Research ArticleDesign of Large Thinned Arrays Using DifferentBiogeography-Based Optimization Migration Models

Sotirios K Goudos1 and John N Sahalos2

1Department of Physics Aristotle University of Thessaloniki Thessaloniki Greece2Department of Electrical and Computer Engineering University of Nicosia Nicosia Cyprus

Correspondence should be addressed to Sotirios K Goudos sgoudophysicsauthgr

Received 6 May 2016 Revised 29 July 2016 Accepted 10 August 2016

Academic Editor Ikmo Park

Copyright copy 2016 S K Goudos and J N Sahalos 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

Array thinning is a common discrete-valued combinatorial optimization problem Evolutionary algorithms are suitable techniquesfor above-mentioned problem Biogeography-Based Optimization (BBO) which is inspired by the science of biogeography is astochastic population-based evolutionary algorithm (EA)The original BBO uses a linear migration model to describe how speciesmigrate from one island to another Other nonlinear migration models have been proposed in the literature In this paper we applyBBO with four different migration models to five different large array design cases Additionally we compare results with five otherpopular algorithms The problems dimensions range from 150 to 300 The results show that BBO with sinusoidal migration modelgenerally performs better than the other algorithms However these results are considered to be indicative and do not generallyapply to all optimization problems in antenna design

1 Introduction

Array thinning is a common discrete-valued combinato-rial optimization problem By array thinning we mean theremoval (turning ldquooffrdquo) of some radiating elements from aperiodic array antenna in order to create an array with lowersidelobe level than that with uniform excitationThe elementsconnected to the feed network are turned ldquoonrdquo while theturned ldquooffrdquo elements are connected to a matched loadArray thinning results in reduction in cost and weight Anexhaustive search for all possible combinations would resultin the best array design however the computational cost willincrease exponentially as the array size increases Thereforearray thinning can be categorized as discrete combinatorialN-P complete optimization problem [1] EAs are suitabletechniques for solving the array thinning problem Theproblemof array thinning has been addressed in the literatureusing different EAs like genetic algorithms (GAs) [2] ParticleSwarm Optimization (PSO) [3] and Ant Colony Optimiza-tion (ACO) [4] The original PSO is inherently real-valuedand operates only in continuous spaces In order to solvebinary-coded combinatorial optimization problems with

PSO several binary versions of this algorithm have beenproposed Binary PSO (BPSO) proposed by Kennedy andEberhart extends the original PSO algorithm using a sigmoidfunction (or S-shaped transfer function) tomap real numbersto bits [5] In [6] a new BPSO that uses V-shaped transferfunction is proposed (VBPSO) The results from [6] showthat VBPSO outperforms the original BPSO and other BPSOvariants with other transfer functions Differential evolution(DE) [7 8] is a population-based stochastic global optimiza-tion algorithm which has been used in several real worldengineering problems Several DE variants or strategies existOppositional differential evolution (ODE) [9] is a DE variantbased on opposition-based learning (OBL) [10] conceptsHarmony Search (HS) [11] is an evolutionary algorithmwhich is inspired by the way that musicians experiment andchange the pitches of their instruments to improvise betterharmonies HS has been applied successfully to antenna arraysynthesis problems [12 13] Ant Colony Optimization (ACO)is a population-based metaheuristic introduced by Dorigoet al [14] and inspired by the behavior of real ants Theauthors in [4 15] have used ACO for thinned array design

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 5359298 11 pageshttpdxdoiorg10115520165359298

2 International Journal of Antennas and Propagation

Biogeography-Based Optimization (BBO) [16] is an EAbased on mathematical models that describe how speciesmigrate from one island to another how new species ariseand how species become extinct The way the problemsolution is found is analogous to naturersquos way of distributingspecies Ma in [17] showed that sinusoidal migration modelsgenerally outperform linear migration models like the onein original BBO algorithm The authors in [18] extend themigration model performance analysis and they propose twononlinear migration models which they call model 7 andmodel 8 BBO has been applied to design problems in elec-tromagnetics including Yagi-Uda synthesis [19] microstripantenna design [20 21] and antenna array synthesis [22ndash24]

The purpose of this paper is to use BBO with differentmigration models in order to design large thinned arraysTo the best of our knowledge this is the first time that BBOwith other than the linear migration mode is applied to anantenna design problem This paper contributes to compareperformance of several state-of-the-art EAs to high dimen-sional thinned array design problems Additionally the papercontributes so that the design cases presented in this papercould also be used as a framework of benchmark functionsfor testing evolutionary algorithms to large array designproblems More specifically we compare the original BBOlinearmigrationmodel with the sinusoidalmodel [17] model7 and model 8 [18] two BPSO variants binary HS ODE andACOWe apply the algorithms to five different design cases oflinear and planar arrays A comparative study of BBOvariantsperformance on benchmark functions is also given More-over a study of the influence on the BBO performance ofthe boundary constraint handling method is also presentedTo the best of our knowledge this is the first time that suchstudy is carried out on BBO in general Numerical resultsshow that BBO with sinusoidal migration model generallyperforms better than or equally with the other migrationmodels Additionally it performs better than the original BBOalgorithm in terms of solution accuracy Additionally resultsshow VBPSO outperforms the initial BPSO algorithm how-ever its performance is inferior to that of the BBO algorithms

This paper is organized as follows We describe theproblem formulation in Section 2 A brief description of theEAs used in this paper is given in Section 3 In Section 4 wepresent the numerical results Finally the conclusion is givenin Section 5

2 Formulation

We consider an119873-element linear array of isotropic elementsThe array factor is expressed as

AF (119906 119868) =119873

sum

119899=1

119868119899119890119895(2120587120582)119899119889119906

(1)

where 120582 is the wavelength 119868119899is the complex excitation of

the 119899th element 119868 is the corresponding vector of elementamplitudes 119889 is distance between two adjacent elements119906 = sin(120599) is the direction cosine and 120599 is the steering anglemeasured from broadside of the array In the thinned arraycase 119868

119899could be only 1 (turned ldquoonrdquo) or 0 (turned ldquooffrdquo)

For a symmetrically excited array (1) becomes

AF (119906 119868) = 2

119872

sum

119899=1

119868119899cos [2120587

120582119889119899119906] + 119868

0 (2)

where 119872 = lfloor1198732rfloor is the largest integer less than or equal to1198732 For an even number of elements 119868

0= 0 and for an odd

number of elements we set 1198680= 1 We assume that 119889 = 05120582

for all cases The fill factor percentage is the percentage ofthe ratio of the turned ldquoonrdquo elements to the total numberof elements The optimization goal is the peak sidelobelevel (SLL) suppression by finding the optimum elementamplitudes This design problem is therefore defined by theminimization of the objective function

1198651(119906 119868) = max

119906isin119878119906

AFdB (119906 119868) (3)

where 119878119906is the set of direction cosines that are outside the

angular range of the main lobeAdditionally we study the effect of maintaining the same

aperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon)

To test the algorithms ability to design planar thinnedarrays we consider a 2119873times 2119872 planar array which lies on the119909-119910 plane The array factor of such an array is given by

AF (119906 V 119868) = 4

119872

sum

119898=1

119873

sum

119899=1

119868119899119898

cos [120587 (2119898 minus 1) 119889119910119906]

sdot cos [120587 (2119899 minus 1) 119889119909V]

(4)

where 119889119909 119889119910are the distance between two adjunct elements

in the 119909 and 119910 direction respectively and 119906 = sin 120599 cos120601V = sin 120599 sin120601 are the direction cosines

The optimization goal here is the sidelobe level (SLL)suppression at two different phi-planes (120601 = 0

∘ 120601 = 90∘)

for a desired fill factor percentage This design problem isexpressed by the minimization of the objective function

1198652(119868) = max PSLL120601=0

∘

dB PSLL120601=90∘

dB (5)

where PSLL120601=0∘

dB PSLL120601=90∘

dB are the calculated peak sidelobelevels at 120601 = 0

∘ and 120601 = 90∘ planes respectively

3 Optimization Algorithms

31 Binary PSO The binary PSO (BPSO) model was pre-sented byKennedy andEberhart and is based on a very simplemodification of the real-valued PSO [5] In binary PSO theparticle positions belong to a119863-dimensional binary space (bitstrings of length119863) while the particle velocities remain real-valued

The 119899th coordinate of each particlersquos position is a bitwhose state is given by

119909119866+1119899119894

=

1 if rand[01]

lt 119879 (119906119866+1119899119894

)

0 otherwise(6)

International Journal of Antennas and Propagation 3

where rand[01]

is a uniformly distributed random number in[0 1] and119879(119909) is a sigmoid limiting transformation thatmapsreal numbers to the interval [0 1] Such a function is definedby

119879 (119909) =1

1 + 119890minus119909(7)

which defines S-shaped transfer functionIn [6] new BPSO variants that use V-shaped transfer

functions are proposed (VBPSO) The BPSO variant thatproduces the best results according to [6] is that with atransfer function expressed by

119879 (119909) =

1003816100381610038161003816100381610038161003816

2

120587arctan(120587

2119909)

1003816100381610038161003816100381610038161003816 (8)

32 Binary Harmony Search Algorithm Harmony Search(HS) [11] is an evolutionary algorithm which is inspired bythe way that musicians experiment and change the pitchesof their instruments to improvise better harmonies Themain control parameter in HS is the Harmony Memory Size(HMS) which determines the number of solutions (harmo-nies) inside the HM and it is equivalent to population size inanother algorithms Another HS parameter is the HarmonyMemory Consideration Rate (HMCR) which determineswhether pitches (decision variables) should be selected fromthe HM or randomly from the predefined range HMCR isa number between 0 and 1 An additional control parameterof HS is the Pitch Adjusting Rate (PAR) which determinesthe probability of adjusting the original value of the selectedpitches from the HM

33 Oppositional Differential Evolution Differential evolu-tion (DE) [7 8] is a population-based stochastic global opti-mization algorithmwhich has been used in several real worldengineering problems Several DE variants or strategies existOne of the DE advantages is that very few control parametershave to be adjusted in each algorithm run In [9] a new DEalgorithm based on opposition-based learning (OBL) [10]oppositional differential evolution (ODE) was introducedThe basic idea of OBL is not only to calculate the fitness of thecurrent individual but also to calculate the fitness of the oppo-site individualThen the algorithm selects the individual withthe lower (higher) fitness value The benefits of using such atechnique are that convergence speedmay be faster and betterapproximation of the global optimum can be found ODEuses an additional control parameter to those of standard DEcalled the jumping rate 119895

119903isin [0 1] which controls in each

generation if the opposite population is created or not

34 Ant Colony Optimization Ant Colony Optimization(ACO) [14 25 26] is a meta-heuristic inspired by theantsrsquo foraging behavior At the core of this behavior isthe indirect communication between the ants by means ofchemical pheromone trails which enables them to find shortpaths between their nest and food sources Ants can sensepheromone When they decide to follow a path they tend tochoose the ones with strong pheromone intensities way backto the nest or to the food source Therefore shorter paths

would accumulate more pheromone than longer ones Thisfeature of real ant colonies is exploited in ACO algorithms inorder to solve combinatorial optimization problems consid-ered to be NP-Hard

35 Biogeography-Based Optimization The mathematicalmodels of biogeography are based on the work of RobertMacArthur and Edward Wilson in the early 1960s Usingthis model it was possible to predict the number of speciesin a habitat The habitat is an area that is geographicallyisolated from other habitats The geographical areas that arewell suited as residences for biological species are said tohave a high habitat suitability index (HSI) Therefore everyhabitat is characterized by the HSI which depends on factorslike rainfall diversity of vegetation diversity of topographicfeatures land area and temperature Each of the featuresthat characterize habitability is known as suitability indexvariables (SIVs) The SIVs are independent variables whileHSI is the dependent variable

Therefore a solution to a 119863-dimensional problem canbe represented as a vector of SIV variables [SIV

1 SIV2

SIV119863] which is a habitat or island The value of HSI of a

habitat is the value of the objective function that correspondsto that solution and it is found by

HSI = 119865 (habitat) = 119865 (SIV1 SIV2 SIV

119863) (9)

Habitats with a high HSI are good solutions of theobjective function while poor solutions are those habitatswith a low HSI The immigration and emigration rates arefunctions of the rank of the given candidate solution Therank of the given candidate solution represents the numberof species in a habitat These are given by the following

(1) Linear migration model (original BBO)

120583119896= 119864(

119896

119878max)

120582119896= 119868(1 minus

119896

119878max)

(10)

(2) Sinusoidal migration model [17]

120583119896=119864

2(minus cos( 119896120587

119878max) + 1)

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(11)

(3) Model 7 [18]

120583119896= 119864(

119896

119878max)

4

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(12)

4 International Journal of Antennas and Propagation

(4) Model 8 [18]

120583119896= 119864(

119896

119878max)

16

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(13)

119868 is the maximum possible immigration rate 119864 is themaximum possible emigration rate 119896 is the rank of the givencandidate solution and 119878max is the maximum number ofspecies (eg population size)The rank of the given candidatesolution or the number of species is obtained by sortingthe solutions from most fit to least fit according to the HSIvalue (eg fitness) BBO uses both mutation and migrationoperators The application of these operators to each SIV ineach solution is decided probabilisticallyThemutation rate119898of a possible solution 119878 is defined to be inversely proportionalto the solution probability and it is given by

119898(119878) = 119898max (1 minus 119875119904

119875max) (14)

where119875119904is the probability that a habitat contains 119878 species and

119898max is a user-defined parameter As with other evolutionaryalgorithms BBO also incorporates elitism This is imple-mented with a user-selected elitism parameter 119901 This meansthat the 119901 best phase vectors remain from one generation tothe other The BBO algorithm is outlined below

(1) Initialize the BBO control parameters Map the prob-lem solutions to habitats (vectors) Set the habitatmodification probability 119875mod the maximum immi-gration rate 119868 the maximum emigration rate 119864the maximum migration rate 119898max and the elitismparameter 119901 (if elitism is desired)

(2) Initialize a random population of119873119875habitats from a

uniform distribution Set the number of generations119866 to one

(3) Evaluate objective function values for each antennaarray of the population

(4) Map the objective function value to the number ofspecies 119878 the immigration rate 120582

119896 and the emigration

rate 120583119896for each solution (antenna array) of the

population(5) Apply themigration operator for each nonelite habitat

based on immigration and emigration rates with theone of the linear sinusoidal or nonlinear migrationmodels

(6) Update the species count probability using

119904

=

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

119878 = 0

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

+ 120582119904minus1

119875119904minus1

1 le 119878 le 119878max minus 1

minus (120582119904+ 120583119904) 119875119904+ 120582119904minus1

119875119904minus1

119878 = 119878max

(15)

