Research ArticleSynthesis of Phase-Only Reconfigurable LinearArrays Using Multiobjective Invasive Weed OptimizationBased on Decomposition

Yan Liu1 Yong-Chang Jiao1 Ya-Ming Zhang2 and Yan-Yan Tan3

1 National Key Laboratory of Antennas and Microwave Technology Xidian University Xirsquoan 710071 China2 School of Electronics and Information Northwestern Polytechnical University Xirsquoan 710072 China3 School of Information Science and Engineering Shandong Normal University Jinan 250014 China

Correspondence should be addressed to Yan Liu liuyan xidian163com

Received 24 April 2014 Revised 25 August 2014 Accepted 26 August 2014 Published 20 October 2014

Academic Editor Stefano Selleri

Copyright copy 2014 Yan Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Synthesis of phase-only reconfigurable array aims at finding a common amplitude distribution and different phase distributions forthe array to form different patterns In this paper the synthesis problem is formulated as amultiobjective optimization problem andsolved by a new proposed algorithmMOEAD-IWO First novel strategies are introduced in invasive weed optimization (IWO) tomake original IWO fit for solving multiobjective optimization problems then the modified IWO is integrated into the frameworkof the recently well proved competitivemultiobjective optimization algorithmMOEAD to form a new competitiveMOEAD-IWOalgorithm At last two sets of experiments are carried out to illustrate the effectiveness of MOEAD-IWO In addition MOEAD-IWO is compared with MOEAD-DE a new version of MOEAD The comparing results show the superiority of MOEAD-IWOand indicate its potential for solving the antenna array synthesis problems

1 Introduction

In actual applications such as satellite communications andradar navigations single antenna array is generally requiredto have the capability of producing a number of radiationpatterns with different shapes so as to save space and reducecost In practice changing excitation phases is much easierthan changing excitation amplitudes in the feeding networkKeeping this in mind the phase-only reconfigurable array isdesigned to radiate multiple radiation patterns using a singlepower divider network and different phase shiftersThanks toits advantages phase-only reconfigurable array attracts muchmore attentions in recent years [1ndash10]

During the past decades a number of design methodsfor synthesizing phase-only reconfigurable arrays have beenproposed [1ndash10] These methods can be divided into twocategories One is local search algorithms like ldquoalternatingprojectionsrdquo method [1ndash3] which are efficient and simplebut sensitive to initial values The other is the evolutionary

algorithms such as genetic algorithm (GA) [4ndash6] particleswarm optimization (PSO) [6ndash8] and differential evolution(DE) [9 10] which have global searching ability It has beenshown that the evolutionary algorithms are more effectiveand flexible for synthesizing phase-only reconfigurable arrays[4ndash10] With the rapid development of high-speed computertechnology more researchers prefer to adopt evolutionaryalgorithms for designing the pattern reconfigurable arrays

For pattern reconfigurable arrays multiple patternsshould meet their design indexes simultaneously and thephase-only synthesis problem can be expressed as a typicalmultiobjective optimization problem with two or more con-flicting objectives

In most existing literatures multiple objectives are usu-ally summed into a single objective function with differ-ent weights and the multiobjective optimization problemis converted into a single-objective optimization problemAlthough the optimization problem is solved easily by usingthis approach someproblems arise inevitablyTheweights for

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 630529 11 pageshttpdxdoiorg1011552014630529

2 International Journal of Antennas and Propagation

different objectives depend on experience or lots of experi-ments thus decision-makersmust have enough prior knowl-edge of the problem Only when the weights are set properlycan the desired patterns be achieved In order to achievemore optimal solutions more experiments need to be doneThis would be a very complex and time-consuming processHence in this paper design of phase-only reconfigurablearrays will be formulated as a multiobjective optimizationproblem (MOP)

In 2007 Zhang and Li [11] proposed a novel method forMOPs calledmultiobjective evolutionary algorithmbased ondecomposition (MOEAD) which makes use of the decom-position methods in mathematics and the optimizationparadigm in evolutionary computation Later a version ofMOEAD employing the differential evolution (DE) operatornamed MOEAD-DE [12ndash14] was proposed and shown toperform well on the MOPs with complicated Pareto setshapes Theoretically any evolutionary operator can be usedin MOEAD for actual problems

Invasive weed optimization (IWO) is a numerical sto-chastic optimization algorithm inspired from weed colo-nization which was proposed by Mehrabian and Lucas in2006 [15] The algorithm imitates seed spatial diffusiongrowth reproduction and competitive exclusion process ofinvasive weed With strong robustness and adaptability IWOconverges to the optimal solution effectively It has beenproved that IWO can be used to successfully solve manysingle-objective optimization problems [16ndash20] and someMOPs [21]

In order to solve the design problem of phase-onlyreconfigurable arrays effectively a new version of MOEADcalled MOEAD-IWO is proposed First in order to improvethe performance of IWO in MOEAD-IWO we presentan adaptive standard deviation which changes not onlywith the increase of evolution generations but also withthe cost function value of each individual This strategyimproves the convergence rate and helps the seeds escapefrom local optimum Then the modified IWO with anadaptive standard deviation is integrated into the frameworkof MOEAD for solving multiobjective problems ThenMOEAD-IWO is formed Taking advantage of the powerfulsearching ability of invasive weeds the overall performanceof the proposed MOEAD-IWO is shown in solving thesynthesis problems The problems are also solved by theexisting MOEAD-DE The superiority of MOEAD-IWOover MOEAD-DE in solving array synthesis problems isdemonstrated

The remainder of this paper is organized as followsSection 2 describes the definitions of multiobjective opti-mization problems Section 3 introduces implementationsteps of MOEAD-IWO In Section 4 basic principle ofpattern reconfigurable arrays is presented In Section 5MOEAD-IWO is used to design the phase-only reconfig-urable linear array Experimental results and discussion aregiven in Section 6 Finally Section 7 concludes this paper

2 Multiobjective Problems

Assuming minimizing 119872 objectives simultaneously a mul-tiobjective optimization problem (MOP) can be stated asfollows

Minimize 997888119865 (

997888119909) = (119891

1(997888119909) 1198912(997888119909) 119891

119872(997888119909))119879

subject to 997888119909 isin Ω

(1)

where Ω is the decision space 997888119909 = (1199091 119909

119873)119879 is the

decision vector 119865 Ω rarr 119877119872 consists of 119872 real-valued

objective functions and 119877119872 is called the objective space The

attainable objective set is defined as the set 119865(119909) | 119909 isin ΩBeing different from single-objective optimization one bestsolution (global minimum) with respect to all objectives maynot exist inMOPs All the objectives in (1) are treatedwith thesame importance and generally no point in Ω can minimizeall the objectives simultaneously The aim of MOPs is to finda set of points which cannot be improved in one objectivewithout impairing at least one other objective These pointsare called the Pareto solution

Definition 1 (Pareto dominance) Let 997888119906 = (119906

1 119906

119872)119879

and 997888V = (V1 V

119872)119879

isin 119877119872 be two vectors 119906 is said to

dominate V denoted by 119906 ≺ V if

forall119894 isin 1 119872 119906119894le V119894and exist119895 isin 1 119872 119906

