Research ArticlePattern Synthesis of Linear Antenna Arrays Using EnhancedFlower Pollination Algorithm

Urvinder Singh and Rohit Salgotra

Department of ECE Thapar University Patiala India

Correspondence should be addressed to Rohit Salgotra cassonova00gmailcom

Received 18 October 2016 Revised 9 December 2016 Accepted 18 January 2017 Published 20 February 2017

Academic Editor Shih Yuan Chen

Copyright copy 2017 Urvinder Singh and Rohit Salgotra 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

In this paper a new variant of flower pollination algorithm (FPA) namely enhanced flower pollination algorithm (EFPA) has beenproposed for the pattern synthesis of nonuniform linear antenna arrays (LAA)The proposed algorithm uses the concept of Cauchymutation in global pollination and enhanced local search to improve the exploration and exploitation tendencies of FPA It also usesdynamic switching to control the rate of exploration and exploitation The algorithm is tested on standard benchmark problemsand has been compared statistically with state of the art to prove its worthiness LAA design is a tricky and difficult electromagneticproblem Hence to check the efficacy of the proposed algorithm it has been used for synthesis of four different LAA with differentsizes Experimental results show that EFPA algorithm provides enhanced performance in terms of side lobe suppression and nullcontrol compared to FPA and other popular algorithms

1 Introduction

Antenna arrays find their application in number of wirelessapplications such as radar sonar mobile TV and satelliteAntenna arrays are favored as these have high directivityreduce power consumption increase spectral efficiency andalso have beam steering capability [1] The design of antennaarrays with desired radiation pattern has found great interestin the literature Antenna arrays design is intricate and non-linear problem Hence number of optimization techniquessuch as genetic algorithm (GA) differential algorithm (DE)particle swarmoptimization (PSO) biogeography based opti-mization (BBO) and many others [2ndash21] have been used tosynthesize these It is required that antenna arrays radiate indesired directions so that these do no add to electromagneticpollution This can be achieved if the energy is maximum inthemain lobe andminimum in side lobesMoreover to avoidinterference from undesired directions and to circumventjamming null placement in radiation pattern has also gainedimportance So overall the main issues in designing theantenna arrays are reducing the side lobe level (SLL) andplacing nulls in the desired directions of radiation pattern

Linear antenna arrays (LAA) consist of number ofantenna elements arranged in a straight line LAA have

become very popular because of their simple geometry andapplications LAA design has been investigated by numerousresearchers using several optimization algorithms in the pastRecioui has used PSO for SLL reduction of LAA [2] Khodierand Christodoulou have synthesized LAA for maximumSLL reduction and null placement They have designedthree arrays for different sizes and showed that PSO givesbetter results than uniform and quadrature programmingmethod (QPM) [3] Rattan et al have applied GA [7] forthe optimization of LAA Their results were better thanthose obtained using PSO [3] DE has been employed byLin et al to design unequally spaced LAA [8] They havediscussed about the impact of angular resolution on the finalresults Dib et al have compared the performance of self-adaptive DE (SADE) and Tagachirsquos method for optimizationof LAA [9] Cengiz and Tokat have used GA memeticalgorithm (MA) and tabu search (TS) to optimize threedifferent LAA [10] Singh et al [11] have applied BBO tosynthesize uniform and nonuniform LAA and found thatBBO is better than PSO and other methods for the designof arrays BBO has also been used by Sharaqa and Dib todesign LAA [12] Ant colony optimization has been used byRajo-lglesias and Quevedo-Teruel for the synthesis of LAA

HindawiInternational Journal of Antennas and PropagationVolume 2017 Article ID 7158752 11 pageshttpsdoiorg10115520177158752

2 International Journal of Antennas and Propagation

for SLL reduction and null placement [13] A multiobjectiveapproach using multiobjective DE (MOEAD-DE) has beenemployed by Pal et al to optimize LAA [14] The authorshave noted that MOEAD-DE gives better trade-off curvesbetween null placement and SLL LAA synthesis using fitnessadaptive differential evolution algorithm (fiADE) has beendone by Chowdhury et al [15] Guney andOnay have utilizedharmony search algorithm (HSA) for SLL minimization andnull placement in different directions of radiation pattern ofLAA [16] A new algorithm based on the breeding strategy ofcuckoos known as cuckoo optimization algorithm (COA) hasbeen employed to optimize three different nonuniform LAA[17] The authors have compared the results with the popularalgorithms like DE PSO firefly algorithm (FA) and foundthat the results provided by COA in terms of SLL reductionand null placement are better than competitive algorithmsMoreover convergence curves of different algorithms arecompared and it is concluded that COA provides faster con-vergence than the other methods Guney and Durmus haveused back scattering algorithm (BSA) for the pattern nullingof LAA [18] Khodier has used cuckoo search (CS) to optimizeantenna arrays [19] An enhanced version of PSO namedas comprehensive learning PSO (CLPSO) has been used todesign three different LAA [20] Recently a novel algorithmwhich mimics the behaviour of flowers known as flowerpollination algorithm (FPA) has been used to synthesize LAA[21] Though number of algorithms have been proposed forsynthesis of LAA these suffer from certain shortcomings likegetting stuck in local minima slow convergence speeds andrequire precise parameter tuning So in this work the authorspropose an enhanced version of FPA known as enhanced FPA(EFPA) The newly proposed algorithm uses the concept ofCauchy distribution to follow large steps in global pollinationenhanced local search and dynamic switch probability tocontrol the rate of local and global pollination It has betterexploration and exploitation capability and is also less likelyto stuck in local minima

The rest of the paper is organized as follows Section 2gives details about the basic FPA algorithm Section 3 pro-poses a new EFPA algorithm Section 4 gives result anddiscussion and Section 5 provides an extensive conclusion

2 Flower Pollination Algorithm

Flowers are the most fascinating plant species and have beendominating earth from the cretaceous period partly fromabout 125 million years Almost 80 of plant species areflowering and it has been made possible by the process ofpollination [22] Pollination in flowers refer to transfer ofpollen grains from one flower to another via pollinators suchas wind diffusion bees birds bats and other animals [23] Ifthe pollination process is through insects it is called bioticpollination and if it takes place via wind or diffusion it iscalled abiotic pollination Overall 200000 varieties of flowerpollinators exist in nature Pollinators such as honey bee hasbeen found to show some specific phenomenon known asflower constancy This phenomenon helps them to visit onlyspecific flower species and by pass others hence maximizingthe transfer of pollens from a particular plant species Thishelps pollinators in finding better food sources andminimize

the exploration or learning cost [24] Pollination can furtherbe classified based upon the pollens of same or differentflowers If a pollinator transfers pollens from one flower toanother it is called cross-pollination while if it does for thesame species it is self-pollination

Based upon the phenomenon of pollination flowerconstancy and behaviour of pollinators flower pollinationalgorithm (FPA) was proposed by Yang [25] He proposedfour sets of rules for governing this algorithm as follows

(1) For long distances the pollination used is biotic cross-pollination the process is called global pollinationand is performed via Levy flights

(2) Abiotic self-pollination process forms the basis oflocal pollination

(3) The reproduction probability of different flowers in-volved in pollination is considered as flower constancy

(4) A switch probability 119901 isin [0 1] is defined forcontrolling local and global pollination

For simplicity a single flower producing only one pollengamete is considered This means a single solution forproblem under test is equivalent to a pollen gamete or aflower From the rules above two key features of global andlocal pollination have been used to formulate FPA

The first step in FPA is global pollination where polli-nators such as insects carry pollen to long distances Thisprocess helps in pollination of the best fit flower or solutionso far and is represented as 119877lowast Rule 1 with a combination offlower constancy is represented as

119909119905+1119894 = 119909119905119894 + 119871 (120582) (119877lowast minus 119909119905119894) (1)

where 119909119905119894 is the 119894th solution of the problem in the 119905th iteration119871 is the step size and Levy flight is generally used to mimicthis phenomenon [25] Levy flight is drawn from a Levydistribution as

119871 sim 120582Γ (120582) sin (1205871205822)120587 11199041+120582 (119904 ≫ 1199040 gt 0) (2)

where Γ(120582) is the standard gamma function with a step sizeof 119904 gt 0

In the second step Rule 2 is used along with flowerconstancy to represent local pollination as

119909119905+1119894 = 119909119905119894 + 120598 (119909119905119895 minus 119909119905119896) (3)

where 119909119905119895 and 119909119905119896 are pollen gametes of same plants butdifferent flowers and 120598 is uniformly distributed in [0 1] Thisphase mimics glower constancy at a local scale or in a limitedsearch space

So we can say that global and local pollination carry outpollination activities at large and small scale respectivelyIn practice we use a switching probability based on Rule 4to define the extent of local and global pollination In thenext section a new enhanced version of flower pollinationalgorithm is proposed

3 Enhanced Flower Pollination Algorithm

In the recent past a large number of researchers have focusedon enhancing the basic capabilities of FPA The algorithm

International Journal of Antennas and Propagation 3

StartInitialize 119899 random flowersDefine switch probability 119901 isin [0 1]Define 119889-dimensional objective function fro problem under testFind current best solution 119877lowast from the initial populationwhile iterations ltmaximum number of iterationsfor 119894 = 1 119899

if rand lt 119901Draw a 119889-dimensional Cauchy step vector obeying Cauchy distributionGlobal Pollination 119909119905+1119894 = 119909119905119894 + 119862(120575)(119877lowast minus 119909119905119894 )elseLocal Pollination 119909119905+1119894 = 119909119905119894 + 119886(119877lowast minus 119909119905119894 ) + 119887(119909119905119895 minus 119909119905119896)end ifevaluate 119909119905+1119894if 119909119905+1119894 is better than 119909119905119894 updateend if

Update 119901 by 119901 = 119901 minus maxiter minus 119905maxiter

times (01)end forUpdate the final best solution

end whileend

Pseudocode 1 Pseudocode of EFPA

due to its linear nature make it suitable for deeper analysisBut it has been proved qualitatively and quantitatively in[26] that the FPA algorithm has a very limited scope foroptimization problems at hand Also the performance of FPAhas not been analyzed to a deeper level and the algorithm isstill to prove its worthiness for becoming a state-of-the-artalgorithm Keeping in view the above analysis a new versionof FPA namely EFPA has been proposedThe newly proposedEFPA aims at providing three different modifications to thebasic FPA These include Cauchy based global pollinationenhanced local pollination based upon experience of currentbest flower pollinator as well as local flowers in proximityand thirdly using dynamic switching probabilities

(I) Cauchy Based Global Pollination In the global pollinationphase instead of using a standard Levy distribution a Cauchybased operator 119862(120575) is used This operator is basically aCauchy random variable with distribution given by

120575 = 12 + 1120587arctan(119909119892) (4)

The Cauchy density function is given by

119891Cauchy(0119892) (119909) = 1120587 1198921198922 + 1199092 (5)

The general equation of global pollination becomes

119909119905+1119894 = 119909119905119894 + 119862 (120575) (119877lowast minus 119909119905119894) (6)

where 119892 is a scale parameter and its value is generally setto 1 The use of Cauchy operator allows for larger mutationby searching the search space at a faster pace and furtheraccounts for avoiding premature convergence

(II) Enhanced Local Pollination The second modification isadded in the local pollination phase Here based upon theexperience of local and current best pollinators the positionof new pollinators is updated Further if the fitness of newposition is greater than the old one each pollinator updatesits position with respect to the previous one The generalequation is given by

119909119905+1119894 = 119909119905119894 + 119886 (119877lowast minus 119909119905119894) + 119887 (119909119905119895 minus 119909119905119896) (7)

where a and b are uniformly distributed random numbers inthe range of [0 1] Also the solutions 119909119905119895 and 119909119905119896 correspondsto 119895th and 119896th flower pollinator in the group with 119895 = 119896 Thisphase enhances the local search capabilities of FPA algorithm

(III) Dynamic Switch Probability In FPA local and global pol-linations are controlled by switching probability 119901 isin [0 1]For a standard state-of-the-art algorithm it is imperative tofollow more global search at the start and as the algorithmprogresses more intensive local search is followed Using thisbasic concept the value of switching probability is selecteddynamicallyThe switch probability is updated by following ageneral formula as

119901 = 119901 minus maxiter minus 119905maxiter

times (01) (8)

The above general equation decreases the value of p linearlywith iterations and hence adds to intensive global search atthe beginning and local search towards the end Heremaxitercorresponds to the maximum number of iterations and t isthe current iteration The pseudocode for the EFPA is givenin Pseudocode 1

4 International Journal of Antennas and Propagation

Table 1 Description of test functions

Test problems Objective function Search range Optimumvalue 119863

Spherefunction

1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100] 0 10

Weierstrassfunction

1198912 (119909) = 119863sum119894=1

119896maxsum119896=0

[119886119896 cos (2120587119887119896 (119909119894 + 05))]minus 119863119896maxsum119896=0

[119886119896 cos(2120587119887119896 sdot 05)] where 119886 = 05 119887 = 3 119896max = 20 [minus05 05] 0 30

Schafferfunction 1198913 (119909) = [ 1119899 minus 1radic119904119894 sdot (sin (50011990415119894 ) + 1)]2 119904119894 = radic1199092119894 + 1199092119894+1 [minus100 100] 0 30

Griewank 1198914 (119909) = 14000119873sum119894=1

1199092119894 minus119873prod119894=1

cos( 119909119894radic119894) + 1 [minus600 600] 0 30

Penalized 1function

1198915(119909) = 120587119899 10sin2 (1205871199101) + 119899minus1sum119894=1

(119910119894 minus 1)2 [1 + 10sin2 (120587119910119894+1)] + (119910119899 minus 1)2+ 119899sum119894=1