(7) Apply the mutation operator according to (14)

(8) Sort the population according to the objective func-tion value from best to worst

(9) Apply elitism by replacing the 119901 worst habitats of theprevious generation with the 119901 best ones

(10) Repeat step (3) until the maximum number of gener-ations 119866max is reached

4 Numerical Results

41 Comparison with Test Functions In order to evaluate thedifferent BBO migration models first we use a set of testproblems from the literature In this paper the test problemsconsist of eight well-known benchmark functions We havechosen two unimodal and six multimodal functions Theseare expressed as follows [27 28]

(A) Unimodal Functions

(1) Sphere Function

1198911 (119909) =

119863minus1

sum

119895=0

1199092

119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 100 119891

1 (0 0 0) = 0 (16)

(2) Rosenbrockrsquos Function

1198912 (119909) =

119863minus2

sum

119895=0

[100 (119909119895+1

minus 1199092

119895)2

+ (119909119895minus 1)2

]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 2048 119891

2 (1 1 1) = 0

(17)

(B) Multimodal Functions

(3) Ackleyrsquos Function

1198913 (119909) = minus20 exp(minus

1

5radic

1

119863

119863minus1

sum

119895=0

1199092

119895)

minus exp( 1

119863

119863minus1

sum

119895=0

cos (2120587119909119895)) + 20 + 119890

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 32 119891

3 (0 0 0) = 0

(18)

(4) Griewanksrsquos Function

1198914 (119909) =

119863minus1

sum

119895=0

1199092

119895

4000minus

119863minus1

prod

119895=0

cos(119909119895

radic119895)

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 600 119891

4 (0 0 0) = 0

(19)

International Journal of Antennas and Propagation 5

Table 1 Comparative results for the test functions The smaller values are in bold

Migration model Test function1198911(119909) 119891

2(119909) 119891

3(119909) 119891

4(119909) 119891

5(119909) 119891

6(119909) 119891

7(119909) 119891

8(119909)

Linear 0 plusmn 0 50592 plusmn 26042 0133 plusmn 0084 0744 plusmn 0295 0 plusmn 0 0012 plusmn 0033 0200 plusmn 0400 3934 plusmn 2437Sinusoidal 0861 plusmn 0552 29844 plusmn 16559 0091 plusmn 0090 0725 plusmn 0205 0 plusmn 0 0003 plusmn 0009 0003 plusmn 0010 2561 plusmn 1894Model 7 0893 plusmn 0503 42243 plusmn 23893 0201 plusmn 0092 0673 plusmn 0142 0 plusmn 0 0018 plusmn 0023 0 plusmn 0 4144 plusmn 3704Model 8 0767 plusmn 0898 70062 plusmn 33001 0131 plusmn 0081 0701 plusmn 0250 0 plusmn 0 0004 plusmn 0014 0 plusmn 0 3209 plusmn 2088

(5) Weierstrass Function

1198915 (119909) =

119863minus1

sum

119895=0

(

119896max

sum

119896=0

[119886119896 cos (2120587119887119896 (119909

119895+ 05))])

minus 119863(

119896max

sum

119896=0

[119886119896 cos (120587119887119896)])

119886 = 05 119887 = 3 119896max = 2010038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 05 119891

5 (0 0 0) = 0

(20)

(6) Rastriginrsquos Function

1198916 (119909) =

119863minus1

sum

119895=0

[1199092

119895minus 10 cos (2120587119909

119895) + 10]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 512 119891

6 (0 0 0) = 0

(21)

(7) Noncontinuous Rastriginrsquos Function

1198917 (119909) =

119863minus1

sum

119895=0

[1199102

119895minus 10 cos (2120587119910

119895) + 10]

119910119895=

119909119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816lt 05

round (2119909119895)

2

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816ge 05

for 119895 = 0 1 119863 minus 1 1198917 (0 0 0) = 0

(22)

(8) Schwefelrsquos Function

1198918 (119909) = 4189829 times 119863 minus

119863minus1

sum

119895=0

[119909119895sin(10038161003816100381610038161003816119909119895

10038161003816100381610038161003816

12

)]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 500 119891

8 (42096 42096 42096) = 0

(23)

All algorithms are executed 50 times The results are com-pared The population size is set to 100 and the maximumnumber of generations is set to 2000 iterationsThe problemsdimension is set to 119863 = 30 For all migration models thehabitat modification probability 119875mod is set to 1 and themaximum mutation rate 119898max is set equal to 0005 Themaximum immigration rate 119868 and the maximum emigrationrate119864 are both set to oneThe elitismparameter119901 is set to twoTable 1 reports the comparative results in terms of mean and

Table 2 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 294Sinusoidal 194Model 7 288Model 8 225

standard deviation values We notice that BBOwith the sinu-soidal migration model performs better in 5 out of the 8 testfunctions Moreover in order to compare the algorithms per-formance on all problems we have conducted the Friedmantest [29] Table 2 shows the average ranking of all algorithmsThe highest ranking is shown in bold It is obvious that thebest average rankingwas obtained by theBBOwith sinusoidalmigration model which outperforms the other models

42 Comparison with ArrayThinning Problems We comparethe four BBO migration modelsrsquo performance with the twoBPSO variants BHS ODE and ACO on different thinnedarray design cases All algorithms are executed 20 times Theresults are compared The population size is set to 200 andthe maximum number of generations is set to 1000 iterationsFor all migrationmodels the habitatmodification probability119875mod is set to 1 and the maximum mutation rate 119898max isset equal to 0005 The maximum immigration rate 119868 and themaximum emigration rate 119864 are both set both to one Theelitism parameter 119901 is set to two In both BPSO algorithmsthe learning factors 119888

1 1198882are both set equal to two and inertia

weight 119908 is linearly decreased from 09 to 04 as in [6] ForBHS the HMCR is set to 099 and the PAR is set to 04 ForODE 119865 = 05 CR = 09 and the jumping rate is set to 03For ACO the initial pheromone value 120591

0is set to 10119890 minus 6 the

pheromone update constant 119876 is set to 20 the explorationconstant 119902

0is set to 1 the global pheromone decay rate 120588

119892is

09 the local pheromone decay rate 120588119897is 05 the pheromone

sensitivity 120572 is 1 and the visibility sensitivity is 120573 is 5The first case is that of a symmetrically excited 300-

element linear array The total number of unknowns forthis case is 150 Table 3 reports the comparative results Wenotice that the sinusoidal model and the model 7 seem tooutperform the other algorithms Bothmodels have obtainedthe same best value Both BPSO variants perform worse thanthe BBOalgorithmsACOandVBPSOperform similarlyTheconvergence rate graph for this case is depicted in Figure 1 AllBBOmodels seem to converge at similar speed faster than theother algorithms The VBPSO converges faster than BPSOIt must be pointed out that the BBO algorithms converge

6 International Journal of Antennas and Propagation

Table 3 Comparative results for 300-element symmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2454 minus2424 minus2443 0096Sinusoidal minus2467 minus2417 minus2441 0146Model 7 minus2467 minus2427 minus2443 0110Model 8 minus2444 minus2351 minus2411 0243VBPSO minus2432 minus2367 minus2405 0175BPSO minus2110 minus1988 minus2035 0326BHS minus2410 minus2279 minus2361 0431ODE minus2406 minus2330 minus2376 0203ACO minus2420 minus2393 minus2406 0084

Table 4 Comparative results for 300-element symmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2445 minus2406 minus2432 0112Sinusoidal minus2459 minus2430 minus2445 0087Model 7 minus2470 minus2427 minus2441 0127Model 8 minus2456 minus2364 minus2418 0257VBPSO minus2441 minus2397 minus2422 0141BPSO minus2150 minus2052 minus2095 0281BHS minus2416 minus2290 minus2379 0413ODE minus2430 minus2351 minus2391 0231ACO minus2336 minus2276 minus2316 0171

at their final value at less than 300 iterations The radiationpattern of the best obtained result is shown in Figure 2 Thebest array found is filled 72 and has a peak SLL ofminus2467 dBAdditionally we study the effect of maintaining the sameaperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon) for a design case The comparative results for this caseare shown in Table 4 The sinusoidal model presents the bestperformance except the best value found where the model 7outperforms the others VBSO is completive with the BBOalgorithms for this case and achieves better performance thanBBOwithmodel 8 ODE performs better than ACO and BHSfor this case Figure 3 shows the convergence rate for thiscase It is obvious that all BBO algorithms converge at similarspeeds while the PSO algorithms converge slower The bestobtained radiation pattern is shown in Figure 4The obtainedpeak SLL value is minus2470 dB slightly smaller than the previouscaseThe array filling percentage is again 72We notice thatthere is not a significant difference if we choose an array with-out constraints or with the same aperture size for this case

Next we consider asymmetric array designs in which theproblem dimension is 300 Therefore we evaluate the algo-rithms ability to solve high dimensional problems It mustbe pointed out that although the number of unknowns hasdoubled compared with the symmetric case the populationsize and iterations remain the same as previously Thereforethe level of difficulty increases for all algorithms Table 5reports the comparative results for this case The algorithms

minus16

minus18

minus20

minus22

minus24

Number of iterations

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Avg

cost

func

tion

10009008007006005004003002001000

Figure 1 Convergence rate graph for the 300-element symmetricthinned array case

Nor

mal

ized

far fi

eld

(dB)

111111111111111111111111111111111111111111111111111111111111111111101111111101011101011111011010001001100111101001110000000000100001010110010000111001

Filled 72

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

Symmetric linear arrayN = 300 elements

Max SLL = minus2467dB

1080604020minus02minus04minus06minus08minus1

Figure 2 Radiation pattern of the best obtained array for the 300-element symmetric thinned array

performance differences are clearer in this case It is obviousthat the sinusoidal model clearly outperforms the others inthis case The results obtained by model 7 and linear modelare quite similar in this case The VBPSO results are betteragain than BPSO BHS ODE and ACO but worse than theBBO algorithms Figure 5 shows the convergence rate graphfor this case Model 7 seems to convergence slightly fasterthan the other algorithms The radiation pattern of the bestobtained array is shown in Figure 6 The obtained array SLLis minus2611 dB and the fill percentage is about 72

In order to further evaluate the algorithms performancewe choose again an asymmetric array case with the sameaperture length Table 6 holds the comparative results Theresults show that sinusoidal model clearly outperforms theother algorithms We notice that VBPSO and BPSO resultsare worse than the BBO algorithms BHS is completive withthe VBPSO algorithm for this case The convergence rategraph of Figure 7 shows that models 7 and 8 converge slightly

International Journal of Antennas and Propagation 7

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

Avg

cost

func

tion

Figure 3 Convergence rate graph for the 300-element symmetricthinned array case with same aperture size

111111111111111111111111111111111111111111111111111111111111111111111011111111001111011111000100110111001001110000101010010001000000001000101001101011

Same aperture length

Nor

mal

ized

far fi

eld

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

1080604020minus02minus04minus06minus08minus1

Filled 72

Symmetric linear arrayN = 300 elements

Max SLL = minus2470dB

Figure 4 Radiation pattern of the best obtained array for the 300-element symmetric thinned array with same aperture length

Table 5 Comparative results for 300-element asymmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2603 minus2565 minus2583 0119Sinusoidal minus2611 minus2573 minus2588 0110Model 7 minus2605 minus2558 minus2582 0146Model 8 minus2557 minus2421 minus2513 0362VBPSO minus2498 minus2349 minus2436 0468BPSO minus2083 minus1998 minus2032 0228BHS minus2532 minus2463 minus2502 0214ODE minus2488 minus2351 minus2419 0424ACO minus2210 minus2155 minus2183 0187

faster than the linear and the sinusoidal model for this caseAgain it is obvious that the BBO algorithms require feweriterations than the other algorithms in order to reach the

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 5 Convergence rate graph for the 300-element asymmetricthinned array

Nor

mal

ized

far fi

eld

(dB)

111110110000010100010000000011001100110100101001010011011100001110100011111101101111011111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111101111111111011101111101101111101001010100101010011100110000010101010100000100000111

Asymmetric linear array

Filled 727

0

minus10

minus20

minus30

minus401080604020minus02minus04minus06minus08minus1

u

N = 300 elements

Max SLL = minus2611dB

Figure 6 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array

Table 6 Comparative results for 300-element asymmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2597 minus2558 minus2578 0113Sinusoidal minus2608 minus2577 minus2591 0107Model 7 minus2587 minus2560 minus2576 0081Model 8 minus2567 minus2446 minus2507 0383VBPSO minus2490 minus2254 minus2392 0658BPSO minus2080 minus1987 minus2021 0276BHS minus2558 minus2392 minus2476 0633ODE minus2492 minus2344 minus2426 0522ACO minus2191 minus2163 minus2177 0008

final values Figure 8 presents the radiation pattern of thebest obtained arrayThe peak SLL is minus2608 dB and the filling

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

Biogeography-Based Optimization (BBO) [16] is an EAbased on mathematical models that describe how speciesmigrate from one island to another how new species ariseand how species become extinct The way the problemsolution is found is analogous to naturersquos way of distributingspecies Ma in [17] showed that sinusoidal migration modelsgenerally outperform linear migration models like the onein original BBO algorithm The authors in [18] extend themigration model performance analysis and they propose twononlinear migration models which they call model 7 andmodel 8 BBO has been applied to design problems in elec-tromagnetics including Yagi-Uda synthesis [19] microstripantenna design [20 21] and antenna array synthesis [22ndash24]

The purpose of this paper is to use BBO with differentmigration models in order to design large thinned arraysTo the best of our knowledge this is the first time that BBOwith other than the linear migration mode is applied to anantenna design problem This paper contributes to compareperformance of several state-of-the-art EAs to high dimen-sional thinned array design problems Additionally the papercontributes so that the design cases presented in this papercould also be used as a framework of benchmark functionsfor testing evolutionary algorithms to large array designproblems More specifically we compare the original BBOlinearmigrationmodel with the sinusoidalmodel [17] model7 and model 8 [18] two BPSO variants binary HS ODE andACOWe apply the algorithms to five different design cases oflinear and planar arrays A comparative study of BBOvariantsperformance on benchmark functions is also given More-over a study of the influence on the BBO performance ofthe boundary constraint handling method is also presentedTo the best of our knowledge this is the first time that suchstudy is carried out on BBO in general Numerical resultsshow that BBO with sinusoidal migration model generallyperforms better than or equally with the other migrationmodels Additionally it performs better than the original BBOalgorithm in terms of solution accuracy Additionally resultsshow VBPSO outperforms the initial BPSO algorithm how-ever its performance is inferior to that of the BBO algorithms