119895lt V119895

(2)

Note that if Pareto dominance does not hold between twovectors then they are considered nondominated with eachother

Definition 2 (Pareto optimality) 997888119909119906isin Ω is said to be Pareto

optimal if and only if

∄997888119909 V isin Ω

997888119865 (

997888119909 V) ≺

997888119865 (

997888119909119906) (3)

Definition 3 (Pareto optimal set) The Pareto optimal setdenoted by PS is defined as

PS = 997888119909119906isin Ω | exist

997888119909 V isin Ω

997888119865 (

997888119909 V) ≺

997888119865 (

997888119909119906) (4)

Definition 4 (Pareto front) For a given MOP 997888119865(

997888119909) and its

Pareto optimal set PS the Pareto front is defined as

PF = 997888119906 =

997888119865 (

997888119909119906) |

997888119909119906isin PS (5)

Instead of searching for a single or just a few optimalsolutions as in solving single-objective problems the goal ofhandling multiobjective problem is to find the Pareto front aswell as the Pareto set of the problem

3 The Main Procedures of MOEAD-IWO

One of the key issues of MOEAD is that a decompositionmethod is used to transform an MOP into a number of

International Journal of Antennas and Propagation 3

single-objective optimization subproblems and MOEADoptimizes these subproblems collectively and simultaneouslyThe objective function of each single-objective subproblemis an aggregation of all the objective functions in MOPAt each generation the population is composed of thebest solutions found so far for each subproblem which isoptimized by using information only from its neighboringsubproblems Neighborhood relations among these single-objective subproblems are defined based on the Euclidiandistances between their weight vectors MOEAD exploitsinformation sharing among subproblems which are neigh-bors to accomplish the optimization task effectively andefficiently MOEAD-IWOmaintains the basic framework ofMOEAD [11] and the IWO algorithm is used for producingnew solutions When MOEAD-IWO is used to optimizeMOP the following issues need to be considered

31 Produce theWeightVectors As in [11] we use the simplex-lattice design method [22] to produce the weight vectorsin which each individual weight takes value from 0119867

1119867 119867119867 where119867 is an integer

32 Decompose an MOP into a Number of Single-Objective Subproblems In theory weighted sum Tcheby-cheff approach boundary intersection and any otherdecomposition approach can be used to convert anMOP intoa number of single-objective subproblems [11] Tchebycheffapproach is mainly employed in our experimental studyLet 1205821 1205822 120582119875 be a set of even spread 119875 weight vectorswhere 120582

119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 is the weight vector for the 119894th

subproblem 120582119894119895ge 0 119895 = 1 2 119872 and sum

119872

119895=1120582119894

119895= 1 119872 is

the number of objectives in (1) 119894 = 1 2 119875 119911lowast representsthe reference point The 119894th single-objective optimizationsubproblem obtained by decomposing the given MOP canbe represented as [23]

Minimize 119892119905119890(119909 | 120582

119894 119911lowast) = max1le119895le119872

120582119894

119895

10038161003816100381610038161003816119891119895(119909) minus 119911

lowast

119895

10038161003816100381610038161003816

subject to 119909 isin Ω

(6)

The Tchebycheff approach searches solutions through min-imizing 119892

119905119890 toward 119911lowast which has an advantage that both

convex and concave Pareto front can be approximated[24] All these 119875 single-objective functions are optimizedsimultaneously in each run If we use a large 119875 and selectthe weight vectors properly all the optimal solutions ofthe single-objective problems from decomposition will wellapproximate the Pareto front The optimization process isaccomplished by the collaboration of the neighbors for eachsubproblem

33 Confirm the Neighbors of Each Single-Objective Sub-problem In MOEAD each subproblem is optimized byusing informationmainly from its neighboring subproblemsThe neighborhood relations among these subproblems aredefined based on the distances between their weight vectorsCompute the Euclidean distances between any two weightvectors and thenwork out the119879 closest weight vectors to each

weight vector For the 119894th subproblem its neighbor is set to119861(119894) = 119894

1 1198942 119894

119879 119894 = 1 2 119875 where 1205821198941 1205821198942 120582119894119879 are

119879 weight vectors close to 120582119894 Single-objective subproblems

which are considered neighbors will have similar fitnesslandscapes and their optimal solutions should be close inthe decision space MOEAD exploits information sharingamong subproblems which are neighbors to accomplish theoptimization task effectively and efficiently

34 The Selection of Evolution Strategy The IWO operatoris integrated into MOEAD-IWO to produce new solutionsIn IWO new solutions (seeds) produced are randomly dis-tributed in119863-dimensional space around their parents (weed)in normal distribution 119873(0 120590

2

iter) the change rule for 120590iter isexpressed as

120590iter = 120590final + (itermax minus iter

itermax)

pow(120590initial minus 120590final) (7)

where 120590initial and 120590final are initial and final standard deviationsand pow is the nonlinear regulatory factor It can be seenfrom (7) that 120590iter decreases with the increase of evolutiongenerations while the 120590iter value of each subproblem inone generation is the same This is not conducive to thealgorithm convergence Thus we modify IWO and presentan adaptive standard deviation in which the value of 120590iterin one generation changes not only with the cost functionvalue of each subproblem but also with the maximum andminimum cost function values in the generation as shown inthe following equation

120590iter119894 =

(1 + 120574FV119894minus FV119886

FVmax minus FV119886

)120590iter FV119894ge FV119886

(1 minus 120574FV119886minus FV119894

FV119886minus FVmin

)120590iter FV119894lt FV119886

(8)

where FV119894is the cost function value of the 119894th subproblem

FVmax FVmin andFV119886 denote themaximumminimum andaverage cost function values among all these subproblemsrespectively and 120574 is the zoom factor controlling the variationrange of the standard deviation which takes value from 0 to1 In this paper we take 120574 = 05

It can be seen that for the minimization problem thesubproblems with lower cost function values have smallerstandard deviation in one generation which enables theseeds distributed around the better subproblems Moreoverthe range of the adaptive standard deviation 120590iter119894 is [1 minus

120574 1 + 120574]120590iter which makes the standard deviation of onesubproblem in the younger generation likely to be larger thanthat in the older generation This will help the new producedseeds escape from local optimum improve the convergencerate and balance the global and local search capabilitieseffectively at the same time

In the present generation the number of seeds producedby the 119894th subproblem is calculated by

119904num = floor(FVmax minus FV

119894

FVmax minus FVmin(119904max minus 119904min) + 119904min) (9)

4 International Journal of Antennas and Propagation

where 119904max and 119904min are the largest and smallest numbersof seeds produced respectively and floor() represents theround-down function It is clear that better populationmembers produce more seeds

Suppose 119909119894= (119909119894

1 119909119894

2 119909

119894

119873)119879 119894 = 1 2 119875 is the

present optimal solution for the 119894th subproblem and eachnew solution produced by 119909

