research article an effective hybrid of bees...

Research ArticleAn Effective Hybrid of Bees Algorithm and DifferentialEvolution Algorithm in Data Clustering

Mohammad Babrdel Bonab1 Siti Zaiton Mohd Hashim1

Nor Erne Nazira Bazin1 and Ahmed Khalaf Zager Alsaedi2

1Faculty of Computing Universiti Teknologi Malaysia Johor Bahru Malaysia2College of Science Misan University Ministry of Higher Education of Iraq Iraq

Correspondence should be addressed to Mohammad Babrdel Bonab bbmohammad2liveutmmy

Received 8 October 2014 Accepted 16 January 2015

Academic Editor Yi-Chung Hu

Copyright copy 2015 Mohammad Babrdel Bonab et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Clustering is one of the most commonly used approaches in data mining and data analysis One clustering technique in clusteringthat gains big attention in clustering related research is k-means clustering such that the observation is grouped into k clusterHowever some obstacles such as the adherence of results to the initial cluster centers or the risk of getting trapped into localoptimality hinder the overall clustering performance The purpose of this research is to minimize the dissimilarity of all points of acluster from gravity center of the cluster with respect to capacity constraints in each cluster such that each element is allocated toonly one cluster This paper proposes an effective combination algorithm to find optimal cluster center for the analysis of data indata mining and a new combination algorithm is proposed to untangle the clustering problem This paper presents a new hybridalgorithm which is based on cluster center initialization algorithm (CCIA) bees algorithm (BA) and differential evolution (DE)known as CCIA-BADE-K aiming at finding the best cluster centerThe proposed algorithm performance is evaluated with standarddata set The evaluation results of the proposed algorithm and its comparison with other alternative algorithms in the literatureconfirm its superior performance and higher efficiency

1 Introduction

Data clustering is one of the most important knowledgediscovery techniques to extract structures from dataset andis widely used in data mining machine learning statisticaldata analysis vector quantization and pattern recognitionThe aim of clustering is to partition data into 119896 clusterso that each cluster contains data which has the mostsimilarity and maximum dissimilarity with the other clus-ters Clustering algorithms can be comprehensively classifiedinto hierarchical partitioning model-based grid-based andconcentration-based clustering algorithms [1ndash3]

Hierarchical clustering algorithm divides a dataset into anumber of levels of nested partitioning In the partitioningalgorithms observations of one dataset decompose into a setof 119896 clusters with most similarity among intra-group mem-bers and least similarity among inter group members [4]

Dissimilarities are evaluated based on attribute values Gen-erally distance criterion is used for data analysis [5]

The 119896-means algorithm is one of the partitional clusteringalgorithm and one of the most popular algorithms usedin many domains The 119896-means algorithm implementationis easy and often practical However results of 119896-meansalgorithm considerably depend on initial state In otherwords its efficiency highly depends on the first initial center[6]

The main purpose of 119896-means clustering algorithm isto minimize the diversity of all objects in a cluster fromtheir cluster centers The initialization problem of 119896-meansalgorithm is considered by heuristic algorithms but it stillrisks being trapped in local optimality Therefore for achiev-ing a better cluster algorithm we should find a solution forovercoming the problem of trap into local optimum [7]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 240419 17 pageshttpdxdoiorg1011552015240419

2 Mathematical Problems in Engineering

There are many studies to overcome this problemFor instance Niknam and Amiri have proposed a hybridapproach based on combining partial swarm optimizationand ant colony optimization with 119896-means algorithm fordata clustering [8] and Nguyen and Cios have proposeda combination technique based on the hybrid of 119896-meansgenetic algorithm and maximization of logarithmic regres-sion expectation [9] Kao et al have presented a combinationalgorithm according to the hybrid of partial swarm optimiza-tion Nelder-Mead simplex search and genetic algorithm [10]Krishna andMurty proposed an algorithm for cluster analysiscalled genetic 119896-means algorithm [11] Zalik proposed anapproach for clusteringwithout preassigning cluster numbers[12] Maulik and Bandyopadhyay haves introduced geneticbased algorithm to solve this problem and evaluate theperformance on real data They define spatial distance-basedmutation according to mutation operator for clustering [13]Laszlo and Mukherjee have proposed another genetic basedapproach that for 119896-means clustering exchanges neighboringcluster centers [14] Fathian et al have presented a techniqueto overcome clustering problem according to honey-beesmating optimization (HBMO) [15ndash17] Shelokar et al havepresented to solve clustering problem based on the ant colonyoptimization [18] Niknam et al have combined to dominatethis problembased on the simulated annealing and ant colonyoptimization [19] Ng and Sung have introduced a techniquebased on the taboo search to find cluster center [20 21]Niknam et al have introduced a hybrid approach based oncombining partial swarm optimization and ant simulatedannealing to solve clustering problem [22 23]

The bees algorithms can be classified in two main cat-egories including foraging-based honeybee algorithms andmarriage-based honeybee algorithm Each of these categorieshave many algorithm such as artificial bee algorithm (ABC)[3 24 25] corporate artificial bee algorithm (CABC) [26]parallel artificial bee algorithm (PABC) [27] bee colonyoptimization (BCO) [28 29] bee algorithm (BA) [30]bee foraging algorithm (BFA) [31] bee swarm optimization(BSO) for first categories [32] Marriage in honey-bees opti-mization (MBO) [32] fast marriage honey-bees optimization(FMBO) [33] and finally modified fast marriage in honey-bees optimization (MFMBO) are in the second category ofbee algorithm [34]

One of the foraging-based algorithms is the bees algo-rithm that is a new population based search algorithmdeveloped by Pham et al in 2006 [30] The algorithmmimics the food foraging behavior of swarms of honeybees(Figure 3) In its basic version the algorithm performs a kindof neighborhood search combined with random search andcan be used for optimization problems [30]

Differential evolution is an evolutionary algorithm (EA)which has been widely used in to optimization problemsmainly in continuous search spaces [35] Differential evo-lution was introduced by Storn and Price in 1995 [36]Global optimization is necessary in fields such as engineeringstatistics and finance but many practical problems haveobjective functions that are nonlinear noisy noncontinuousand multidimensional or have many local minima and con-straints Such problems are difficult if not impossible to solve

analytically Differential evolution can be used to find approx-imate solutions to such problems Differential evolutionalso includes genetic algorithms evolutionary strategies andevolutionary programming Differential evolution encodessolutions as vectors and new solution compared to its parentIf the candidate is better than its parents it replaces the parentin the population Differential evolution can be applied innumerical optimization [37 38]

In this paper a hybrid evolutionary technique is used inorder to solve the 119896-means problem The proposed algorithmhelps clustering technique to escape from being trapped inlocal optimum Our algorithm takes the benefits of bothalgorithms Also in this survey some standard datasets areused for testing the proposed algorithm To obtain the bestcluster centers in proposed algorithm the advantages of BA(bees algorithm) and DE (differential evolution) are usedwith a data preprocessing technique called CCIA (clustercenter initialization algorithm) for data analysis Throughexperiments the proposed CCIA-BADE-K algorithm hasshown that this algorithm efficiently selects the exact clustercenters

The main contribution of this paper is the introductionof a novel combination of evolutionary algorithm accordingto bees algorithm and differential evolution to overcomedata analysis problem and hybrid with CCIA (cluster centerinitialization algorithm) preprocessing technique

The rest of this paper is arranged as follows in Section 2the data clustering issue is introduced In Sections 3 and4 the classic principles of the DE and BA evolutionaryalgorithm are discussed In Section 5 the suggested approachis introduced In Section 6 experimental results of proposedalgorithm are shown and compared with PSO-ANT SAACO GA ACO-SA TS HBMO PSO and 119896-means onbenchmark data and finally Section 7 presents the concludingremarks

2 Data Clustering

Clustering is defined as grouping similar objects either phys-ically or in abstract The groups inside one cluster have themost similarity with each other and the maximum diversitywith other groupsrsquo objects [39]

Definition 1 Suppose the set of 119883 = 1199091 1199092 119909

119899 contain-

ing 119899 objects The purpose of clustering is to group objects in119896 clusters as 119862 = 119888

1 1198882 119888

119896 while each cluster satisfies the

following conditions [40]

(1) 1198621cup 1198622cup sdot sdot sdot cup 119862

119896= 119883

(2) 119862119894= 0 119894 = 1 119896

(3) 119862119894cap 119862119895= 0

According to the mentioned definition the possiblemodes for clustering 119899 objects in 119896 clusters are obtained asfollows

NW (119899 119896) =1

119896

119896

sum

119894=1

(minus1)119894(119896

119894) (119896 minus 119894)

119899 (1)