119906 (119909119894 10 100 4)[minus50 50] 0 30

119910119894 = 1 + 119909119894+14 119906(119909119894 119886 119896 119898) =

119896 (119909119894 minus 119886)119898 119909119894 gt 1198860 minus119886 lt 119909119894 lt 119886119896 (minus119909119894 minus 119886)119898 119909119894 lt minus119886

Ackleyfunction

1198916 (119909) = minus20 exp(minus02radic 1119863119863sum119894=1

1199092119894) minus exp( 1119863119863sum119894=1

cos (2120587119909119894)) + 20 + 119890 [minus100 100] 0 30

Rosenbrockfunction

1198917 =119873minus1sum119894=1

(100 (119909119894+1 minus 1199092119894 ) 2 + (119909119894 minus 1) 2) [minus100 100] 0 30

4 Results and Discussion

41 Results of Benchmark Functions The performance ofEFPA is evaluated by applying it to two set of optimizationproblems One set includes test functions and other is thereal-world application of antenna array designThe algorithmis also compared with other states of the art to prove its com-petitiveness The simulations are performed on Intel Core i3personal computer with Windows 10 operating system usingMATLAB version 7100 (R2010a) EFPA is tested on sevenwell-known benchmark problems [27] as shown in Table 1with search range and optimal solution For comparison pur-poses artificial bee colony (ABC) [28] DE [29] bat algorithm(BA) [30] and FPA [25] are used The maximum number offunction evaluations for each algorithm is set to 30 times 1000with 30 as the population size and 1000 as the maximumnumber of iterationsThe parameter settings of all algorithmsare given in Table 2 All the algorithms are run 50 times andresults in terms of best worst mean and standard deviationare reported Since because of the stochastic nature of algo-rithm Wilcoxon rank-sum test has also been performed tosignificantly prove the better performance of EFPA The per-formance in terms of time-complexity has also been shownin Table 5 to prove its competitiveness Time-complexity isbasically taken for each of the algorithms in the benchmark

problems and it has been calculated for one run for eachalgorithm

The comparison of comparison for various test functionsis given in Table 3 The best and worst values give the mostoptimal and least optimal solution among the 50 solutionsfrom the maximum number of runs Mean gives the averagevalues of results and is used when the best and worst valuesdo not give comparable results But mean may give sameresults for widely different results So standard deviationis also used to calculate results For functions 1198911 1198914 11989151198916 and 1198917 it can be seen that EFPA performs much betterthan other algorithms except for 1198912 where ABC gives betterresults For function 1198913 ABC and EFPA gives same resultsFurther statistical analysis from the Table 4 verify the resultsIn Table 5 time-complexity for each algorithm is shownHere time-complexity show that EFPA takes comparativelyless time for solving the benchmark problem for most ofthe benchmark functions in comparison to FPA BA andABC ForDE only four functions give better time-complexitywhile for the rest three again EFPA is better Overall we cansay that EFPA is superior to ABC DE FPA and BA In thenext subsection EFPA is applied for synthesis of LAA

42 Results of LAA A symmetrical LAA consists of 2119873elements with N elements on both sides of the origin and

International Journal of Antennas and Propagation 5

Table 2 Parameter settings of various algorithms

Algorithm Parameters Values

DE

119865 15CR 08

Population size 30Maximum iterations 1000Stopping criteria Max iteration

ABC

Colony size 30Number of food sources Colony size2

Limit 100Maximum cycles 1000Stopping criteria Max iteration

FPA

Population size 30Probability switch 06Maximum iterations 1000Stopping criteria Max iteration

BA

Population size 50Loudness 05Pulse rate 05

Maximum number of iterations 200Stopping criteria Max iteration

EFPA

Population size 30

Probability switchDynamic and

linearlydecreasing

Maximum iterations 1000Stopping criteria Max iteration

middot middot middot middot middot middot120601

2 n N minus 1 Nx

Figure 1 Geometry of a linear antenna array

arranged in a straight line as shown in Figure 1 The 119899thelement is excited with a current of amplitude 119868119899 and phase120593119899The position of 119899th element is represented by119909119899The arrayfactor (AF) of such an antenna is given by [1]

AF (0) = 2 119873sum119899=1

119868119899 cos [119896119909119899 cos (0) + 120593119899] (9)

Here 119896 is the wave number and is given by 2120587120582 Thesymmetric nature of the array reduces the computational costas only N elements are needed to be optimized Moreoverthese symmetric arrays give symmetric radiation patternwhich is desired for number of applications

The AF for LAA in (9) is a function of excitation currents(both 119868119899 and 120593119899) and positions 119909119899 If current excitations areheld uniform (ie 119868119899 = 1 and 120593119899 = 0∘) then the LAA is

function of 119909119899 only and array is called nonuniform LAAHence for nonuniform LAA AF reduces to

AF (120601) = 2 119873sum119899=1

cos [119896119909119899 cos (120601)] (10)

The design of nonuniform LAA has gained importance inthe past and several researchers have synthesized it usingdifferent optimization techniques [3 7 9 11ndash21] The arrayfactor of a nonuniform array is aperiodic by virtue of nonuni-form spacing between the elements This aperiodic nature ofarrays helps in obtaining lower SLL with lesser number ofelements for a given antenna size Furthermore the uniformelement excitations help in reducing the feed network costand complexity However the relationship between the arrayfactor and element spacing is nonconvex and nonlinear Sothis makes the design of nonuniform arrays difficult

For directing antenna arrays energy in a specific direc-tion it is desired that its radiation pattern should have lowside lobes Moreover nulls are also required in the directionof interferenceThe desired radiation pattern can be achievedby suitably adjusting the positions of antenna elements fornonuniform LAA The adjustment of positions is a trickyand nonlinear problem So optimization algorithms are wellsuited for this type of problems For applying optimization toantenna problem it is imperative to define a fitness functionwhich is given as

fit = SLL + 120572 lowastmax 0 1003816100381610038161003816BW minus BW1198891003816100381610038161003816 minus 1 + 120572

lowast 119870sum119896=1

max 0AFdB (120601119896) minusNudB (11)

In the above fitness function SLL is the side lobe levelAFdB is the array factor in decibels BW and BW119889 are thecalculated and desired beamwidth of the main lobe 120601119896 givesthe direction of the 119896th null NudB is the required null depthin dB 119870 is the number of the required nulls and 120572 is a verylarge numberThe fitness function given in (11) uses a penaltymethod which penalizes any antenna design which does notmeet the required constraints of beamwidth and nulls (ifdesired) This is achieved by taking value 120572 to be very largeThe BW here is obtained computationally from the radiationpattern data

421 Linear Antenna Optimization This subsection presentsthe application of EFPA to optimization of nonuniformLAA In the first example a 12-element nonuniform antennais taken for optimization The goal of optimization is tominimize the SLL in the region 120601 isin [98∘ 180∘] Because ofsymmetry there are only six element positions which are tobe optimized The number of generations for EFPA are takenas 1000 The population size taken as 20 in order to makefair comparison between the results of FPA [21] and EFPAThe total number of function evaluations are 20 lowast 1000 =2000 for both the FPA [21] and EFPA As the aim of theoptimization is to suppress the SLL only so the value of 120572 istaken as 0 in the fitness equation of (11) The EFPA algorithmis run for 30 times and best results are listed in this work

6 International Journal of Antennas and Propagation

Table 3 Comparison of results for various test functions

Objective function Algorithm Best Worst Mean Standard deviation

1198911 (119909)DE 443119864 + 03 297119864 + 04 154119864 + 04 622119864 + 03ABC 274119864 + 03 367119864 + 04 274119864 + 04 593119864 + 05BA 808119864 + 03 354119864 + 04 196119864 + 04 701119864 + 03FPA 100119864 + 02 100119864 + 03 361119864 + 02 166119864 + 02EFPA 335119864 minus 15 472119864 minus 07 182119864 minus 08 685119864 minus 08

1198912 (119909)DE 164119864 + 01 382119864 + 01 254119864 + 01 451119864 + 00ABC 253119864 + 00 590119864 + 00 390119864 + 00 840119864 minus 01BA 251119864 + 01 430119864 + 01 336119864 + 01 424119864 + 00FPA 383119864 + 01 572119864 + 01 512119864 + 01 464119864 + 00EFPA 134119864 + 01 266119864 + 01 200119864 + 01 310119864 + 00

1198913 (119909)DE 795119864 minus 05 229119864 minus 01 464119864 minus 02 537119864 minus 02ABC 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00BA 306119864 minus 14 347119864 minus 01 895119864 minus 02 922119864 minus 02FPA 000119864 minus 00 198119864 minus 12 103119864 minus 13 339119864 minus 13EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

1198914 (119909)DE 489119864 + 01 368119864 + 02 154119864 + 02 719119864 + 01ABC 106119864 + 00 176119864 + 00 125119864 + 00 173119864 minus 01BA 294119864 + 00 118119864 + 01 577119864 + 00 163119864 + 00FPA 571119864 minus 01 104119864 + 00 925119864 minus 01 976119864 minus 02EFPA 000119864 minus 00 145119864 minus 01 290119864 minus 03 206119864 minus 02

1198915 (119909)DE 178119864 + 06 692119864 + 08 224119864 + 08 227119864 + 08ABC 349119864 minus 02 277119864 + 07 107119864 + 08 891119864 + 07BA 340119864 + 05 555119864 + 07 158119864 + 07 153119864 + 07FPA 281119864 + 00 111119864 + 01 681119864 + 00 187119864 + 00EFPA 125119864 + 00 31119864 + 01 893119864 + 00 640119864 + 00

1198916 (119909)DE 200119864 + 01 200119864 + 01 200119864 + 01 000119864 + 00ABC 200119864 + 01 202119864 + 01 200119864 + 01 530119864 minus 03BA 197119864 + 01 200119864 + 01 199119864 + 01 292119864 minus 02FPA 205119864 + 01 211119864 + 01 210119864 + 01 837119864 minus 02EFPA 729119864 + 00 133119864 + 01 995119864 + 00 129119864 + 00

1198917 (119909)DE 421119864 + 08 378119864 + 10 863119864 + 09 100119864 + 10ABC 218119864 + 04 361119864 + 05 296119864 + 04 539119864 + 04BA 862119864 + 08 129119864 + 10 307119864 + 09 210119864 + 09FPA 148119864 + 05 649119864 + 06 230119864 + 06 172119864 + 06EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

Table 4 119901-test values of various algorithms

Objective function BA FPA ABC DE EFPA1198911 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 706119864 minus 18 NA1198912 (119909) 706119864 minus 18 706119864 minus 18 NA 325119864 minus 09 706119864 minus 181198913 (119909) 331119864 minus 20 689119864 minus 17 NA 331119864 minus 20 NA1198914 (119909) 473119864 minus 20 473119864 minus 20 473119864 minus 20 473119864 minus 20 NA1198915 (119909) 706119864 minus 18 770119864 minus 02 421119864 minus 04 706119864 minus 18 NA1198916 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 331119864 minus 20 NA1198917 (119909) 331119864 minus 20 106119864 minus 17 331119864 minus 20 331119864 minus 20 NA

International Journal of Antennas and Propagation 7

Table 5 Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds)

Function DE ABC BA FPA EFPA1198911 (119909) 1284464 2527247 1190053 2328157 19659271198912 (119909) 1598023 1562962 1902841 1522211 14801681198913 (119909) 1185347 2470673 1047533 2183295 19005151198914 (119909) 3022755 5167423 4165333 2183153 20305021198915 (119909) 2848487 5988428 2974728 3755994 34984111198916 (119909) 1667705 3186360 2261759 2346810 23835991198917 (119909) 1693011 3265792 2087382 3384049 2999886

Table 6 Geometry of 12-element LAA (with respect to 1205822)obtained using EFPA

Element 1 2 3 4 5 6Position 03765 09734 17376 25051 34827 46744

The optimum positions obtained using EFPA are shown inTable 6The performance of EFPA is contrasted against otherpopular algorithms in terms of SLL in Table 7TheminimumSLL obtained by EFPA isminus2107 dB and is better thanGA [10]FiADE [15] TS [10] MA [10] FPA [21] and PSO [15] Thecomparison of convergence curve between EFPA and FPAis shown in Figure 2 which clearly shows faster convergenceof EFPA than the FPA [21] The FPA converged in about 700iterations whereas EFPA took only 225 iterations to convergeto the final solution This is due to enhanced explorationdue to Cauchy operator and better exploitation in EFPATheradiation pattern of EFPA antenna array is shown in Figure 3For comparison the radiation patterns of GA [10] FiADE[15] and PSO [15] are also shown in Figure 3

In the next example a 22-element nonuniform LAA isoptimized for reducing SLL in the region 120601 isin [98∘ 180∘] andalso imposing a null in the direction of 99∘ For achievingthe required pattern the value of NudB is set to minus60 dB Itis established fact that as SLL is decreased the width of themain lobe in radiation pattern increases So for this designthe desired beamwidth BW119889 is taken as 18∘ The value of 120572is taken as 106 in fitness function The optimum positionsare shown in Table 8 The results are compared in Table 9which clearly indicate better performance of EFPA as againstBSA [18] MOEADE [14] HSA [16] PSO [15] GA [10] MA[10] TS [10] and FPA [21] The SLL of EFPA antenna arrayis minus2631 dB which is much lower than that obtained usingother algorithms Furthermore the null in the direction of99∘ is minus7558 dBwhich is less than the desired value of minus60 dBand deeper than MOEADE [14] GA [7] PSO [3] and MA[10] In order to compare the performance of EFPA and FPAin terms of convergence speed FPA is also run for the sameobjective with same population size The convergence graphsof EFPA and FPA are plotted in the Figure 4The convergencecharacteristics clearly substantiates the superior performanceof EPFA in terms of faster convergence speed The radiationplots of EFPA HSA [16] BSA [18] and MOEADE [14]optimized arrays are shown in Figure 5

