Commun. Nonlinear Sci. Numer. Simul., vol. 17,pp. 4831–4845, May 2012.

[21] A. Hatamlou, ‘‘Black hole: A new heuristic optimization approach for dataclustering,’’ Inf. Sci., vol. 222, pp. 175–184, Feb. 2012.

[22] X.-S. Yang and S. Deb, ‘‘Cuckoo search via Lévy flights,’’ in Proc. WorldCongr. Nature Biologically Inspired Comput., 2014, pp. 210–214.

[23] X.-S. Yang, ‘‘A new metaheuristic bat-inspired algorithm,’’ in NatureInspired Cooperative Strategies for Optimization (NICSO). Berlin,Germany: Springer, 2010, pp. 65–74.

[24] X.-S. Yang, ‘‘Engineering optimizations via nature-inspired virtual beealgorithms,’’ in Proc. Int. Work-Conf. Interplay Between Natural Artif.Comput., 2005, pp. 317–323.

[25] Z. W. Geem, J. H. Kim, and G. V. Loganathan, ‘‘A new heuristic opti-mization algorithm: Harmony search,’’ J. Simul., vol. 76, no. 2, pp. 60–68,Feb. 2001.

VOLUME 7, 2019 142095

H. A. Abdulwahab et al.: Enhanced Version of BH Algorithm via Levy Flight

[26] E. Pashaei and N. Aydin, ‘‘Binary black hole algorithm for feature selec-tion and classification on biological data,’’ Appl. Soft Comput., vol. 56,pp. 94–106, Jul. 2017.

[27] H. Bouchekara, ‘‘Optimal power flow using black-hole-based optimizationapproach,’’ Appl. Soft Comput., vol. 24, pp. 879–888, Nov. 2014.

[28] R. Azizipanah-Abarghooee, T. Niknam, F. Bavafa, and M. Zare, ‘‘Short-term scheduling of thermal power systems using hybrid gradient basedmodified teaching–learning optimizer with black hole algorithm,’’ Electr.Power Syst. Res., vol. 108, pp. 16–34, Mar. 2014.

[29] M. Nemati and H. Momeni, ‘‘Black holes algorithm with fuzzy Hawkingradiation,’’ Int. J. Sci. Technol. Res., vol. 3, no. 6, pp. 85–88, 2014.

[30] H. R. E. H. Bouchekara, ‘‘Optimal design of electromagnetic devices usinga black-hole-based optimization technique,’’ IEEE Trans. Magn., vol. 49,no. 12, pp. 5709–5714, Dec. 2013.

[31] M. Doraghinejad and H. Nezamabadi-pour, ‘‘Black hole: A new operatorfor gravitational search algorithm,’’ Int. J. Comput. Intell. Syst., vol. 7,no. 5, pp. 809–826, 2014.

[32] M. Eskandarzadehalamdary, B. Masoumi, and O. Sojodishijani, ‘‘A newhybrid algorithm based on black hole optimization and bisecting k-meansfor cluster analysis,’’ in Proc. 22nd Iranian Conf. Elect. Eng., 2014,pp. 1075–1079.

[33] S. Yaghoobi, S. Hemayat, and H. Mojallali, ‘‘Image gray-level enhance-ment using Black Hole algorithm,’’ in Proc. 2nd Int. Conf. Pattern Recog-nit. Image Anal., Mar. 2015, pp. 1–5.

[34] E. Pashaei, M. Ozen, and N. Aydin, ‘‘An application of black hole algo-rithm and decision tree for medical problem,’’ in Proc. IEEE 15th Int. Conf.Bioinf. Bioeng., Nov. 2015, pp. 1–6.

[35] K. Premalatha and R. Balamurugan, ‘‘A nature inspired swarm basedstellar-mass black hole for engineering optimization,’’ in Proc. IEEE Int.Conf. Elect., Comput. Commun. Technol., Mar. 2015, pp. 1–8.

[36] K. Lenin, B. R. Reddy, and M. S. Kalavathi, ‘‘Dwindling of active powerloss by enhanced black hole algorithm,’’ Int. J. Res. Electron. Commun.Tech., vol. 1, no. 4, pp. 11–15, 2014.

[37] D. Rodrigues, C. C. O. Ramos, A. N. De Souza, and J. P. Papa, ‘‘Blackhole algorithm for non-technical losses characterization,’’ in Proc. IEEE6th Latin Amer. Symp. Circuits Syst., Feb. 2015, pp. 1–4.

[38] Y. Chen, ‘‘Research and simulation on Levy flight model for DTN,’’ inProc. 3rd International Congress on Image and Signal Process., Oct. 2010,pp. 4421–4423.