Mathematical Problems in Engineering 3

(1) Input Data SET (119883 = 1199091 1199092 119909

119899) Attribute Set (119860 = 119860

1 1198602 119860

119902) Cluster Number (119870)

(2) Output Clusters Seed-Set (SC = sc1 sc2 sc

119867)

(3) Begin(4) While (forall119860

119895isin 119860)

(41) Compute Standard Deviation (120590119895) and Mean (120583

119895)

(42) Compute Cluster Center (e = 1 2 119896)119883119890= 119885119890lowast 120590119895+ 120583119895119885119890=2 lowast 119890 minus 1

2 lowast 119896(43) Execute 119896-means on this attribute(44) Allocate cluster labels obtained from Step (43) to every data pattern

(5) Find unique patterns (119867 ge 119896) and clustering each data with obtained patterns(6) Return SC(7) End

Algorithm 1 Pseudocode of CCIA algorithm

In most approaches the cluster number that is 119896 isspecified by an expert Relation (1) implies that even with agiven 119896 finding the optimum solution for clustering is notso simple Moreover the number of possible solutions forclustering with 119899 objects in 119896 clusters increases by the order of119896119899119896 So obtaining the best mode for clustering 119899 objects in

119896 clusters is an intricate NP-complete problem which needsto be settled by optimization approaches [5]

21 The 119870-Means Algorithm There have been many algo-rithms suggested for addressing the clustering problem andamong them the 119896-means algorithm which is one of themost famous and most practical algorithms [41] In thismethod besides the input datasets 119896 samples are introducedinto the algorithm as the initial centers of 119896 clusters Theserepresenting 119896rsquos are usually the first 119896 data samples [39]The way these 119896 representatives are chosen influences theperformance of 119870-means algorithm [42] The four stages ofthis algorithm are shown as follows

Stage I Choose 119896 data items randomly from 119883 = 1199091 1199092

119909119899 as cluster centers of (119898

1 1198982 119898

119896)

Stage II Based on relation (2) add every data item to arelevant cluster For example if the following relation (2)holds the object 119909

119894from the set of 119883 = 119909

1 1199092 119909

119899 is

added to the cluster 119888119895

10038171003817100381710038171003817119909119894minus 119898119895

10038171003817100381710038171003817lt10038171003817100381710038171003817119909119894minus 119898119901

100381710038171003817100381710038171 le 119901 le 119896 119895 = 119901 (2)

Stage III Now based on the clustering of Stage II thenew cluster centers (119898lowast

1 119898lowast

2 119898

lowast

119896) are calculated by using

relation (3) as follows (119899119894is the number of objects in the

cluster 119894)

119898lowast

119894=1

119899119894

sum

119909119895isin119862119894

1199091198951 le 119894 le 119896 (3)

Stage IV If the cluster centers are changed repeat thealgorithm from Stage II otherwise do the clustering based onthe resulted centers

The performance of the 119896-means clustering algorithmrelies on initial centers and this is a major challenge in thisalgorithm Random selection of initial cluster centers makesthis algorithm yield different results for different runs overthe same datasets which is considered as one of the potentialdisadvantages of this algorithm [43] This mix is not sensitiveto center initialization but it still has tendency towards localoptimality In this algorithm strong ties among data pointsand the nearest data centers cause cluster centers not to exitfrom their local dense ranges [44]

The algorithm of bees first developed by Karaboga andBasturk [3] and Pham et al in 2006 [30] is a new swarm-based algorithm to search solutions independently The algo-rithm was inspired by the behavior of food foraging fromswarms of honeybees In classic edition the algorithm usedrandom search to find neighborhood to solve optimizationproblems and issues

22 Algorithm for Finding Cluster Initial Centers In thisstudy with regards to efficiency purposes all data objectsare first clustered using 119896-means algorithm to find the initialcluster centers to be used in the solutions based on all theirattributes Based on the generated clusters the pattern for anobject is produced from each attribute at any stage

Objects with the same patterns are located in one clusterand hence all objects are clustered The obtained clustersin this stage will be more than the original number ofclusters For more information refer to paper [6] In thispaper clustering is completed in two stages The first stageis performed as discussed above and in the second stagesimilar clusters are integrated with each other until achievinga given number of clusters Algorithm 1 shows the proposedapproach for initial clustering of data objects and the achievedcluster centers are called seed points

As can be observed in Algorithm 1 for every attribute ofdata objects a cluster label is generated and this label is addedto the data object pattern Objects with identical patterns areplaced in one cluster To produce data object labels based oneach attribute first the mean and standard deviation of thatattribute are computed for all data objects Thereafter basedon the mean and standard deviation the range of attribute


(a) Dance floor

120572

(b) Dance languages

Figure 1 Waggle dance of the honeybee

values are broken into 119896 identical intervals so that the tail ofeach interval appears as an initial cluster center Thus usingthe initial centers all data objects are clustered by the119896-meansmethod

23 Fitness Function To calculate the fitness of each solutionthe distance between the centers of clusters and each datawill be used To do this first a set of cluster centers will begenerated randomly and then clustering of the numeratorwill be conducted based on (2) Now according to centersobtained in the interaction step the new centers of theclusters and fitness of solutions based on (3) will be calculated[40]

Fitness (119862) =119896

sum

119894=1

sum

119909119895isin119862119894

10038171003817100381710038171003817119909119895minus 119898lowast

119894

10038171003817100381710038171003817 (4)

3 The Dance Language of Bees

For honeybees finding nectar is essential to survival Beeslead others to specific sources of food and then scout beesstart to identify the visited resources by making movementsas ldquodancingrdquo These dances are very careful and fast indifferent directions Dancers try to give information abouta food resource by specifying the direction distance andquality of the visited food source [45]

31 Describing the Dance Language There are two kindof dance for Observed bees including ldquoround dancerdquo andldquowaggle dancerdquo [46] When a food resource is less than fiftymeters away they do round dance and when a food resourceis greater than fifty meters away they perform waggle dance(Figure 1)

There are some concepts in this dance in which the anglebetween vertical and waggle run is equal to the angle betweenthe sun and food resource Dance ldquotempordquo shows the distanceof food resource (Figure 2) A slower dance tempomeans that

120572

120572

Figure 2 Communication and information sharing

a food resource is farther and vice versa [47] Another conceptis the duration of dance and a longer dance duration meansthat a food resource is rich and better [45] Audiences areother bees which follow the dancer In this algorithm thereare two kinds of bees SCOUTS are bees that find new foodsources and perform the dance RECRUTTS are bees thatfollow the scout bees dance and then forage One of the firstpeople that translate the waggle dance mining was Austrianetiologist Karl von Frisch

Distance between flowers and hive is demonstrated by theduration of the waggle dance The flowers that are fartherfrom the hive have longer waggle dance duration Eachhundred meters distance between flowers from the hive isshown in the waggle dance phase close to 75 milliseconds

32 Bee in Nature A colony of honeybees can extend itselfover long distances (more than 10 km) and in multiple


Waggle dance

Waggle dance

Waggle danceFood source

Food source

Food source

Food source

Food source

1205721

1205722

1205723

12057241205725

Figure 3 The intelligent foraging behavior of honeybee colony

(1) Initialize population with random solutions (119899 scout bees are placed randomly in the search space)(2) Evaluate fitness of the population(3) While (Repeat optimization cycles for the specified number)(4) Select sites for neighborhood search (Bee that have the highest fitness are chosen as ldquoselectedrdquo and

sites visited by them are chosen for neighborhood search)(5) Recruit bees for selected sites (more bees for best 119890 sites) and evaluate fitness(6) Select the fittest bee from each patch (For each patch only the bee with the highest fitness will be

selected to form the next bee population)(7) Assign remaining bees to search randomly and evaluate their fitness(8) End While

Algorithm 2 Pseudocode of Basic Bee Algorithm

directions simultaneously to exploit a large number of foodsources In principle flower patches with plentiful amount ofnectar or pollen that can be collected with less effort shouldbe visited by more bees whereas patches with less nectar orpollen should receive fewer bees [35 47]

The foraging process begins in a colony with the scoutbees being sent out to search for promising flower patchesScout bees move randomly from one patch to anotherDuring the harvest season a colony continues its explorationkeeping a percentage of the population as scout bees Whenthe scout bees return to the hive those that found a patchwhich is rated above a certain quality threshold (measured asa combination of some constituents such as sugar content)deposit their nectar or pollen and go to the ldquodance floorrdquoto perform a dance known as ldquowaggle dancerdquo [46] Thewaggle dance is essential for colony communication andcontains three pieces of information regarding a flower patch

the direction in which it will be found its distance from thehive and its quality rating (or fitness)This information helpsthe colony to send its bees to flower patches precisely withoutusing guides or maps [45] After the waggle dance on thedancing floor the dancers (ie scout bee) go back to theflower patch with follower bees that are waiting inside thehive More follower bees are sent to patches that are mostpromising [48 49] The flowchart of bee algorithm is shownin Figure 4 [50]