In order to show the capability of EFPA in synthesiz-ing the radiation pattern with multiple nulls a 28-element

FPAEFPA

minus22

minus21

minus20

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

Fitn

ess

200 400 600 800 10000Iterations

Figure 2 Convergence plots of 12-element LAA

GAFiADE

PSOEFPA

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 3 Radiation plot of 12-element EFPA optimized LAA

nonuniform LAA is optimized for SLL reduction in theregion 120601 isin [100∘ 180∘] and having nulls in the directionsof 120∘ 1225∘ and 125∘ The desired beamwidth is set to thevalue of 835∘ with tolerance of plusmn1∘ The optimum geometry

8 International Journal of Antennas and Propagation

Table 7 Performance of different algorithms for 12-element LAA

Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPASLL (dB) minus1896 minus1910 minus1877 minus1790 minus20764 minus1840 minus2107

Table 8 Geometry of 22-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11Position 04937 07965 12153 21313 25003 31327 39654 45877 55435 66391 80603

Table 9 Performance of different algorithms for 22-element LAA

Algorithm BSA [18] TS [10] MOEAD-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPASLL (dB) minus2354 minus1717 minus2093 minus2328 minus2068 minus1573 minus1811 minus2381 minus2631Null (dB) (99∘) minus10461 minus6794 minus6964 minus1033 minus4994 minus5429 minus7392 minus10171 minus7558

Table 10 Geometry of 28-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14Position 05880 12142 25291 28291 41205 49374 61006 69646 81374 95186 107399 124399 141399 158399

FPAEFPA

minus25

minus20

minus15

minus10

minus5

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 4 Convergence plots of 22-element LAA

obtained using EFPA is shown in Table 10The comparison ofconvergence curves between EFPA and BBO [17] CS [17] DE[17] FA [17] and COA [17] is given in Figure 6 which showsthat EFPA reaches the optimum solution in less number ofiterations The comparison of results in terms of SLL andnull values achieved is given in Table 11 which again clearlyindicates the superior performance of EFPA as against othermethods The SLL of EFPA is minus2290 dB which is lower by1 dB 924 dB 93 dB 104 dB 826 dB 924 dB 127 dB and030 dB than BSA [18] QPM [3] COA [17] ACO [13] GA[7] CLPSO [20] and FPA [21] respectively The null valuesare also equal to or less than minus60 dB Figure 7 shows the

BSAHSA

MOEAEFPA

Null

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 5 Radiation plot of 22-element EFPA optimized LAA

comparison of QPM [3] HSA [16] BSA [18] and EFPAoptimized array radiation plots

The last design example illustrates the use of EFPA tooptimize the element positions for minimization of SLLand null placement for 32-element LAA The SLL is to beminimized in the region120601 isin [93∘ 180∘] andnull is to be placedin the direction of 99∘ The required beamwidth BW119889 is 71

∘The plot of convergence for EFPA is shown in Figure 8 Alsofor the sake of comparison the convergence plots of BBO[17] CS [17] DE [17] FA [17] and COA [17] are shown in thesame figureThe element positions for EFPA optimized array

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

International Journal of Antennas and Propagation 3

StartInitialize 119899 random flowersDefine switch probability 119901 isin [0 1]Define 119889-dimensional objective function fro problem under testFind current best solution 119877lowast from the initial populationwhile iterations ltmaximum number of iterationsfor 119894 = 1 119899

if rand lt 119901Draw a 119889-dimensional Cauchy step vector obeying Cauchy distributionGlobal Pollination 119909119905+1119894 = 119909119905119894 + 119862(120575)(119877lowast minus 119909119905119894 )elseLocal Pollination 119909119905+1119894 = 119909119905119894 + 119886(119877lowast minus 119909119905119894 ) + 119887(119909119905119895 minus 119909119905119896)end ifevaluate 119909119905+1119894if 119909119905+1119894 is better than 119909119905119894 updateend if

Update 119901 by 119901 = 119901 minus maxiter minus 119905maxiter

times (01)end forUpdate the final best solution

end whileend

Pseudocode 1 Pseudocode of EFPA

due to its linear nature make it suitable for deeper analysisBut it has been proved qualitatively and quantitatively in[26] that the FPA algorithm has a very limited scope foroptimization problems at hand Also the performance of FPAhas not been analyzed to a deeper level and the algorithm isstill to prove its worthiness for becoming a state-of-the-artalgorithm Keeping in view the above analysis a new versionof FPA namely EFPA has been proposedThe newly proposedEFPA aims at providing three different modifications to thebasic FPA These include Cauchy based global pollinationenhanced local pollination based upon experience of currentbest flower pollinator as well as local flowers in proximityand thirdly using dynamic switching probabilities

(I) Cauchy Based Global Pollination In the global pollinationphase instead of using a standard Levy distribution a Cauchybased operator 119862(120575) is used This operator is basically aCauchy random variable with distribution given by

120575 = 12 + 1120587arctan(119909119892) (4)

The Cauchy density function is given by

119891Cauchy(0119892) (119909) = 1120587 1198921198922 + 1199092 (5)

The general equation of global pollination becomes

119909119905+1119894 = 119909119905119894 + 119862 (120575) (119877lowast minus 119909119905119894) (6)

where 119892 is a scale parameter and its value is generally setto 1 The use of Cauchy operator allows for larger mutationby searching the search space at a faster pace and furtheraccounts for avoiding premature convergence

(II) Enhanced Local Pollination The second modification isadded in the local pollination phase Here based upon theexperience of local and current best pollinators the positionof new pollinators is updated Further if the fitness of newposition is greater than the old one each pollinator updatesits position with respect to the previous one The generalequation is given by

119909119905+1119894 = 119909119905119894 + 119886 (119877lowast minus 119909119905119894) + 119887 (119909119905119895 minus 119909119905119896) (7)

where a and b are uniformly distributed random numbers inthe range of [0 1] Also the solutions 119909119905119895 and 119909119905119896 correspondsto 119895th and 119896th flower pollinator in the group with 119895 = 119896 Thisphase enhances the local search capabilities of FPA algorithm

(III) Dynamic Switch Probability In FPA local and global pol-linations are controlled by switching probability 119901 isin [0 1]For a standard state-of-the-art algorithm it is imperative tofollow more global search at the start and as the algorithmprogresses more intensive local search is followed Using thisbasic concept the value of switching probability is selecteddynamicallyThe switch probability is updated by following ageneral formula as

119901 = 119901 minus maxiter minus 119905maxiter

times (01) (8)

The above general equation decreases the value of p linearlywith iterations and hence adds to intensive global search atthe beginning and local search towards the end Heremaxitercorresponds to the maximum number of iterations and t isthe current iteration The pseudocode for the EFPA is givenin Pseudocode 1

4 International Journal of Antennas and Propagation

Table 1 Description of test functions

Test problems Objective function Search range Optimumvalue 119863

Spherefunction

1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100] 0 10

Weierstrassfunction

1198912 (119909) = 119863sum119894=1

119896maxsum119896=0

[119886119896 cos (2120587119887119896 (119909119894 + 05))]minus 119863119896maxsum119896=0

[119886119896 cos(2120587119887119896 sdot 05)] where 119886 = 05 119887 = 3 119896max = 20 [minus05 05] 0 30

Schafferfunction 1198913 (119909) = [ 1119899 minus 1radic119904119894 sdot (sin (50011990415119894 ) + 1)]2 119904119894 = radic1199092119894 + 1199092119894+1 [minus100 100] 0 30

Griewank 1198914 (119909) = 14000119873sum119894=1

1199092119894 minus119873prod119894=1

cos( 119909119894radic119894) + 1 [minus600 600] 0 30

Penalized 1function

1198915(119909) = 120587119899 10sin2 (1205871199101) + 119899minus1sum119894=1

(119910119894 minus 1)2 [1 + 10sin2 (120587119910119894+1)] + (119910119899 minus 1)2+ 119899sum119894=1

119906 (119909119894 10 100 4)[minus50 50] 0 30

119910119894 = 1 + 119909119894+14 119906(119909119894 119886 119896 119898) =

119896 (119909119894 minus 119886)119898 119909119894 gt 1198860 minus119886 lt 119909119894 lt 119886119896 (minus119909119894 minus 119886)119898 119909119894 lt minus119886

Ackleyfunction

1198916 (119909) = minus20 exp(minus02radic 1119863119863sum119894=1

1199092119894) minus exp( 1119863119863sum119894=1

cos (2120587119909119894)) + 20 + 119890 [minus100 100] 0 30

Rosenbrockfunction

1198917 =119873minus1sum119894=1

(100 (119909119894+1 minus 1199092119894 ) 2 + (119909119894 minus 1) 2) [minus100 100] 0 30

4 Results and Discussion

41 Results of Benchmark Functions The performance ofEFPA is evaluated by applying it to two set of optimizationproblems One set includes test functions and other is thereal-world application of antenna array designThe algorithmis also compared with other states of the art to prove its com-petitiveness The simulations are performed on Intel Core i3personal computer with Windows 10 operating system usingMATLAB version 7100 (R2010a) EFPA is tested on sevenwell-known benchmark problems [27] as shown in Table 1with search range and optimal solution For comparison pur-poses artificial bee colony (ABC) [28] DE [29] bat algorithm(BA) [30] and FPA [25] are used The maximum number offunction evaluations for each algorithm is set to 30 times 1000with 30 as the population size and 1000 as the maximumnumber of iterationsThe parameter settings of all algorithmsare given in Table 2 All the algorithms are run 50 times andresults in terms of best worst mean and standard deviationare reported Since because of the stochastic nature of algo-rithm Wilcoxon rank-sum test has also been performed tosignificantly prove the better performance of EFPA The per-formance in terms of time-complexity has also been shownin Table 5 to prove its competitiveness Time-complexity isbasically taken for each of the algorithms in the benchmark

problems and it has been calculated for one run for eachalgorithm

The comparison of comparison for various test functionsis given in Table 3 The best and worst values give the mostoptimal and least optimal solution among the 50 solutionsfrom the maximum number of runs Mean gives the averagevalues of results and is used when the best and worst valuesdo not give comparable results But mean may give sameresults for widely different results So standard deviationis also used to calculate results For functions 1198911 1198914 11989151198916 and 1198917 it can be seen that EFPA performs much betterthan other algorithms except for 1198912 where ABC gives betterresults For function 1198913 ABC and EFPA gives same resultsFurther statistical analysis from the Table 4 verify the resultsIn Table 5 time-complexity for each algorithm is shownHere time-complexity show that EFPA takes comparativelyless time for solving the benchmark problem for most ofthe benchmark functions in comparison to FPA BA andABC ForDE only four functions give better time-complexitywhile for the rest three again EFPA is better Overall we cansay that EFPA is superior to ABC DE FPA and BA In thenext subsection EFPA is applied for synthesis of LAA

42 Results of LAA A symmetrical LAA consists of 2119873elements with N elements on both sides of the origin and

International Journal of Antennas and Propagation 5

Table 2 Parameter settings of various algorithms

Algorithm Parameters Values

DE

119865 15CR 08

Population size 30Maximum iterations 1000Stopping criteria Max iteration

ABC

Colony size 30Number of food sources Colony size2

Limit 100Maximum cycles 1000Stopping criteria Max iteration

FPA

Population size 30Probability switch 06Maximum iterations 1000Stopping criteria Max iteration

BA

Population size 50Loudness 05Pulse rate 05

Maximum number of iterations 200Stopping criteria Max iteration

EFPA

Population size 30

Probability switchDynamic and

linearlydecreasing

Maximum iterations 1000Stopping criteria Max iteration

middot middot middot middot middot middot120601

2 n N minus 1 Nx

Figure 1 Geometry of a linear antenna array

arranged in a straight line as shown in Figure 1 The 119899thelement is excited with a current of amplitude 119868119899 and phase120593119899The position of 119899th element is represented by119909119899The arrayfactor (AF) of such an antenna is given by [1]

AF (0) = 2 119873sum119899=1

119868119899 cos [119896119909119899 cos (0) + 120593119899] (9)

Here 119896 is the wave number and is given by 2120587120582 Thesymmetric nature of the array reduces the computational costas only N elements are needed to be optimized Moreoverthese symmetric arrays give symmetric radiation patternwhich is desired for number of applications

The AF for LAA in (9) is a function of excitation currents(both 119868119899 and 120593119899) and positions 119909119899 If current excitations areheld uniform (ie 119868119899 = 1 and 120593119899 = 0∘) then the LAA is

function of 119909119899 only and array is called nonuniform LAAHence for nonuniform LAA AF reduces to

AF (120601) = 2 119873sum119899=1

cos [119896119909119899 cos (120601)] (10)

The design of nonuniform LAA has gained importance inthe past and several researchers have synthesized it usingdifferent optimization techniques [3 7 9 11ndash21] The arrayfactor of a nonuniform array is aperiodic by virtue of nonuni-form spacing between the elements This aperiodic nature ofarrays helps in obtaining lower SLL with lesser number ofelements for a given antenna size Furthermore the uniformelement excitations help in reducing the feed network costand complexity However the relationship between the arrayfactor and element spacing is nonconvex and nonlinear Sothis makes the design of nonuniform arrays difficult