[39] G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy,P. A. Prince, and H. E. Stanley, ‘‘Lévy flight search patterns of wanderingalbatrosses,’’ Nature, vol. 381, no. 6581, pp. 413–415, 1996.

[40] M. A. Pereyra and H. Batatia, ‘‘A Levy flight model for ultrasound in skintissues,’’ in Proc. IEEE Int. Ultrason. Symp., Oct. 2010, pp. 2327–2331.

[41] G. Terdik and T. Gyires, ‘‘Lévy flights and fractal modeling of Internettraffic,’’ IEEE/ACM Trans. Netw., vol. 17, no. 1, pp. 120–129, Feb. 2009.

[42] D. K. Sutantyo, S. Kernbach, P. Levi, andV. A. Nepomnyashchikh, ‘‘Multi-robot searching algorithm using Lévy flight and artificial potential field,’’in Proc. IEEE Saf. Secur. Rescue Robot., Jul. 2010, pp. 1–6.

[43] I. Rhee, M. Shin, S. Hong, K. Lee, S. J. Kim, and S. Chong, ‘‘On the Levy-walk nature of human mobility,’’ IEEE/ACM Trans. Netw., vol. 19, no. 3,pp. 630–643, Jun. 2011.

[44] M. F. Tasgetiren, Y. C. Liang, M. Sevkli, and G. Gencyilmaz,‘‘A particle swarm optimization algorithm for makespan and total flowtimeminimization in the permutation flowshop sequencing problem,’’ Eur.J. Oper. Res., vol. 177, no. 3, pp. 1930–1947, 2007.

[45] X.-S. Yang and S. Deb, ‘‘Multiobjective cuckoo search for design opti-mization,’’ Comput. Oper. Res., vol. 40, no. 6, pp. 1616–1624, Jun. 2013.

[46] X.-S. Yang, ‘‘Firefly algorithm, Lévy flights and global optimization,’’ inResearch and Development in Intelligent Systems XXVI. London, U.K.:Springer, 2010, pp. 209–218.

[47] M. R. Anderberg, ‘‘Cluster analysis for applications,’’ Office AssistantStudy Support Kirtland AFB N MEX, 1973.

[48] A. K. Jain, ‘‘Data clustering: 50 years beyond K-means,’’ Pattern Recognit.Lett., vol. 31, no. 8, pp. 651–666, 2010.

[49] J. A. Hartigan, Clustering Algorithms. Hoboken, NJ, USA: Wiley, 1975.[50] M. B. Dowlatshahi and H. Nezamabadi-Pour, ‘‘GGSA: A grouping grav-

itational search algorithm for data clustering,’’ Eng. Appl. Artif. Intell.,vol. 36, pp. 114–121, Nov. 2014.

[51] X. Xiao, E. R. Dow, R. Eberhart, Z. B. Miled, and R. J. Oppelt, ‘‘Geneclustering using self-organizing maps and particle swarm optimization,’’in Proc. Int. Parallel Distrib. Process. Symp., Mar. 2003, P. 10.

[52] P. S. Shelokar, V. K. Jayaraman, and B. D. Kulkarni, ‘‘An ant colonyapproach for clustering,’’ Anal. Chim. Acta, vol. 509, no. 2, pp. 187–195,2004.

[53] J. Senthilnath, V. Das, S. N. Omkar, and V. Mani, ‘‘Clustering usinglevy flight cuckoo search,’’ in Proc. 7th Int. Conf. Bio-Inspired Comput.,Theories Appl., 2012, pp. 65–75.

[54] V. Singh and M. M. Sood, ‘‘Krill Herd clustering algorithm using dbscantechnique,’’ Int. J. Comput. Sci. Eng. Technol, vol. 4, no. 3, pp. 197–200,2013.

[55] R. Jensi and G. W. Jiji, ‘‘MBA-LF: A new data clustering method usingmodified bat algorithm and levy flight,’’ ICTACT J. Soft Comput., vol. 6,no. 1, pp. 1093–1101, 2015.

[56] S. I. Boushaki, N. Kamel, and O. Bendjeghaba, ‘‘A new quantum chaoticcuckoo search algorithm for data clustering,’’ Expert Syst. Appl., vol. 96,pp. 358–372, Apr. 2018.

[57] F. Zabihi and B. Nasiri, ‘‘A novel history-driven artificial bee colonyalgorithm for data clustering,’’ Appl. Soft Comput., vol. 71, pp. 226–241,Oct. 2018.

[58] S. Kumar, D. Datta, and S. K. Singh, ‘‘Black hole algorithm and itsapplications,’’ inComputational Intelligence Applications inModeling andControl. Cham, Switzerland: Springer, 2015, pp. 147–170.