The Basic Bee Algorithm is shown as in Algorithm 2 [51]

4 Differential Evolution

Differential evolution is a type of standard genetic algorithmDifferential evolution algorithm evaluates the initial popu-lation by using probability motion and observation modelsand population evolution is performed by using evolution


(1) Begin(2) First 119895

0is selected randomly between 1 and119863

(3) 1198950is added to set 119869

(4) For all values of 119895 the following operations are repeated(a) One random number is generated such as rand

119895that has uniform distribution between zero and one

(b) If 119903119895is less than or equal to 119875cr then number of 119895 is added to 119869 set

(5) End

Algorithm 3 Pseudocode of the binomial crossover

Evaluate the fitness of the population

Determine the size of neighborhood (patch size)

Recruit bees for selected sites(more bees for the best sites)

Select the fittest bees from each patch

Allocate randomly instead of remaining bees

New population of scout bees

Nei

ghbo

rhoo

dse

arch

Repe

at o

ptim

izat

ion

cycle

s for

the s

peci

fied

num

ber

Initialize a population of n scout bees

Select m sites for neighborhood search

Figure 4 The flowchart of bee algorithm

operators [52] The main idea in the differential evolutionalgorithm is to generate a new solution for each solution byusing one constant member and two random members Ineach generation the best member of population is selectedand then the difference between each member of populationand the best member is calculated Two randommembers arethen selected and the difference between them is calculatedCoefficient of this difference is added to 119894th member and thusa new member is created The cost of each new member iscalculated and if the cost value of the new member is less the119894th member is replaced instead of 119894th member otherwise theprevious value can be kept in the next generation [35]

Differential evolution is one of the population basedpopular algorithms that uses point floating (real coded) forpresentation as follows [53]

119885119894(119905) = [119885

1198941(119905) 119885

1198942(119905) 119885

119894119863(119905)] (5)

where 119905 is the number of generation (iteration) 119894 refers tomembers (population) and 119889 is the number of optimizationparameters Now in each generation (or each iteration of

algorithm) to perform changes on members of population119885119894(119905) one donor vector 119884

119894(119905) is formed The various methods

of DE are used to determine how to make the donorvector The first kind of DE named 1randDE generates 119894thmember 119884

119894(119905) in which three members of current generation

(1199031 1199032 1199033) are chosen randomly as

1199031= 1199032= 1199033isin 1 2 119863 (6)

Then the difference between two vectors from threeselected vectors are calculated andmultiplied by 119865 coefficientand with the third vector added [53] Therefore donor vector119884119894(119905) is obtained Calculation process of donor vector for 119895th

element from 119894th vector can be demonstrated as follows [54]

119884119894119895(119905) = 119885

1199031119895(119905) + 119865 (119885

1199032119895(119905) minus 119885

1199033119895(119905)) (7)

To increase the exploration of algorithm a crossover oper-ation is then performed Differential algorithm has generallytwo kinds of crossover exponential and binomial [55] Inthis paper to save time the binomial mode has been usedTo apply the binomial crossover it requires that set of 119869 isconstituted as in Algorithm 3

Therefore for each target vector 119885119894(119905) there is a trial

vector as follows [56]

119877119894119895(119905)

=

119884119894119895(119905) If rand

119895119894le 119875cr or 119895 = 119895

0119894 = 1 2 119873

119885119894119895(119905) If rand

119895119894gt 119875cr and 119895 = 119895

0119895 = 1 2 119863

(8)

where 119895 is equal to 119895 = 1 2 119863 and rand119895is uniform

distribution number between [0 1] Set of 119869 is guaranteedwhere there is at least one difference between 119877

119894(119905) and 119885

119894(119905)

In the next step the selection process is performed betweentarget vector and trial vector as follows

119885119894(119905 + 1)

=

119877119894(119905) If 119891 (119877

119894(119905)) le 119891 (119885

119894(119905)) 119894 = 1 2 119873

119885119894(119905) otherwise

(9)

where 119891(sdot) is a function that should be the minimum Inthis paper to escape from premature convergence two newstrategies of merging have been studied In the basic DEare used difference vector of 119885

1199032119895(119905) minus 119885

1199033119895(119905) multiplied 119865


(1) Define algorithm parameter(2) Generate and evaluate initial population or solutions(3) For all members of population per form the following steps

(a) With mutation operator create a new trial solution(b) By using the crossover generate new solution and evaluate them(c) Replace new solution with current solution if new solution is better than current solution otherwise

the current solution is retained(4) Return to step three if termination condition is not achieved

Algorithm 4 Pseudocode of differential evolution

Z1(t) Z2(t) Z3(t) Z4(t) Znminus1(t) Zn(t)

f(Z1) f(Z2) f(Z3) f(Z4) f(Znminus1) f(Zn)

Zr1(t) Zr2(t) Zr3(t)

di = Zr2(t) minus Zr3(t)

Zr1(t) + F(di)

Y1(t) Y2(t) Y3(t) Y4(t) Ynminus1(t) Yn(t)

f(Y1) f(Y2) f(Y3) f(Y4) f(Ynminus1) f(Yn)

Z1(t + 1) Z2(t + 1) Z3(t + 1) Z4(t + 1) Znminus1(t + 1) Zn(t + 1)middot middot middot

middot middot middot


Crossover

Selection

Figure 5 The process of differential evolution

where119865 is control parameter between 04 and one [55 57] Toimprove the convergence feature in the DE this paper makesthe following proposal

119865 = 05 (Rnd + 1) (10)

where Rnd is uniform distribution number between zero andone Generally the DE algorithm steps are as in Algorithm 4

In Figure 5 the process of differential evolution is illus-trated

5 Proposed Algorithm

As noted in the former sections studies conducted on the BAmethod have shown that this algorithm can be a powerfulapproach with enough performance to handle different typesof nonlinear problems in various fields However it can bepossibly trapped into local optimum Lately several ideas

have been used to reduce this problem by hybrid differentevolutionary techniques such as partial swarm optimizationgenetic algorithm and simulating annealing In most popu-lation based evolutionary algorithms in each iteration newmembers are generated and then the movement operationsare applied to explore new positions based on providingbetter opportunities To increase the diversity of algorithmin the differential evolution algorithm all members have apossibility to win the global optimum and move to that sideThe ability of the best particle to local search also dependson the other particles by selecting the two other particles andcalculating the difference between them This situation maylead to local convergence

In this proposed algorithm to escape from randomselecting of the global best particle we used competencyselection for choosing the global best particle If particle isbetter than the other solutions then the probability of beingselected is greater


Begin(a) Find seed cluster center (preprocessing)(b) Create an initial Bees population randomly with 119899 Scout Bees(c) Calculate the objective function for each individual(d) Sort and update best site ever found(e) Select the elite sites non-elite sites and non-selected site (three site groups)(f) Determine number of recruited bees for each kind of site(g) While (iteration lt 100)

(I) For each selected kind of sites calculate the neighborhoods

(1) For each recruited bees Mutation

(2) Choose target site and base site from this group(3) Random choice of two sites from this group(4) Calculate weighted difference site(5) Add to base selected site

Crossover(6) Perform crossover operation with Crossover Probability(7) Evaluate the trial site that is generated

update site(8) If trial site is less than target site(9) Select trial site instead of target site(10) else(11) Select target site(12) End if

(II) End (for of recruited bees)(h) End (for of selected Sites)(i) Sort and update best site ever found

End

Algorithm 5 Pseudocode of proposed CCIA-BADE-K algorithm

The basic idea behind the proposed algorithm is that oursolutions are grouped based on the beesrsquo algorithm

On the other hand in this algorithm new approach isproposed to the movement and selects the recruiting beesfor selecting sites This algorithm classified the bees intothree groups and named them elite sites nonelite sitesand nonselected site To increase diversity the two modesfor movement based on the differential evolution algorithmoperator as parallel mode and serial mode were used Thesuggested algorithm tries to use the advantage of thesealgorithms to find the best cluster center and to improvesimulation results In other words in this algorithm first apreprocessing technique is performed on the data and thenthe proposed hybrid algorithm is used to find the best clustercenter for 119896-means problem

The flowchart and pseudocode of the combined algo-rithm calledCCIA-BA-DE are illustrated inAlgorithm 5 andFigure 6

6 Application of CCIA-BA-BE on Clustering

The application of CCIA-BADE-K algorithm on the clus-tering problem in this section is presented To perform theCCIA-BADE-K algorithm to find best cluster centers thefollowing steps should be repeated and taken