For directing antenna arrays energy in a specific direc-tion it is desired that its radiation pattern should have lowside lobes Moreover nulls are also required in the directionof interferenceThe desired radiation pattern can be achievedby suitably adjusting the positions of antenna elements fornonuniform LAA The adjustment of positions is a trickyand nonlinear problem So optimization algorithms are wellsuited for this type of problems For applying optimization toantenna problem it is imperative to define a fitness functionwhich is given as

fit = SLL + 120572 lowastmax 0 1003816100381610038161003816BW minus BW1198891003816100381610038161003816 minus 1 + 120572

lowast 119870sum119896=1

max 0AFdB (120601119896) minusNudB (11)

In the above fitness function SLL is the side lobe levelAFdB is the array factor in decibels BW and BW119889 are thecalculated and desired beamwidth of the main lobe 120601119896 givesthe direction of the 119896th null NudB is the required null depthin dB 119870 is the number of the required nulls and 120572 is a verylarge numberThe fitness function given in (11) uses a penaltymethod which penalizes any antenna design which does notmeet the required constraints of beamwidth and nulls (ifdesired) This is achieved by taking value 120572 to be very largeThe BW here is obtained computationally from the radiationpattern data

421 Linear Antenna Optimization This subsection presentsthe application of EFPA to optimization of nonuniformLAA In the first example a 12-element nonuniform antennais taken for optimization The goal of optimization is tominimize the SLL in the region 120601 isin [98∘ 180∘] Because ofsymmetry there are only six element positions which are tobe optimized The number of generations for EFPA are takenas 1000 The population size taken as 20 in order to makefair comparison between the results of FPA [21] and EFPAThe total number of function evaluations are 20 lowast 1000 =2000 for both the FPA [21] and EFPA As the aim of theoptimization is to suppress the SLL only so the value of 120572 istaken as 0 in the fitness equation of (11) The EFPA algorithmis run for 30 times and best results are listed in this work

6 International Journal of Antennas and Propagation

Table 3 Comparison of results for various test functions

Objective function Algorithm Best Worst Mean Standard deviation

1198911 (119909)DE 443119864 + 03 297119864 + 04 154119864 + 04 622119864 + 03ABC 274119864 + 03 367119864 + 04 274119864 + 04 593119864 + 05BA 808119864 + 03 354119864 + 04 196119864 + 04 701119864 + 03FPA 100119864 + 02 100119864 + 03 361119864 + 02 166119864 + 02EFPA 335119864 minus 15 472119864 minus 07 182119864 minus 08 685119864 minus 08

1198912 (119909)DE 164119864 + 01 382119864 + 01 254119864 + 01 451119864 + 00ABC 253119864 + 00 590119864 + 00 390119864 + 00 840119864 minus 01BA 251119864 + 01 430119864 + 01 336119864 + 01 424119864 + 00FPA 383119864 + 01 572119864 + 01 512119864 + 01 464119864 + 00EFPA 134119864 + 01 266119864 + 01 200119864 + 01 310119864 + 00

1198913 (119909)DE 795119864 minus 05 229119864 minus 01 464119864 minus 02 537119864 minus 02ABC 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00BA 306119864 minus 14 347119864 minus 01 895119864 minus 02 922119864 minus 02FPA 000119864 minus 00 198119864 minus 12 103119864 minus 13 339119864 minus 13EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

1198914 (119909)DE 489119864 + 01 368119864 + 02 154119864 + 02 719119864 + 01ABC 106119864 + 00 176119864 + 00 125119864 + 00 173119864 minus 01BA 294119864 + 00 118119864 + 01 577119864 + 00 163119864 + 00FPA 571119864 minus 01 104119864 + 00 925119864 minus 01 976119864 minus 02EFPA 000119864 minus 00 145119864 minus 01 290119864 minus 03 206119864 minus 02

1198915 (119909)DE 178119864 + 06 692119864 + 08 224119864 + 08 227119864 + 08ABC 349119864 minus 02 277119864 + 07 107119864 + 08 891119864 + 07BA 340119864 + 05 555119864 + 07 158119864 + 07 153119864 + 07FPA 281119864 + 00 111119864 + 01 681119864 + 00 187119864 + 00EFPA 125119864 + 00 31119864 + 01 893119864 + 00 640119864 + 00

1198916 (119909)DE 200119864 + 01 200119864 + 01 200119864 + 01 000119864 + 00ABC 200119864 + 01 202119864 + 01 200119864 + 01 530119864 minus 03BA 197119864 + 01 200119864 + 01 199119864 + 01 292119864 minus 02FPA 205119864 + 01 211119864 + 01 210119864 + 01 837119864 minus 02EFPA 729119864 + 00 133119864 + 01 995119864 + 00 129119864 + 00

1198917 (119909)DE 421119864 + 08 378119864 + 10 863119864 + 09 100119864 + 10ABC 218119864 + 04 361119864 + 05 296119864 + 04 539119864 + 04BA 862119864 + 08 129119864 + 10 307119864 + 09 210119864 + 09FPA 148119864 + 05 649119864 + 06 230119864 + 06 172119864 + 06EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

Table 4 119901-test values of various algorithms

Objective function BA FPA ABC DE EFPA1198911 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 706119864 minus 18 NA1198912 (119909) 706119864 minus 18 706119864 minus 18 NA 325119864 minus 09 706119864 minus 181198913 (119909) 331119864 minus 20 689119864 minus 17 NA 331119864 minus 20 NA1198914 (119909) 473119864 minus 20 473119864 minus 20 473119864 minus 20 473119864 minus 20 NA1198915 (119909) 706119864 minus 18 770119864 minus 02 421119864 minus 04 706119864 minus 18 NA1198916 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 331119864 minus 20 NA1198917 (119909) 331119864 minus 20 106119864 minus 17 331119864 minus 20 331119864 minus 20 NA

International Journal of Antennas and Propagation 7

Table 5 Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds)

Function DE ABC BA FPA EFPA1198911 (119909) 1284464 2527247 1190053 2328157 19659271198912 (119909) 1598023 1562962 1902841 1522211 14801681198913 (119909) 1185347 2470673 1047533 2183295 19005151198914 (119909) 3022755 5167423 4165333 2183153 20305021198915 (119909) 2848487 5988428 2974728 3755994 34984111198916 (119909) 1667705 3186360 2261759 2346810 23835991198917 (119909) 1693011 3265792 2087382 3384049 2999886

Table 6 Geometry of 12-element LAA (with respect to 1205822)obtained using EFPA

Element 1 2 3 4 5 6Position 03765 09734 17376 25051 34827 46744

The optimum positions obtained using EFPA are shown inTable 6The performance of EFPA is contrasted against otherpopular algorithms in terms of SLL in Table 7TheminimumSLL obtained by EFPA isminus2107 dB and is better thanGA [10]FiADE [15] TS [10] MA [10] FPA [21] and PSO [15] Thecomparison of convergence curve between EFPA and FPAis shown in Figure 2 which clearly shows faster convergenceof EFPA than the FPA [21] The FPA converged in about 700iterations whereas EFPA took only 225 iterations to convergeto the final solution This is due to enhanced explorationdue to Cauchy operator and better exploitation in EFPATheradiation pattern of EFPA antenna array is shown in Figure 3For comparison the radiation patterns of GA [10] FiADE[15] and PSO [15] are also shown in Figure 3

In the next example a 22-element nonuniform LAA isoptimized for reducing SLL in the region 120601 isin [98∘ 180∘] andalso imposing a null in the direction of 99∘ For achievingthe required pattern the value of NudB is set to minus60 dB Itis established fact that as SLL is decreased the width of themain lobe in radiation pattern increases So for this designthe desired beamwidth BW119889 is taken as 18∘ The value of 120572is taken as 106 in fitness function The optimum positionsare shown in Table 8 The results are compared in Table 9which clearly indicate better performance of EFPA as againstBSA [18] MOEADE [14] HSA [16] PSO [15] GA [10] MA[10] TS [10] and FPA [21] The SLL of EFPA antenna arrayis minus2631 dB which is much lower than that obtained usingother algorithms Furthermore the null in the direction of99∘ is minus7558 dBwhich is less than the desired value of minus60 dBand deeper than MOEADE [14] GA [7] PSO [3] and MA[10] In order to compare the performance of EFPA and FPAin terms of convergence speed FPA is also run for the sameobjective with same population size The convergence graphsof EFPA and FPA are plotted in the Figure 4The convergencecharacteristics clearly substantiates the superior performanceof EPFA in terms of faster convergence speed The radiationplots of EFPA HSA [16] BSA [18] and MOEADE [14]optimized arrays are shown in Figure 5

In order to show the capability of EFPA in synthesiz-ing the radiation pattern with multiple nulls a 28-element

FPAEFPA

minus22

minus21

minus20

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

Fitn

ess

200 400 600 800 10000Iterations

Figure 2 Convergence plots of 12-element LAA

GAFiADE

PSOEFPA

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 3 Radiation plot of 12-element EFPA optimized LAA

nonuniform LAA is optimized for SLL reduction in theregion 120601 isin [100∘ 180∘] and having nulls in the directionsof 120∘ 1225∘ and 125∘ The desired beamwidth is set to thevalue of 835∘ with tolerance of plusmn1∘ The optimum geometry

8 International Journal of Antennas and Propagation

Table 7 Performance of different algorithms for 12-element LAA

Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPASLL (dB) minus1896 minus1910 minus1877 minus1790 minus20764 minus1840 minus2107

Table 8 Geometry of 22-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11Position 04937 07965 12153 21313 25003 31327 39654 45877 55435 66391 80603

Table 9 Performance of different algorithms for 22-element LAA

Algorithm BSA [18] TS [10] MOEAD-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPASLL (dB) minus2354 minus1717 minus2093 minus2328 minus2068 minus1573 minus1811 minus2381 minus2631Null (dB) (99∘) minus10461 minus6794 minus6964 minus1033 minus4994 minus5429 minus7392 minus10171 minus7558

Table 10 Geometry of 28-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14Position 05880 12142 25291 28291 41205 49374 61006 69646 81374 95186 107399 124399 141399 158399

FPAEFPA

minus25

minus20

minus15

minus10

minus5

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 4 Convergence plots of 22-element LAA

obtained using EFPA is shown in Table 10The comparison ofconvergence curves between EFPA and BBO [17] CS [17] DE[17] FA [17] and COA [17] is given in Figure 6 which showsthat EFPA reaches the optimum solution in less number ofiterations The comparison of results in terms of SLL andnull values achieved is given in Table 11 which again clearlyindicates the superior performance of EFPA as against othermethods The SLL of EFPA is minus2290 dB which is lower by1 dB 924 dB 93 dB 104 dB 826 dB 924 dB 127 dB and030 dB than BSA [18] QPM [3] COA [17] ACO [13] GA[7] CLPSO [20] and FPA [21] respectively The null valuesare also equal to or less than minus60 dB Figure 7 shows the

BSAHSA

MOEAEFPA

Null

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 5 Radiation plot of 22-element EFPA optimized LAA

comparison of QPM [3] HSA [16] BSA [18] and EFPAoptimized array radiation plots

The last design example illustrates the use of EFPA tooptimize the element positions for minimization of SLLand null placement for 32-element LAA The SLL is to beminimized in the region120601 isin [93∘ 180∘] andnull is to be placedin the direction of 99∘ The required beamwidth BW119889 is 71

∘The plot of convergence for EFPA is shown in Figure 8 Alsofor the sake of comparison the convergence plots of BBO[17] CS [17] DE [17] FA [17] and COA [17] are shown in thesame figureThe element positions for EFPA optimized array

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

4 International Journal of Antennas and Propagation

Table 1 Description of test functions

Test problems Objective function Search range Optimumvalue 119863

Spherefunction

1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100] 0 10

Weierstrassfunction

1198912 (119909) = 119863sum119894=1

119896maxsum119896=0

[119886119896 cos (2120587119887119896 (119909119894 + 05))]minus 119863119896maxsum119896=0

[119886119896 cos(2120587119887119896 sdot 05)] where 119886 = 05 119887 = 3 119896max = 20 [minus05 05] 0 30

Schafferfunction 1198913 (119909) = [ 1119899 minus 1radic119904119894 sdot (sin (50011990415119894 ) + 1)]2 119904119894 = radic1199092119894 + 1199092119894+1 [minus100 100] 0 30

Griewank 1198914 (119909) = 14000119873sum119894=1

1199092119894 minus119873prod119894=1

cos( 119909119894radic119894) + 1 [minus600 600] 0 30

Penalized 1function

1198915(119909) = 120587119899 10sin2 (1205871199101) + 119899minus1sum119894=1

(119910119894 minus 1)2 [1 + 10sin2 (120587119910119894+1)] + (119910119899 minus 1)2+ 119899sum119894=1

119906 (119909119894 10 100 4)[minus50 50] 0 30

119910119894 = 1 + 119909119894+14 119906(119909119894 119886 119896 119898) =

119896 (119909119894 minus 119886)119898 119909119894 gt 1198860 minus119886 lt 119909119894 lt 119886119896 (minus119909119894 minus 119886)119898 119909119894 lt minus119886

Ackleyfunction

1198916 (119909) = minus20 exp(minus02radic 1119863119863sum119894=1

1199092119894) minus exp( 1119863119863sum119894=1

cos (2120587119909119894)) + 20 + 119890 [minus100 100] 0 30

Rosenbrockfunction

1198917 =119873minus1sum119894=1

(100 (119909119894+1 minus 1199092119894 ) 2 + (119909119894 minus 1) 2) [minus100 100] 0 30