This paper is organized as follows We describe theproblem formulation in Section 2 A brief description of theEAs used in this paper is given in Section 3 In Section 4 wepresent the numerical results Finally the conclusion is givenin Section 5

2 Formulation

We consider an119873-element linear array of isotropic elementsThe array factor is expressed as

AF (119906 119868) =119873

sum

119899=1

119868119899119890119895(2120587120582)119899119889119906

(1)

where 120582 is the wavelength 119868119899is the complex excitation of

the 119899th element 119868 is the corresponding vector of elementamplitudes 119889 is distance between two adjacent elements119906 = sin(120599) is the direction cosine and 120599 is the steering anglemeasured from broadside of the array In the thinned arraycase 119868

119899could be only 1 (turned ldquoonrdquo) or 0 (turned ldquooffrdquo)

For a symmetrically excited array (1) becomes

AF (119906 119868) = 2

119872

sum

119899=1

119868119899cos [2120587

120582119889119899119906] + 119868

0 (2)

where 119872 = lfloor1198732rfloor is the largest integer less than or equal to1198732 For an even number of elements 119868

0= 0 and for an odd

number of elements we set 1198680= 1 We assume that 119889 = 05120582

for all cases The fill factor percentage is the percentage ofthe ratio of the turned ldquoonrdquo elements to the total numberof elements The optimization goal is the peak sidelobelevel (SLL) suppression by finding the optimum elementamplitudes This design problem is therefore defined by theminimization of the objective function

1198651(119906 119868) = max

119906isin119878119906

AFdB (119906 119868) (3)

where 119878119906is the set of direction cosines that are outside the

angular range of the main lobeAdditionally we study the effect of maintaining the same

aperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon)

To test the algorithms ability to design planar thinnedarrays we consider a 2119873times 2119872 planar array which lies on the119909-119910 plane The array factor of such an array is given by

AF (119906 V 119868) = 4

119872

sum

119898=1

119873

sum

119899=1

119868119899119898

cos [120587 (2119898 minus 1) 119889119910119906]

sdot cos [120587 (2119899 minus 1) 119889119909V]

(4)

where 119889119909 119889119910are the distance between two adjunct elements

in the 119909 and 119910 direction respectively and 119906 = sin 120599 cos120601V = sin 120599 sin120601 are the direction cosines

The optimization goal here is the sidelobe level (SLL)suppression at two different phi-planes (120601 = 0

∘ 120601 = 90∘)

for a desired fill factor percentage This design problem isexpressed by the minimization of the objective function

1198652(119868) = max PSLL120601=0

∘

dB PSLL120601=90∘

dB (5)

where PSLL120601=0∘

dB PSLL120601=90∘

dB are the calculated peak sidelobelevels at 120601 = 0

∘ and 120601 = 90∘ planes respectively

3 Optimization Algorithms

31 Binary PSO The binary PSO (BPSO) model was pre-sented byKennedy andEberhart and is based on a very simplemodification of the real-valued PSO [5] In binary PSO theparticle positions belong to a119863-dimensional binary space (bitstrings of length119863) while the particle velocities remain real-valued

The 119899th coordinate of each particlersquos position is a bitwhose state is given by

119909119866+1119899119894

=

1 if rand[01]

lt 119879 (119906119866+1119899119894

)

0 otherwise(6)

International Journal of Antennas and Propagation 3

where rand[01]

is a uniformly distributed random number in[0 1] and119879(119909) is a sigmoid limiting transformation thatmapsreal numbers to the interval [0 1] Such a function is definedby

119879 (119909) =1

1 + 119890minus119909(7)

which defines S-shaped transfer functionIn [6] new BPSO variants that use V-shaped transfer

functions are proposed (VBPSO) The BPSO variant thatproduces the best results according to [6] is that with atransfer function expressed by

119879 (119909) =

1003816100381610038161003816100381610038161003816

2

120587arctan(120587

2119909)

1003816100381610038161003816100381610038161003816 (8)

32 Binary Harmony Search Algorithm Harmony Search(HS) [11] is an evolutionary algorithm which is inspired bythe way that musicians experiment and change the pitchesof their instruments to improvise better harmonies Themain control parameter in HS is the Harmony Memory Size(HMS) which determines the number of solutions (harmo-nies) inside the HM and it is equivalent to population size inanother algorithms Another HS parameter is the HarmonyMemory Consideration Rate (HMCR) which determineswhether pitches (decision variables) should be selected fromthe HM or randomly from the predefined range HMCR isa number between 0 and 1 An additional control parameterof HS is the Pitch Adjusting Rate (PAR) which determinesthe probability of adjusting the original value of the selectedpitches from the HM

33 Oppositional Differential Evolution Differential evolu-tion (DE) [7 8] is a population-based stochastic global opti-mization algorithmwhich has been used in several real worldengineering problems Several DE variants or strategies existOne of the DE advantages is that very few control parametershave to be adjusted in each algorithm run In [9] a new DEalgorithm based on opposition-based learning (OBL) [10]oppositional differential evolution (ODE) was introducedThe basic idea of OBL is not only to calculate the fitness of thecurrent individual but also to calculate the fitness of the oppo-site individualThen the algorithm selects the individual withthe lower (higher) fitness value The benefits of using such atechnique are that convergence speedmay be faster and betterapproximation of the global optimum can be found ODEuses an additional control parameter to those of standard DEcalled the jumping rate 119895

119903isin [0 1] which controls in each

generation if the opposite population is created or not

34 Ant Colony Optimization Ant Colony Optimization(ACO) [14 25 26] is a meta-heuristic inspired by theantsrsquo foraging behavior At the core of this behavior isthe indirect communication between the ants by means ofchemical pheromone trails which enables them to find shortpaths between their nest and food sources Ants can sensepheromone When they decide to follow a path they tend tochoose the ones with strong pheromone intensities way backto the nest or to the food source Therefore shorter paths

would accumulate more pheromone than longer ones Thisfeature of real ant colonies is exploited in ACO algorithms inorder to solve combinatorial optimization problems consid-ered to be NP-Hard

35 Biogeography-Based Optimization The mathematicalmodels of biogeography are based on the work of RobertMacArthur and Edward Wilson in the early 1960s Usingthis model it was possible to predict the number of speciesin a habitat The habitat is an area that is geographicallyisolated from other habitats The geographical areas that arewell suited as residences for biological species are said tohave a high habitat suitability index (HSI) Therefore everyhabitat is characterized by the HSI which depends on factorslike rainfall diversity of vegetation diversity of topographicfeatures land area and temperature Each of the featuresthat characterize habitability is known as suitability indexvariables (SIVs) The SIVs are independent variables whileHSI is the dependent variable

Therefore a solution to a 119863-dimensional problem canbe represented as a vector of SIV variables [SIV

1 SIV2

SIV119863] which is a habitat or island The value of HSI of a

habitat is the value of the objective function that correspondsto that solution and it is found by

HSI = 119865 (habitat) = 119865 (SIV1 SIV2 SIV

119863) (9)

Habitats with a high HSI are good solutions of theobjective function while poor solutions are those habitatswith a low HSI The immigration and emigration rates arefunctions of the rank of the given candidate solution Therank of the given candidate solution represents the numberof species in a habitat These are given by the following

(1) Linear migration model (original BBO)

120583119896= 119864(

119896

119878max)

120582119896= 119868(1 minus

119896

119878max)

(10)

(2) Sinusoidal migration model [17]

120583119896=119864

2(minus cos( 119896120587

119878max) + 1)

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(11)

(3) Model 7 [18]

120583119896= 119864(

119896

119878max)

4

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(12)

4 International Journal of Antennas and Propagation

(4) Model 8 [18]

120583119896= 119864(

119896

119878max)

16

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(13)

119868 is the maximum possible immigration rate 119864 is themaximum possible emigration rate 119896 is the rank of the givencandidate solution and 119878max is the maximum number ofspecies (eg population size)The rank of the given candidatesolution or the number of species is obtained by sortingthe solutions from most fit to least fit according to the HSIvalue (eg fitness) BBO uses both mutation and migrationoperators The application of these operators to each SIV ineach solution is decided probabilisticallyThemutation rate119898of a possible solution 119878 is defined to be inversely proportionalto the solution probability and it is given by

119898(119878) = 119898max (1 minus 119875119904

119875max) (14)

where119875119904is the probability that a habitat contains 119878 species and

119898max is a user-defined parameter As with other evolutionaryalgorithms BBO also incorporates elitism This is imple-mented with a user-selected elitism parameter 119901 This meansthat the 119901 best phase vectors remain from one generation tothe other The BBO algorithm is outlined below

(1) Initialize the BBO control parameters Map the prob-lem solutions to habitats (vectors) Set the habitatmodification probability 119875mod the maximum immi-gration rate 119868 the maximum emigration rate 119864the maximum migration rate 119898max and the elitismparameter 119901 (if elitism is desired)

(2) Initialize a random population of119873119875habitats from a

uniform distribution Set the number of generations119866 to one

(3) Evaluate objective function values for each antennaarray of the population

(4) Map the objective function value to the number ofspecies 119878 the immigration rate 120582

119896 and the emigration

rate 120583119896for each solution (antenna array) of the

population(5) Apply themigration operator for each nonelite habitat

based on immigration and emigration rates with theone of the linear sinusoidal or nonlinear migrationmodels

(6) Update the species count probability using

119904

=

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

119878 = 0

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

+ 120582119904minus1

119875119904minus1

1 le 119878 le 119878max minus 1

minus (120582119904+ 120583119904) 119875119904+ 120582119904minus1

119875119904minus1

119878 = 119878max

(15)

(7) Apply the mutation operator according to (14)

(8) Sort the population according to the objective func-tion value from best to worst

(9) Apply elitism by replacing the 119901 worst habitats of theprevious generation with the 119901 best ones

(10) Repeat step (3) until the maximum number of gener-ations 119866max is reached

4 Numerical Results

41 Comparison with Test Functions In order to evaluate thedifferent BBO migration models first we use a set of testproblems from the literature In this paper the test problemsconsist of eight well-known benchmark functions We havechosen two unimodal and six multimodal functions Theseare expressed as follows [27 28]

(A) Unimodal Functions

(1) Sphere Function

1198911 (119909) =

119863minus1

sum

119895=0

1199092

119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 100 119891

1 (0 0 0) = 0 (16)

(2) Rosenbrockrsquos Function

1198912 (119909) =

119863minus2

sum

119895=0

[100 (119909119895+1

minus 1199092

119895)2

+ (119909119895minus 1)2

]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 2048 119891

2 (1 1 1) = 0

(17)

(B) Multimodal Functions

(3) Ackleyrsquos Function

1198913 (119909) = minus20 exp(minus

1

5radic

1

119863

119863minus1

sum

119895=0

1199092

119895)

minus exp( 1

119863

119863minus1

sum

119895=0

cos (2120587119909119895)) + 20 + 119890

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 32 119891

3 (0 0 0) = 0

(18)

(4) Griewanksrsquos Function

1198914 (119909) =

119863minus1

sum

119895=0

1199092

119895

4000minus

119863minus1

prod

119895=0

cos(119909119895

radic119895)

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 600 119891

4 (0 0 0) = 0

(19)

International Journal of Antennas and Propagation 5

Table 1 Comparative results for the test functions The smaller values are in bold

Migration model Test function1198911(119909) 119891

2(119909) 119891

3(119909) 119891

4(119909) 119891

5(119909) 119891

6(119909) 119891

7(119909) 119891

8(119909)

Linear 0 plusmn 0 50592 plusmn 26042 0133 plusmn 0084 0744 plusmn 0295 0 plusmn 0 0012 plusmn 0033 0200 plusmn 0400 3934 plusmn 2437Sinusoidal 0861 plusmn 0552 29844 plusmn 16559 0091 plusmn 0090 0725 plusmn 0205 0 plusmn 0 0003 plusmn 0009 0003 plusmn 0010 2561 plusmn 1894Model 7 0893 plusmn 0503 42243 plusmn 23893 0201 plusmn 0092 0673 plusmn 0142 0 plusmn 0 0018 plusmn 0023 0 plusmn 0 4144 plusmn 3704Model 8 0767 plusmn 0898 70062 plusmn 33001 0131 plusmn 0081 0701 plusmn 0250 0 plusmn 0 0004 plusmn 0014 0 plusmn 0 3209 plusmn 2088

(5) Weierstrass Function

1198915 (119909) =

119863minus1

sum

119895=0

(

119896max

sum

119896=0

[119886119896 cos (2120587119887119896 (119909

119895+ 05))])

minus 119863(

119896max

sum

119896=0

[119886119896 cos (120587119887119896)])

119886 = 05 119887 = 3 119896max = 2010038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 05 119891

5 (0 0 0) = 0

(20)

(6) Rastriginrsquos Function

1198916 (119909) =

119863minus1

sum

119895=0

[1199092

119895minus 10 cos (2120587119909

119895) + 10]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 512 119891

6 (0 0 0) = 0

(21)

(7) Noncontinuous Rastriginrsquos Function

1198917 (119909) =

119863minus1

sum

119895=0

[1199102

119895minus 10 cos (2120587119910

119895) + 10]

119910119895=

119909119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816lt 05

round (2119909119895)

2

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816ge 05

for 119895 = 0 1 119863 minus 1 1198917 (0 0 0) = 0

(22)

(8) Schwefelrsquos Function

1198918 (119909) = 4189829 times 119863 minus

119863minus1

sum

119895=0

[119909119895sin(10038161003816100381610038161003816119909119895

10038161003816100381610038161003816

12

)]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 500 119891

8 (42096 42096 42096) = 0

(23)

All algorithms are executed 50 times The results are com-pared The population size is set to 100 and the maximumnumber of generations is set to 2000 iterationsThe problemsdimension is set to 119863 = 30 For all migration models thehabitat modification probability 119875mod is set to 1 and themaximum mutation rate 119898max is set equal to 0005 Themaximum immigration rate 119868 and the maximum emigrationrate119864 are both set to oneThe elitismparameter119901 is set to twoTable 1 reports the comparative results in terms of mean and

Table 2 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 294Sinusoidal 194Model 7 288Model 8 225

standard deviation values We notice that BBOwith the sinu-soidal migration model performs better in 5 out of the 8 testfunctions Moreover in order to compare the algorithms per-formance on all problems we have conducted the Friedmantest [29] Table 2 shows the average ranking of all algorithmsThe highest ranking is shown in bold It is obvious that thebest average rankingwas obtained by theBBOwith sinusoidalmigration model which outperforms the other models