Step 1 (generate the seed cluster center) This step is apreprocessing step to find the seed cluster center to choosethe best interval for each cluster

Step 2 (generate the initial beesrsquo population randomly) Inother words generate initial solutions to find the best clustercenters statistically

Population =

[[[[[[[

[

center1198791

center1198792

center119879119899Scout

]]]]]]]

]

(11)

where center is a vector with 119896 cluster and each vector has 119901dimension

center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)

where 119862119895is cluster center of 119895 for 119894th scout bee and 119901 is the

number of dimension for each cluster center In fact each


Find seed cluster center (preprocessing)

Load dataset

Calculate the objective function for each individual

Sort the solutions and select scout bees for each groups

YesNo


Calculate weighted difference siteAdd to base selected site

Perform crossover operation with crossover probability

Select trial site instead of target site N

eigh

borh

ood

sear

ch

Calculate the fitness function for trial site

Select target site

No

NoThe convergence

condition is satisfiedYes

Yes

Stop and plot the resultsYes

No

i = i + 1

All i group are selected

i = 1

Select m sites for neighborhood search in ith group

Choose target site or base site from ith groupChoose randomly two sites from ith group

If trial site lt target

All m sites are selected

Generate a population with n scout bees

Figure 6 Flowchart of proposed algorithm

solution in the algorithm is a matrix with 119896times119901 119888min119894

and 119888max119894

are values of theminimumandmaximum for each dimension(each feature of center)

Step 3 (calculate the objective function for each individual)Calculate the cost function for each solution (each site) in thisalgorithm

Step 4 (sort the solutions and select scout bees for eachgroups) The sorting of the site is carried out based on theobjective function value

Step 5 (select the first group of sites) Finding the newsolutions is performed by selecting the group of sites Thereare three groups of sites in which the first group or elitesites are evaluated to find the neighbors of the selected sitefollowed by nonelite site and finally nonselected sites To findthe neighbors of each group of sites either the serial modeor parallel mode may be used This algorithm used parallelmodel

Step 6 (select the number of bees for each site) Numbers ofbees for each site depend on their group and are consideredas competence more bees for better site If the site is richthen more bees are allocated to this site In other words ifthe solution is better it is rated as more important than theother sites

Step 7 (performing the differential evolution operator (muta-tion)) In this step the target site is chosen from the groupsites and then two other sites from this group are selectedrandomly to calculate the weighted difference between themAfter calculating this difference it is added to base trial siteas shown in the following equation

V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)


minus6 minus4 minus2 0 2 4 6 8

minus4

minus2

0

2

4

6

X1

X2

(a) First artificial dataset


minus4

minus2

0

2

4

6

X1

X2

(b) Clustered dataset after with cluster centers

Figure 7 Used artificial dataset one

where 1199091199031119866

is the target site119882 is the weight and the 1199091199032119866

1199091199033119866

are the selected sites from targetrsquos group V119894119866+1

is the trialsolution for comparison purposes

Step 8 (perform crossover operation with crossover proba-bility) The recombination step incorporates successful solu-tions from the previous generation The trial vectors 119906

119894119866+1is

developed from the elements of the target vector 119909119894119866

and theelements of donor vector V

119894119866+1 Elements of the donor vector

enter the trial vector with probability CR

119906119895119894119866+1

=

V119895119894119866+1

If rand119895119894le CR or 119895 = 119868rand 119894 = 1 2 119873

119909119895119894119866

If rand119895119894gt CR and 119895 = 119868rand 119895 = 1 2 119871

(14)

where rand119895119894sim 119880[0 1] and 119868rand is a random integer from

[1 2 119871] and 119868rand ensures that 119906119894119866+1

= 119909119894119866

Step 9 (calculate the cost function for trial site) In theselection step target vector 119909

119894119866is compared with trial vector

119906119894119866+1

There are two modes to calculate the new site asfollows

119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)

Trial vector 119906119894(119866) is compared to target vector 119909

119894(119866) To

use greedy criterion if 119906119894(119866) is better than the 119909

119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909

119894(119866) ldquosurviverdquo and 119906

119894(119866)

are discarded

Step 10 If not all sites from this group are selected go toStep 6 and select another site from this group otherwise goto the next step

Table 1 Table type styles

Datasetname

Dataset attribute

Dataset size Clusternumber

Attributenumber

Iris 150 (50 50 50) 3 4Wine 178 (59 71 48) 3 13CMC 1473 (629 334 510) 3 9Glass 214 (70 17 76 13 9 29) 6 9

Step 11 If not all groups are selected go to Step 5 and selectthe next group otherwise go to the next step

Step 12 (check the termination criteria) If the current num-ber of iteration does not reach the maximum number ofiterations go to Step 4 and start next generation otherwisego to the next step

7 Evaluation

To evaluate the accuracy and efficiency of the proposedalgorithm experiments have been performed on two artificialdatasets four real-life datasets and four standard datasetsto determine the correctness of clustering algorithms Thiscollection includes Iris Glass Wine and ContraceptiveMethod Choice (CMC) datasets that have been chosen fromstandard UCI dataset

The suggested algorithm is coded by an appropriateprogramming language and is run on an i5 computer with260GHzmicroprocessor speed and 4GBmain memory Formeasuring the performance of the proposed algorithm thebenchmarks data items of Table 1 are used

The execution results of the proposed algorithm overthe selected datasets as well as the comparison figuresrelative to 119870-means PSO and K-NM-PSO results in [10] aretabulated in Table 2 As easily seen in Table 2 the suggested


Table 2 The obtained results from implementing the suggested algorithm over selected datasets

Dataset 119870-means [10] PSO [10] K-NM-PSO [10]Proposed Alg

Result CPUTime (S)

Iris 9733 9666 9666 965403 sim15Wine 1655568 1629400 1629200 1629225 sim30CMC 554220 553850 553240 553222 sim57Glass 21568 27129 19968 2104318 sim34

Table 3 The results of implementing the algorithms over Iris testdata for 100 runs

Method Result CPUtime (S)Best Average Worst

PSO-ACO-K 96650 96650 96650 sim16PSO-ACO 96654 96654 96674 sim17PSO 968942 97232 97897 sim30SA 97457 99957 10201 sim32TS 97365 97868 98569 sim135GA 113986 125197 139778 sim140ACO 97100 97171 97808 sim75HBMO 96752 96953 97757 sim82PSO SA 9666 9667 96678 sim17ACO-SA 96660 96731 96863 sim25119896-Means 97333 10605 12045 04MY proposed ALG 965403 965412 965438 sim15

Table 4 The results of implementing the algorithms over Wine testdata for 100 runs



algorithm provides superior results relative to 119870-means andPSO algorithms The real-life datasets compared with severaloptimization algorithms are included

For better study and analysis of the proposed approachthe execution results of the proposed approach along withHBMO PSO ACO-SA PSO-ACO ACO PSO-SA TS GASA and 119896-means clustering algorithm results as reported

Table 5 The results of implementing the algorithms over CMC testdata for 100 runs



Table 6 The results of implementing the algorithms over Glass testdata for 100 runs


PSO-ACO-K 19953 19953 19953 sim31PSO-ACO 19957 19961 20001 sim35PSO 27057 27571 28352 sim400SA 27516 28219 28718 sim410TS 27987 28379 28647 sim410GA 27837 28232 28677 sim410ACO 26972 27346 28008 sim395HBMO 24573 24771 24954 sim390PSO SA 20014 20145 20245 sim38ACO-SA 20071 20189 20276 sim49119896-Means 21574 2355 25538 sim1MY proposed ALG 210431 21554 21693 sim34

in [8] are tabulated in Tables 3ndash6 It is worth mentioningthat the investigated algorithms of [8] are implemented withMATLAB 71 using a Pentium IV system of 28GHz CPUspeed and 512MB main memory

Frist artificial dataset includes (119899 = 800 119896 = 4 119889 = 2)where 119899 is the number of instance 119896 is the number of clustersand 119889 is the number of dimensionsThe instances were drawn


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3

(a) Second artificial dataset

minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3

(b) Clustered dataset after and with cluster centers

Figure 8 Used artificial dataset two

4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10

Sepal length with other dimensions (cm)

Sepal width with other dimensions (cm)

Petal length with other dimensions (cm)

Petal width with other dimensions (cm)

Figure 9 The scatter plot to show nonlinear relationship between variables for Iris dataset

for four absolute classes where each of these groups wasdistributed as

Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)

where Σ and 120583 are covariance matrix and vector respectively[10]The first artificial dataset is demonstrated in Figure 7(a)Figure 7(b) illustrated the clustered data after applyingCCIA-BADE-K algorithm on data