119894 is 119910 = (1199101 1199102 119910

119873)119879 in

which each element 119910119896is generated as follows

119910119896= 119909119894

119896+ 119873(0 120590

2

iter119894) 119896 = 1 2 119873 (10)

Then 119904num new solutions are used to update the 119894th subprob-lem and its neighborhoods

35 Metrics for Comparing Nondominated Sets Several met-rics have been proposed to compare the performance ofdifferent multiobjective evolutionary algorithms quantita-tively such as generational distances (GD) [25] invertedgenerational distance (IGD) [26] and spread [27] Whenusing the above three metrics we need to know the truePareto front of the optimization problem first However inactual engineering problem we cannot obtain the true Paretofront [28] Then in this section three metrics are adoptedto make a quantitative assessment of the performance ofMOEAD-DE and MOEAD-IWO

Before giving the performance metrics we first presentnormalization of vectors in objective space Assume we have119901nondominated sets119885

1 1198852 119885

119901 First we set119885 = 119885

119894 119894 =

1 2 119901Then we define 119891min119894

= min119891119894(119909) 119865(119909) = 119911 isin 119885

and 119891max119894

= max119891119894(119909) 119865(119909) = 119911 isin 119885 which correspond to

the minimum and maximum values for the 119894th coordinate ofthe objective space Then for all coordinates 119894 = 1 2 119899we calculate all points according to

119891119894(119909) =

119891119894(119909) minus 119891

min119894

119891max119894

minus 119891min119894

(11)

The normalized vector of a vector 119911 is denoted by 119911 the set ofnormalized vectors is shown as 119885

351 119863-Metric for Accuracy [29] Suppose we have twonormalized nondominated sets 119885

1and 119885

2 forall119886 isin 119885

1 we

search for exist119887 isin 11988521

sube 1198852such that 119887 lt 119886 Then we calculate

Euclidean distances from 119886 to all points 119887 isin 11988521 So the 119863-

metric is defined as

119863(1198851 1198852) = sum

119886isin1198851

10038171003817100381710038171003817119886 minus 119887

10038171003817100381710038171003817max

radic119899100381610038161003816100381610038161198851

10038161003816100381610038161003816

(12)

where 119886 minus 119887max = max119886 minus 119887 119886 isin 1198851 119887 isin 119885

21 and 119899 is

the dimension of the objective space The 119863-metric returnsa value among 0 and 1 where the smaller the better Themaximum distance from a dominating point is taken as abasis for accuracy

352 Δ-Metric for Uniformity [29] For a given normalizednondominated set119885 we define119889

119894to be the Euclidean distance

between two consecutive vectors 119894 = 1 2 (|119885| minus 1) Let119889 = sum

|119885|minus1

119894=1119889119894(|119885| minus 1)Then we define

Δ (119885) =

|119885|minus1

sum

119894=1

10038161003816100381610038161003816119889119894minus 119889

10038161003816100381610038161003816

radic119899 (1003816100381610038161003816100381611988510038161003816100381610038161003816minus 1)

(13)

Note that 0 le Δ(119885) le 1 where 0 is the best

353 nabla-Metric for Extent [29] Given a nondominated set 119885we define

nabla (119885) =

119899

prod

119894=1

10038161003816100381610038161003816119891max119894

minus 119891min119894

10038161003816100381610038161003816 (14)

where 119899 is the dimension of the objective space A bigger valuespans a larger portion of the hypervolume and therefore isalways better

4 Basic Principle of Pattern ReconfigurableArray Antennas

Design of phase-only reconfigurable antenna array aimsat finding a common amplitude distribution and differentphase distributions such that the array can produce multipledifferent patterns

Consider a linear equispaced array with119873 elements If119872different patterns need to be produced only by changing theexcitation phases of the array under the common excitationamplitude distribution the optimization variable 119909 is a vectorwith 119872119873 + 119873 elements where 119909

119899(119899 = 1 2 119873) is the

excitation amplitude for the 119899th antenna element denotedby 119868119899and 119909

119898119873+119899(119898 = 1 2 119872 119899 = 1 2 119873) is the

excitation phase for the 119899th antenna element and the 119898thpattern denoted by 120593

119898119899 Then the complex excitation of the

119899th element in the119898th pattern is

119894119898119899

= 119868119899sdot 119890119895120593119898119899 = 119909

119899119890119895119909119899+119898119873 (15)

It can be seen from (15) that in the process of optimiza-tion the common excitation amplitude is used for119872 patternsall the time and only phases of the excitation are differentThe119898th pattern produced by the antenna array for far field isgiven by

119865119898(119906) =

EF (119906)

119865max

119873

sum

119899=1

119894119898119899

sdot 1198901198952120587119899119889119906120582

(16)

where 119898 = 1 2 119872 119889 is the spacing between arrayelements 120582 is the wavelength in free space 119906 = sin 120579 120579 isthe angle from ray direction to normal of array axis EF(120579) iselement factor (EF(120579) = 1 for isotropic source) and 119865max ispeak value of far field pattern

In this paper patterns we need to reconfigure by thephase-only reconfigurable linear arrays are presented below

(1) A flat-top beam and a pencil beam the designproblem is expressed as

minimize 119865 (119909) = (1198911(119909) 119891

2(119909))119879

(17)

International Journal of Antennas and Propagation 5

Table 1 Design objectives and simulated results for Problem 1

Design parametersFlat-top beam Pencil beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus2546 minus2613 minus3000 minus2956 minus3050

Half-power beam width(HPBW in 120579-space) 24∘ 242∘ 24∘ 68∘ 68∘ 68∘

Beam width at SLL(BW in 120579-space) 40∘ 416∘ 402∘ 20∘ 20∘ 20∘

Ripple(in dB 78∘ le 120579 le 102

∘) 050 050 049 NA NA NA

Table 2 Design objectives and simulated results for Problem 2

Design parametersFlat-top beam Cosecant-squared beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus240106 minus249659 minus2500 minus240735 minus249060

Beam width at SLL(BW in 120579-space) 084 088 086 084 088 086

Ripple(in dB)

060|119906| le 032

05998 05994 0600 le 119906 le 064

08588 05341

The cost functions of each single pattern in (17) are

1198911(119909) = Cost

119891=

4

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(18)

1198912(119909) = Cost

119901=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(19)

(2) A flat-top beam and a cosecant beam the designproblem is expressed as

minimize 119865 (119909) = (1198913(119909) 119891

4(119909))119879

(20)

The cost functions of each single pattern in (20) are

1198913(119909) = Cost

119891=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(21)

1198914(119909) = Cost

119888=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(22)

where Cost119891 Cost

119888 and Cost

119901represent the cost functions of

the flat-top beam cosecant-squared beam and pencil beamrespectively and119876

119904119889and119876

119904(119909) are the desired and calculated

values for each design specification we use Desired andcalculated values of each design index in (18) (19) (21) and(22) are summarized in Tables 1 and 2 The lower the costvalue is the closer the calculated value approaches the desiredvalue When the calculated values of all the indexes are lessthan the corresponding desired values the cost value is set tozero