42 Comparison with ArrayThinning Problems We comparethe four BBO migration modelsrsquo performance with the twoBPSO variants BHS ODE and ACO on different thinnedarray design cases All algorithms are executed 20 times Theresults are compared The population size is set to 200 andthe maximum number of generations is set to 1000 iterationsFor all migrationmodels the habitatmodification probability119875mod is set to 1 and the maximum mutation rate 119898max isset equal to 0005 The maximum immigration rate 119868 and themaximum emigration rate 119864 are both set both to one Theelitism parameter 119901 is set to two In both BPSO algorithmsthe learning factors 119888

1 1198882are both set equal to two and inertia

weight 119908 is linearly decreased from 09 to 04 as in [6] ForBHS the HMCR is set to 099 and the PAR is set to 04 ForODE 119865 = 05 CR = 09 and the jumping rate is set to 03For ACO the initial pheromone value 120591

0is set to 10119890 minus 6 the

pheromone update constant 119876 is set to 20 the explorationconstant 119902

0is set to 1 the global pheromone decay rate 120588

119892is

09 the local pheromone decay rate 120588119897is 05 the pheromone

sensitivity 120572 is 1 and the visibility sensitivity is 120573 is 5The first case is that of a symmetrically excited 300-

element linear array The total number of unknowns forthis case is 150 Table 3 reports the comparative results Wenotice that the sinusoidal model and the model 7 seem tooutperform the other algorithms Bothmodels have obtainedthe same best value Both BPSO variants perform worse thanthe BBOalgorithmsACOandVBPSOperform similarlyTheconvergence rate graph for this case is depicted in Figure 1 AllBBOmodels seem to converge at similar speed faster than theother algorithms The VBPSO converges faster than BPSOIt must be pointed out that the BBO algorithms converge

6 International Journal of Antennas and Propagation

Table 3 Comparative results for 300-element symmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2454 minus2424 minus2443 0096Sinusoidal minus2467 minus2417 minus2441 0146Model 7 minus2467 minus2427 minus2443 0110Model 8 minus2444 minus2351 minus2411 0243VBPSO minus2432 minus2367 minus2405 0175BPSO minus2110 minus1988 minus2035 0326BHS minus2410 minus2279 minus2361 0431ODE minus2406 minus2330 minus2376 0203ACO minus2420 minus2393 minus2406 0084

Table 4 Comparative results for 300-element symmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2445 minus2406 minus2432 0112Sinusoidal minus2459 minus2430 minus2445 0087Model 7 minus2470 minus2427 minus2441 0127Model 8 minus2456 minus2364 minus2418 0257VBPSO minus2441 minus2397 minus2422 0141BPSO minus2150 minus2052 minus2095 0281BHS minus2416 minus2290 minus2379 0413ODE minus2430 minus2351 minus2391 0231ACO minus2336 minus2276 minus2316 0171

at their final value at less than 300 iterations The radiationpattern of the best obtained result is shown in Figure 2 Thebest array found is filled 72 and has a peak SLL ofminus2467 dBAdditionally we study the effect of maintaining the sameaperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon) for a design case The comparative results for this caseare shown in Table 4 The sinusoidal model presents the bestperformance except the best value found where the model 7outperforms the others VBSO is completive with the BBOalgorithms for this case and achieves better performance thanBBOwithmodel 8 ODE performs better than ACO and BHSfor this case Figure 3 shows the convergence rate for thiscase It is obvious that all BBO algorithms converge at similarspeeds while the PSO algorithms converge slower The bestobtained radiation pattern is shown in Figure 4The obtainedpeak SLL value is minus2470 dB slightly smaller than the previouscaseThe array filling percentage is again 72We notice thatthere is not a significant difference if we choose an array with-out constraints or with the same aperture size for this case

Next we consider asymmetric array designs in which theproblem dimension is 300 Therefore we evaluate the algo-rithms ability to solve high dimensional problems It mustbe pointed out that although the number of unknowns hasdoubled compared with the symmetric case the populationsize and iterations remain the same as previously Thereforethe level of difficulty increases for all algorithms Table 5reports the comparative results for this case The algorithms

minus16

minus18

minus20

minus22

minus24

Number of iterations

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Avg

cost

func

tion

10009008007006005004003002001000

Figure 1 Convergence rate graph for the 300-element symmetricthinned array case

Nor

mal

ized

far fi

eld

(dB)

111111111111111111111111111111111111111111111111111111111111111111101111111101011101011111011010001001100111101001110000000000100001010110010000111001

Filled 72

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

Symmetric linear arrayN = 300 elements

Max SLL = minus2467dB

1080604020minus02minus04minus06minus08minus1

Figure 2 Radiation pattern of the best obtained array for the 300-element symmetric thinned array

performance differences are clearer in this case It is obviousthat the sinusoidal model clearly outperforms the others inthis case The results obtained by model 7 and linear modelare quite similar in this case The VBPSO results are betteragain than BPSO BHS ODE and ACO but worse than theBBO algorithms Figure 5 shows the convergence rate graphfor this case Model 7 seems to convergence slightly fasterthan the other algorithms The radiation pattern of the bestobtained array is shown in Figure 6 The obtained array SLLis minus2611 dB and the fill percentage is about 72

In order to further evaluate the algorithms performancewe choose again an asymmetric array case with the sameaperture length Table 6 holds the comparative results Theresults show that sinusoidal model clearly outperforms theother algorithms We notice that VBPSO and BPSO resultsare worse than the BBO algorithms BHS is completive withthe VBPSO algorithm for this case The convergence rategraph of Figure 7 shows that models 7 and 8 converge slightly

International Journal of Antennas and Propagation 7

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

Avg

cost

func

tion

Figure 3 Convergence rate graph for the 300-element symmetricthinned array case with same aperture size

111111111111111111111111111111111111111111111111111111111111111111111011111111001111011111000100110111001001110000101010010001000000001000101001101011

Same aperture length

Nor

mal

ized

far fi

eld

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

1080604020minus02minus04minus06minus08minus1

Filled 72

Symmetric linear arrayN = 300 elements

Max SLL = minus2470dB

Figure 4 Radiation pattern of the best obtained array for the 300-element symmetric thinned array with same aperture length

Table 5 Comparative results for 300-element asymmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2603 minus2565 minus2583 0119Sinusoidal minus2611 minus2573 minus2588 0110Model 7 minus2605 minus2558 minus2582 0146Model 8 minus2557 minus2421 minus2513 0362VBPSO minus2498 minus2349 minus2436 0468BPSO minus2083 minus1998 minus2032 0228BHS minus2532 minus2463 minus2502 0214ODE minus2488 minus2351 minus2419 0424ACO minus2210 minus2155 minus2183 0187

faster than the linear and the sinusoidal model for this caseAgain it is obvious that the BBO algorithms require feweriterations than the other algorithms in order to reach the

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 5 Convergence rate graph for the 300-element asymmetricthinned array

Nor

mal

ized

far fi

eld

(dB)

111110110000010100010000000011001100110100101001010011011100001110100011111101101111011111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111101111111111011101111101101111101001010100101010011100110000010101010100000100000111

Asymmetric linear array

Filled 727

0

minus10

minus20

minus30

minus401080604020minus02minus04minus06minus08minus1

u

N = 300 elements

Max SLL = minus2611dB

Figure 6 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array

Table 6 Comparative results for 300-element asymmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2597 minus2558 minus2578 0113Sinusoidal minus2608 minus2577 minus2591 0107Model 7 minus2587 minus2560 minus2576 0081Model 8 minus2567 minus2446 minus2507 0383VBPSO minus2490 minus2254 minus2392 0658BPSO minus2080 minus1987 minus2021 0276BHS minus2558 minus2392 minus2476 0633ODE minus2492 minus2344 minus2426 0522ACO minus2191 minus2163 minus2177 0008

final values Figure 8 presents the radiation pattern of thebest obtained arrayThe peak SLL is minus2608 dB and the filling

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 3

where rand[01]

is a uniformly distributed random number in[0 1] and119879(119909) is a sigmoid limiting transformation thatmapsreal numbers to the interval [0 1] Such a function is definedby

119879 (119909) =1

1 + 119890minus119909(7)

which defines S-shaped transfer functionIn [6] new BPSO variants that use V-shaped transfer

functions are proposed (VBPSO) The BPSO variant thatproduces the best results according to [6] is that with atransfer function expressed by

119879 (119909) =

1003816100381610038161003816100381610038161003816

2

120587arctan(120587

2119909)

1003816100381610038161003816100381610038161003816 (8)

32 Binary Harmony Search Algorithm Harmony Search(HS) [11] is an evolutionary algorithm which is inspired bythe way that musicians experiment and change the pitchesof their instruments to improvise better harmonies Themain control parameter in HS is the Harmony Memory Size(HMS) which determines the number of solutions (harmo-nies) inside the HM and it is equivalent to population size inanother algorithms Another HS parameter is the HarmonyMemory Consideration Rate (HMCR) which determineswhether pitches (decision variables) should be selected fromthe HM or randomly from the predefined range HMCR isa number between 0 and 1 An additional control parameterof HS is the Pitch Adjusting Rate (PAR) which determinesthe probability of adjusting the original value of the selectedpitches from the HM

33 Oppositional Differential Evolution Differential evolu-tion (DE) [7 8] is a population-based stochastic global opti-mization algorithmwhich has been used in several real worldengineering problems Several DE variants or strategies existOne of the DE advantages is that very few control parametershave to be adjusted in each algorithm run In [9] a new DEalgorithm based on opposition-based learning (OBL) [10]oppositional differential evolution (ODE) was introducedThe basic idea of OBL is not only to calculate the fitness of thecurrent individual but also to calculate the fitness of the oppo-site individualThen the algorithm selects the individual withthe lower (higher) fitness value The benefits of using such atechnique are that convergence speedmay be faster and betterapproximation of the global optimum can be found ODEuses an additional control parameter to those of standard DEcalled the jumping rate 119895

119903isin [0 1] which controls in each

generation if the opposite population is created or not

34 Ant Colony Optimization Ant Colony Optimization(ACO) [14 25 26] is a meta-heuristic inspired by theantsrsquo foraging behavior At the core of this behavior isthe indirect communication between the ants by means ofchemical pheromone trails which enables them to find shortpaths between their nest and food sources Ants can sensepheromone When they decide to follow a path they tend tochoose the ones with strong pheromone intensities way backto the nest or to the food source Therefore shorter paths

would accumulate more pheromone than longer ones Thisfeature of real ant colonies is exploited in ACO algorithms inorder to solve combinatorial optimization problems consid-ered to be NP-Hard

35 Biogeography-Based Optimization The mathematicalmodels of biogeography are based on the work of RobertMacArthur and Edward Wilson in the early 1960s Usingthis model it was possible to predict the number of speciesin a habitat The habitat is an area that is geographicallyisolated from other habitats The geographical areas that arewell suited as residences for biological species are said tohave a high habitat suitability index (HSI) Therefore everyhabitat is characterized by the HSI which depends on factorslike rainfall diversity of vegetation diversity of topographicfeatures land area and temperature Each of the featuresthat characterize habitability is known as suitability indexvariables (SIVs) The SIVs are independent variables whileHSI is the dependent variable

Therefore a solution to a 119863-dimensional problem canbe represented as a vector of SIV variables [SIV

1 SIV2

SIV119863] which is a habitat or island The value of HSI of a

habitat is the value of the objective function that correspondsto that solution and it is found by

HSI = 119865 (habitat) = 119865 (SIV1 SIV2 SIV

119863) (9)

Habitats with a high HSI are good solutions of theobjective function while poor solutions are those habitatswith a low HSI The immigration and emigration rates arefunctions of the rank of the given candidate solution Therank of the given candidate solution represents the numberof species in a habitat These are given by the following

(1) Linear migration model (original BBO)

120583119896= 119864(

119896

119878max)

120582119896= 119868(1 minus

119896

119878max)

(10)

(2) Sinusoidal migration model [17]

120583119896=119864

2(minus cos( 119896120587

119878max) + 1)

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(11)

(3) Model 7 [18]

120583119896= 119864(

119896

119878max)

4

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(12)

4 International Journal of Antennas and Propagation

(4) Model 8 [18]

120583119896= 119864(

119896

119878max)

16

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(13)

119868 is the maximum possible immigration rate 119864 is themaximum possible emigration rate 119896 is the rank of the givencandidate solution and 119878max is the maximum number ofspecies (eg population size)The rank of the given candidatesolution or the number of species is obtained by sortingthe solutions from most fit to least fit according to the HSIvalue (eg fitness) BBO uses both mutation and migrationoperators The application of these operators to each SIV ineach solution is decided probabilisticallyThemutation rate119898of a possible solution 119878 is defined to be inversely proportionalto the solution probability and it is given by

119898(119878) = 119898max (1 minus 119875119904

119875max) (14)

where119875119904is the probability that a habitat contains 119878 species and

119898max is a user-defined parameter As with other evolutionaryalgorithms BBO also incorporates elitism This is imple-mented with a user-selected elitism parameter 119901 This meansthat the 119901 best phase vectors remain from one generation tothe other The BBO algorithm is outlined below

(1) Initialize the BBO control parameters Map the prob-lem solutions to habitats (vectors) Set the habitatmodification probability 119875mod the maximum immi-gration rate 119868 the maximum emigration rate 119864the maximum migration rate 119898max and the elitismparameter 119901 (if elitism is desired)

(2) Initialize a random population of119873119875habitats from a

uniform distribution Set the number of generations119866 to one

(3) Evaluate objective function values for each antennaarray of the population

(4) Map the objective function value to the number ofspecies 119878 the immigration rate 120582

119896 and the emigration

rate 120583119896for each solution (antenna array) of the

population(5) Apply themigration operator for each nonelite habitat

based on immigration and emigration rates with theone of the linear sinusoidal or nonlinear migrationmodels

(6) Update the species count probability using

119904

=

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

119878 = 0

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

+ 120582119904minus1

119875119904minus1

1 le 119878 le 119878max minus 1

minus (120582119904+ 120583119904) 119875119904+ 120582119904minus1

119875119904minus1

119878 = 119878max

(15)

(7) Apply the mutation operator according to (14)

(8) Sort the population according to the objective func-tion value from best to worst

(9) Apply elitism by replacing the 119901 worst habitats of theprevious generation with the 119901 best ones

(10) Repeat step (3) until the maximum number of gener-ations 119866max is reached

4 Numerical Results

41 Comparison with Test Functions In order to evaluate thedifferent BBO migration models first we use a set of testproblems from the literature In this paper the test problemsconsist of eight well-known benchmark functions We havechosen two unimodal and six multimodal functions Theseare expressed as follows [27 28]

(A) Unimodal Functions

(1) Sphere Function

1198911 (119909) =

119863minus1

sum

119895=0

1199092

119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 100 119891

1 (0 0 0) = 0 (16)