Second artificial dataset includes (119899 = 800 119896 = 4 119889 = 3)where 119899 is the number of instance 119896 is the number of clustersand 119889 is the number of dimensionsThe instances were drawn






4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10

Figure 10 Clustered scatter plot to show nonlinear relationship between variables for Iris dataset


Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)

where Σ and 120583 are covariance matrix and vector respectively[10]The second artificial dataset is demonstrated in Figure 8Figure 8 shows clusters after applying proposed algorithm onthe artificial dataset

In Tables 3ndash6 best worst and average results are reportedfor 100 runs respectively The resulting figures represent thedistance of every data from the cluster center to which itbelongs and is computed by using relation (4) As observedin the table regarding the execution time the proposedalgorithm generates acceptable solutions

To clarify the issue in Figure 10 the scatterplot (scatter-graph) is illustrated The scatter-graph is one kind of mathe-matic diagram which shows the values for a dataset for twovariables using Cartesian coordinates In this diagram data isdemonstrated as a set of spotsThis type of diagram is knownas a scatter diagram or scatter-gram This kind of diagramis also used to display relation between response variableswith control variables when a variable is below the control ofthe experimenter One of the strongest aspects of the scatter-diagram is the ability to show nonlinear relationship between

variables In Figure 9 the scatter-diagram of Iris dataset isdisplayed and in Figure 10 the clustered Iris data on thescatter-diagram is shown

In Table 4 best worst and average results ofWine datasetare reported for 100 runs The resulting figures represent thedistance of every data from the cluster center

In Figure 11 best cost and average best costs of resultsfor all datasets are reported for 100 runs The resultingfigures represent the distance of every data from the clustercenter by using relation (4) Figure 11(a) is related to the bestcost and mean of best cost for Iris dataset and Figure 11(b)illustrated the best cost and mean of best cost for Winedataset Figure 11(c) reported best cost and mean of bestcost for CMC datasets and finally Figure 11(d) demonstratedmean value of best cost and best cost of Glass dataset

According to the reported results in Tables 3 to 6the proposed method over Iris CMC and Wine Datasetsprovides the best results in comparisonwith othermentionedalgorithms According to Table 6 the suggested algorithmover Glass dataset provides more acceptable results thanthe alternative algorithms The reason for this behavior isjustified by the fact that as data objects increase in numberthe efficiency of the alternative algorithms decreaseswhile thedeficiency of the suggested algorithm highlights more

8 Image Segmentation

In Section 7 it was shown that the proposed CCIA-BADE-K algorithm is one of the best methods for data clusteringFor further investigation of the performance of algorithm


0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost

Best cost 965403 in 100 iterations for Iris dataset

(a) Best cost and mean of best cost for Iris dataset


16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost

(b) Best cost and mean of best cost for Wine dataset

5500

6000

6500

7000

7500

8000Best cost 55322215 in 100 iterations for Iris dataset

0 20 40 60 80 100Iterations

Best

cost

Best costMean

(c) Best cost and mean of best cost for CMC dataset

200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean

(d) Best cost and mean of best cost for Glass dataset

Figure 11 Best cost and mean of best cost in 100 iterations

the algorithm was tested on one standard image and oneindustrial image Each digital image in RGB space is formedby three-color components consisting of red green andblue Each of these three alone is a grayscale image and thenumerical value of each pixel is between 1 and 255 Imagehistogram is a chart that is made by the number of pixelson an image that is determined based on the brightness level[58] To obtain a histogram of image it is enough to scrollthe whole pixel of image and to calculate the number ofpixels for each brightness level The normalized histogram isobtained by dividing the total number of histogram value toeach value of pixels Normalizing the histogram causes thehistogram value to be in [0 1] interval Figures 12 and 13show that image samples in this paper are shown for imagesegmentation In Figure 12 the color grayscale and clusteredmodes of these images are shown and in Figure 13 histogram

diagrams for these four images are shown Furthermore thesesegmentation charts will be used to detect segmentation animage

9 Concluding Remarks

In this paper a new technique based on a combination of beesalgorithm and differential evolution algorithm with 119896-meanswas presented In the proposed algorithm bee algorithmwas assigned to perform globally and differential evolutionalgorithm was assigned to implement local searching on 119896-means problem which is responsible for the task of find-ing the best cluster centers The new proposed algorithmCCIA-BADE-K applies abilities of both algorithms and byremoving shortcomings of each algorithm it tries to useits own strengths to cover other algorithm defects as well


(a) Color image of raisins (b) Grayscale image of raisins (c) Clustered raisins

(d) Color image of Lena (e) Grayscale image of Lena (f) Clustered LenaFigure 12 Images used for image segmentation with proposed algorithm

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250

(a) Histogram of raisins image

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250

(b) Best cluster centers of raisins image histogram

0

500

1000

1500

2000

2500

0 50 100 150 200 250

(c) Histogram of Lena image

0

500

1000

1500

2000

2500

0 50 100 150 200 250

(d) Best cluster centers of Lena image histogramFigure 13 Sample used images for image segmentation in the grayscale mode


as to find best cluster centers that is the proposed seedcluster center algorithm Experimental results showed thatthe CCIA-BADE-K algorithm enjoys acceptable results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to express their cordial thanks tothe Ministry of Education (MoE) University TechnologyMalaysia (UTM) for the Research University Grant noQJ130000252806H90 The authors are also grateful to SoftComputing Research Group (SCRG) for their support andincisive comments in making this study a success

References

[1] G Gan C Ma and J Wu Data Clustering Theory Algorithmsand Applications vol 20 SIAM 2007

[2] J Han M Kamber and J Pei Data Mining Concepts andTechniques Morgan Kaufmann 2006

[3] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[4] E Alpaydin Introduction to Machine Learning MIT Press2004

[5] S Bandyopadhyay and U Maulik ldquoAn evolutionary techniquebased on K-means algorithm for optimal clustering in R119873rdquoInformation Sciences vol 146 no 1ndash4 pp 221ndash237 2002

[6] S S Khan and A Ahmad ldquoCluster center initialization algo-rithm for K-means clusteringrdquo Pattern Recognition Letters vol25 no 11 pp 1293ndash1302 2004

[7] G Hamerly and C Elkan ldquoAlternatives to the k-means algo-rithm that find better clusteringsrdquo in Proceedings of the 11thInternational Conference on Information and Knowledge Man-agement (CIKM rsquo02) pp 600ndash607McLeanVaUSANovember2002

[8] T Niknam and B Amiri ldquoAn efficient hybrid approach basedon PSO ACO and k-means for cluster analysisrdquo Applied SoftComputing Journal vol 10 no 1 pp 183ndash197 2010

[9] C D Nguyen and K J Cios ldquoGAKREM a novel hybridclustering algorithmrdquo Information Sciences vol 178 no 22 pp4205ndash4227 2008

[10] Y-T Kao E Zahara and I-W Kao ldquoA hybridized approach todata clusteringrdquo Expert Systems with Applications vol 34 no 3pp 1754ndash1762 2008

[11] K Krishna and M N Murty ldquoGenetic K-means algorithmrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 29 no 3 pp 433ndash439 1999

[12] K R Zalik ldquoAn efficient k1015840-means clustering algorithmrdquoPatternRecognition Letters vol 29 no 9 pp 1385ndash1391 2008

[13] U Maulik and S Bandyopadhyay ldquoGenetic algorithm-basedclustering techniquerdquo Pattern Recognition vol 33 no 9 pp1455ndash1465 2000

[14] M Laszlo and S Mukherjee ldquoA genetic algorithm thatexchanges neighboring centers for 119896-means clusteringrdquo PatternRecognition Letters vol 28 no 16 pp 2359ndash2366 2007

[15] M Fathian and B Amiri ldquoA honeybee-mating approach forcluster analysisrdquo The International Journal of Advanced Manu-facturing Technology vol 38 no 7-8 pp 809ndash821 2008

[16] A Afshar O Bozorg Haddad M A Marino and B J AdamsldquoHoney-bee mating optimization (HBMO) algorithm for opti-mal reservoir operationrdquo Journal of the Franklin Institute vol344 no 5 pp 452ndash462 2007

[17] M Fathian B Amiri and A Maroosi ldquoApplication of honey-bee mating optimization algorithm on clusteringrdquo AppliedMathematics and Computation vol 190 no 2 pp 1502ndash15132007

[18] P S Shelokar V K Jayaraman and B D Kulkarni ldquoAn antcolony approach for clusteringrdquo Analytica Chimica Acta vol509 no 2 pp 187ndash195 2004

[19] T Niknam B B Firouzi and M Nayeripour ldquoAn efficienthybrid evolutionary algorithm for cluster analysisrdquo WorldApplied Sciences Journal vol 4 no 2 pp 300ndash307 2008