[59] A. P. Piotrowski, J. J. Napiorkowski, and P. M. Rowinski, ‘‘How novelis the ‘novel’ black hole optimization approach?’’ Inf. Sci., vol. 267,pp. 191–200, May 2014.

[60] A. V. Chechkin, R. Metzler, J. Klafter, and V. Y. Gonchar, ‘‘Introductionto the theory of Lévy flights,’’ Anomalous Transp., Found. Appl., vol. 49,no. 2, pp. 431–451, 2008.

[61] N. S. Jaddi, J. Alvankarian, and S. Abdullah, ‘‘Kidney-inspired algo-rithm for optimization problems,’’Commun. Nonlinear Sci. Numer. Simul.,vol. 42, pp. 358–369, Jan. 2017.

[62] L. Zhang, L. Shan, and J.Wang, ‘‘Optimal feature selection using distance-based discrete firefly algorithmwith mutual information criterion,’’NeuralComput. Appl., vol. 28, no. 9, pp. 2795–2808, Sep. 2017.

[63] B. Niu, Y. Zhu, X. He, and H. Wu, ‘‘MCPSO: A multi-swarm cooper-ative particle swarm optimizer,’’ Appl. Math. Comput., vol. 185, no. 2,pp. 1050–1062, 2007.

[64] O. K. Erol and I. Eksin, ‘‘A new optimization method: Big bang–bigcrunch,’’ Adv. Eng. Softw., vol. 37, no. 2, pp. 106–111, 2006.

[65] D. Karaboga and B. Basturk, ‘‘A powerful and efficient algorithm fornumerical function optimization: Artificial bee colony (ABC) algorithm,’’J. Global Optim., vol. 39, no. 3, pp. 459–471, Apr. 2007.

[66] M. Zambrano-Bigiarini, M. Clerc, and R. Rojas, ‘‘Standard particle swarmoptimisation 2011 at CEC-2013: A baseline for future pso improvements,’’in Proc. IEEE Congr. Evol. Comput., Jun. 2013, pp. 2337–2344.

[67] E. Rashedi, H. Nezamabadi-Pour, and S. Saryazdi, ‘‘GSA: A gravitationalsearch algorithm,’’ J. Inf. Sci., vol. 179, no. 13, pp. 2232–2248, 2009.

[68] S.-C. Chu, P.-W. Tsai, and J.-S. Pan, ‘‘Cat swarm optimization,’’ in Proc.Pacific Rim Int. Conf. Artif. Intell., 2006, pp. 854–858.

[69] J. Kennedy and R. Eberhart, ‘‘Particle swarm optimization,’’ in Proc.IEEE Int. Conf. Neural Netw., Piscataway, NY, USA, Dec. 1995,pp. pp. 1942–1948.

[70] T. A. Runkler, ‘‘Ant colony optimization of clustering models,’’ Int.J. Intell. Syst., vol. 20, no. 12, pp. 1233–1251, 2005.

[71] A. Hatamlou, S. Abdullah, and H. Nezamabadi-Pour, ‘‘Application ofgravitational search algorithm on data clustering,’’ in Proc. Int. Conf.Rough Sets Knowl. Technol., 2011, pp. 337–346.

[72] A. Hatamlou, S. Abdullah, and M. Hatamlou, ‘‘Data clustering using bigbang–big crunch algorithm,’’ in International Conference on InnovativeComputing Technology. Berlin, Germany: Springer, 2011, pp. 383–388.

[73] V. Kumar, J. K. Chhabra, and D. Kumar, ‘‘Grey wolf algorithm-basedclustering technique,’’ J. Intell. Syst., vol. 26, no. 1, pp. 153–168,2017.

[74] R. Jensi and G. W. Jiji, ‘‘An improved krill herd algorithm with globalexploration capability for solving numerical function optimization prob-lems and its application to data clustering,’’ Appl. Soft Comput., vol. 46,pp. 230–245, Sep. 2016.

[75] H. Ghafarzadeh and A. Bouyer, ‘‘An efficient hybrid clustering methodusing an artificial bee colony algorithm and Mantegna Lévy distribution,’’Int. J. Artif. Intell. Tools, vol. 25, no. 2, 2016, Art. no. 1550034.

[76] A. Bouyer and A. Hatamlou, ‘‘An efficient hybrid clustering methodbased on improved cuckoo optimization and modified particle swarmoptimization algorithms,’’ Appl. Soft Comput., vol. 67, pp. 172–182,Jun. 2018.

142096 VOLUME 7, 2019