5 MOEAD-IWO for Synthesis of Phase-OnlyReconfigurable Linear Array

Suppose 119875 is the population size that is to say we need tooptimize 119875 subproblems in a single run 1205821 1205822 120582119875 isa set of weight vectors The objective functions of all thepatterns are summed with assigned weights For each single-objective subproblem the objective function is described asfollows

119892119905119890(119909 | 120582

119894 119911lowast) = max1le119898le119872

120582119894

119898

1003816100381610038161003816119891119898 (119909) minus 119911lowast

119898

1003816100381610038161003816 (23)

where 120582119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 119894 = 1 2 119875 MOEAD-IWO

is used tominimize119875 objective functions (23) simultaneouslyin a single run and the neighborhood of 120582119894 is the severalweight vectors close to 120582

119894At each generation MOEAD-IWOmaintains the follow-

ing information

(i) a population of 119875 points 1199091 1199092 119909

119875isin Ω where

Ω is the feasible solution space of the excitationamplitude and phases

(ii) FV1 FV2 FV119875 where FV119894 = 119892119905119890(119909119894

| 120582119894) 119894 =

1 2 119875

(iii) Gen the current generation number

(iv) 119911 = (1199111 1199112 119911

119872)119879 where 119911

119898is the best value found

so far for objective 119891119898

6 International Journal of Antennas and Propagation

The algorithm is outlined as follows

Input

(i) The multiobjective problem (17) or (20)(ii) A stopping criterion(iii) 119875 the population size (the number of subproblems)(iv) 119879 the number of the neighbors for each subproblem(v) A uniform set of 119875 weight vectors 1205821 1205822 120582119875

Output

(i) Approximation to the PS 1199091 1199092 119909119875(ii) Approximation to the PF 119865(1199091) 119865(1199092) 119865(119909119875)

Step 1 (initialization) Consider the following

Step 11 Generate the weight vectors 1205821 1205822 120582119875

Step 12 Compute the Euclidian distances between any twoweight vectors to each weigh vector For each 119894 = 1 2 119875set 119861(119894) = 119894

1 1198942 119894

119879 where 120582

1198941 1205821198942 120582

119894119879 are 119879 weight

vectors close to 120582119894

Step 13 Generate an initial population 1199091 1199092 119909

119875 byuniformly random sampling from the search spaceΩ

Set FV119894 = 119892119905119890(119909119894| 120582119894)

Step 14 Initialize 119911 = (1199111 1199112 119911

119872)119879 by setting 119911

119898=

min1le119894le119875

119891119898(119909119894)

Step 2 (update) For 119894 = 1 2 119875 do the following

Step 21 (selection of matingupdate range) Generate a num-ber rand from [0 1] randomly Then set

119878 = 119861 (119894) if rand lt 120575

1 2 119875 otherwise(24)

120575 is the probability that parent individuals are selected fromthe neighborhood

Step 22 (reproduction) Calculate the number of new solu-tions 119904num using (9) and produce new solutions 119910

119896(119896 =

1 2 119904num) using (8)

Step 23 (repair) If an element in 119910119896 is out of the boundary of

Ω its value is reset to the boundary value

Step 24 (update of 119911) For each119898 = 1 2 119872 if119891119898(119910

min) lt

119911119898 in which 119910

min is the minimum solution among 119910119896 119896 isin

1 2 119904num then set 119911119898

= 119891119898(119910

min)

Step 25 (update of solutions) Set 119888 = 0 and then do thefollowing

(1) If 119888 = 120578119903or 119878 is empty go to Step 3 Otherwise

randomly pick an index 119895 from 119878

(2) If min119892119905119890(1199101 | 120582119895) 119892119905119890(1199102 | 120582119895) 119892119905119890(119910119904num | 120582119895) le

119892119905119890(119909119895| 120582119895) then set 119909119895 = 119910

min FV119895 = 119892119905119890(119910

min|

120582119895) in which 119910

min is the minimum solution among119910119896 119896 isin 1 2 119904num Set 119888 = 119888 + 1

(3) Delete 119895 from 119878 and go to (1)

Step 3 (stopping criteria) If stopping criteria are satisfiedthen stop and output 119909

1 1199092 119909

119875 and 119865(119909

1) 119865(119909

2)

119865(119909119875) Otherwise go to Step 2

In the population optimization variables for each indi-vidual are the common excitation amplitudes for all patternsand different phases for forming different patterns Theoutput of the algorithm is a Pareto set in which each Paretooptimal solution corresponds to a phase-only reconfigurablearray In the optimization results different weight coefficientscorresponding to various patterns form a weight vector eachPareto optimal solution corresponds to a weight vector anddifferent Pareto optimal solutions have different weight vec-tors In actual applications decision-makers select a desiredsolution from the approximated PF or obey some standardsfor choosing the best compromise solution

6 Experiments Results

To demonstrate whether the proposed MOEAD-IWO out-performs MOEAD-DE in the design of phase-only patternreconfigurable linear arrays we carry out two experiments

61 Parameter Setting In MOEAD the population size andweight vectors are controlled by an integer 119867 1205821 1205822 120582119875are the weight vectors in which each individual weight takesa value from 0119867 1119867 119867119867 therefore the populationsize is 119875 = 119862

119872minus1

119867+119872minus1 where 119872 is the number of objectives In

the experiments population size 119875 is set to 201 (119867 = 200) for2-objective instances For the other control parameters we set119879 = 01119875 120578

119903= 001119875 and 120575 = 07 [30] which are the same

for MOEAD-DE and MOEAD-IWO For DE operator weset the crossover rate CR = 10 the scalar factor 119865 = 05120578 = 20 and 119901

119898= 1119899 in mutation operators [12 30] For

IWO operator we set 120590initial = 01 120590final = 00002 pow =

3 119904max = 3 and 119904min = 1 To be fair in the comparisonboth algorithms stop after reaching the maximum functionevaluations 500000 for all the tests In order to restrict theinfluence of random effects both algorithms run 20 timesindependently of each problem and the best results arepresented in this section Source codes for MOEAD-DE aretaken from [31]

62 Determining the Best Compromise Solution When facinga set of nondominated solutions a decision-makerrsquos judg-ment may have fuzzy or imprecise goals for each objectivefunction so useful quantitative measures for assessing thequality of each nondominated solution are needed In presentliteratures an effective method based on the concept of fuzzysets is proposed [25] and successfully used in some MOPs[32ndash34] in which a membership function 120583

119894is defined for

each objective 119891119894by taking account of the minimum and

International Journal of Antennas and Propagation 7

Table 3 The best compromise solutions for the first instance by MOEAD-IWO and MOEAD-DE (DRR = 4)

1198911(flat-top beam) 119891

2(pencil beam) (120582

1 1205822) Cost = 120582

11198911+ 12058221198912

MOEAD-DE with TE 00069 00059 (05350 04650) 00064MOEAD-DE with WS 00086 00084 (05100 04900) 00085MOEAD-IWO with TE 00037 00056 (05100 04900) 00046MOEAD-IWO with WS 00062 00060 (05450 04550) 00061