[20] M K Ng and J CWong ldquoClustering categorical data sets usingtabu search techniquesrdquo Pattern Recognition vol 35 no 12 pp2783ndash2790 2002

[21] C S Sung and H W Jin ldquoA tabu-search-based heuristic forclusteringrdquo Pattern Recognition vol 33 no 5 pp 849ndash8582000

[22] T Niknam B Amiri J Olamaei and A Arefi ldquoAn efficienthybrid evolutionary optimization algorithm based on PSO andSA for clusteringrdquo Journal of Zhejiang University SCIENCE Avol 10 no 4 pp 512ndash519 2009

[23] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[24] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

[25] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep tr06 Erciyes University EngineeringFaculty Computer Engineering Department 2005

[26] W Zou Y Zhu H Chen and X Sui ldquoA clustering approachusing cooperative artificial bee colony algorithmrdquo DiscreteDynamics in Nature and Society vol 2010 Article ID 45979616 pages 2010

[27] H Narasimhan ldquoParallel artificial bee colony (PABC) algo-rithmrdquo in Proceedings of the World Congress on Nature ampBiologically Inspired Computing (NABIC rsquo09) pp 306ndash311 IEEECoimbatore India December 2009

[28] D Teodorovic ldquoBee colony optimization (BCO)rdquo in Innovationsin Swarm Intelligence C Lim L Jain and S Dehuri Eds vol248 pp 39ndash60 Springer Berlin Germany 2009

[29] D Teodorovic P Lucic G Markovic and M Dellrsquo Orco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering (NEUREL rsquo06) pp 151ndash156 2006

[30] D Pham A Ghanbarzadeh E Koc S Otri S Rahim andM Zaidi ldquoThe bees algorithmmdasha novel tool for complexoptimisation problemsrdquo in Proceedings of the 2nd VirtualInternational Conference on Intelligent ProductionMachines andSystems (IPROMS rsquo06) pp 454ndash459 2006


[31] R Akbari A Mohammadi and K Ziarati ldquoA novel bee swarmoptimization algorithm for numerical function optimizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 10 pp 3142ndash3155 2010

[32] H Drias S Sadeg and S Yahi ldquoCooperative bees swarm forsolving the maximum weighted satisfiability problemrdquo in Com-putational Intelligence and Bioinspired Systems J CabestanyA Prieto and F Sandoval Eds vol 3512 of Lecture Notes inComputer Science pp 318ndash325 Springer Berlin Germany 2005

[33] C Yang J Chen and X Tu ldquoAlgorithm of fast marriage inhoney bees optimization and convergence analysisrdquo in Proceed-ings of the IEEE International Conference on Automation andLogistics (ICAL rsquo07) pp 1794ndash1799 Jinan China August 2007

[34] M T Vakil-Baghmisheh and M Salim ldquoA modified fast mar-riage in honey bee optimization algorithmrdquo inProceedings of the5th International Symposium on Telecommunications (IST rsquo10)pp 950ndash955 December 2010

[35] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[36] R Storn and K PriceDifferential EvolutionmdashA Simple and Effi-cient Adaptive Scheme for Global Optimization over ContinuousSpaces ICSI Berkeley Calif USA 1995

[37] V Feoktistov ldquoDifferential evolutionrdquo in Differential Evolutionvol 5 pp 1ndash24 Springer New York NY USA 2006

[38] K V Price R M Storn and J A Lampinen ldquoThe differentialevolution algorithmrdquo in Differential Evolution pp 37ndash134Springer Berlin Germany 2005

[39] A K Jain M N Murty and P J Flynn ldquoData clustering areviewrdquo ACM Computing Surveys vol 31 no 3 pp 264ndash3231999

[40] R J Kuo E Suryani and A Yasid ldquoAutomatic clustering com-bining differential evolution algorithmand k-means algorithmrdquoin Proceedings of the Institute of Industrial Engineers AsianConference 2013 Y-K Lin Y-C Tsao and S-W Lin Eds pp1207ndash1215 Springer Singapore 2013

[41] W Kwedlo ldquoA clustering method combining differential evolu-tion with the K-means algorithmrdquo Pattern Recognition Lettersvol 32 no 12 pp 1613ndash1621 2011

[42] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[43] M Babrdelbonab S Z M H M Hashim and N E NBazin ldquoData analysis by combining the modified k-means andimperialist competitive algorithmrdquo Jurnal Teknologi vol 70 no5 2014

[44] P Berkhin ldquoA survey of clustering data mining techniquesrdquo inGrouping Multidimensional Data J Kogan C Nicholas andMTeboulle Eds pp 25ndash71 Springer Berlin Germany 2006

[45] J R Riley U Greggers A D Smith D R Reynolds and RMenzel ldquoThe flight paths of honeybees recruited by the waggledancerdquo Nature vol 435 no 7039 pp 205ndash207 2005

[46] C Gruter M S Balbuena and W M Farina ldquoInformationalconflicts created by the waggle dancerdquo Proceedings of the RoyalSociety B Biological Sciences vol 275 no 1640 pp 1321ndash13272008

[47] A Dornhaus and L Chittka ldquoWhy do honey bees dancerdquoBehavioral Ecology and Sociobiology vol 55 no 4 pp 395ndash4012004

[48] K O Jones and A Bouffet ldquoComparison of bees algorithm antcolony optimisation and particle swarm optimisation for PIDcontroller tuningrdquo in Proceedings of the 9th International Con-ference on Computer Systems and Technologies and Workshopfor PhD Students in Computing (CompSysTech rsquo08) GabrovoBulgaria June 2008

[49] D T Pham and M Kalyoncu ldquoOptimisation of a fuzzy logiccontroller for a flexible single-link robot arm using the BeesAlgorithmrdquo in Proceedimgs of the 7th IEEE International Confer-ence on Industrial Informatics (INDIN 09) pp 475ndash480 CardiffWales June 2009

[50] D T Pham S Otri A Ghanbarzadeh and E Koc ldquoApplica-tion of the bees algorithm to the training of learning vectorquantisation networks for control chart pattern recognitionrdquoin Proceedings of the 2nd Information and CommunicationTechnologies (ICTTA rsquo06) vol 1 pp 1624ndash1629 DamascusSyria 2006

[51] L Ozbakir A Baykasoglu and P Tapkan ldquoBees algorithmfor generalized assignment problemrdquo Applied Mathematics andComputation vol 215 no 11 pp 3782ndash3795 2010

[52] P Rocca G Oliveri and A Massa ldquoDifferential evolution asapplied to electromagneticsrdquo IEEE Antennas and PropagationMagazine vol 53 no 1 pp 38ndash49 2011

[53] R Mallipeddi P N Suganthan Q K Pan and M F TasgetirenldquoDifferential evolution algorithm with ensemble of parametersand mutation strategiesrdquo Applied Soft Computing Journal vol11 no 2 pp 1679ndash1696 2011

[54] R Storn ldquoOn the usage of differential evolution for functionoptimizationrdquo in Proceedings of the Biennial Conference of theNorth American Fuzzy Information Processing Society (NAFIPSrsquo96) pp 519ndash523 June 1996

[55] U K Chakraborty Advances in Differential Evolution SpringerBerlin Germany 2008

[56] G Liu Y Li X Nie and H Zheng ldquoA novel clustering-baseddifferential evolution with 2 multi-parent crossovers for globaloptimizationrdquoApplied Soft Computing Journal vol 12 no 2 pp663ndash681 2012

[57] Z Cai W Gong C X Ling and H Zhang ldquoA clustering-based differential evolution for global optimizationrdquo AppliedSoft Computing Journal vol 11 no 1 pp 1363ndash1379 2011

[58] M Abbasgholipour M Omid A Keyhani and S S MohtasebildquoColor image segmentation with genetic algorithm in a raisinsorting system based onmachine vision in variable conditionsrdquoExpert Systems with Applications vol 38 no 4 pp 3671ndash36782011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


There are many studies to overcome this problemFor instance Niknam and Amiri have proposed a hybridapproach based on combining partial swarm optimizationand ant colony optimization with 119896-means algorithm fordata clustering [8] and Nguyen and Cios have proposeda combination technique based on the hybrid of 119896-meansgenetic algorithm and maximization of logarithmic regres-sion expectation [9] Kao et al have presented a combinationalgorithm according to the hybrid of partial swarm optimiza-tion Nelder-Mead simplex search and genetic algorithm [10]Krishna andMurty proposed an algorithm for cluster analysiscalled genetic 119896-means algorithm [11] Zalik proposed anapproach for clusteringwithout preassigning cluster numbers[12] Maulik and Bandyopadhyay haves introduced geneticbased algorithm to solve this problem and evaluate theperformance on real data They define spatial distance-basedmutation according to mutation operator for clustering [13]Laszlo and Mukherjee have proposed another genetic basedapproach that for 119896-means clustering exchanges neighboringcluster centers [14] Fathian et al have presented a techniqueto overcome clustering problem according to honey-beesmating optimization (HBMO) [15ndash17] Shelokar et al havepresented to solve clustering problem based on the ant colonyoptimization [18] Niknam et al have combined to dominatethis problembased on the simulated annealing and ant colonyoptimization [19] Ng and Sung have introduced a techniquebased on the taboo search to find cluster center [20 21]Niknam et al have introduced a hybrid approach based oncombining partial swarm optimization and ant simulatedannealing to solve clustering problem [22 23]