4 Results and Discussion

41 Results of Benchmark Functions The performance ofEFPA is evaluated by applying it to two set of optimizationproblems One set includes test functions and other is thereal-world application of antenna array designThe algorithmis also compared with other states of the art to prove its com-petitiveness The simulations are performed on Intel Core i3personal computer with Windows 10 operating system usingMATLAB version 7100 (R2010a) EFPA is tested on sevenwell-known benchmark problems [27] as shown in Table 1with search range and optimal solution For comparison pur-poses artificial bee colony (ABC) [28] DE [29] bat algorithm(BA) [30] and FPA [25] are used The maximum number offunction evaluations for each algorithm is set to 30 times 1000with 30 as the population size and 1000 as the maximumnumber of iterationsThe parameter settings of all algorithmsare given in Table 2 All the algorithms are run 50 times andresults in terms of best worst mean and standard deviationare reported Since because of the stochastic nature of algo-rithm Wilcoxon rank-sum test has also been performed tosignificantly prove the better performance of EFPA The per-formance in terms of time-complexity has also been shownin Table 5 to prove its competitiveness Time-complexity isbasically taken for each of the algorithms in the benchmark

problems and it has been calculated for one run for eachalgorithm

The comparison of comparison for various test functionsis given in Table 3 The best and worst values give the mostoptimal and least optimal solution among the 50 solutionsfrom the maximum number of runs Mean gives the averagevalues of results and is used when the best and worst valuesdo not give comparable results But mean may give sameresults for widely different results So standard deviationis also used to calculate results For functions 1198911 1198914 11989151198916 and 1198917 it can be seen that EFPA performs much betterthan other algorithms except for 1198912 where ABC gives betterresults For function 1198913 ABC and EFPA gives same resultsFurther statistical analysis from the Table 4 verify the resultsIn Table 5 time-complexity for each algorithm is shownHere time-complexity show that EFPA takes comparativelyless time for solving the benchmark problem for most ofthe benchmark functions in comparison to FPA BA andABC ForDE only four functions give better time-complexitywhile for the rest three again EFPA is better Overall we cansay that EFPA is superior to ABC DE FPA and BA In thenext subsection EFPA is applied for synthesis of LAA

42 Results of LAA A symmetrical LAA consists of 2119873elements with N elements on both sides of the origin and

International Journal of Antennas and Propagation 5

Table 2 Parameter settings of various algorithms

Algorithm Parameters Values

DE

119865 15CR 08

Population size 30Maximum iterations 1000Stopping criteria Max iteration

ABC

Colony size 30Number of food sources Colony size2

Limit 100Maximum cycles 1000Stopping criteria Max iteration

FPA

Population size 30Probability switch 06Maximum iterations 1000Stopping criteria Max iteration

BA

Population size 50Loudness 05Pulse rate 05

Maximum number of iterations 200Stopping criteria Max iteration

EFPA

Population size 30

Probability switchDynamic and

linearlydecreasing

Maximum iterations 1000Stopping criteria Max iteration

middot middot middot middot middot middot120601

2 n N minus 1 Nx

Figure 1 Geometry of a linear antenna array

arranged in a straight line as shown in Figure 1 The 119899thelement is excited with a current of amplitude 119868119899 and phase120593119899The position of 119899th element is represented by119909119899The arrayfactor (AF) of such an antenna is given by [1]

AF (0) = 2 119873sum119899=1

119868119899 cos [119896119909119899 cos (0) + 120593119899] (9)

Here 119896 is the wave number and is given by 2120587120582 Thesymmetric nature of the array reduces the computational costas only N elements are needed to be optimized Moreoverthese symmetric arrays give symmetric radiation patternwhich is desired for number of applications

The AF for LAA in (9) is a function of excitation currents(both 119868119899 and 120593119899) and positions 119909119899 If current excitations areheld uniform (ie 119868119899 = 1 and 120593119899 = 0∘) then the LAA is

function of 119909119899 only and array is called nonuniform LAAHence for nonuniform LAA AF reduces to

AF (120601) = 2 119873sum119899=1

cos [119896119909119899 cos (120601)] (10)

The design of nonuniform LAA has gained importance inthe past and several researchers have synthesized it usingdifferent optimization techniques [3 7 9 11ndash21] The arrayfactor of a nonuniform array is aperiodic by virtue of nonuni-form spacing between the elements This aperiodic nature ofarrays helps in obtaining lower SLL with lesser number ofelements for a given antenna size Furthermore the uniformelement excitations help in reducing the feed network costand complexity However the relationship between the arrayfactor and element spacing is nonconvex and nonlinear Sothis makes the design of nonuniform arrays difficult

For directing antenna arrays energy in a specific direc-tion it is desired that its radiation pattern should have lowside lobes Moreover nulls are also required in the directionof interferenceThe desired radiation pattern can be achievedby suitably adjusting the positions of antenna elements fornonuniform LAA The adjustment of positions is a trickyand nonlinear problem So optimization algorithms are wellsuited for this type of problems For applying optimization toantenna problem it is imperative to define a fitness functionwhich is given as

fit = SLL + 120572 lowastmax 0 1003816100381610038161003816BW minus BW1198891003816100381610038161003816 minus 1 + 120572

lowast 119870sum119896=1

max 0AFdB (120601119896) minusNudB (11)

In the above fitness function SLL is the side lobe levelAFdB is the array factor in decibels BW and BW119889 are thecalculated and desired beamwidth of the main lobe 120601119896 givesthe direction of the 119896th null NudB is the required null depthin dB 119870 is the number of the required nulls and 120572 is a verylarge numberThe fitness function given in (11) uses a penaltymethod which penalizes any antenna design which does notmeet the required constraints of beamwidth and nulls (ifdesired) This is achieved by taking value 120572 to be very largeThe BW here is obtained computationally from the radiationpattern data

421 Linear Antenna Optimization This subsection presentsthe application of EFPA to optimization of nonuniformLAA In the first example a 12-element nonuniform antennais taken for optimization The goal of optimization is tominimize the SLL in the region 120601 isin [98∘ 180∘] Because ofsymmetry there are only six element positions which are tobe optimized The number of generations for EFPA are takenas 1000 The population size taken as 20 in order to makefair comparison between the results of FPA [21] and EFPAThe total number of function evaluations are 20 lowast 1000 =2000 for both the FPA [21] and EFPA As the aim of theoptimization is to suppress the SLL only so the value of 120572 istaken as 0 in the fitness equation of (11) The EFPA algorithmis run for 30 times and best results are listed in this work

6 International Journal of Antennas and Propagation

Table 3 Comparison of results for various test functions

Objective function Algorithm Best Worst Mean Standard deviation

1198911 (119909)DE 443119864 + 03 297119864 + 04 154119864 + 04 622119864 + 03ABC 274119864 + 03 367119864 + 04 274119864 + 04 593119864 + 05BA 808119864 + 03 354119864 + 04 196119864 + 04 701119864 + 03FPA 100119864 + 02 100119864 + 03 361119864 + 02 166119864 + 02EFPA 335119864 minus 15 472119864 minus 07 182119864 minus 08 685119864 minus 08

1198912 (119909)DE 164119864 + 01 382119864 + 01 254119864 + 01 451119864 + 00ABC 253119864 + 00 590119864 + 00 390119864 + 00 840119864 minus 01BA 251119864 + 01 430119864 + 01 336119864 + 01 424119864 + 00FPA 383119864 + 01 572119864 + 01 512119864 + 01 464119864 + 00EFPA 134119864 + 01 266119864 + 01 200119864 + 01 310119864 + 00

1198913 (119909)DE 795119864 minus 05 229119864 minus 01 464119864 minus 02 537119864 minus 02ABC 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00BA 306119864 minus 14 347119864 minus 01 895119864 minus 02 922119864 minus 02FPA 000119864 minus 00 198119864 minus 12 103119864 minus 13 339119864 minus 13EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

1198914 (119909)DE 489119864 + 01 368119864 + 02 154119864 + 02 719119864 + 01ABC 106119864 + 00 176119864 + 00 125119864 + 00 173119864 minus 01BA 294119864 + 00 118119864 + 01 577119864 + 00 163119864 + 00FPA 571119864 minus 01 104119864 + 00 925119864 minus 01 976119864 minus 02EFPA 000119864 minus 00 145119864 minus 01 290119864 minus 03 206119864 minus 02

1198915 (119909)DE 178119864 + 06 692119864 + 08 224119864 + 08 227119864 + 08ABC 349119864 minus 02 277119864 + 07 107119864 + 08 891119864 + 07BA 340119864 + 05 555119864 + 07 158119864 + 07 153119864 + 07FPA 281119864 + 00 111119864 + 01 681119864 + 00 187119864 + 00EFPA 125119864 + 00 31119864 + 01 893119864 + 00 640119864 + 00

1198916 (119909)DE 200119864 + 01 200119864 + 01 200119864 + 01 000119864 + 00ABC 200119864 + 01 202119864 + 01 200119864 + 01 530119864 minus 03BA 197119864 + 01 200119864 + 01 199119864 + 01 292119864 minus 02FPA 205119864 + 01 211119864 + 01 210119864 + 01 837119864 minus 02EFPA 729119864 + 00 133119864 + 01 995119864 + 00 129119864 + 00

1198917 (119909)DE 421119864 + 08 378119864 + 10 863119864 + 09 100119864 + 10ABC 218119864 + 04 361119864 + 05 296119864 + 04 539119864 + 04BA 862119864 + 08 129119864 + 10 307119864 + 09 210119864 + 09FPA 148119864 + 05 649119864 + 06 230119864 + 06 172119864 + 06EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

Table 4 119901-test values of various algorithms

Objective function BA FPA ABC DE EFPA1198911 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 706119864 minus 18 NA1198912 (119909) 706119864 minus 18 706119864 minus 18 NA 325119864 minus 09 706119864 minus 181198913 (119909) 331119864 minus 20 689119864 minus 17 NA 331119864 minus 20 NA1198914 (119909) 473119864 minus 20 473119864 minus 20 473119864 minus 20 473119864 minus 20 NA1198915 (119909) 706119864 minus 18 770119864 minus 02 421119864 minus 04 706119864 minus 18 NA1198916 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 331119864 minus 20 NA1198917 (119909) 331119864 minus 20 106119864 minus 17 331119864 minus 20 331119864 minus 20 NA

International Journal of Antennas and Propagation 7

Table 5 Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds)

Function DE ABC BA FPA EFPA1198911 (119909) 1284464 2527247 1190053 2328157 19659271198912 (119909) 1598023 1562962 1902841 1522211 14801681198913 (119909) 1185347 2470673 1047533 2183295 19005151198914 (119909) 3022755 5167423 4165333 2183153 20305021198915 (119909) 2848487 5988428 2974728 3755994 34984111198916 (119909) 1667705 3186360 2261759 2346810 23835991198917 (119909) 1693011 3265792 2087382 3384049 2999886

Table 6 Geometry of 12-element LAA (with respect to 1205822)obtained using EFPA

Element 1 2 3 4 5 6Position 03765 09734 17376 25051 34827 46744

The optimum positions obtained using EFPA are shown inTable 6The performance of EFPA is contrasted against otherpopular algorithms in terms of SLL in Table 7TheminimumSLL obtained by EFPA isminus2107 dB and is better thanGA [10]FiADE [15] TS [10] MA [10] FPA [21] and PSO [15] Thecomparison of convergence curve between EFPA and FPAis shown in Figure 2 which clearly shows faster convergenceof EFPA than the FPA [21] The FPA converged in about 700iterations whereas EFPA took only 225 iterations to convergeto the final solution This is due to enhanced explorationdue to Cauchy operator and better exploitation in EFPATheradiation pattern of EFPA antenna array is shown in Figure 3For comparison the radiation patterns of GA [10] FiADE[15] and PSO [15] are also shown in Figure 3

In the next example a 22-element nonuniform LAA isoptimized for reducing SLL in the region 120601 isin [98∘ 180∘] andalso imposing a null in the direction of 99∘ For achievingthe required pattern the value of NudB is set to minus60 dB Itis established fact that as SLL is decreased the width of themain lobe in radiation pattern increases So for this designthe desired beamwidth BW119889 is taken as 18∘ The value of 120572is taken as 106 in fitness function The optimum positionsare shown in Table 8 The results are compared in Table 9which clearly indicate better performance of EFPA as againstBSA [18] MOEADE [14] HSA [16] PSO [15] GA [10] MA[10] TS [10] and FPA [21] The SLL of EFPA antenna arrayis minus2631 dB which is much lower than that obtained usingother algorithms Furthermore the null in the direction of99∘ is minus7558 dBwhich is less than the desired value of minus60 dBand deeper than MOEADE [14] GA [7] PSO [3] and MA[10] In order to compare the performance of EFPA and FPAin terms of convergence speed FPA is also run for the sameobjective with same population size The convergence graphsof EFPA and FPA are plotted in the Figure 4The convergencecharacteristics clearly substantiates the superior performanceof EPFA in terms of faster convergence speed The radiationplots of EFPA HSA [16] BSA [18] and MOEADE [14]optimized arrays are shown in Figure 5

In order to show the capability of EFPA in synthesiz-ing the radiation pattern with multiple nulls a 28-element

FPAEFPA

minus22

minus21

minus20

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

Fitn

ess

200 400 600 800 10000Iterations

Figure 2 Convergence plots of 12-element LAA

GAFiADE

PSOEFPA

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 3 Radiation plot of 12-element EFPA optimized LAA

nonuniform LAA is optimized for SLL reduction in theregion 120601 isin [100∘ 180∘] and having nulls in the directionsof 120∘ 1225∘ and 125∘ The desired beamwidth is set to thevalue of 835∘ with tolerance of plusmn1∘ The optimum geometry