Table 4 The best compromise solutions for the second instance by MOEAD-IWO and MOEAD-DE (DRR = 5)

1198913(flat-top beam) 119891

4(cosecant beam) (120582

3 1205824) Cost = 120582

31198913+ 12058241198914

MOEAD-DE with TE 00043 00578 (05200 04800) 00300MOEAD-DE with WS 00762 01702 (04400 05600) 01288MOEAD-IWO with TE 00015 00511 (04750 05250) 00275MOEAD-IWO with WS 00016 00516 (04650 05350) 00284

maximum values of each objective function together with therate of increase of membership satisfaction As in [33 34]we assume that 120583

119894is a strictly monotonic decreasing and

continuous function defined as

120583119894=

1 if 119891119894le 119891

min119894

119891max119894

minus 119891119894

119891max119894

minus 119891min119894

if 119891min119894

le 119891119894le 119891

max119894

0 if 119891119894ge 119891

max119894

(25)

where 119891max119894

and 119891min119894

are the maximum and minimumvalues of the 119894th objective solutions among all nondominatedsolutions respectively Value 120583

119894indicates how much a non-

dominated solution satisfies the 119894th objective so the sum of120583119894(119894 = 1 2 119898) for all the objectives can be calculated

for measuring the ldquoaccomplishmentrdquo of each solution insatisfying the 119872 objectives and then a membership function119877119906(119906 = 1 2 119873

119904) can be defined for nondominated

solutions as follows

119877119906=

sum119872

119894=1120583119894(119906)

sum119873119904

V=1sum119872

119894=1120583119894(V)

(26)

where119873119904represents the number of nondominated solutions

The function 119877119906can be treated as a membership function

for the nondominated solutions in a fuzzy set Value 119877119906

represents the quality of a solution among all nondominatedsolutions and the one with the maximum value of 120583119902 is thebest compromise solution

63 Results Effectiveness of the MOEAD-IWO algorithmis verified through the design of two examples In theoptimization process excitation amplitudes range from 0 to1 and phases are restricted from minus180 to 180 degrees

631 Problem 1 Consider a 20-element 05120582-equispacedsymmetrical array with EF(120579) = sin(120579) for generating a flat-top beam and a pencil beamThe flat-top beam is realized bythe common excitation amplitudes and unknown phases andthe phases for forming pencil beam are set to zeroTherefore

there are 10 common excitation amplitude variables and 10phase variables for forming flat-top beam to be optimized inobjective function (17) The maximum dynamic range ratio(DRR) of excitation amplitudes is controlled with 4

The best compromise solutions obtained by MOEAD-IWO and MOEAD-DE are shown in Figure 1 and Table 1 Itcan be seen that for flat-top beam the peak SLL obtainedby MOEAD-IWO is 067 dB lower than that obtained byMOEAD-DE The BW value obtained by MOEAD-DE is16∘ wider than the objective value while that obtained byMOEAD-IWO is just 02∘ wider than the design objectiveFor the pencil beam the pattern obtained byMOEAD-IWOhas 094 dB lower peak SLL value when other indexes meetthe design objectives well The excitation amplitudes and thephases for the synthesized radiation patterns obtained byMOEAD-IWO are shown in Figure 2

632 Problem 2 Consider a 05120582-equispaced linear arraywith 25 isotropic elements to generate a flat-top beam and acosecant-squared beam The dimension of vector 119909 in objec-tive function (20) is 75 including 25 common amplitudesand 50 unknown phases The maximum DRR of excitationamplitudes is controlled with DRR = 5

Radiation patterns of the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE are shownin Figure 4 and Table 2 and the excitation amplitudes andthe phases for the synthesized radiation patterns obtainedby MOEAD-IWO are shown in Figure 5 It can be seenfrom Table 2 that for the flat-top beam the ripples obtainedby these two algorithms meet the design objectives wellthe peak SLL value obtained by MOEAD-IWO is 09553 dBlower than that obtained by MOEAD-DE and the BWvalues obtained by MOEAD-DE and MOEAD-IWO are088 and 086 respectively For the cosecant beam the BWobtained by MOEAD-IWO is smaller than that obtainedby MOEAD-DE and the ripple and peak SLL valuesobtained by MOEAD-IWO are 03247 dB and 08325 dBlower than those obtained by MOEAD-DE respectively

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

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2 International Journal of Antennas and Propagation

different objectives depend on experience or lots of experi-ments thus decision-makersmust have enough prior knowl-edge of the problem Only when the weights are set properlycan the desired patterns be achieved In order to achievemore optimal solutions more experiments need to be doneThis would be a very complex and time-consuming processHence in this paper design of phase-only reconfigurablearrays will be formulated as a multiobjective optimizationproblem (MOP)

In 2007 Zhang and Li [11] proposed a novel method forMOPs calledmultiobjective evolutionary algorithmbased ondecomposition (MOEAD) which makes use of the decom-position methods in mathematics and the optimizationparadigm in evolutionary computation Later a version ofMOEAD employing the differential evolution (DE) operatornamed MOEAD-DE [12ndash14] was proposed and shown toperform well on the MOPs with complicated Pareto setshapes Theoretically any evolutionary operator can be usedin MOEAD for actual problems

Invasive weed optimization (IWO) is a numerical sto-chastic optimization algorithm inspired from weed colo-nization which was proposed by Mehrabian and Lucas in2006 [15] The algorithm imitates seed spatial diffusiongrowth reproduction and competitive exclusion process ofinvasive weed With strong robustness and adaptability IWOconverges to the optimal solution effectively It has beenproved that IWO can be used to successfully solve manysingle-objective optimization problems [16ndash20] and someMOPs [21]

In order to solve the design problem of phase-onlyreconfigurable arrays effectively a new version of MOEADcalled MOEAD-IWO is proposed First in order to improvethe performance of IWO in MOEAD-IWO we presentan adaptive standard deviation which changes not onlywith the increase of evolution generations but also withthe cost function value of each individual This strategyimproves the convergence rate and helps the seeds escapefrom local optimum Then the modified IWO with anadaptive standard deviation is integrated into the frameworkof MOEAD for solving multiobjective problems ThenMOEAD-IWO is formed Taking advantage of the powerfulsearching ability of invasive weeds the overall performanceof the proposed MOEAD-IWO is shown in solving thesynthesis problems The problems are also solved by theexisting MOEAD-DE The superiority of MOEAD-IWOover MOEAD-DE in solving array synthesis problems isdemonstrated

The remainder of this paper is organized as followsSection 2 describes the definitions of multiobjective opti-mization problems Section 3 introduces implementationsteps of MOEAD-IWO In Section 4 basic principle ofpattern reconfigurable arrays is presented In Section 5MOEAD-IWO is used to design the phase-only reconfig-urable linear array Experimental results and discussion aregiven in Section 6 Finally Section 7 concludes this paper

2 Multiobjective Problems

Assuming minimizing 119872 objectives simultaneously a mul-tiobjective optimization problem (MOP) can be stated asfollows