The bees algorithms can be classified in two main cat-egories including foraging-based honeybee algorithms andmarriage-based honeybee algorithm Each of these categorieshave many algorithm such as artificial bee algorithm (ABC)[3 24 25] corporate artificial bee algorithm (CABC) [26]parallel artificial bee algorithm (PABC) [27] bee colonyoptimization (BCO) [28 29] bee algorithm (BA) [30]bee foraging algorithm (BFA) [31] bee swarm optimization(BSO) for first categories [32] Marriage in honey-bees opti-mization (MBO) [32] fast marriage honey-bees optimization(FMBO) [33] and finally modified fast marriage in honey-bees optimization (MFMBO) are in the second category ofbee algorithm [34]

One of the foraging-based algorithms is the bees algo-rithm that is a new population based search algorithmdeveloped by Pham et al in 2006 [30] The algorithmmimics the food foraging behavior of swarms of honeybees(Figure 3) In its basic version the algorithm performs a kindof neighborhood search combined with random search andcan be used for optimization problems [30]

Differential evolution is an evolutionary algorithm (EA)which has been widely used in to optimization problemsmainly in continuous search spaces [35] Differential evo-lution was introduced by Storn and Price in 1995 [36]Global optimization is necessary in fields such as engineeringstatistics and finance but many practical problems haveobjective functions that are nonlinear noisy noncontinuousand multidimensional or have many local minima and con-straints Such problems are difficult if not impossible to solve

analytically Differential evolution can be used to find approx-imate solutions to such problems Differential evolutionalso includes genetic algorithms evolutionary strategies andevolutionary programming Differential evolution encodessolutions as vectors and new solution compared to its parentIf the candidate is better than its parents it replaces the parentin the population Differential evolution can be applied innumerical optimization [37 38]

In this paper a hybrid evolutionary technique is used inorder to solve the 119896-means problem The proposed algorithmhelps clustering technique to escape from being trapped inlocal optimum Our algorithm takes the benefits of bothalgorithms Also in this survey some standard datasets areused for testing the proposed algorithm To obtain the bestcluster centers in proposed algorithm the advantages of BA(bees algorithm) and DE (differential evolution) are usedwith a data preprocessing technique called CCIA (clustercenter initialization algorithm) for data analysis Throughexperiments the proposed CCIA-BADE-K algorithm hasshown that this algorithm efficiently selects the exact clustercenters

The main contribution of this paper is the introductionof a novel combination of evolutionary algorithm accordingto bees algorithm and differential evolution to overcomedata analysis problem and hybrid with CCIA (cluster centerinitialization algorithm) preprocessing technique

The rest of this paper is arranged as follows in Section 2the data clustering issue is introduced In Sections 3 and4 the classic principles of the DE and BA evolutionaryalgorithm are discussed In Section 5 the suggested approachis introduced In Section 6 experimental results of proposedalgorithm are shown and compared with PSO-ANT SAACO GA ACO-SA TS HBMO PSO and 119896-means onbenchmark data and finally Section 7 presents the concludingremarks

2 Data Clustering

Clustering is defined as grouping similar objects either phys-ically or in abstract The groups inside one cluster have themost similarity with each other and the maximum diversitywith other groupsrsquo objects [39]

Definition 1 Suppose the set of 119883 = 1199091 1199092 119909

119899 contain-

ing 119899 objects The purpose of clustering is to group objects in119896 clusters as 119862 = 119888

1 1198882 119888

119896 while each cluster satisfies the

following conditions [40]

(1) 1198621cup 1198622cup sdot sdot sdot cup 119862

119896= 119883

(2) 119862119894= 0 119894 = 1 119896

(3) 119862119894cap 119862119895= 0

According to the mentioned definition the possiblemodes for clustering 119899 objects in 119896 clusters are obtained asfollows

NW (119899 119896) =1

119896

119896

sum

119894=1

(minus1)119894(119896

119894) (119896 minus 119894)

119899 (1)


(1) Input Data SET (119883 = 1199091 1199092 119909

119899) Attribute Set (119860 = 119860

1 1198602 119860



119867)


119895isin 119860)


119895)










1 1198982 119898

119896)


119894from the set of 119883 = 119909

1 1199092 119909

119899 is


10038171003817100381710038171003817119909119894minus 119898119895

10038171003817100381710038171003817lt10038171003817100381710038171003817119909119894minus 119898119901

100381710038171003817100381710038171 le 119901 le 119896 119895 = 119901 (2)


1 119898lowast

2 119898

lowast



cluster 119894)

119898lowast

119894=1

119899119894

sum

119909119895isin119862119894

1199091198951 le 119894 le 119896 (3)








(a) Dance floor

120572

(b) Dance languages




Fitness (119862) =119896

sum

119894=1

sum

119909119895isin119862119894

10038171003817100381710038171003817119909119895minus 119898lowast

119894

10038171003817100381710038171003817 (4)





120572

120572






Waggle dance

Waggle dance


Food source

Food source

Food source

Food source

1205721

1205722

1205723

12057241205725















(3) 1198950is added to set 119869




(5) End








Nei

ghbo

rhoo

dse

arch

Repe

at o

ptim

izat

ion

cycle

s for

the s

peci

fied

num

ber






119885119894(119905) = [119885

1198941(119905) 119885

1198942(119905) 119885

119894119863(119905)] (5)







1199031= 1199032= 1199033isin 1 2 119863 (6)



119884119894119895(119905) = 119885

1199031119895(119905) + 119865 (119885

1199032119895(119905) minus 119885

1199033119895(119905)) (7)




119877119894119895(119905)

=

119884119894119895(119905) If rand

119895119894le 119875cr or 119895 = 119895

0119894 = 1 2 119873

119885119894119895(119905) If rand

119895119894gt 119875cr and 119895 = 119895

0119895 = 1 2 119863

(8)



119894(119905) and 119885

119894(119905)


119885119894(119905 + 1)

=

119877119894(119905) If 119891 (119877

119894(119905)) le 119891 (119885

119894(119905)) 119894 = 1 2 119873

119885119894(119905) otherwise

(9)


1199032119895(119905) minus 119885

1199033119895(119905) multiplied 119865










Zr1(t) + F(di)






Crossover

Selection



119865 = 05 (Rnd + 1) (10)















End









Population =

[[[[[[[

[

center1198791

center1198792


]]]]]]]

]

(11)


center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)





Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1) Input Data SET (119883 = 1199091 1199092 119909

119899) Attribute Set (119860 = 119860

1 1198602 119860



119867)


119895isin 119860)


119895)










1 1198982 119898

119896)


119894from the set of 119883 = 119909

1 1199092 119909

119899 is


10038171003817100381710038171003817119909119894minus 119898119895

10038171003817100381710038171003817lt10038171003817100381710038171003817119909119894minus 119898119901

100381710038171003817100381710038171 le 119901 le 119896 119895 = 119901 (2)


1 119898lowast

2 119898

lowast



cluster 119894)

119898lowast

119894=1

119899119894

sum

119909119895isin119862119894

1199091198951 le 119894 le 119896 (3)








(a) Dance floor

120572

(b) Dance languages




Fitness (119862) =119896

sum

119894=1

sum

119909119895isin119862119894

10038171003817100381710038171003817119909119895minus 119898lowast

119894

10038171003817100381710038171003817 (4)





120572

120572






Waggle dance

Waggle dance


Food source

Food source

Food source

Food source

1205721

1205722

1205723

12057241205725















(3) 1198950is added to set 119869




(5) End








Nei

ghbo

rhoo

dse

arch

Repe

at o

ptim

izat

ion

cycle

s for

the s

peci

fied

num

ber






119885119894(119905) = [119885

1198941(119905) 119885

1198942(119905) 119885

119894119863(119905)] (5)







1199031= 1199032= 1199033isin 1 2 119863 (6)



119884119894119895(119905) = 119885

1199031119895(119905) + 119865 (119885

1199032119895(119905) minus 119885

1199033119895(119905)) (7)




119877119894119895(119905)