(2) Rosenbrockrsquos Function

1198912 (119909) =

119863minus2

sum

119895=0

[100 (119909119895+1

minus 1199092

119895)2

+ (119909119895minus 1)2

]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 2048 119891

2 (1 1 1) = 0

(17)

(B) Multimodal Functions

(3) Ackleyrsquos Function

1198913 (119909) = minus20 exp(minus

1

5radic

1

119863

119863minus1

sum

119895=0

1199092

119895)

minus exp( 1

119863

119863minus1

sum

119895=0

cos (2120587119909119895)) + 20 + 119890

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 32 119891

3 (0 0 0) = 0

(18)

(4) Griewanksrsquos Function

1198914 (119909) =

119863minus1

sum

119895=0

1199092

119895

4000minus

119863minus1

prod

119895=0

cos(119909119895

radic119895)

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 600 119891

4 (0 0 0) = 0

(19)

International Journal of Antennas and Propagation 5

Table 1 Comparative results for the test functions The smaller values are in bold

Migration model Test function1198911(119909) 119891

2(119909) 119891

3(119909) 119891

4(119909) 119891

5(119909) 119891

6(119909) 119891

7(119909) 119891

8(119909)

Linear 0 plusmn 0 50592 plusmn 26042 0133 plusmn 0084 0744 plusmn 0295 0 plusmn 0 0012 plusmn 0033 0200 plusmn 0400 3934 plusmn 2437Sinusoidal 0861 plusmn 0552 29844 plusmn 16559 0091 plusmn 0090 0725 plusmn 0205 0 plusmn 0 0003 plusmn 0009 0003 plusmn 0010 2561 plusmn 1894Model 7 0893 plusmn 0503 42243 plusmn 23893 0201 plusmn 0092 0673 plusmn 0142 0 plusmn 0 0018 plusmn 0023 0 plusmn 0 4144 plusmn 3704Model 8 0767 plusmn 0898 70062 plusmn 33001 0131 plusmn 0081 0701 plusmn 0250 0 plusmn 0 0004 plusmn 0014 0 plusmn 0 3209 plusmn 2088

(5) Weierstrass Function

1198915 (119909) =

119863minus1

sum

119895=0

(

119896max

sum

119896=0

[119886119896 cos (2120587119887119896 (119909

119895+ 05))])

minus 119863(

119896max

sum

119896=0

[119886119896 cos (120587119887119896)])

119886 = 05 119887 = 3 119896max = 2010038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 05 119891

5 (0 0 0) = 0

(20)

(6) Rastriginrsquos Function

1198916 (119909) =

119863minus1

sum

119895=0

[1199092

119895minus 10 cos (2120587119909

119895) + 10]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 512 119891

6 (0 0 0) = 0

(21)

(7) Noncontinuous Rastriginrsquos Function

1198917 (119909) =

119863minus1

sum

119895=0

[1199102

119895minus 10 cos (2120587119910

119895) + 10]

119910119895=

119909119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816lt 05

round (2119909119895)

2

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816ge 05

for 119895 = 0 1 119863 minus 1 1198917 (0 0 0) = 0

(22)

(8) Schwefelrsquos Function

1198918 (119909) = 4189829 times 119863 minus

119863minus1

sum

119895=0

[119909119895sin(10038161003816100381610038161003816119909119895

10038161003816100381610038161003816

12

)]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 500 119891

8 (42096 42096 42096) = 0

(23)

All algorithms are executed 50 times The results are com-pared The population size is set to 100 and the maximumnumber of generations is set to 2000 iterationsThe problemsdimension is set to 119863 = 30 For all migration models thehabitat modification probability 119875mod is set to 1 and themaximum mutation rate 119898max is set equal to 0005 Themaximum immigration rate 119868 and the maximum emigrationrate119864 are both set to oneThe elitismparameter119901 is set to twoTable 1 reports the comparative results in terms of mean and

Table 2 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 294Sinusoidal 194Model 7 288Model 8 225

standard deviation values We notice that BBOwith the sinu-soidal migration model performs better in 5 out of the 8 testfunctions Moreover in order to compare the algorithms per-formance on all problems we have conducted the Friedmantest [29] Table 2 shows the average ranking of all algorithmsThe highest ranking is shown in bold It is obvious that thebest average rankingwas obtained by theBBOwith sinusoidalmigration model which outperforms the other models

42 Comparison with ArrayThinning Problems We comparethe four BBO migration modelsrsquo performance with the twoBPSO variants BHS ODE and ACO on different thinnedarray design cases All algorithms are executed 20 times Theresults are compared The population size is set to 200 andthe maximum number of generations is set to 1000 iterationsFor all migrationmodels the habitatmodification probability119875mod is set to 1 and the maximum mutation rate 119898max isset equal to 0005 The maximum immigration rate 119868 and themaximum emigration rate 119864 are both set both to one Theelitism parameter 119901 is set to two In both BPSO algorithmsthe learning factors 119888

1 1198882are both set equal to two and inertia

weight 119908 is linearly decreased from 09 to 04 as in [6] ForBHS the HMCR is set to 099 and the PAR is set to 04 ForODE 119865 = 05 CR = 09 and the jumping rate is set to 03For ACO the initial pheromone value 120591

0is set to 10119890 minus 6 the

pheromone update constant 119876 is set to 20 the explorationconstant 119902

0is set to 1 the global pheromone decay rate 120588

119892is

09 the local pheromone decay rate 120588119897is 05 the pheromone

sensitivity 120572 is 1 and the visibility sensitivity is 120573 is 5The first case is that of a symmetrically excited 300-

element linear array The total number of unknowns forthis case is 150 Table 3 reports the comparative results Wenotice that the sinusoidal model and the model 7 seem tooutperform the other algorithms Bothmodels have obtainedthe same best value Both BPSO variants perform worse thanthe BBOalgorithmsACOandVBPSOperform similarlyTheconvergence rate graph for this case is depicted in Figure 1 AllBBOmodels seem to converge at similar speed faster than theother algorithms The VBPSO converges faster than BPSOIt must be pointed out that the BBO algorithms converge

6 International Journal of Antennas and Propagation

Table 3 Comparative results for 300-element symmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2454 minus2424 minus2443 0096Sinusoidal minus2467 minus2417 minus2441 0146Model 7 minus2467 minus2427 minus2443 0110Model 8 minus2444 minus2351 minus2411 0243VBPSO minus2432 minus2367 minus2405 0175BPSO minus2110 minus1988 minus2035 0326BHS minus2410 minus2279 minus2361 0431ODE minus2406 minus2330 minus2376 0203ACO minus2420 minus2393 minus2406 0084

Table 4 Comparative results for 300-element symmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2445 minus2406 minus2432 0112Sinusoidal minus2459 minus2430 minus2445 0087Model 7 minus2470 minus2427 minus2441 0127Model 8 minus2456 minus2364 minus2418 0257VBPSO minus2441 minus2397 minus2422 0141BPSO minus2150 minus2052 minus2095 0281BHS minus2416 minus2290 minus2379 0413ODE minus2430 minus2351 minus2391 0231ACO minus2336 minus2276 minus2316 0171

at their final value at less than 300 iterations The radiationpattern of the best obtained result is shown in Figure 2 Thebest array found is filled 72 and has a peak SLL ofminus2467 dBAdditionally we study the effect of maintaining the sameaperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon) for a design case The comparative results for this caseare shown in Table 4 The sinusoidal model presents the bestperformance except the best value found where the model 7outperforms the others VBSO is completive with the BBOalgorithms for this case and achieves better performance thanBBOwithmodel 8 ODE performs better than ACO and BHSfor this case Figure 3 shows the convergence rate for thiscase It is obvious that all BBO algorithms converge at similarspeeds while the PSO algorithms converge slower The bestobtained radiation pattern is shown in Figure 4The obtainedpeak SLL value is minus2470 dB slightly smaller than the previouscaseThe array filling percentage is again 72We notice thatthere is not a significant difference if we choose an array with-out constraints or with the same aperture size for this case

Next we consider asymmetric array designs in which theproblem dimension is 300 Therefore we evaluate the algo-rithms ability to solve high dimensional problems It mustbe pointed out that although the number of unknowns hasdoubled compared with the symmetric case the populationsize and iterations remain the same as previously Thereforethe level of difficulty increases for all algorithms Table 5reports the comparative results for this case The algorithms

minus16

minus18

minus20

minus22

minus24

Number of iterations

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Avg

cost

func

tion

10009008007006005004003002001000

Figure 1 Convergence rate graph for the 300-element symmetricthinned array case

Nor

mal

ized

far fi

eld

(dB)

111111111111111111111111111111111111111111111111111111111111111111101111111101011101011111011010001001100111101001110000000000100001010110010000111001

Filled 72

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

Symmetric linear arrayN = 300 elements

Max SLL = minus2467dB

1080604020minus02minus04minus06minus08minus1

Figure 2 Radiation pattern of the best obtained array for the 300-element symmetric thinned array

performance differences are clearer in this case It is obviousthat the sinusoidal model clearly outperforms the others inthis case The results obtained by model 7 and linear modelare quite similar in this case The VBPSO results are betteragain than BPSO BHS ODE and ACO but worse than theBBO algorithms Figure 5 shows the convergence rate graphfor this case Model 7 seems to convergence slightly fasterthan the other algorithms The radiation pattern of the bestobtained array is shown in Figure 6 The obtained array SLLis minus2611 dB and the fill percentage is about 72

In order to further evaluate the algorithms performancewe choose again an asymmetric array case with the sameaperture length Table 6 holds the comparative results Theresults show that sinusoidal model clearly outperforms theother algorithms We notice that VBPSO and BPSO resultsare worse than the BBO algorithms BHS is completive withthe VBPSO algorithm for this case The convergence rategraph of Figure 7 shows that models 7 and 8 converge slightly

International Journal of Antennas and Propagation 7

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

Avg

cost

func

tion

Figure 3 Convergence rate graph for the 300-element symmetricthinned array case with same aperture size

111111111111111111111111111111111111111111111111111111111111111111111011111111001111011111000100110111001001110000101010010001000000001000101001101011

Same aperture length

Nor

mal

ized

far fi

eld

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

1080604020minus02minus04minus06minus08minus1

Filled 72

Symmetric linear arrayN = 300 elements

Max SLL = minus2470dB

Figure 4 Radiation pattern of the best obtained array for the 300-element symmetric thinned array with same aperture length

Table 5 Comparative results for 300-element asymmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2603 minus2565 minus2583 0119Sinusoidal minus2611 minus2573 minus2588 0110Model 7 minus2605 minus2558 minus2582 0146Model 8 minus2557 minus2421 minus2513 0362VBPSO minus2498 minus2349 minus2436 0468BPSO minus2083 minus1998 minus2032 0228BHS minus2532 minus2463 minus2502 0214ODE minus2488 minus2351 minus2419 0424ACO minus2210 minus2155 minus2183 0187

faster than the linear and the sinusoidal model for this caseAgain it is obvious that the BBO algorithms require feweriterations than the other algorithms in order to reach the

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 5 Convergence rate graph for the 300-element asymmetricthinned array

Nor

mal

ized

far fi

eld

(dB)

111110110000010100010000000011001100110100101001010011011100001110100011111101101111011111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111101111111111011101111101101111101001010100101010011100110000010101010100000100000111

Asymmetric linear array

Filled 727

0

minus10

minus20

minus30

minus401080604020minus02minus04minus06minus08minus1

u

N = 300 elements

Max SLL = minus2611dB

Figure 6 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array

Table 6 Comparative results for 300-element asymmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2597 minus2558 minus2578 0113Sinusoidal minus2608 minus2577 minus2591 0107Model 7 minus2587 minus2560 minus2576 0081Model 8 minus2567 minus2446 minus2507 0383VBPSO minus2490 minus2254 minus2392 0658BPSO minus2080 minus1987 minus2021 0276BHS minus2558 minus2392 minus2476 0633ODE minus2492 minus2344 minus2426 0522ACO minus2191 minus2163 minus2177 0008

final values Figure 8 presents the radiation pattern of thebest obtained arrayThe peak SLL is minus2608 dB and the filling

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

(4) Model 8 [18]

120583119896= 119864(

119896

119878max)

16

120582119896=

119868

2(cos( 119896120587

119878max) + 1)

(13)

119868 is the maximum possible immigration rate 119864 is themaximum possible emigration rate 119896 is the rank of the givencandidate solution and 119878max is the maximum number ofspecies (eg population size)The rank of the given candidatesolution or the number of species is obtained by sortingthe solutions from most fit to least fit according to the HSIvalue (eg fitness) BBO uses both mutation and migrationoperators The application of these operators to each SIV ineach solution is decided probabilisticallyThemutation rate119898of a possible solution 119878 is defined to be inversely proportionalto the solution probability and it is given by

119898(119878) = 119898max (1 minus 119875119904

119875max) (14)

where119875119904is the probability that a habitat contains 119878 species and

119898max is a user-defined parameter As with other evolutionaryalgorithms BBO also incorporates elitism This is imple-mented with a user-selected elitism parameter 119901 This meansthat the 119901 best phase vectors remain from one generation tothe other The BBO algorithm is outlined below

(1) Initialize the BBO control parameters Map the prob-lem solutions to habitats (vectors) Set the habitatmodification probability 119875mod the maximum immi-gration rate 119868 the maximum emigration rate 119864the maximum migration rate 119898max and the elitismparameter 119901 (if elitism is desired)

(2) Initialize a random population of119873119875habitats from a

uniform distribution Set the number of generations119866 to one

(3) Evaluate objective function values for each antennaarray of the population

(4) Map the objective function value to the number ofspecies 119878 the immigration rate 120582

119896 and the emigration

rate 120583119896for each solution (antenna array) of the

population(5) Apply themigration operator for each nonelite habitat

based on immigration and emigration rates with theone of the linear sinusoidal or nonlinear migrationmodels

(6) Update the species count probability using

119904

=

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

119878 = 0

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1

119875119904+1

+ 120582119904minus1

119875119904minus1

1 le 119878 le 119878max minus 1

minus (120582119904+ 120583119904) 119875119904+ 120582119904minus1

119875119904minus1

119878 = 119878max

(15)

(7) Apply the mutation operator according to (14)

(8) Sort the population according to the objective func-tion value from best to worst

(9) Apply elitism by replacing the 119901 worst habitats of theprevious generation with the 119901 best ones

(10) Repeat step (3) until the maximum number of gener-ations 119866max is reached

4 Numerical Results

41 Comparison with Test Functions In order to evaluate thedifferent BBO migration models first we use a set of testproblems from the literature In this paper the test problemsconsist of eight well-known benchmark functions We havechosen two unimodal and six multimodal functions Theseare expressed as follows [27 28]