Minimize 997888119865 (

997888119909) = (119891

1(997888119909) 1198912(997888119909) 119891

119872(997888119909))119879

subject to 997888119909 isin Ω

(1)

where Ω is the decision space 997888119909 = (1199091 119909

119873)119879 is the

decision vector 119865 Ω rarr 119877119872 consists of 119872 real-valued

objective functions and 119877119872 is called the objective space The

attainable objective set is defined as the set 119865(119909) | 119909 isin ΩBeing different from single-objective optimization one bestsolution (global minimum) with respect to all objectives maynot exist inMOPs All the objectives in (1) are treatedwith thesame importance and generally no point in Ω can minimizeall the objectives simultaneously The aim of MOPs is to finda set of points which cannot be improved in one objectivewithout impairing at least one other objective These pointsare called the Pareto solution

Definition 1 (Pareto dominance) Let 997888119906 = (119906

1 119906

119872)119879

and 997888V = (V1 V

119872)119879

isin 119877119872 be two vectors 119906 is said to

dominate V denoted by 119906 ≺ V if

forall119894 isin 1 119872 119906119894le V119894and exist119895 isin 1 119872 119906

119895lt V119895

(2)

Note that if Pareto dominance does not hold between twovectors then they are considered nondominated with eachother

Definition 2 (Pareto optimality) 997888119909119906isin Ω is said to be Pareto

optimal if and only if

∄997888119909 V isin Ω

997888119865 (

997888119909 V) ≺

997888119865 (

997888119909119906) (3)

Definition 3 (Pareto optimal set) The Pareto optimal setdenoted by PS is defined as

PS = 997888119909119906isin Ω | exist

997888119909 V isin Ω

997888119865 (

997888119909 V) ≺

997888119865 (

997888119909119906) (4)

Definition 4 (Pareto front) For a given MOP 997888119865(

997888119909) and its

Pareto optimal set PS the Pareto front is defined as

PF = 997888119906 =

997888119865 (

997888119909119906) |

997888119909119906isin PS (5)

Instead of searching for a single or just a few optimalsolutions as in solving single-objective problems the goal ofhandling multiobjective problem is to find the Pareto front aswell as the Pareto set of the problem

3 The Main Procedures of MOEAD-IWO

One of the key issues of MOEAD is that a decompositionmethod is used to transform an MOP into a number of

International Journal of Antennas and Propagation 3

single-objective optimization subproblems and MOEADoptimizes these subproblems collectively and simultaneouslyThe objective function of each single-objective subproblemis an aggregation of all the objective functions in MOPAt each generation the population is composed of thebest solutions found so far for each subproblem which isoptimized by using information only from its neighboringsubproblems Neighborhood relations among these single-objective subproblems are defined based on the Euclidiandistances between their weight vectors MOEAD exploitsinformation sharing among subproblems which are neigh-bors to accomplish the optimization task effectively andefficiently MOEAD-IWOmaintains the basic framework ofMOEAD [11] and the IWO algorithm is used for producingnew solutions When MOEAD-IWO is used to optimizeMOP the following issues need to be considered

31 Produce theWeightVectors As in [11] we use the simplex-lattice design method [22] to produce the weight vectorsin which each individual weight takes value from 0119867

1119867 119867119867 where119867 is an integer

32 Decompose an MOP into a Number of Single-Objective Subproblems In theory weighted sum Tcheby-cheff approach boundary intersection and any otherdecomposition approach can be used to convert anMOP intoa number of single-objective subproblems [11] Tchebycheffapproach is mainly employed in our experimental studyLet 1205821 1205822 120582119875 be a set of even spread 119875 weight vectorswhere 120582

119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 is the weight vector for the 119894th

subproblem 120582119894119895ge 0 119895 = 1 2 119872 and sum

119872

119895=1120582119894

119895= 1 119872 is

the number of objectives in (1) 119894 = 1 2 119875 119911lowast representsthe reference point The 119894th single-objective optimizationsubproblem obtained by decomposing the given MOP canbe represented as [23]

Minimize 119892119905119890(119909 | 120582

119894 119911lowast) = max1le119895le119872

120582119894

119895

10038161003816100381610038161003816119891119895(119909) minus 119911

lowast

119895

10038161003816100381610038161003816

subject to 119909 isin Ω

(6)

The Tchebycheff approach searches solutions through min-imizing 119892

119905119890 toward 119911lowast which has an advantage that both

convex and concave Pareto front can be approximated[24] All these 119875 single-objective functions are optimizedsimultaneously in each run If we use a large 119875 and selectthe weight vectors properly all the optimal solutions ofthe single-objective problems from decomposition will wellapproximate the Pareto front The optimization process isaccomplished by the collaboration of the neighbors for eachsubproblem

33 Confirm the Neighbors of Each Single-Objective Sub-problem In MOEAD each subproblem is optimized byusing informationmainly from its neighboring subproblemsThe neighborhood relations among these subproblems aredefined based on the distances between their weight vectorsCompute the Euclidean distances between any two weightvectors and thenwork out the119879 closest weight vectors to each

weight vector For the 119894th subproblem its neighbor is set to119861(119894) = 119894

1 1198942 119894

119879 119894 = 1 2 119875 where 1205821198941 1205821198942 120582119894119879 are

119879 weight vectors close to 120582119894 Single-objective subproblems

which are considered neighbors will have similar fitnesslandscapes and their optimal solutions should be close inthe decision space MOEAD exploits information sharingamong subproblems which are neighbors to accomplish theoptimization task effectively and efficiently

34 The Selection of Evolution Strategy The IWO operatoris integrated into MOEAD-IWO to produce new solutionsIn IWO new solutions (seeds) produced are randomly dis-tributed in119863-dimensional space around their parents (weed)in normal distribution 119873(0 120590

2

iter) the change rule for 120590iter isexpressed as

120590iter = 120590final + (itermax minus iter

itermax)

pow(120590initial minus 120590final) (7)

where 120590initial and 120590final are initial and final standard deviationsand pow is the nonlinear regulatory factor It can be seenfrom (7) that 120590iter decreases with the increase of evolutiongenerations while the 120590iter value of each subproblem inone generation is the same This is not conducive to thealgorithm convergence Thus we modify IWO and presentan adaptive standard deviation in which the value of 120590iterin one generation changes not only with the cost functionvalue of each subproblem but also with the maximum andminimum cost function values in the generation as shown inthe following equation

120590iter119894 =

(1 + 120574FV119894minus FV119886

FVmax minus FV119886

)120590iter FV119894ge FV119886

(1 minus 120574FV119886minus FV119894

FV119886minus FVmin

)120590iter FV119894lt FV119886

(8)

where FV119894is the cost function value of the 119894th subproblem

FVmax FVmin andFV119886 denote themaximumminimum andaverage cost function values among all these subproblemsrespectively and 120574 is the zoom factor controlling the variationrange of the standard deviation which takes value from 0 to1 In this paper we take 120574 = 05