=

119884119894119895(119905) If rand

119895119894le 119875cr or 119895 = 119895

0119894 = 1 2 119873

119885119894119895(119905) If rand

119895119894gt 119875cr and 119895 = 119895

0119895 = 1 2 119863

(8)



119894(119905) and 119885

119894(119905)


119885119894(119905 + 1)

=

119877119894(119905) If 119891 (119877

119894(119905)) le 119891 (119885

119894(119905)) 119894 = 1 2 119873

119885119894(119905) otherwise

(9)


1199032119895(119905) minus 119885

1199033119895(119905) multiplied 119865










Zr1(t) + F(di)






Crossover

Selection



119865 = 05 (Rnd + 1) (10)















End









Population =

[[[[[[[

[

center1198791

center1198792


]]]]]]]

]

(11)


center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)





Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) Dance floor

120572

(b) Dance languages




Fitness (119862) =119896

sum

119894=1

sum

119909119895isin119862119894

10038171003817100381710038171003817119909119895minus 119898lowast

119894

10038171003817100381710038171003817 (4)





120572

120572






Waggle dance

Waggle dance


Food source

Food source

Food source

Food source

1205721

1205722

1205723

12057241205725















(3) 1198950is added to set 119869




(5) End








Nei

ghbo

rhoo

dse

arch

Repe

at o

ptim

izat

ion

cycle

s for

the s

peci

fied

num

ber






119885119894(119905) = [119885

1198941(119905) 119885

1198942(119905) 119885

119894119863(119905)] (5)







1199031= 1199032= 1199033isin 1 2 119863 (6)



119884119894119895(119905) = 119885

1199031119895(119905) + 119865 (119885

1199032119895(119905) minus 119885

1199033119895(119905)) (7)




119877119894119895(119905)

=

119884119894119895(119905) If rand

119895119894le 119875cr or 119895 = 119895

0119894 = 1 2 119873

119885119894119895(119905) If rand

119895119894gt 119875cr and 119895 = 119895

0119895 = 1 2 119863

(8)



119894(119905) and 119885

119894(119905)


119885119894(119905 + 1)

=

119877119894(119905) If 119891 (119877

119894(119905)) le 119891 (119885

119894(119905)) 119894 = 1 2 119873

119885119894(119905) otherwise

(9)


1199032119895(119905) minus 119885

1199033119895(119905) multiplied 119865










Zr1(t) + F(di)






Crossover

Selection



119865 = 05 (Rnd + 1) (10)















End









Population =

[[[[[[[

[

center1198791

center1198792


]]]]]]]

]

(11)


center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)





Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Waggle dance

Waggle dance


Food source

Food source

Food source

Food source

1205721

1205722

1205723

12057241205725















(3) 1198950is added to set 119869




(5) End








Nei

ghbo

rhoo

dse

arch

Repe

at o

ptim

izat

ion

cycle

s for

the s

peci

fied

num

ber






119885119894(119905) = [119885

1198941(119905) 119885

1198942(119905) 119885

119894119863(119905)] (5)







1199031= 1199032= 1199033isin 1 2 119863 (6)



119884119894119895(119905) = 119885

1199031119895(119905) + 119865 (119885

1199032119895(119905) minus 119885

1199033119895(119905)) (7)




119877119894119895(119905)

=

119884119894119895(119905) If rand

119895119894le 119875cr or 119895 = 119895

0119894 = 1 2 119873

119885119894119895(119905) If rand

119895119894gt 119875cr and 119895 = 119895

0119895 = 1 2 119863

(8)



119894(119905) and 119885

119894(119905)


119885119894(119905 + 1)

=

119877119894(119905) If 119891 (119877

119894(119905)) le 119891 (119885

119894(119905)) 119894 = 1 2 119873

119885119894(119905) otherwise

(9)


1199032119895(119905) minus 119885

1199033119895(119905) multiplied 119865










Zr1(t) + F(di)






Crossover

Selection



119865 = 05 (Rnd + 1) (10)















End









Population =

[[[[[[[

[

center1198791

center1198792


]]]]]]]

]

(11)


center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)





Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(3) 1198950is added to set 119869




(5) End








Nei

ghbo

rhoo

dse

arch

Repe

at o

ptim

izat

ion

cycle

s for

the s

peci

fied

num

ber






119885119894(119905) = [119885

1198941(119905) 119885

1198942(119905) 119885

119894119863(119905)] (5)







1199031= 1199032= 1199033isin 1 2 119863 (6)



119884119894119895(119905) = 119885

1199031119895(119905) + 119865 (119885

1199032119895(119905) minus 119885

1199033119895(119905)) (7)




119877119894119895(119905)

=

119884119894119895(119905) If rand

119895119894le 119875cr or 119895 = 119895

0119894 = 1 2 119873

119885119894119895(119905) If rand

119895119894gt 119875cr and 119895 = 119895

0119895 = 1 2 119863

(8)



119894(119905) and 119885

119894(119905)


119885119894(119905 + 1)

=

119877119894(119905) If 119891 (119877

119894(119905)) le 119891 (119885

119894(119905)) 119894 = 1 2 119873

119885119894(119905) otherwise

(9)


1199032119895(119905) minus 119885

1199033119895(119905) multiplied 119865










Zr1(t) + F(di)






Crossover

Selection



119865 = 05 (Rnd + 1) (10)















End









Population =

[[[[[[[

[

center1198791

center1198792


]]]]]]]

]

(11)


center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)





Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Zr1(t) + F(di)






Crossover

Selection



119865 = 05 (Rnd + 1) (10)















End









Population =

[[[[[[[

[

center1198791

center1198792


]]]]]]]

]

(11)


center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)





Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















End









Population =

[[[[[[[

[

center1198791

center1198792


]]]]]]]

]

(11)


center119894= [1198621 1198622 119862

119896] 119894 = 1 2 119899Scout

119862119895= [1198881 1198882 119888

119901] 119895 = 1 2 119896

119862min119895lt 119862119895lt 119862

max119895

(12)





Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Load dataset



YesNo





eigh

borh

ood

sear

ch


Select target site

No

NoThe convergence


Yes


No

i = i + 1


i = 1








and 119888max119894







V119894119866+1

= 1199091199031119866

+119882(1199091199032119866

minus 1199091199033119866)

1199031= 1199032= 1199033isin 1 2 119873

(13)



minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus4

minus2

0

2

4

6

X1

X2



minus4

minus2

0

2

4

6

X1

X2



where 1199091199031119866


1199091199033119866




119894119866+1is





119906119895119894119866+1

=

V119895119894119866+1


119909119895119894119866


(14)



= 119909119894119866



119906119894119866+1


119909119894119866+1

=

119906119894119866+1

If 119891 (119906119894119866+1

) le 119891 (119909119894119866) 119894 = 1 2 119873

119909119894119866

otherwise(15)


119894(119866) To


119894(119866) then

replace 119909119894(119866)with 119906

119894(119866) otherwise 119909


119894(119866)

are discarded



Datasetname

Dataset attribute


Attributenumber




7 Evaluation







Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Result CPUTime (S)



















minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3


minus4

minus2

0

2

4

6

minus6minus4

minus20

24

6

minus8

minus4minus2

02

4

X1

X2

X3



4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10







Art1(120583 = (

119898119894

119898119894

) Σ = [05 005

005 05]) 119894 = 1 2 3 4

1198981= minus4 119898

2= minus1 119898

3= 2 119898

4= 5

(16)








4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














4 5 6 7 82

3

4

5

4 5 6 7 80

5

10

4 5 6 7 80

1

2

3

4

6

8

0

5

10

2 25 3 35 4 452 25 3 35 4 452 25 3 35 4 450

1

2

3

4

6

8

2

3

4

5

0 2 4 6 80 2 4 6 80 2 4 6 80

1

2

3

0 05 1 15 2 25 0 05 1 15 2 25 0 05 1 15 2 254

6

8

2

3

4

5

0

5

10



Art2(120583 = (

119898119894

minus119898119894

119898119894

)Σ =[[

[

05 005 005

005 05 005

005 005 05

]]

]

)

119894 = 1 2 3 4 1198981= minus3 119898

2= 0 119898

3= 3 119898

4= 6

(17)











0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 10090

100

110

120

130

140

150

Iterations

Best

cost




16

165

17

175

18

185

19

195times104

0 20 40 60 80 100Iterations

Best

cost


5500

6000

6500

7000

7500


0 20 40 60 80 100Iterations

Best

cost

Best costMean


200

250

300

350

400

450


0 20 40 60 80 100Iterations

Best

cost

Best costMean










0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

100

200

300

400

500

600

700

800

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250


0

500

1000

1500

2000

2500

0 50 100 150 200 250






Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an effective hybrid of bees...

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