8 International Journal of Antennas and Propagation

Table 7 Performance of different algorithms for 12-element LAA

Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPASLL (dB) minus1896 minus1910 minus1877 minus1790 minus20764 minus1840 minus2107

Table 8 Geometry of 22-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11Position 04937 07965 12153 21313 25003 31327 39654 45877 55435 66391 80603

Table 9 Performance of different algorithms for 22-element LAA

Algorithm BSA [18] TS [10] MOEAD-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPASLL (dB) minus2354 minus1717 minus2093 minus2328 minus2068 minus1573 minus1811 minus2381 minus2631Null (dB) (99∘) minus10461 minus6794 minus6964 minus1033 minus4994 minus5429 minus7392 minus10171 minus7558

Table 10 Geometry of 28-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14Position 05880 12142 25291 28291 41205 49374 61006 69646 81374 95186 107399 124399 141399 158399

FPAEFPA

minus25

minus20

minus15

minus10

minus5

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 4 Convergence plots of 22-element LAA

obtained using EFPA is shown in Table 10The comparison ofconvergence curves between EFPA and BBO [17] CS [17] DE[17] FA [17] and COA [17] is given in Figure 6 which showsthat EFPA reaches the optimum solution in less number ofiterations The comparison of results in terms of SLL andnull values achieved is given in Table 11 which again clearlyindicates the superior performance of EFPA as against othermethods The SLL of EFPA is minus2290 dB which is lower by1 dB 924 dB 93 dB 104 dB 826 dB 924 dB 127 dB and030 dB than BSA [18] QPM [3] COA [17] ACO [13] GA[7] CLPSO [20] and FPA [21] respectively The null valuesare also equal to or less than minus60 dB Figure 7 shows the

BSAHSA

MOEAEFPA

Null

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 5 Radiation plot of 22-element EFPA optimized LAA

comparison of QPM [3] HSA [16] BSA [18] and EFPAoptimized array radiation plots

The last design example illustrates the use of EFPA tooptimize the element positions for minimization of SLLand null placement for 32-element LAA The SLL is to beminimized in the region120601 isin [93∘ 180∘] andnull is to be placedin the direction of 99∘ The required beamwidth BW119889 is 71

∘The plot of convergence for EFPA is shown in Figure 8 Alsofor the sake of comparison the convergence plots of BBO[17] CS [17] DE [17] FA [17] and COA [17] are shown in thesame figureThe element positions for EFPA optimized array

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

International Journal of Antennas and Propagation 5

Table 2 Parameter settings of various algorithms

Algorithm Parameters Values

DE

119865 15CR 08

Population size 30Maximum iterations 1000Stopping criteria Max iteration

ABC

Colony size 30Number of food sources Colony size2

Limit 100Maximum cycles 1000Stopping criteria Max iteration

FPA

Population size 30Probability switch 06Maximum iterations 1000Stopping criteria Max iteration

BA

Population size 50Loudness 05Pulse rate 05

Maximum number of iterations 200Stopping criteria Max iteration

EFPA

Population size 30

Probability switchDynamic and

linearlydecreasing

Maximum iterations 1000Stopping criteria Max iteration

middot middot middot middot middot middot120601

2 n N minus 1 Nx

Figure 1 Geometry of a linear antenna array

arranged in a straight line as shown in Figure 1 The 119899thelement is excited with a current of amplitude 119868119899 and phase120593119899The position of 119899th element is represented by119909119899The arrayfactor (AF) of such an antenna is given by [1]

AF (0) = 2 119873sum119899=1

119868119899 cos [119896119909119899 cos (0) + 120593119899] (9)

Here 119896 is the wave number and is given by 2120587120582 Thesymmetric nature of the array reduces the computational costas only N elements are needed to be optimized Moreoverthese symmetric arrays give symmetric radiation patternwhich is desired for number of applications

The AF for LAA in (9) is a function of excitation currents(both 119868119899 and 120593119899) and positions 119909119899 If current excitations areheld uniform (ie 119868119899 = 1 and 120593119899 = 0∘) then the LAA is

function of 119909119899 only and array is called nonuniform LAAHence for nonuniform LAA AF reduces to

AF (120601) = 2 119873sum119899=1

cos [119896119909119899 cos (120601)] (10)

The design of nonuniform LAA has gained importance inthe past and several researchers have synthesized it usingdifferent optimization techniques [3 7 9 11ndash21] The arrayfactor of a nonuniform array is aperiodic by virtue of nonuni-form spacing between the elements This aperiodic nature ofarrays helps in obtaining lower SLL with lesser number ofelements for a given antenna size Furthermore the uniformelement excitations help in reducing the feed network costand complexity However the relationship between the arrayfactor and element spacing is nonconvex and nonlinear Sothis makes the design of nonuniform arrays difficult

For directing antenna arrays energy in a specific direc-tion it is desired that its radiation pattern should have lowside lobes Moreover nulls are also required in the directionof interferenceThe desired radiation pattern can be achievedby suitably adjusting the positions of antenna elements fornonuniform LAA The adjustment of positions is a trickyand nonlinear problem So optimization algorithms are wellsuited for this type of problems For applying optimization toantenna problem it is imperative to define a fitness functionwhich is given as

fit = SLL + 120572 lowastmax 0 1003816100381610038161003816BW minus BW1198891003816100381610038161003816 minus 1 + 120572

lowast 119870sum119896=1

max 0AFdB (120601119896) minusNudB (11)

In the above fitness function SLL is the side lobe levelAFdB is the array factor in decibels BW and BW119889 are thecalculated and desired beamwidth of the main lobe 120601119896 givesthe direction of the 119896th null NudB is the required null depthin dB 119870 is the number of the required nulls and 120572 is a verylarge numberThe fitness function given in (11) uses a penaltymethod which penalizes any antenna design which does notmeet the required constraints of beamwidth and nulls (ifdesired) This is achieved by taking value 120572 to be very largeThe BW here is obtained computationally from the radiationpattern data

421 Linear Antenna Optimization This subsection presentsthe application of EFPA to optimization of nonuniformLAA In the first example a 12-element nonuniform antennais taken for optimization The goal of optimization is tominimize the SLL in the region 120601 isin [98∘ 180∘] Because ofsymmetry there are only six element positions which are tobe optimized The number of generations for EFPA are takenas 1000 The population size taken as 20 in order to makefair comparison between the results of FPA [21] and EFPAThe total number of function evaluations are 20 lowast 1000 =2000 for both the FPA [21] and EFPA As the aim of theoptimization is to suppress the SLL only so the value of 120572 istaken as 0 in the fitness equation of (11) The EFPA algorithmis run for 30 times and best results are listed in this work

6 International Journal of Antennas and Propagation

Table 3 Comparison of results for various test functions

Objective function Algorithm Best Worst Mean Standard deviation

1198911 (119909)DE 443119864 + 03 297119864 + 04 154119864 + 04 622119864 + 03ABC 274119864 + 03 367119864 + 04 274119864 + 04 593119864 + 05BA 808119864 + 03 354119864 + 04 196119864 + 04 701119864 + 03FPA 100119864 + 02 100119864 + 03 361119864 + 02 166119864 + 02EFPA 335119864 minus 15 472119864 minus 07 182119864 minus 08 685119864 minus 08

1198912 (119909)DE 164119864 + 01 382119864 + 01 254119864 + 01 451119864 + 00ABC 253119864 + 00 590119864 + 00 390119864 + 00 840119864 minus 01BA 251119864 + 01 430119864 + 01 336119864 + 01 424119864 + 00FPA 383119864 + 01 572119864 + 01 512119864 + 01 464119864 + 00EFPA 134119864 + 01 266119864 + 01 200119864 + 01 310119864 + 00

1198913 (119909)DE 795119864 minus 05 229119864 minus 01 464119864 minus 02 537119864 minus 02ABC 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00BA 306119864 minus 14 347119864 minus 01 895119864 minus 02 922119864 minus 02FPA 000119864 minus 00 198119864 minus 12 103119864 minus 13 339119864 minus 13EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

1198914 (119909)DE 489119864 + 01 368119864 + 02 154119864 + 02 719119864 + 01ABC 106119864 + 00 176119864 + 00 125119864 + 00 173119864 minus 01BA 294119864 + 00 118119864 + 01 577119864 + 00 163119864 + 00FPA 571119864 minus 01 104119864 + 00 925119864 minus 01 976119864 minus 02EFPA 000119864 minus 00 145119864 minus 01 290119864 minus 03 206119864 minus 02

1198915 (119909)DE 178119864 + 06 692119864 + 08 224119864 + 08 227119864 + 08ABC 349119864 minus 02 277119864 + 07 107119864 + 08 891119864 + 07BA 340119864 + 05 555119864 + 07 158119864 + 07 153119864 + 07FPA 281119864 + 00 111119864 + 01 681119864 + 00 187119864 + 00EFPA 125119864 + 00 31119864 + 01 893119864 + 00 640119864 + 00

1198916 (119909)DE 200119864 + 01 200119864 + 01 200119864 + 01 000119864 + 00ABC 200119864 + 01 202119864 + 01 200119864 + 01 530119864 minus 03BA 197119864 + 01 200119864 + 01 199119864 + 01 292119864 minus 02FPA 205119864 + 01 211119864 + 01 210119864 + 01 837119864 minus 02EFPA 729119864 + 00 133119864 + 01 995119864 + 00 129119864 + 00

1198917 (119909)DE 421119864 + 08 378119864 + 10 863119864 + 09 100119864 + 10ABC 218119864 + 04 361119864 + 05 296119864 + 04 539119864 + 04BA 862119864 + 08 129119864 + 10 307119864 + 09 210119864 + 09FPA 148119864 + 05 649119864 + 06 230119864 + 06 172119864 + 06EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

Table 4 119901-test values of various algorithms

Objective function BA FPA ABC DE EFPA1198911 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 706119864 minus 18 NA1198912 (119909) 706119864 minus 18 706119864 minus 18 NA 325119864 minus 09 706119864 minus 181198913 (119909) 331119864 minus 20 689119864 minus 17 NA 331119864 minus 20 NA1198914 (119909) 473119864 minus 20 473119864 minus 20 473119864 minus 20 473119864 minus 20 NA1198915 (119909) 706119864 minus 18 770119864 minus 02 421119864 minus 04 706119864 minus 18 NA1198916 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 331119864 minus 20 NA1198917 (119909) 331119864 minus 20 106119864 minus 17 331119864 minus 20 331119864 minus 20 NA

International Journal of Antennas and Propagation 7

Table 5 Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds)

Function DE ABC BA FPA EFPA1198911 (119909) 1284464 2527247 1190053 2328157 19659271198912 (119909) 1598023 1562962 1902841 1522211 14801681198913 (119909) 1185347 2470673 1047533 2183295 19005151198914 (119909) 3022755 5167423 4165333 2183153 20305021198915 (119909) 2848487 5988428 2974728 3755994 34984111198916 (119909) 1667705 3186360 2261759 2346810 23835991198917 (119909) 1693011 3265792 2087382 3384049 2999886

Table 6 Geometry of 12-element LAA (with respect to 1205822)obtained using EFPA

Element 1 2 3 4 5 6Position 03765 09734 17376 25051 34827 46744

The optimum positions obtained using EFPA are shown inTable 6The performance of EFPA is contrasted against otherpopular algorithms in terms of SLL in Table 7TheminimumSLL obtained by EFPA isminus2107 dB and is better thanGA [10]FiADE [15] TS [10] MA [10] FPA [21] and PSO [15] Thecomparison of convergence curve between EFPA and FPAis shown in Figure 2 which clearly shows faster convergenceof EFPA than the FPA [21] The FPA converged in about 700iterations whereas EFPA took only 225 iterations to convergeto the final solution This is due to enhanced explorationdue to Cauchy operator and better exploitation in EFPATheradiation pattern of EFPA antenna array is shown in Figure 3For comparison the radiation patterns of GA [10] FiADE[15] and PSO [15] are also shown in Figure 3

In the next example a 22-element nonuniform LAA isoptimized for reducing SLL in the region 120601 isin [98∘ 180∘] andalso imposing a null in the direction of 99∘ For achievingthe required pattern the value of NudB is set to minus60 dB Itis established fact that as SLL is decreased the width of themain lobe in radiation pattern increases So for this designthe desired beamwidth BW119889 is taken as 18∘ The value of 120572is taken as 106 in fitness function The optimum positionsare shown in Table 8 The results are compared in Table 9which clearly indicate better performance of EFPA as againstBSA [18] MOEADE [14] HSA [16] PSO [15] GA [10] MA[10] TS [10] and FPA [21] The SLL of EFPA antenna arrayis minus2631 dB which is much lower than that obtained usingother algorithms Furthermore the null in the direction of99∘ is minus7558 dBwhich is less than the desired value of minus60 dBand deeper than MOEADE [14] GA [7] PSO [3] and MA[10] In order to compare the performance of EFPA and FPAin terms of convergence speed FPA is also run for the sameobjective with same population size The convergence graphsof EFPA and FPA are plotted in the Figure 4The convergencecharacteristics clearly substantiates the superior performanceof EPFA in terms of faster convergence speed The radiationplots of EFPA HSA [16] BSA [18] and MOEADE [14]optimized arrays are shown in Figure 5

In order to show the capability of EFPA in synthesiz-ing the radiation pattern with multiple nulls a 28-element