(A) Unimodal Functions

(1) Sphere Function

1198911 (119909) =

119863minus1

sum

119895=0

1199092

119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 100 119891

1 (0 0 0) = 0 (16)

(2) Rosenbrockrsquos Function

1198912 (119909) =

119863minus2

sum

119895=0

[100 (119909119895+1

minus 1199092

119895)2

+ (119909119895minus 1)2

]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 2048 119891

2 (1 1 1) = 0

(17)

(B) Multimodal Functions

(3) Ackleyrsquos Function

1198913 (119909) = minus20 exp(minus

1

5radic

1

119863

119863minus1

sum

119895=0

1199092

119895)

minus exp( 1

119863

119863minus1

sum

119895=0

cos (2120587119909119895)) + 20 + 119890

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 32 119891

3 (0 0 0) = 0

(18)

(4) Griewanksrsquos Function

1198914 (119909) =

119863minus1

sum

119895=0

1199092

119895

4000minus

119863minus1

prod

119895=0

cos(119909119895

radic119895)

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 600 119891

4 (0 0 0) = 0

(19)

International Journal of Antennas and Propagation 5

Table 1 Comparative results for the test functions The smaller values are in bold

Migration model Test function1198911(119909) 119891

2(119909) 119891

3(119909) 119891

4(119909) 119891

5(119909) 119891

6(119909) 119891

7(119909) 119891

8(119909)

Linear 0 plusmn 0 50592 plusmn 26042 0133 plusmn 0084 0744 plusmn 0295 0 plusmn 0 0012 plusmn 0033 0200 plusmn 0400 3934 plusmn 2437Sinusoidal 0861 plusmn 0552 29844 plusmn 16559 0091 plusmn 0090 0725 plusmn 0205 0 plusmn 0 0003 plusmn 0009 0003 plusmn 0010 2561 plusmn 1894Model 7 0893 plusmn 0503 42243 plusmn 23893 0201 plusmn 0092 0673 plusmn 0142 0 plusmn 0 0018 plusmn 0023 0 plusmn 0 4144 plusmn 3704Model 8 0767 plusmn 0898 70062 plusmn 33001 0131 plusmn 0081 0701 plusmn 0250 0 plusmn 0 0004 plusmn 0014 0 plusmn 0 3209 plusmn 2088

(5) Weierstrass Function

1198915 (119909) =

119863minus1

sum

119895=0

(

119896max

sum

119896=0

[119886119896 cos (2120587119887119896 (119909

119895+ 05))])

minus 119863(

119896max

sum

119896=0

[119886119896 cos (120587119887119896)])

119886 = 05 119887 = 3 119896max = 2010038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 05 119891

5 (0 0 0) = 0

(20)

(6) Rastriginrsquos Function

1198916 (119909) =

119863minus1

sum

119895=0

[1199092

119895minus 10 cos (2120587119909

119895) + 10]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 512 119891

6 (0 0 0) = 0

(21)

(7) Noncontinuous Rastriginrsquos Function

1198917 (119909) =

119863minus1

sum

119895=0

[1199102

119895minus 10 cos (2120587119910

119895) + 10]

119910119895=

119909119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816lt 05

round (2119909119895)

2

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816ge 05

for 119895 = 0 1 119863 minus 1 1198917 (0 0 0) = 0

(22)

(8) Schwefelrsquos Function

1198918 (119909) = 4189829 times 119863 minus

119863minus1

sum

119895=0

[119909119895sin(10038161003816100381610038161003816119909119895

10038161003816100381610038161003816

12

)]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 500 119891

8 (42096 42096 42096) = 0

(23)

All algorithms are executed 50 times The results are com-pared The population size is set to 100 and the maximumnumber of generations is set to 2000 iterationsThe problemsdimension is set to 119863 = 30 For all migration models thehabitat modification probability 119875mod is set to 1 and themaximum mutation rate 119898max is set equal to 0005 Themaximum immigration rate 119868 and the maximum emigrationrate119864 are both set to oneThe elitismparameter119901 is set to twoTable 1 reports the comparative results in terms of mean and

Table 2 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 294Sinusoidal 194Model 7 288Model 8 225

standard deviation values We notice that BBOwith the sinu-soidal migration model performs better in 5 out of the 8 testfunctions Moreover in order to compare the algorithms per-formance on all problems we have conducted the Friedmantest [29] Table 2 shows the average ranking of all algorithmsThe highest ranking is shown in bold It is obvious that thebest average rankingwas obtained by theBBOwith sinusoidalmigration model which outperforms the other models

42 Comparison with ArrayThinning Problems We comparethe four BBO migration modelsrsquo performance with the twoBPSO variants BHS ODE and ACO on different thinnedarray design cases All algorithms are executed 20 times Theresults are compared The population size is set to 200 andthe maximum number of generations is set to 1000 iterationsFor all migrationmodels the habitatmodification probability119875mod is set to 1 and the maximum mutation rate 119898max isset equal to 0005 The maximum immigration rate 119868 and themaximum emigration rate 119864 are both set both to one Theelitism parameter 119901 is set to two In both BPSO algorithmsthe learning factors 119888

1 1198882are both set equal to two and inertia

weight 119908 is linearly decreased from 09 to 04 as in [6] ForBHS the HMCR is set to 099 and the PAR is set to 04 ForODE 119865 = 05 CR = 09 and the jumping rate is set to 03For ACO the initial pheromone value 120591

0is set to 10119890 minus 6 the

pheromone update constant 119876 is set to 20 the explorationconstant 119902

0is set to 1 the global pheromone decay rate 120588

119892is

09 the local pheromone decay rate 120588119897is 05 the pheromone

sensitivity 120572 is 1 and the visibility sensitivity is 120573 is 5The first case is that of a symmetrically excited 300-

element linear array The total number of unknowns forthis case is 150 Table 3 reports the comparative results Wenotice that the sinusoidal model and the model 7 seem tooutperform the other algorithms Bothmodels have obtainedthe same best value Both BPSO variants perform worse thanthe BBOalgorithmsACOandVBPSOperform similarlyTheconvergence rate graph for this case is depicted in Figure 1 AllBBOmodels seem to converge at similar speed faster than theother algorithms The VBPSO converges faster than BPSOIt must be pointed out that the BBO algorithms converge

6 International Journal of Antennas and Propagation

Table 3 Comparative results for 300-element symmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2454 minus2424 minus2443 0096Sinusoidal minus2467 minus2417 minus2441 0146Model 7 minus2467 minus2427 minus2443 0110Model 8 minus2444 minus2351 minus2411 0243VBPSO minus2432 minus2367 minus2405 0175BPSO minus2110 minus1988 minus2035 0326BHS minus2410 minus2279 minus2361 0431ODE minus2406 minus2330 minus2376 0203ACO minus2420 minus2393 minus2406 0084

Table 4 Comparative results for 300-element symmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2445 minus2406 minus2432 0112Sinusoidal minus2459 minus2430 minus2445 0087Model 7 minus2470 minus2427 minus2441 0127Model 8 minus2456 minus2364 minus2418 0257VBPSO minus2441 minus2397 minus2422 0141BPSO minus2150 minus2052 minus2095 0281BHS minus2416 minus2290 minus2379 0413ODE minus2430 minus2351 minus2391 0231ACO minus2336 minus2276 minus2316 0171

at their final value at less than 300 iterations The radiationpattern of the best obtained result is shown in Figure 2 Thebest array found is filled 72 and has a peak SLL ofminus2467 dBAdditionally we study the effect of maintaining the sameaperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon) for a design case The comparative results for this caseare shown in Table 4 The sinusoidal model presents the bestperformance except the best value found where the model 7outperforms the others VBSO is completive with the BBOalgorithms for this case and achieves better performance thanBBOwithmodel 8 ODE performs better than ACO and BHSfor this case Figure 3 shows the convergence rate for thiscase It is obvious that all BBO algorithms converge at similarspeeds while the PSO algorithms converge slower The bestobtained radiation pattern is shown in Figure 4The obtainedpeak SLL value is minus2470 dB slightly smaller than the previouscaseThe array filling percentage is again 72We notice thatthere is not a significant difference if we choose an array with-out constraints or with the same aperture size for this case

Next we consider asymmetric array designs in which theproblem dimension is 300 Therefore we evaluate the algo-rithms ability to solve high dimensional problems It mustbe pointed out that although the number of unknowns hasdoubled compared with the symmetric case the populationsize and iterations remain the same as previously Thereforethe level of difficulty increases for all algorithms Table 5reports the comparative results for this case The algorithms

minus16

minus18

minus20

minus22

minus24

Number of iterations

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Avg

cost

func

tion

10009008007006005004003002001000

Figure 1 Convergence rate graph for the 300-element symmetricthinned array case

Nor

mal

ized

far fi

eld

(dB)

111111111111111111111111111111111111111111111111111111111111111111101111111101011101011111011010001001100111101001110000000000100001010110010000111001

Filled 72

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

Symmetric linear arrayN = 300 elements

Max SLL = minus2467dB

1080604020minus02minus04minus06minus08minus1

Figure 2 Radiation pattern of the best obtained array for the 300-element symmetric thinned array

performance differences are clearer in this case It is obviousthat the sinusoidal model clearly outperforms the others inthis case The results obtained by model 7 and linear modelare quite similar in this case The VBPSO results are betteragain than BPSO BHS ODE and ACO but worse than theBBO algorithms Figure 5 shows the convergence rate graphfor this case Model 7 seems to convergence slightly fasterthan the other algorithms The radiation pattern of the bestobtained array is shown in Figure 6 The obtained array SLLis minus2611 dB and the fill percentage is about 72

In order to further evaluate the algorithms performancewe choose again an asymmetric array case with the sameaperture length Table 6 holds the comparative results Theresults show that sinusoidal model clearly outperforms theother algorithms We notice that VBPSO and BPSO resultsare worse than the BBO algorithms BHS is completive withthe VBPSO algorithm for this case The convergence rategraph of Figure 7 shows that models 7 and 8 converge slightly

International Journal of Antennas and Propagation 7

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

Avg

cost

func

tion

Figure 3 Convergence rate graph for the 300-element symmetricthinned array case with same aperture size

111111111111111111111111111111111111111111111111111111111111111111111011111111001111011111000100110111001001110000101010010001000000001000101001101011

Same aperture length

Nor

mal

ized

far fi

eld

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

1080604020minus02minus04minus06minus08minus1

Filled 72

Symmetric linear arrayN = 300 elements

Max SLL = minus2470dB

Figure 4 Radiation pattern of the best obtained array for the 300-element symmetric thinned array with same aperture length

Table 5 Comparative results for 300-element asymmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2603 minus2565 minus2583 0119Sinusoidal minus2611 minus2573 minus2588 0110Model 7 minus2605 minus2558 minus2582 0146Model 8 minus2557 minus2421 minus2513 0362VBPSO minus2498 minus2349 minus2436 0468BPSO minus2083 minus1998 minus2032 0228BHS minus2532 minus2463 minus2502 0214ODE minus2488 minus2351 minus2419 0424ACO minus2210 minus2155 minus2183 0187

faster than the linear and the sinusoidal model for this caseAgain it is obvious that the BBO algorithms require feweriterations than the other algorithms in order to reach the

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 5 Convergence rate graph for the 300-element asymmetricthinned array

Nor

mal

ized

far fi

eld

(dB)

111110110000010100010000000011001100110100101001010011011100001110100011111101101111011111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111101111111111011101111101101111101001010100101010011100110000010101010100000100000111

Asymmetric linear array

Filled 727

0

minus10

minus20

minus30

minus401080604020minus02minus04minus06minus08minus1

u

N = 300 elements

Max SLL = minus2611dB

Figure 6 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array

Table 6 Comparative results for 300-element asymmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2597 minus2558 minus2578 0113Sinusoidal minus2608 minus2577 minus2591 0107Model 7 minus2587 minus2560 minus2576 0081Model 8 minus2567 minus2446 minus2507 0383VBPSO minus2490 minus2254 minus2392 0658BPSO minus2080 minus1987 minus2021 0276BHS minus2558 minus2392 minus2476 0633ODE minus2492 minus2344 minus2426 0522ACO minus2191 minus2163 minus2177 0008

final values Figure 8 presents the radiation pattern of thebest obtained arrayThe peak SLL is minus2608 dB and the filling

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

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International Journal of Antennas and Propagation 5

Table 1 Comparative results for the test functions The smaller values are in bold

Migration model Test function1198911(119909) 119891

2(119909) 119891

3(119909) 119891

4(119909) 119891

5(119909) 119891

6(119909) 119891

7(119909) 119891

8(119909)

Linear 0 plusmn 0 50592 plusmn 26042 0133 plusmn 0084 0744 plusmn 0295 0 plusmn 0 0012 plusmn 0033 0200 plusmn 0400 3934 plusmn 2437Sinusoidal 0861 plusmn 0552 29844 plusmn 16559 0091 plusmn 0090 0725 plusmn 0205 0 plusmn 0 0003 plusmn 0009 0003 plusmn 0010 2561 plusmn 1894Model 7 0893 plusmn 0503 42243 plusmn 23893 0201 plusmn 0092 0673 plusmn 0142 0 plusmn 0 0018 plusmn 0023 0 plusmn 0 4144 plusmn 3704Model 8 0767 plusmn 0898 70062 plusmn 33001 0131 plusmn 0081 0701 plusmn 0250 0 plusmn 0 0004 plusmn 0014 0 plusmn 0 3209 plusmn 2088

(5) Weierstrass Function

1198915 (119909) =

119863minus1

sum

119895=0

(

119896max

sum

119896=0

[119886119896 cos (2120587119887119896 (119909

119895+ 05))])

minus 119863(

119896max

sum

119896=0

[119886119896 cos (120587119887119896)])

119886 = 05 119887 = 3 119896max = 2010038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 05 119891

5 (0 0 0) = 0

(20)

(6) Rastriginrsquos Function

1198916 (119909) =

119863minus1

sum

119895=0

[1199092

119895minus 10 cos (2120587119909

119895) + 10]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 512 119891

6 (0 0 0) = 0

(21)

(7) Noncontinuous Rastriginrsquos Function

1198917 (119909) =

119863minus1

sum

119895=0

[1199102

119895minus 10 cos (2120587119910

119895) + 10]

119910119895=

119909119895

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816lt 05

round (2119909119895)

2

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816ge 05

for 119895 = 0 1 119863 minus 1 1198917 (0 0 0) = 0

(22)

(8) Schwefelrsquos Function

1198918 (119909) = 4189829 times 119863 minus

119863minus1

sum

119895=0

[119909119895sin(10038161003816100381610038161003816119909119895

10038161003816100381610038161003816

12

)]