It can be seen that for the minimization problem thesubproblems with lower cost function values have smallerstandard deviation in one generation which enables theseeds distributed around the better subproblems Moreoverthe range of the adaptive standard deviation 120590iter119894 is [1 minus

120574 1 + 120574]120590iter which makes the standard deviation of onesubproblem in the younger generation likely to be larger thanthat in the older generation This will help the new producedseeds escape from local optimum improve the convergencerate and balance the global and local search capabilitieseffectively at the same time

In the present generation the number of seeds producedby the 119894th subproblem is calculated by

119904num = floor(FVmax minus FV

119894

FVmax minus FVmin(119904max minus 119904min) + 119904min) (9)

4 International Journal of Antennas and Propagation

where 119904max and 119904min are the largest and smallest numbersof seeds produced respectively and floor() represents theround-down function It is clear that better populationmembers produce more seeds

Suppose 119909119894= (119909119894

1 119909119894

2 119909

119894

119873)119879 119894 = 1 2 119875 is the

present optimal solution for the 119894th subproblem and eachnew solution produced by 119909

119894 is 119910 = (1199101 1199102 119910

119873)119879 in

which each element 119910119896is generated as follows

119910119896= 119909119894

119896+ 119873(0 120590

2

iter119894) 119896 = 1 2 119873 (10)

Then 119904num new solutions are used to update the 119894th subprob-lem and its neighborhoods

35 Metrics for Comparing Nondominated Sets Several met-rics have been proposed to compare the performance ofdifferent multiobjective evolutionary algorithms quantita-tively such as generational distances (GD) [25] invertedgenerational distance (IGD) [26] and spread [27] Whenusing the above three metrics we need to know the truePareto front of the optimization problem first However inactual engineering problem we cannot obtain the true Paretofront [28] Then in this section three metrics are adoptedto make a quantitative assessment of the performance ofMOEAD-DE and MOEAD-IWO

Before giving the performance metrics we first presentnormalization of vectors in objective space Assume we have119901nondominated sets119885

1 1198852 119885

119901 First we set119885 = 119885

119894 119894 =

1 2 119901Then we define 119891min119894

= min119891119894(119909) 119865(119909) = 119911 isin 119885

and 119891max119894

= max119891119894(119909) 119865(119909) = 119911 isin 119885 which correspond to

the minimum and maximum values for the 119894th coordinate ofthe objective space Then for all coordinates 119894 = 1 2 119899we calculate all points according to

119891119894(119909) =

119891119894(119909) minus 119891

min119894

119891max119894

minus 119891min119894

(11)

The normalized vector of a vector 119911 is denoted by 119911 the set ofnormalized vectors is shown as 119885

351 119863-Metric for Accuracy [29] Suppose we have twonormalized nondominated sets 119885

1and 119885

2 forall119886 isin 119885

1 we

search for exist119887 isin 11988521

sube 1198852such that 119887 lt 119886 Then we calculate

Euclidean distances from 119886 to all points 119887 isin 11988521 So the 119863-

metric is defined as

119863(1198851 1198852) = sum

119886isin1198851

10038171003817100381710038171003817119886 minus 119887

10038171003817100381710038171003817max

radic119899100381610038161003816100381610038161198851

10038161003816100381610038161003816

(12)

where 119886 minus 119887max = max119886 minus 119887 119886 isin 1198851 119887 isin 119885

21 and 119899 is

the dimension of the objective space The 119863-metric returnsa value among 0 and 1 where the smaller the better Themaximum distance from a dominating point is taken as abasis for accuracy

352 Δ-Metric for Uniformity [29] For a given normalizednondominated set119885 we define119889

119894to be the Euclidean distance

between two consecutive vectors 119894 = 1 2 (|119885| minus 1) Let119889 = sum

|119885|minus1

119894=1119889119894(|119885| minus 1)Then we define

Δ (119885) =

|119885|minus1

sum

119894=1

10038161003816100381610038161003816119889119894minus 119889

10038161003816100381610038161003816

radic119899 (1003816100381610038161003816100381611988510038161003816100381610038161003816minus 1)

(13)

Note that 0 le Δ(119885) le 1 where 0 is the best

353 nabla-Metric for Extent [29] Given a nondominated set 119885we define

nabla (119885) =

119899

prod

119894=1

10038161003816100381610038161003816119891max119894

minus 119891min119894

10038161003816100381610038161003816 (14)

where 119899 is the dimension of the objective space A bigger valuespans a larger portion of the hypervolume and therefore isalways better

4 Basic Principle of Pattern ReconfigurableArray Antennas

Design of phase-only reconfigurable antenna array aimsat finding a common amplitude distribution and differentphase distributions such that the array can produce multipledifferent patterns

Consider a linear equispaced array with119873 elements If119872different patterns need to be produced only by changing theexcitation phases of the array under the common excitationamplitude distribution the optimization variable 119909 is a vectorwith 119872119873 + 119873 elements where 119909

119899(119899 = 1 2 119873) is the

excitation amplitude for the 119899th antenna element denotedby 119868119899and 119909

119898119873+119899(119898 = 1 2 119872 119899 = 1 2 119873) is the

excitation phase for the 119899th antenna element and the 119898thpattern denoted by 120593

119898119899 Then the complex excitation of the

119899th element in the119898th pattern is

119894119898119899

= 119868119899sdot 119890119895120593119898119899 = 119909

119899119890119895119909119899+119898119873 (15)

It can be seen from (15) that in the process of optimiza-tion the common excitation amplitude is used for119872 patternsall the time and only phases of the excitation are differentThe119898th pattern produced by the antenna array for far field isgiven by

119865119898(119906) =

EF (119906)

119865max

119873

sum

119899=1

119894119898119899

sdot 1198901198952120587119899119889119906120582

(16)

where 119898 = 1 2 119872 119889 is the spacing between arrayelements 120582 is the wavelength in free space 119906 = sin 120579 120579 isthe angle from ray direction to normal of array axis EF(120579) iselement factor (EF(120579) = 1 for isotropic source) and 119865max ispeak value of far field pattern

In this paper patterns we need to reconfigure by thephase-only reconfigurable linear arrays are presented below

(1) A flat-top beam and a pencil beam the designproblem is expressed as

minimize 119865 (119909) = (1198911(119909) 119891

2(119909))119879

(17)

International Journal of Antennas and Propagation 5

Table 1 Design objectives and simulated results for Problem 1

Design parametersFlat-top beam Pencil beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus2546 minus2613 minus3000 minus2956 minus3050

Half-power beam width(HPBW in 120579-space) 24∘ 242∘ 24∘ 68∘ 68∘ 68∘

Beam width at SLL(BW in 120579-space) 40∘ 416∘ 402∘ 20∘ 20∘ 20∘

Ripple(in dB 78∘ le 120579 le 102

∘) 050 050 049 NA NA NA

Table 2 Design objectives and simulated results for Problem 2

Design parametersFlat-top beam Cosecant-squared beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus240106 minus249659 minus2500 minus240735 minus249060

Beam width at SLL(BW in 120579-space) 084 088 086 084 088 086

Ripple(in dB)