FPAEFPA

minus22

minus21

minus20

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

Fitn

ess

200 400 600 800 10000Iterations

Figure 2 Convergence plots of 12-element LAA

GAFiADE

PSOEFPA

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 3 Radiation plot of 12-element EFPA optimized LAA

nonuniform LAA is optimized for SLL reduction in theregion 120601 isin [100∘ 180∘] and having nulls in the directionsof 120∘ 1225∘ and 125∘ The desired beamwidth is set to thevalue of 835∘ with tolerance of plusmn1∘ The optimum geometry

8 International Journal of Antennas and Propagation

Table 7 Performance of different algorithms for 12-element LAA

Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPASLL (dB) minus1896 minus1910 minus1877 minus1790 minus20764 minus1840 minus2107

Table 8 Geometry of 22-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11Position 04937 07965 12153 21313 25003 31327 39654 45877 55435 66391 80603

Table 9 Performance of different algorithms for 22-element LAA

Algorithm BSA [18] TS [10] MOEAD-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPASLL (dB) minus2354 minus1717 minus2093 minus2328 minus2068 minus1573 minus1811 minus2381 minus2631Null (dB) (99∘) minus10461 minus6794 minus6964 minus1033 minus4994 minus5429 minus7392 minus10171 minus7558

Table 10 Geometry of 28-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14Position 05880 12142 25291 28291 41205 49374 61006 69646 81374 95186 107399 124399 141399 158399

FPAEFPA

minus25

minus20

minus15

minus10

minus5

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 4 Convergence plots of 22-element LAA

obtained using EFPA is shown in Table 10The comparison ofconvergence curves between EFPA and BBO [17] CS [17] DE[17] FA [17] and COA [17] is given in Figure 6 which showsthat EFPA reaches the optimum solution in less number ofiterations The comparison of results in terms of SLL andnull values achieved is given in Table 11 which again clearlyindicates the superior performance of EFPA as against othermethods The SLL of EFPA is minus2290 dB which is lower by1 dB 924 dB 93 dB 104 dB 826 dB 924 dB 127 dB and030 dB than BSA [18] QPM [3] COA [17] ACO [13] GA[7] CLPSO [20] and FPA [21] respectively The null valuesare also equal to or less than minus60 dB Figure 7 shows the

BSAHSA

MOEAEFPA

Null

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 5 Radiation plot of 22-element EFPA optimized LAA

comparison of QPM [3] HSA [16] BSA [18] and EFPAoptimized array radiation plots

The last design example illustrates the use of EFPA tooptimize the element positions for minimization of SLLand null placement for 32-element LAA The SLL is to beminimized in the region120601 isin [93∘ 180∘] andnull is to be placedin the direction of 99∘ The required beamwidth BW119889 is 71

∘The plot of convergence for EFPA is shown in Figure 8 Alsofor the sake of comparison the convergence plots of BBO[17] CS [17] DE [17] FA [17] and COA [17] are shown in thesame figureThe element positions for EFPA optimized array

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

6 International Journal of Antennas and Propagation

Table 3 Comparison of results for various test functions

Objective function Algorithm Best Worst Mean Standard deviation

1198911 (119909)DE 443119864 + 03 297119864 + 04 154119864 + 04 622119864 + 03ABC 274119864 + 03 367119864 + 04 274119864 + 04 593119864 + 05BA 808119864 + 03 354119864 + 04 196119864 + 04 701119864 + 03FPA 100119864 + 02 100119864 + 03 361119864 + 02 166119864 + 02EFPA 335119864 minus 15 472119864 minus 07 182119864 minus 08 685119864 minus 08

1198912 (119909)DE 164119864 + 01 382119864 + 01 254119864 + 01 451119864 + 00ABC 253119864 + 00 590119864 + 00 390119864 + 00 840119864 minus 01BA 251119864 + 01 430119864 + 01 336119864 + 01 424119864 + 00FPA 383119864 + 01 572119864 + 01 512119864 + 01 464119864 + 00EFPA 134119864 + 01 266119864 + 01 200119864 + 01 310119864 + 00

1198913 (119909)DE 795119864 minus 05 229119864 minus 01 464119864 minus 02 537119864 minus 02ABC 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00BA 306119864 minus 14 347119864 minus 01 895119864 minus 02 922119864 minus 02FPA 000119864 minus 00 198119864 minus 12 103119864 minus 13 339119864 minus 13EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

1198914 (119909)DE 489119864 + 01 368119864 + 02 154119864 + 02 719119864 + 01ABC 106119864 + 00 176119864 + 00 125119864 + 00 173119864 minus 01BA 294119864 + 00 118119864 + 01 577119864 + 00 163119864 + 00FPA 571119864 minus 01 104119864 + 00 925119864 minus 01 976119864 minus 02EFPA 000119864 minus 00 145119864 minus 01 290119864 minus 03 206119864 minus 02

1198915 (119909)DE 178119864 + 06 692119864 + 08 224119864 + 08 227119864 + 08ABC 349119864 minus 02 277119864 + 07 107119864 + 08 891119864 + 07BA 340119864 + 05 555119864 + 07 158119864 + 07 153119864 + 07FPA 281119864 + 00 111119864 + 01 681119864 + 00 187119864 + 00EFPA 125119864 + 00 31119864 + 01 893119864 + 00 640119864 + 00

1198916 (119909)DE 200119864 + 01 200119864 + 01 200119864 + 01 000119864 + 00ABC 200119864 + 01 202119864 + 01 200119864 + 01 530119864 minus 03BA 197119864 + 01 200119864 + 01 199119864 + 01 292119864 minus 02FPA 205119864 + 01 211119864 + 01 210119864 + 01 837119864 minus 02EFPA 729119864 + 00 133119864 + 01 995119864 + 00 129119864 + 00

1198917 (119909)DE 421119864 + 08 378119864 + 10 863119864 + 09 100119864 + 10ABC 218119864 + 04 361119864 + 05 296119864 + 04 539119864 + 04BA 862119864 + 08 129119864 + 10 307119864 + 09 210119864 + 09FPA 148119864 + 05 649119864 + 06 230119864 + 06 172119864 + 06EFPA 000119864 minus 00 000119864 minus 00 000119864 minus 00 000119864 minus 00

Table 4 119901-test values of various algorithms

Objective function BA FPA ABC DE EFPA1198911 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 706119864 minus 18 NA1198912 (119909) 706119864 minus 18 706119864 minus 18 NA 325119864 minus 09 706119864 minus 181198913 (119909) 331119864 minus 20 689119864 minus 17 NA 331119864 minus 20 NA1198914 (119909) 473119864 minus 20 473119864 minus 20 473119864 minus 20 473119864 minus 20 NA1198915 (119909) 706119864 minus 18 770119864 minus 02 421119864 minus 04 706119864 minus 18 NA1198916 (119909) 706119864 minus 18 706119864 minus 18 706119864 minus 18 331119864 minus 20 NA1198917 (119909) 331119864 minus 20 106119864 minus 17 331119864 minus 20 331119864 minus 20 NA

International Journal of Antennas and Propagation 7

Table 5 Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds)

Function DE ABC BA FPA EFPA1198911 (119909) 1284464 2527247 1190053 2328157 19659271198912 (119909) 1598023 1562962 1902841 1522211 14801681198913 (119909) 1185347 2470673 1047533 2183295 19005151198914 (119909) 3022755 5167423 4165333 2183153 20305021198915 (119909) 2848487 5988428 2974728 3755994 34984111198916 (119909) 1667705 3186360 2261759 2346810 23835991198917 (119909) 1693011 3265792 2087382 3384049 2999886

Table 6 Geometry of 12-element LAA (with respect to 1205822)obtained using EFPA

Element 1 2 3 4 5 6Position 03765 09734 17376 25051 34827 46744

The optimum positions obtained using EFPA are shown inTable 6The performance of EFPA is contrasted against otherpopular algorithms in terms of SLL in Table 7TheminimumSLL obtained by EFPA isminus2107 dB and is better thanGA [10]FiADE [15] TS [10] MA [10] FPA [21] and PSO [15] Thecomparison of convergence curve between EFPA and FPAis shown in Figure 2 which clearly shows faster convergenceof EFPA than the FPA [21] The FPA converged in about 700iterations whereas EFPA took only 225 iterations to convergeto the final solution This is due to enhanced explorationdue to Cauchy operator and better exploitation in EFPATheradiation pattern of EFPA antenna array is shown in Figure 3For comparison the radiation patterns of GA [10] FiADE[15] and PSO [15] are also shown in Figure 3

In the next example a 22-element nonuniform LAA isoptimized for reducing SLL in the region 120601 isin [98∘ 180∘] andalso imposing a null in the direction of 99∘ For achievingthe required pattern the value of NudB is set to minus60 dB Itis established fact that as SLL is decreased the width of themain lobe in radiation pattern increases So for this designthe desired beamwidth BW119889 is taken as 18∘ The value of 120572is taken as 106 in fitness function The optimum positionsare shown in Table 8 The results are compared in Table 9which clearly indicate better performance of EFPA as againstBSA [18] MOEADE [14] HSA [16] PSO [15] GA [10] MA[10] TS [10] and FPA [21] The SLL of EFPA antenna arrayis minus2631 dB which is much lower than that obtained usingother algorithms Furthermore the null in the direction of99∘ is minus7558 dBwhich is less than the desired value of minus60 dBand deeper than MOEADE [14] GA [7] PSO [3] and MA[10] In order to compare the performance of EFPA and FPAin terms of convergence speed FPA is also run for the sameobjective with same population size The convergence graphsof EFPA and FPA are plotted in the Figure 4The convergencecharacteristics clearly substantiates the superior performanceof EPFA in terms of faster convergence speed The radiationplots of EFPA HSA [16] BSA [18] and MOEADE [14]optimized arrays are shown in Figure 5

In order to show the capability of EFPA in synthesiz-ing the radiation pattern with multiple nulls a 28-element

FPAEFPA

minus22

minus21

minus20

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

Fitn

ess

200 400 600 800 10000Iterations

Figure 2 Convergence plots of 12-element LAA

GAFiADE

PSOEFPA

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 3 Radiation plot of 12-element EFPA optimized LAA

nonuniform LAA is optimized for SLL reduction in theregion 120601 isin [100∘ 180∘] and having nulls in the directionsof 120∘ 1225∘ and 125∘ The desired beamwidth is set to thevalue of 835∘ with tolerance of plusmn1∘ The optimum geometry

8 International Journal of Antennas and Propagation

Table 7 Performance of different algorithms for 12-element LAA

Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPASLL (dB) minus1896 minus1910 minus1877 minus1790 minus20764 minus1840 minus2107

Table 8 Geometry of 22-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11Position 04937 07965 12153 21313 25003 31327 39654 45877 55435 66391 80603

Table 9 Performance of different algorithms for 22-element LAA

Algorithm BSA [18] TS [10] MOEAD-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPASLL (dB) minus2354 minus1717 minus2093 minus2328 minus2068 minus1573 minus1811 minus2381 minus2631Null (dB) (99∘) minus10461 minus6794 minus6964 minus1033 minus4994 minus5429 minus7392 minus10171 minus7558

Table 10 Geometry of 28-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14Position 05880 12142 25291 28291 41205 49374 61006 69646 81374 95186 107399 124399 141399 158399

FPAEFPA

minus25

minus20

minus15

minus10

minus5

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 4 Convergence plots of 22-element LAA

obtained using EFPA is shown in Table 10The comparison ofconvergence curves between EFPA and BBO [17] CS [17] DE[17] FA [17] and COA [17] is given in Figure 6 which showsthat EFPA reaches the optimum solution in less number ofiterations The comparison of results in terms of SLL andnull values achieved is given in Table 11 which again clearlyindicates the superior performance of EFPA as against othermethods The SLL of EFPA is minus2290 dB which is lower by1 dB 924 dB 93 dB 104 dB 826 dB 924 dB 127 dB and030 dB than BSA [18] QPM [3] COA [17] ACO [13] GA[7] CLPSO [20] and FPA [21] respectively The null valuesare also equal to or less than minus60 dB Figure 7 shows the

BSAHSA

MOEAEFPA

Null

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 5 Radiation plot of 22-element EFPA optimized LAA

comparison of QPM [3] HSA [16] BSA [18] and EFPAoptimized array radiation plots

The last design example illustrates the use of EFPA tooptimize the element positions for minimization of SLLand null placement for 32-element LAA The SLL is to beminimized in the region120601 isin [93∘ 180∘] andnull is to be placedin the direction of 99∘ The required beamwidth BW119889 is 71

∘The plot of convergence for EFPA is shown in Figure 8 Alsofor the sake of comparison the convergence plots of BBO[17] CS [17] DE [17] FA [17] and COA [17] are shown in thesame figureThe element positions for EFPA optimized array

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

International Journal of Antennas and Propagation 7

Table 5 Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds)

Function DE ABC BA FPA EFPA1198911 (119909) 1284464 2527247 1190053 2328157 19659271198912 (119909) 1598023 1562962 1902841 1522211 14801681198913 (119909) 1185347 2470673 1047533 2183295 19005151198914 (119909) 3022755 5167423 4165333 2183153 20305021198915 (119909) 2848487 5988428 2974728 3755994 34984111198916 (119909) 1667705 3186360 2261759 2346810 23835991198917 (119909) 1693011 3265792 2087382 3384049 2999886

Table 6 Geometry of 12-element LAA (with respect to 1205822)obtained using EFPA

Element 1 2 3 4 5 6Position 03765 09734 17376 25051 34827 46744

The optimum positions obtained using EFPA are shown inTable 6The performance of EFPA is contrasted against otherpopular algorithms in terms of SLL in Table 7TheminimumSLL obtained by EFPA isminus2107 dB and is better thanGA [10]FiADE [15] TS [10] MA [10] FPA [21] and PSO [15] Thecomparison of convergence curve between EFPA and FPAis shown in Figure 2 which clearly shows faster convergenceof EFPA than the FPA [21] The FPA converged in about 700iterations whereas EFPA took only 225 iterations to convergeto the final solution This is due to enhanced explorationdue to Cauchy operator and better exploitation in EFPATheradiation pattern of EFPA antenna array is shown in Figure 3For comparison the radiation patterns of GA [10] FiADE[15] and PSO [15] are also shown in Figure 3