10038161003816100381610038161003816119909119895

10038161003816100381610038161003816le 500 119891

8 (42096 42096 42096) = 0

(23)

All algorithms are executed 50 times The results are com-pared The population size is set to 100 and the maximumnumber of generations is set to 2000 iterationsThe problemsdimension is set to 119863 = 30 For all migration models thehabitat modification probability 119875mod is set to 1 and themaximum mutation rate 119898max is set equal to 0005 Themaximum immigration rate 119868 and the maximum emigrationrate119864 are both set to oneThe elitismparameter119901 is set to twoTable 1 reports the comparative results in terms of mean and

Table 2 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 294Sinusoidal 194Model 7 288Model 8 225

standard deviation values We notice that BBOwith the sinu-soidal migration model performs better in 5 out of the 8 testfunctions Moreover in order to compare the algorithms per-formance on all problems we have conducted the Friedmantest [29] Table 2 shows the average ranking of all algorithmsThe highest ranking is shown in bold It is obvious that thebest average rankingwas obtained by theBBOwith sinusoidalmigration model which outperforms the other models

42 Comparison with ArrayThinning Problems We comparethe four BBO migration modelsrsquo performance with the twoBPSO variants BHS ODE and ACO on different thinnedarray design cases All algorithms are executed 20 times Theresults are compared The population size is set to 200 andthe maximum number of generations is set to 1000 iterationsFor all migrationmodels the habitatmodification probability119875mod is set to 1 and the maximum mutation rate 119898max isset equal to 0005 The maximum immigration rate 119868 and themaximum emigration rate 119864 are both set both to one Theelitism parameter 119901 is set to two In both BPSO algorithmsthe learning factors 119888

1 1198882are both set equal to two and inertia

weight 119908 is linearly decreased from 09 to 04 as in [6] ForBHS the HMCR is set to 099 and the PAR is set to 04 ForODE 119865 = 05 CR = 09 and the jumping rate is set to 03For ACO the initial pheromone value 120591

0is set to 10119890 minus 6 the

pheromone update constant 119876 is set to 20 the explorationconstant 119902

0is set to 1 the global pheromone decay rate 120588

119892is

09 the local pheromone decay rate 120588119897is 05 the pheromone

sensitivity 120572 is 1 and the visibility sensitivity is 120573 is 5The first case is that of a symmetrically excited 300-

element linear array The total number of unknowns forthis case is 150 Table 3 reports the comparative results Wenotice that the sinusoidal model and the model 7 seem tooutperform the other algorithms Bothmodels have obtainedthe same best value Both BPSO variants perform worse thanthe BBOalgorithmsACOandVBPSOperform similarlyTheconvergence rate graph for this case is depicted in Figure 1 AllBBOmodels seem to converge at similar speed faster than theother algorithms The VBPSO converges faster than BPSOIt must be pointed out that the BBO algorithms converge

6 International Journal of Antennas and Propagation

Table 3 Comparative results for 300-element symmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2454 minus2424 minus2443 0096Sinusoidal minus2467 minus2417 minus2441 0146Model 7 minus2467 minus2427 minus2443 0110Model 8 minus2444 minus2351 minus2411 0243VBPSO minus2432 minus2367 minus2405 0175BPSO minus2110 minus1988 minus2035 0326BHS minus2410 minus2279 minus2361 0431ODE minus2406 minus2330 minus2376 0203ACO minus2420 minus2393 minus2406 0084

Table 4 Comparative results for 300-element symmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2445 minus2406 minus2432 0112Sinusoidal minus2459 minus2430 minus2445 0087Model 7 minus2470 minus2427 minus2441 0127Model 8 minus2456 minus2364 minus2418 0257VBPSO minus2441 minus2397 minus2422 0141BPSO minus2150 minus2052 minus2095 0281BHS minus2416 minus2290 minus2379 0413ODE minus2430 minus2351 minus2391 0231ACO minus2336 minus2276 minus2316 0171

at their final value at less than 300 iterations The radiationpattern of the best obtained result is shown in Figure 2 Thebest array found is filled 72 and has a peak SLL ofminus2467 dBAdditionally we study the effect of maintaining the sameaperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon) for a design case The comparative results for this caseare shown in Table 4 The sinusoidal model presents the bestperformance except the best value found where the model 7outperforms the others VBSO is completive with the BBOalgorithms for this case and achieves better performance thanBBOwithmodel 8 ODE performs better than ACO and BHSfor this case Figure 3 shows the convergence rate for thiscase It is obvious that all BBO algorithms converge at similarspeeds while the PSO algorithms converge slower The bestobtained radiation pattern is shown in Figure 4The obtainedpeak SLL value is minus2470 dB slightly smaller than the previouscaseThe array filling percentage is again 72We notice thatthere is not a significant difference if we choose an array with-out constraints or with the same aperture size for this case

Next we consider asymmetric array designs in which theproblem dimension is 300 Therefore we evaluate the algo-rithms ability to solve high dimensional problems It mustbe pointed out that although the number of unknowns hasdoubled compared with the symmetric case the populationsize and iterations remain the same as previously Thereforethe level of difficulty increases for all algorithms Table 5reports the comparative results for this case The algorithms

minus16

minus18

minus20

minus22

minus24

Number of iterations

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Avg

cost

func

tion

10009008007006005004003002001000

Figure 1 Convergence rate graph for the 300-element symmetricthinned array case

Nor

mal

ized

far fi

eld

(dB)

111111111111111111111111111111111111111111111111111111111111111111101111111101011101011111011010001001100111101001110000000000100001010110010000111001

Filled 72

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

Symmetric linear arrayN = 300 elements

Max SLL = minus2467dB

1080604020minus02minus04minus06minus08minus1

Figure 2 Radiation pattern of the best obtained array for the 300-element symmetric thinned array

performance differences are clearer in this case It is obviousthat the sinusoidal model clearly outperforms the others inthis case The results obtained by model 7 and linear modelare quite similar in this case The VBPSO results are betteragain than BPSO BHS ODE and ACO but worse than theBBO algorithms Figure 5 shows the convergence rate graphfor this case Model 7 seems to convergence slightly fasterthan the other algorithms The radiation pattern of the bestobtained array is shown in Figure 6 The obtained array SLLis minus2611 dB and the fill percentage is about 72

In order to further evaluate the algorithms performancewe choose again an asymmetric array case with the sameaperture length Table 6 holds the comparative results Theresults show that sinusoidal model clearly outperforms theother algorithms We notice that VBPSO and BPSO resultsare worse than the BBO algorithms BHS is completive withthe VBPSO algorithm for this case The convergence rategraph of Figure 7 shows that models 7 and 8 converge slightly

International Journal of Antennas and Propagation 7

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

Avg

cost

func

tion

Figure 3 Convergence rate graph for the 300-element symmetricthinned array case with same aperture size

111111111111111111111111111111111111111111111111111111111111111111111011111111001111011111000100110111001001110000101010010001000000001000101001101011

Same aperture length

Nor

mal

ized

far fi

eld

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

1080604020minus02minus04minus06minus08minus1

Filled 72

Symmetric linear arrayN = 300 elements

Max SLL = minus2470dB

Figure 4 Radiation pattern of the best obtained array for the 300-element symmetric thinned array with same aperture length

Table 5 Comparative results for 300-element asymmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2603 minus2565 minus2583 0119Sinusoidal minus2611 minus2573 minus2588 0110Model 7 minus2605 minus2558 minus2582 0146Model 8 minus2557 minus2421 minus2513 0362VBPSO minus2498 minus2349 minus2436 0468BPSO minus2083 minus1998 minus2032 0228BHS minus2532 minus2463 minus2502 0214ODE minus2488 minus2351 minus2419 0424ACO minus2210 minus2155 minus2183 0187

faster than the linear and the sinusoidal model for this caseAgain it is obvious that the BBO algorithms require feweriterations than the other algorithms in order to reach the

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 5 Convergence rate graph for the 300-element asymmetricthinned array

Nor

mal

ized

far fi

eld

(dB)

111110110000010100010000000011001100110100101001010011011100001110100011111101101111011111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111101111111111011101111101101111101001010100101010011100110000010101010100000100000111

Asymmetric linear array

Filled 727

0

minus10

minus20

minus30

minus401080604020minus02minus04minus06minus08minus1

u

N = 300 elements

Max SLL = minus2611dB

Figure 6 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array

Table 6 Comparative results for 300-element asymmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2597 minus2558 minus2578 0113Sinusoidal minus2608 minus2577 minus2591 0107Model 7 minus2587 minus2560 minus2576 0081Model 8 minus2567 minus2446 minus2507 0383VBPSO minus2490 minus2254 minus2392 0658BPSO minus2080 minus1987 minus2021 0276BHS minus2558 minus2392 minus2476 0633ODE minus2492 minus2344 minus2426 0522ACO minus2191 minus2163 minus2177 0008

final values Figure 8 presents the radiation pattern of thebest obtained arrayThe peak SLL is minus2608 dB and the filling

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

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Chemical EngineeringInternational Journal of Antennas and

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6 International Journal of Antennas and Propagation

Table 3 Comparative results for 300-element symmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2454 minus2424 minus2443 0096Sinusoidal minus2467 minus2417 minus2441 0146Model 7 minus2467 minus2427 minus2443 0110Model 8 minus2444 minus2351 minus2411 0243VBPSO minus2432 minus2367 minus2405 0175BPSO minus2110 minus1988 minus2035 0326BHS minus2410 minus2279 minus2361 0431ODE minus2406 minus2330 minus2376 0203ACO minus2420 minus2393 minus2406 0084

Table 4 Comparative results for 300-element symmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2445 minus2406 minus2432 0112Sinusoidal minus2459 minus2430 minus2445 0087Model 7 minus2470 minus2427 minus2441 0127Model 8 minus2456 minus2364 minus2418 0257VBPSO minus2441 minus2397 minus2422 0141BPSO minus2150 minus2052 minus2095 0281BHS minus2416 minus2290 minus2379 0413ODE minus2430 minus2351 minus2391 0231ACO minus2336 minus2276 minus2316 0171

at their final value at less than 300 iterations The radiationpattern of the best obtained result is shown in Figure 2 Thebest array found is filled 72 and has a peak SLL ofminus2467 dBAdditionally we study the effect of maintaining the sameaperture length as the original uniform array for this caseWetherefore force the first and the last element to be one (turnedon) for a design case The comparative results for this caseare shown in Table 4 The sinusoidal model presents the bestperformance except the best value found where the model 7outperforms the others VBSO is completive with the BBOalgorithms for this case and achieves better performance thanBBOwithmodel 8 ODE performs better than ACO and BHSfor this case Figure 3 shows the convergence rate for thiscase It is obvious that all BBO algorithms converge at similarspeeds while the PSO algorithms converge slower The bestobtained radiation pattern is shown in Figure 4The obtainedpeak SLL value is minus2470 dB slightly smaller than the previouscaseThe array filling percentage is again 72We notice thatthere is not a significant difference if we choose an array with-out constraints or with the same aperture size for this case

Next we consider asymmetric array designs in which theproblem dimension is 300 Therefore we evaluate the algo-rithms ability to solve high dimensional problems It mustbe pointed out that although the number of unknowns hasdoubled compared with the symmetric case the populationsize and iterations remain the same as previously Thereforethe level of difficulty increases for all algorithms Table 5reports the comparative results for this case The algorithms

minus16

minus18

minus20

minus22

minus24

Number of iterations

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Avg

cost

func

tion

10009008007006005004003002001000

Figure 1 Convergence rate graph for the 300-element symmetricthinned array case

Nor

mal

ized

far fi

eld

(dB)

111111111111111111111111111111111111111111111111111111111111111111101111111101011101011111011010001001100111101001110000000000100001010110010000111001

Filled 72

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

Symmetric linear arrayN = 300 elements

Max SLL = minus2467dB

1080604020minus02minus04minus06minus08minus1

Figure 2 Radiation pattern of the best obtained array for the 300-element symmetric thinned array

performance differences are clearer in this case It is obviousthat the sinusoidal model clearly outperforms the others inthis case The results obtained by model 7 and linear modelare quite similar in this case The VBPSO results are betteragain than BPSO BHS ODE and ACO but worse than theBBO algorithms Figure 5 shows the convergence rate graphfor this case Model 7 seems to convergence slightly fasterthan the other algorithms The radiation pattern of the bestobtained array is shown in Figure 6 The obtained array SLLis minus2611 dB and the fill percentage is about 72

In order to further evaluate the algorithms performancewe choose again an asymmetric array case with the sameaperture length Table 6 holds the comparative results Theresults show that sinusoidal model clearly outperforms theother algorithms We notice that VBPSO and BPSO resultsare worse than the BBO algorithms BHS is completive withthe VBPSO algorithm for this case The convergence rategraph of Figure 7 shows that models 7 and 8 converge slightly

International Journal of Antennas and Propagation 7

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

Avg

cost

func

tion

Figure 3 Convergence rate graph for the 300-element symmetricthinned array case with same aperture size

111111111111111111111111111111111111111111111111111111111111111111111011111111001111011111000100110111001001110000101010010001000000001000101001101011

Same aperture length

Nor

mal

ized

far fi

eld

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

1080604020minus02minus04minus06minus08minus1

Filled 72

Symmetric linear arrayN = 300 elements

Max SLL = minus2470dB

Figure 4 Radiation pattern of the best obtained array for the 300-element symmetric thinned array with same aperture length

Table 5 Comparative results for 300-element asymmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2603 minus2565 minus2583 0119Sinusoidal minus2611 minus2573 minus2588 0110Model 7 minus2605 minus2558 minus2582 0146Model 8 minus2557 minus2421 minus2513 0362VBPSO minus2498 minus2349 minus2436 0468BPSO minus2083 minus1998 minus2032 0228BHS minus2532 minus2463 minus2502 0214ODE minus2488 minus2351 minus2419 0424ACO minus2210 minus2155 minus2183 0187

faster than the linear and the sinusoidal model for this caseAgain it is obvious that the BBO algorithms require feweriterations than the other algorithms in order to reach the

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 5 Convergence rate graph for the 300-element asymmetricthinned array

Nor

mal

ized

far fi

eld

(dB)

111110110000010100010000000011001100110100101001010011011100001110100011111101101111011111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111101111111111011101111101101111101001010100101010011100110000010101010100000100000111