In the next example a 22-element nonuniform LAA isoptimized for reducing SLL in the region 120601 isin [98∘ 180∘] andalso imposing a null in the direction of 99∘ For achievingthe required pattern the value of NudB is set to minus60 dB Itis established fact that as SLL is decreased the width of themain lobe in radiation pattern increases So for this designthe desired beamwidth BW119889 is taken as 18∘ The value of 120572is taken as 106 in fitness function The optimum positionsare shown in Table 8 The results are compared in Table 9which clearly indicate better performance of EFPA as againstBSA [18] MOEADE [14] HSA [16] PSO [15] GA [10] MA[10] TS [10] and FPA [21] The SLL of EFPA antenna arrayis minus2631 dB which is much lower than that obtained usingother algorithms Furthermore the null in the direction of99∘ is minus7558 dBwhich is less than the desired value of minus60 dBand deeper than MOEADE [14] GA [7] PSO [3] and MA[10] In order to compare the performance of EFPA and FPAin terms of convergence speed FPA is also run for the sameobjective with same population size The convergence graphsof EFPA and FPA are plotted in the Figure 4The convergencecharacteristics clearly substantiates the superior performanceof EPFA in terms of faster convergence speed The radiationplots of EFPA HSA [16] BSA [18] and MOEADE [14]optimized arrays are shown in Figure 5

In order to show the capability of EFPA in synthesiz-ing the radiation pattern with multiple nulls a 28-element

FPAEFPA

minus22

minus21

minus20

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

Fitn

ess

200 400 600 800 10000Iterations

Figure 2 Convergence plots of 12-element LAA

GAFiADE

PSOEFPA

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 3 Radiation plot of 12-element EFPA optimized LAA

nonuniform LAA is optimized for SLL reduction in theregion 120601 isin [100∘ 180∘] and having nulls in the directionsof 120∘ 1225∘ and 125∘ The desired beamwidth is set to thevalue of 835∘ with tolerance of plusmn1∘ The optimum geometry

8 International Journal of Antennas and Propagation

Table 7 Performance of different algorithms for 12-element LAA

Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPASLL (dB) minus1896 minus1910 minus1877 minus1790 minus20764 minus1840 minus2107

Table 8 Geometry of 22-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11Position 04937 07965 12153 21313 25003 31327 39654 45877 55435 66391 80603

Table 9 Performance of different algorithms for 22-element LAA

Algorithm BSA [18] TS [10] MOEAD-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPASLL (dB) minus2354 minus1717 minus2093 minus2328 minus2068 minus1573 minus1811 minus2381 minus2631Null (dB) (99∘) minus10461 minus6794 minus6964 minus1033 minus4994 minus5429 minus7392 minus10171 minus7558

Table 10 Geometry of 28-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14Position 05880 12142 25291 28291 41205 49374 61006 69646 81374 95186 107399 124399 141399 158399

FPAEFPA

minus25

minus20

minus15

minus10

minus5

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 4 Convergence plots of 22-element LAA

obtained using EFPA is shown in Table 10The comparison ofconvergence curves between EFPA and BBO [17] CS [17] DE[17] FA [17] and COA [17] is given in Figure 6 which showsthat EFPA reaches the optimum solution in less number ofiterations The comparison of results in terms of SLL andnull values achieved is given in Table 11 which again clearlyindicates the superior performance of EFPA as against othermethods The SLL of EFPA is minus2290 dB which is lower by1 dB 924 dB 93 dB 104 dB 826 dB 924 dB 127 dB and030 dB than BSA [18] QPM [3] COA [17] ACO [13] GA[7] CLPSO [20] and FPA [21] respectively The null valuesare also equal to or less than minus60 dB Figure 7 shows the

BSAHSA

MOEAEFPA

Null

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 5 Radiation plot of 22-element EFPA optimized LAA

comparison of QPM [3] HSA [16] BSA [18] and EFPAoptimized array radiation plots

The last design example illustrates the use of EFPA tooptimize the element positions for minimization of SLLand null placement for 32-element LAA The SLL is to beminimized in the region120601 isin [93∘ 180∘] andnull is to be placedin the direction of 99∘ The required beamwidth BW119889 is 71

∘The plot of convergence for EFPA is shown in Figure 8 Alsofor the sake of comparison the convergence plots of BBO[17] CS [17] DE [17] FA [17] and COA [17] are shown in thesame figureThe element positions for EFPA optimized array

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

8 International Journal of Antennas and Propagation

Table 7 Performance of different algorithms for 12-element LAA

Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPASLL (dB) minus1896 minus1910 minus1877 minus1790 minus20764 minus1840 minus2107

Table 8 Geometry of 22-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11Position 04937 07965 12153 21313 25003 31327 39654 45877 55435 66391 80603

Table 9 Performance of different algorithms for 22-element LAA

Algorithm BSA [18] TS [10] MOEAD-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPASLL (dB) minus2354 minus1717 minus2093 minus2328 minus2068 minus1573 minus1811 minus2381 minus2631Null (dB) (99∘) minus10461 minus6794 minus6964 minus1033 minus4994 minus5429 minus7392 minus10171 minus7558

Table 10 Geometry of 28-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14Position 05880 12142 25291 28291 41205 49374 61006 69646 81374 95186 107399 124399 141399 158399

FPAEFPA

minus25

minus20

minus15

minus10

minus5

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 4 Convergence plots of 22-element LAA

obtained using EFPA is shown in Table 10The comparison ofconvergence curves between EFPA and BBO [17] CS [17] DE[17] FA [17] and COA [17] is given in Figure 6 which showsthat EFPA reaches the optimum solution in less number ofiterations The comparison of results in terms of SLL andnull values achieved is given in Table 11 which again clearlyindicates the superior performance of EFPA as against othermethods The SLL of EFPA is minus2290 dB which is lower by1 dB 924 dB 93 dB 104 dB 826 dB 924 dB 127 dB and030 dB than BSA [18] QPM [3] COA [17] ACO [13] GA[7] CLPSO [20] and FPA [21] respectively The null valuesare also equal to or less than minus60 dB Figure 7 shows the

BSAHSA

MOEAEFPA

Null

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 5 Radiation plot of 22-element EFPA optimized LAA

comparison of QPM [3] HSA [16] BSA [18] and EFPAoptimized array radiation plots

The last design example illustrates the use of EFPA tooptimize the element positions for minimization of SLLand null placement for 32-element LAA The SLL is to beminimized in the region120601 isin [93∘ 180∘] andnull is to be placedin the direction of 99∘ The required beamwidth BW119889 is 71

∘The plot of convergence for EFPA is shown in Figure 8 Alsofor the sake of comparison the convergence plots of BBO[17] CS [17] DE [17] FA [17] and COA [17] are shown in thesame figureThe element positions for EFPA optimized array

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

International Journal of Antennas and Propagation 9

Table 11 Performance of different algorithms for 28-element LAA

Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2190 minus1324 minus136 minus2186 minus1464 minus1360 minus2163 minus2260 minus2290Null (dB)(120∘) minus8249 minus4849 minus5274 minus6008 minus5274 minus5925 minus6004 minus7845 minus60Null (dB)(1225∘) minus9359 minus4835 minus5166 minus6005 minus5920 minus7553 minus6001 minus9651 minus60Null (dB)(125∘) minus8049 minus893 minus6146 minus6010 minus4358 minus6252 minus6000 minus8258 minus6766

Table 12 Geometry of 32-element (with respect to 1205822) LAA obtained using EFPA

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Position 05268 11101 22759 29840 38883 45802 57817 66474 77093 88370 97490 110885 126565 143043 160043 177043

Table 13 Performance of different algorithms for 32-element LAA

Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPASLL (dB) minus2050 minus1880 minus1951 minus1773 minus2283 minus1624 minus2275 minus2310 minus2373Null (dB)(99∘) minus10750 minus6220 minus8808 minus3474 minus6263 minus6294 minus60 minus13060 minus6000

BBOCSDE

FAEFPA

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

Fitn

ess

500 1000 1500 20000Iterations

Figure 6 Convergence plots of 28-element LAA

are shown in Table 12 Table 13 compares the performanceof different optimization methods for design of 32-elementLAA The SLL of proposed EFPA antenna array is minus2373 dBwhich is least in Table 13 The reduction is considerable ascompared to BSA [18] QPM [3] PSO [3] HSA [16] and GA[7] methods (Figure 9)

5 Conclusions

FPA is a novel nature inspired algorithm which mimics thepollination of flowers But it has some weaknesses like poor

QPMHSA

BSAEFPA

Nulls

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 7 Radiation plot of 28-element EFPA optimized LAA

exploitation capability and slow convergence speed So toovercome these problems an improved version of FPA calledEFPAhas been proposed in this paperThreemodifications inthe basic FPA have been proposed in the enhanced versionThe modifications in EFPA help in providing better searchcapability and help it to escape local minima The proposedalgorithm has been tested on seven benchmark functionsThe results show that EFPA is able to find global optimaof most of the benchmark functions Moreover the EFPAis utilized for pattern synthesis of nonuniform LAA Fourdifferent antenna arrays of different sizes and different designconstraints have been taken When EFPA is used for SLL

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

10 International Journal of Antennas and Propagation

minus20

minus15

minus10

minus5

0

BBOCSDE

FACOAEFPA

500 1000 1500 20000Iterations

Fitn

ess

Figure 8 Convergence plots of 32-element LAA

CSHSA

PSOEFPA

Null

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Nor

mal

ised

AF

(dB)

100 110 120 130 140 150 160 170 18090120601 (deg)

Figure 9 Radiation plot of 32-element EFPA optimized LAA

minimization only it reaches the optimum result in lessnumber of iterations than FPA For multiobjective designwith SLL suppression null and beamwidth control EFPAobtains better performance than the results of well-knownalgorithms reported in literature The results indicate thepotential ability of EFPA to be used for optimization for otherantenna designs

Competing Interests

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

References

[1] C Balanis Antenna Theory-Analysis and Design John Wiley ampSons 2nd edition 1997

[2] A Recioui ldquoSidelobe level reduction in linear array patternsynthesis using particle swarm optimizationrdquo Journal of Opti-mization Theory and Applications vol 153 no 2 pp 497ndash5122012

[3] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[4] M Shihab Y Najjar N Dib and M Khodier ldquoDesign ofnon-uniform circular antenna arrays using particle swarmoptimizationrdquo Journal of Electrical Engineering vol 59 no 4pp 216ndash220 2008

[5] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo ProgressIn Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[6] NDib andA Sharaqa ldquoSynthesis of thinned concentric circularantenna arrays using teaching-learning-based optimizationrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 24 no 4 pp 443ndash450 2014

[7] M Rattan M S Patterh and B S Sohi ldquoSynthesis of aperiodicliner antenna arrays using genetic algorithmrdquo in Proceedings ofthe 19th International Conference on Applied Electromagneticsand Communications (ICECom rsquo07) pp 1ndash4 September 2007

[8] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[9] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[10] Y Cengiz and H Tokat ldquoLinear antenna array design with useof genetic memetic and tabu search optimization algorithmsrdquoProgress in Electromagnetics Research C vol 1 pp 63ndash72 2008

[11] 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

[12] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[13] 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

[14] S Pal B-Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress In Electromagnetics Research B no 21 pp 87ndash1112010

[15] A Chowdhury R Giri A Ghosh S Das A Abraham and VSnasel ldquoLinear antenna array synthesis using fitness-adaptivedifferential evolution algorithmrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10)mdashIEEE Congress on Evolutionary Computation (CEC rsquo10) July2010

[16] 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

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

International Journal of Antennas and Propagation 11

[17] U Singh and M Rattan ldquoDesign of linear and circular antennaarrays using cuckoo optimization algorithmrdquo PIER C vol 46pp 1ndash11 2014

[18] K Guney and A Durmus ldquoPattern nulling of linear antennaarrays using backtracking search optimization algorithmrdquo Inter-national Journal of Antennas and Propagation vol 2015 ArticleID 713080 10 pages 2015

[19] M Khodier ldquoOptimisation of antenna arrays using the cuckoosearch algorithmrdquo IET Microwaves Antennas amp Propagationvol 7 no 6 pp 458ndash464 2013

[20] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[21] U Singh and R Salgotra ldquoSynthesis of linear antenna arrayusing flower pollination algorithmrdquo Neural Computing andApplications 2016

[22] M Walker ldquoHow flowers conquered the worldrdquo BBC EarthNews July 2009 httpnewsbbccoukearthhiearth newsnewsid 81430008143095stm

[23] B Glover Understanding Flowers and Flowering An IntegratedAppraoch Oxford University Press 2007

[24] N M Waser ldquoFlower constancy definition cause and mea-surementrdquoTheAmerican Naturalist vol 127 no 5 pp 593ndash6031986

[25] X S Yang ldquoFlower pollination algorithm for global optimiza-tionrdquo in Unconventional Computation and Natural Computa-tion pp 240ndash249 Springer Berlin Germany 2012

[26] A Draa ldquoOn the performances of the flower pollinationalgorithmmdashqualitative and quantitative analysesrdquo Applied SoftComputing vol 34 pp 349ndash371 2015

[27] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal ofMathematicalModelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[28] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR-06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[29] 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

[30] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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