Asymmetric linear array

Filled 727

0

minus10

minus20

minus30

minus401080604020minus02minus04minus06minus08minus1

u

N = 300 elements

Max SLL = minus2611dB

Figure 6 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array

Table 6 Comparative results for 300-element asymmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2597 minus2558 minus2578 0113Sinusoidal minus2608 minus2577 minus2591 0107Model 7 minus2587 minus2560 minus2576 0081Model 8 minus2567 minus2446 minus2507 0383VBPSO minus2490 minus2254 minus2392 0658BPSO minus2080 minus1987 minus2021 0276BHS minus2558 minus2392 minus2476 0633ODE minus2492 minus2344 minus2426 0522ACO minus2191 minus2163 minus2177 0008

final values Figure 8 presents the radiation pattern of thebest obtained arrayThe peak SLL is minus2608 dB and the filling

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

Avg

cost

func

tion

Figure 3 Convergence rate graph for the 300-element symmetricthinned array case with same aperture size

111111111111111111111111111111111111111111111111111111111111111111111011111111001111011111000100110111001001110000101010010001000000001000101001101011

Same aperture length

Nor

mal

ized

far fi

eld

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

u

1080604020minus02minus04minus06minus08minus1

Filled 72

Symmetric linear arrayN = 300 elements

Max SLL = minus2470dB

Figure 4 Radiation pattern of the best obtained array for the 300-element symmetric thinned array with same aperture length

Table 5 Comparative results for 300-element asymmetric thinnedarray The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2603 minus2565 minus2583 0119Sinusoidal minus2611 minus2573 minus2588 0110Model 7 minus2605 minus2558 minus2582 0146Model 8 minus2557 minus2421 minus2513 0362VBPSO minus2498 minus2349 minus2436 0468BPSO minus2083 minus1998 minus2032 0228BHS minus2532 minus2463 minus2502 0214ODE minus2488 minus2351 minus2419 0424ACO minus2210 minus2155 minus2183 0187

faster than the linear and the sinusoidal model for this caseAgain it is obvious that the BBO algorithms require feweriterations than the other algorithms in order to reach the

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 5 Convergence rate graph for the 300-element asymmetricthinned array

Nor

mal

ized

far fi

eld

(dB)

111110110000010100010000000011001100110100101001010011011100001110100011111101101111011111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111101111111111011101111101101111101001010100101010011100110000010101010100000100000111

Asymmetric linear array

Filled 727

0

minus10

minus20

minus30

minus401080604020minus02minus04minus06minus08minus1

u

N = 300 elements

Max SLL = minus2611dB

Figure 6 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array

Table 6 Comparative results for 300-element asymmetric thinnedarray with same aperture size The smaller values are in bold

Migration model Best Worst Mean St devLinear minus2597 minus2558 minus2578 0113Sinusoidal minus2608 minus2577 minus2591 0107Model 7 minus2587 minus2560 minus2576 0081Model 8 minus2567 minus2446 minus2507 0383VBPSO minus2490 minus2254 minus2392 0658BPSO minus2080 minus1987 minus2021 0276BHS minus2558 minus2392 minus2476 0633ODE minus2492 minus2344 minus2426 0522ACO minus2191 minus2163 minus2177 0008

final values Figure 8 presents the radiation pattern of thebest obtained arrayThe peak SLL is minus2608 dB and the filling

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

Linear modelSinusoidal modelModel 7Model 8BPSO

VBPSOBHSODEACO

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

Avg

cost

func

tion

Figure 7 Convergence rate graph for the 300-element asymmetricthinned array with same aperture size

101000100011010000100101001001100011100010100111011000111111111011111101110111111111111111011111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111101001111101110111011101110000001011011001001010001011000001000000100100100110001101111

Asymmetric linear array

Filled 727

Same aperture length

u

1080604020minus02minus04minus06minus08minus1

Nor

mal

ized

far fi

eld

(dB)

0

minus10

minus20

minus30

minus40

Max SLL = minus2608dB

N = 300 elements

Figure 8 Radiation pattern of the best obtained array for the 300-element asymmetric thinned array with same aperture length

percentage is 73 Again the results are quite similar with theprevious case with the peak SLL slightly higher than previous

The final example is that of planar thinned array Weconsider a 50times20planar array All elements are equally spacedalong 119909- and 119910-axis at half-wavelength distance Again apopulation of 200 vectors is selected for all algorithms Thetotal number of generations is set to 500 Table 7 holds thecomparative results for the planar array case The sinusoidalmodel obtains the best value However the model 7 resultsare better than the results of the other algorithms models interms of best mean value The convergence rate graph forthis case is shown in Figure 9 The BBO algorithms seem toconverge at similar speed faster than the other algorithmsThe 3D radiation pattern of the best obtained array is shownin Figure 10(a) Figure 10(b) shows the array pattern at the twophi-planes For120601 = 0

∘ the PSLL isminus3396 dBwhile for120601 = 90∘

the PSLL is minus3385 dB The filling percentage is 476

Number of iterations10009008007006005004003002001000

minus16

minus18

minus20

minus22

minus24

minus26

minus28

minus30

minus32

minus34

Avg

cost

func

tion

LinearSinusoidalModel 7Model 8BPSO

VBPSOBHSODEACO

Figure 9 Convergence rate graph for the planar thinned arraydesign case

Table 7 Comparative results for 1000-element planar thinned arrayThe smaller values are in bold

Migration model Best Worst Mean St devLinear minus3234 minus3080 minus3138 0421Sinusoidal minus3385 minus3011 minus3213 1076Model 7 minus3344 minus3115 minus3227 0710Model 8 minus3230 minus2934 minus3055 0899VPBSO minus3155 minus2844 minus2955 1107BPSO minus2187 minus2089 minus2127 0311BHS minus3196 minus2778 minus2984 1290ODE minus2463 minus2290 minus2340 0454ACO minus2519 minus2398 minus2458 0346

Overall BBO with sinusoidal model has been ranked firstregarding mean and standard deviation values in 3 out ofthe 5 array design cases presented Additionally BBO withsinusoidal model in 4 out of the 5 design cases producedthe best result The results obtained by the BBO with model7 outperformed the linear model in 2 out of the 5 casesIn all cases BBO with migration model 8 seems to performworse than the other models The VBPSO has outperformedthe original BPSO in all cases and the BBO with migrationmodel 8 in 2 out of the 5 cases In order to compare thealgorithms performance on all problems we have conductedthe Friedman test [29] Table 8 shows the average rankingof all algorithms The highest ranking is shown in bold Itis obvious that the best average ranking was obtained by theBBOwith sinusoidalmigrationmodelwhich outperforms theother algorithms VBPSO and BHS perform similarly whileODE outperforms ACO

Similar to the other evolutionary algorithms (EAs) suchas differential evolution (DE) Harmony Search (HS) and

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

0

0

minus20

minus40

minus60

1105

050 0minus05 minus05

minus10

minus20

minus30

minus40

minus50

minus60

minus70minus1 minus1

u

|AF|

(dB)

(a)

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

far fi

eld

(dB)

Planar array

Filled 476

120579 (deg)

0

minus10

minus20

minus30

minus40

minus50

minus60

N = 1000 elements

120601 = 0∘

120601 = 90∘

120601 = 0∘ max SLL = minus3396dB120601 = 90∘ max SLL = minus3385dB

(b)

Figure 10 Best obtained 1000-element planar array (a) 3D radiationpattern of the (b) far-field patterns at 120601 = 0

∘ and 120601 = 90∘ planes

Table 8 Average algorithm rankings obtained by Friedman test

Algorithm Average rankingLinear 21Sinusoidal 18Model 7 21Model 8 44VBPSO 58BPSO 88BHS 58ODE 66ACO 72

Particle Swarm Optimization (PSO) in the BBO approachthere is a way of sharing information between solutions[16] This feature makes BBO suitable for the same typesof problems that the other algorithms are used for namelyhigh dimensional data Additionally BBO has some uniquefeatures that are different from those found in the otherevolutionary algorithms For example quite different fromDE and PSO from one generation to the next the set ofthe BBOrsquos solutions is maintained and improved using themigration model where the emigration and immigrationrates are determined by the fitness of each solution BBOdiffers from PSO in the fact that PSO solutions do not changedirectly the velocities change The BBO solutions share

directly their attributes using the migration models Thesedifferences can make BBO outperform other algorithms[16 17 30] It must be pointed out that if PSO or DE areconstrained to discrete space then the next generation willnot necessarily be discrete [30] However this is not truefor BBO if BBO is constrained to a discrete space then thenext generation will also be discrete to the same space Asthe authors in [30] suggest this indicates that BBO couldperformbetter than other EAs on combinatorial optimizationproblems which makes BBO suitable for application to theantenna array thinning problems

43 Boundary Conditions Constraint Handling MethodsStudy In this subsection we apply different boundary con-ditions handling methods for the BBO with the sinusoidalmodel in order to find if the settings used in the previous sec-tions could be improved The boundary constraint handlingmethods that we will test include the following [31 32]

(1) Reflection Method

119909V119895=

2119909119871119895

minus 119909119895

if 119909119895lt 119909119871119895

2119909119880119895

minus 119909119895

if 119909119895gt 119909119880119895

(24)

where 119909V119895is a valid value 119909

119895is the value which violates the

bound constraint and 119909119871119895

and 119909119880119895

are the lower and upperbounds for the 119895th variable respectively

(2) Projection Method

119909V119895=

119909119871119895

if 119909119895lt 119909119871119895

119909119880119895

if 119909119895gt 119909119880119895

(25)

In this case the variables that violate the bound con-straints are trimmed to the lower and upper bounds respec-tively

(3) Reinitialization by Position In this case each variable thatviolates the constraints is randomly reinitialized with

119909V119895= rand

119895[01](119909119880119895

minus 119909119871119895) + 119909119871119895 (26)

where rand119895[01]

is a uniformly distributed random numberbetween 0 and 1

(4) Reinitialize All In this case if at least one of the solutionvariables violates the boundaries a complete new vector isgenerated within the allowed boundaries using (26)

(5) Conservatism This technique was proposed to workparticularly with DE In this case the infeasible solution isrejected and it is replaced by the original feasible vector

The test function we use in all cases is a symmetricallythinned array with 100 elements We run each case for 50independent trials for each different boundary conditionsetting The population size is set to 200 and the number ofiterations to 1000 The best value the worst value the meanand the standard deviation at the last generation are presented

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

Table 9 Comparative results using different boundary conditionsconstraint handling methods The smaller values are in bold

Boundary condition Best Worst Mean St devProjection minus2111 minus2050 minus2080 0138Reflection minus2102 minus2041 minus2077 0143Reinitialization minus2128 minus2049 minus2076 0155Reinitialize all minus2108 minus2041 minus2073 0159Conservatism minus2133 minus2038 minus2078 0189

here Table 9 holds the comparative results for all methodsWe notice that the projection method in this case hasobtained the bestmean worst and standard deviation valuesThe conservatism method has obtained the best objectivefunction value The mean value results for all methods areclose while the best obtained values seem to differ

5 Conclusion

In this paper we have addressed the problem of designingthinned arrays using the BBO algorithm We have comparedperformance of four different BBO migration models withother popular EAs The results showed that BBO withsinusoidal model is highly competitive for the array thinningproblem BBO with sinusoidal model outperformed theother migration models in general All the BBO algorithmsconverge faster and produce better results than the otheralgorithms However these results are considered to beindicative and cannot be generalized in all array designproblems Further tests should be carried out to test theBBO performance on other array design problems The BBOalgorithm is a powerful and efficient optimizer especially incombinatorial optimization problems In our future work wewill study further the performance of BBOmigration modelsto different antenna design problems

Competing Interests

The authors declare that they have no competing interests

References

[1] R Jain and G S Mani ldquoSolving lsquoantenna array thinningproblemrsquo using genetic algorithmrdquoApplied Computational Intel-ligence and Soft Computing vol 2012 Article ID 946398 14pages 2012

[2] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[3] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 I pp 556ndash5672007

[4] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[5] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and CyberneticsComputational Cybernetics and Simulation pp 4104ndash4108IEEE Orlando Fla USA 1997

[6] S Mirjalili and A Lewis ldquoS-shaped versus V-shaped transferfunctions for binary Particle Swarm Optimizationrdquo Swarm andEvolutionary Computation vol 9 pp 1ndash14 2013

[7] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo 1995 httpciteseeristpsueduarticlestorn95differen-tialhtml

[8] 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

[9] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[10] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation (CIMCA rsquo05) and International Conferenceon Intelligent Agents Web Technologies and Internet Commerce(IAWTIC rsquo05) vol 1 pp 695ndash701 Vienna Austria November2005

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] S-H Yang and J-F Kiang ldquoOptimization of sparse lineararrays using harmony search algorithmsrdquo IEEE Transactions onAntennas and Propagation vol 63 no 11 pp 4732ndash4738 2015

[13] K Guney and M Onay ldquoOptimal synthesis of linear antennaarrays using a harmony search algorithmrdquo Expert Systems withApplications vol 38 no 12 pp 15455ndash15462 2011

[14] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[15] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[16] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[17] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[18] WGuo LWang andQWu ldquoAn analysis of themigration ratesfor biogeography-based optimizationrdquo Information Sciencesvol 254 pp 111ndash140 2014

[19] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[20] M R Lohokare S S Pattnaik S Devi K M Bakwad andJ G Joshi ldquoParameter calculation of rectangular microstripantenna using biogeography-based optimizationrdquo in Proceed-ings of the Applied Electromagnetics Conference (AEMC rsquo09) pp1ndash4 Kolkata India December 2009

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 11

[21] M R Lohokare S S Pattnaik S Devi B K Panigrahi KM Bakwad and J G Joshi ldquoModified BBO and calculation ofresonant frequency of circular microstrip antennardquo in Proceed-ings of the World Congress on Nature and Biologically InspiredComputing (NABIC rsquo09) pp 487ndash492 IEEE Coimbatore IndiaDecember 2009

[22] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[23] A Sharaqa and N Dib ldquoDesign of linear and circular antennaarrays using biogeography based optimizationrdquo in Proceedingsof the 2011 1st IEEE Jordan Conference on Applied ElectricalEngineering and Computing Technologies (AEECT rsquo11) pp 1ndash6Amman Jordan December 2011

[24] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[25] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[26] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[27] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[28] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[30] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[31] J Arabas A Szczepankiewicz and T Wroniak ldquoExperimen-tal comparison of methods to handle boundary constraintsin differential evolutionrdquo in Parallel Problem Solving fromNature PPSNXI 11th International Conference Krakow PolandSeptember 11ndash15 2010 Proceedings Part II R Schaefer C CottaJ Kołodziej andG Rudolph Eds pp 411ndash420 Springer BerlinGermany 2010

[32] E Juarez-Castillo N Perez-Castro and E Mezura-Montes ldquoAnovel boundary constraint-handling technique for constrainednumerical optimization problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo15) pp 2034ndash2041 Sendai Japan May 2015

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of