060|119906| le 032

05998 05994 0600 le 119906 le 064

08588 05341

The cost functions of each single pattern in (17) are

1198911(119909) = Cost

119891=

4

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(18)

1198912(119909) = Cost

119901=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(19)

(2) A flat-top beam and a cosecant beam the designproblem is expressed as

minimize 119865 (119909) = (1198913(119909) 119891

4(119909))119879

(20)

The cost functions of each single pattern in (20) are

1198913(119909) = Cost

119891=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(21)

1198914(119909) = Cost

119888=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(22)

where Cost119891 Cost

119888 and Cost

119901represent the cost functions of

the flat-top beam cosecant-squared beam and pencil beamrespectively and119876

119904119889and119876

119904(119909) are the desired and calculated

values for each design specification we use Desired andcalculated values of each design index in (18) (19) (21) and(22) are summarized in Tables 1 and 2 The lower the costvalue is the closer the calculated value approaches the desiredvalue When the calculated values of all the indexes are lessthan the corresponding desired values the cost value is set tozero

5 MOEAD-IWO for Synthesis of Phase-OnlyReconfigurable Linear Array

Suppose 119875 is the population size that is to say we need tooptimize 119875 subproblems in a single run 1205821 1205822 120582119875 isa set of weight vectors The objective functions of all thepatterns are summed with assigned weights For each single-objective subproblem the objective function is described asfollows

119892119905119890(119909 | 120582

119894 119911lowast) = max1le119898le119872

120582119894

119898

1003816100381610038161003816119891119898 (119909) minus 119911lowast

119898

1003816100381610038161003816 (23)

where 120582119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 119894 = 1 2 119875 MOEAD-IWO

is used tominimize119875 objective functions (23) simultaneouslyin a single run and the neighborhood of 120582119894 is the severalweight vectors close to 120582

119894At each generation MOEAD-IWOmaintains the follow-

ing information

(i) a population of 119875 points 1199091 1199092 119909

119875isin Ω where

Ω is the feasible solution space of the excitationamplitude and phases

(ii) FV1 FV2 FV119875 where FV119894 = 119892119905119890(119909119894

| 120582119894) 119894 =

1 2 119875

(iii) Gen the current generation number

(iv) 119911 = (1199111 1199112 119911

119872)119879 where 119911

119898is the best value found

so far for objective 119891119898

6 International Journal of Antennas and Propagation

The algorithm is outlined as follows

Input

(i) The multiobjective problem (17) or (20)(ii) A stopping criterion(iii) 119875 the population size (the number of subproblems)(iv) 119879 the number of the neighbors for each subproblem(v) A uniform set of 119875 weight vectors 1205821 1205822 120582119875

Output

(i) Approximation to the PS 1199091 1199092 119909119875(ii) Approximation to the PF 119865(1199091) 119865(1199092) 119865(119909119875)

Step 1 (initialization) Consider the following

Step 11 Generate the weight vectors 1205821 1205822 120582119875

Step 12 Compute the Euclidian distances between any twoweight vectors to each weigh vector For each 119894 = 1 2 119875set 119861(119894) = 119894

1 1198942 119894

119879 where 120582

1198941 1205821198942 120582

119894119879 are 119879 weight

vectors close to 120582119894

Step 13 Generate an initial population 1199091 1199092 119909

119875 byuniformly random sampling from the search spaceΩ

Set FV119894 = 119892119905119890(119909119894| 120582119894)

Step 14 Initialize 119911 = (1199111 1199112 119911

119872)119879 by setting 119911

119898=

min1le119894le119875

119891119898(119909119894)

Step 2 (update) For 119894 = 1 2 119875 do the following

Step 21 (selection of matingupdate range) Generate a num-ber rand from [0 1] randomly Then set

119878 = 119861 (119894) if rand lt 120575

1 2 119875 otherwise(24)

120575 is the probability that parent individuals are selected fromthe neighborhood

Step 22 (reproduction) Calculate the number of new solu-tions 119904num using (9) and produce new solutions 119910

119896(119896 =

1 2 119904num) using (8)

Step 23 (repair) If an element in 119910119896 is out of the boundary of

Ω its value is reset to the boundary value

Step 24 (update of 119911) For each119898 = 1 2 119872 if119891119898(119910

min) lt

119911119898 in which 119910

min is the minimum solution among 119910119896 119896 isin

1 2 119904num then set 119911119898

= 119891119898(119910

min)

Step 25 (update of solutions) Set 119888 = 0 and then do thefollowing

(1) If 119888 = 120578119903or 119878 is empty go to Step 3 Otherwise

randomly pick an index 119895 from 119878

(2) If min119892119905119890(1199101 | 120582119895) 119892119905119890(1199102 | 120582119895) 119892119905119890(119910119904num | 120582119895) le

119892119905119890(119909119895| 120582119895) then set 119909119895 = 119910

min FV119895 = 119892119905119890(119910

min|

120582119895) in which 119910

min is the minimum solution among119910119896 119896 isin 1 2 119904num Set 119888 = 119888 + 1

(3) Delete 119895 from 119878 and go to (1)

Step 3 (stopping criteria) If stopping criteria are satisfiedthen stop and output 119909

1 1199092 119909

119875 and 119865(119909

1) 119865(119909

2)

119865(119909119875) Otherwise go to Step 2

In the population optimization variables for each indi-vidual are the common excitation amplitudes for all patternsand different phases for forming different patterns Theoutput of the algorithm is a Pareto set in which each Paretooptimal solution corresponds to a phase-only reconfigurablearray In the optimization results different weight coefficientscorresponding to various patterns form a weight vector eachPareto optimal solution corresponds to a weight vector anddifferent Pareto optimal solutions have different weight vec-tors In actual applications decision-makers select a desiredsolution from the approximated PF or obey some standardsfor choosing the best compromise solution

6 Experiments Results

To demonstrate whether the proposed MOEAD-IWO out-performs MOEAD-DE in the design of phase-only patternreconfigurable linear arrays we carry out two experiments

61 Parameter Setting In MOEAD the population size andweight vectors are controlled by an integer 119867 1205821 1205822 120582119875are the weight vectors in which each individual weight takesa value from 0119867 1119867 119867119867 therefore the populationsize is 119875 = 119862

119872minus1

119867+119872minus1 where 119872 is the number of objectives In

the experiments population size 119875 is set to 201 (119867 = 200) for2-objective instances For the other control parameters we set119879 = 01119875 120578

119903= 001119875 and 120575 = 07 [30] which are the same

for MOEAD-DE and MOEAD-IWO For DE operator weset the crossover rate CR = 10 the scalar factor 119865 = 05120578 = 20 and 119901

119898= 1119899 in mutation operators [12 30] For

IWO operator we set 120590initial = 01 120590final = 00002 pow =

3 119904max = 3 and 119904min = 1 To be fair in the comparisonboth algorithms stop after reaching the maximum functionevaluations 500000 for all the tests In order to restrict theinfluence of random effects both algorithms run 20 timesindependently of each problem and the best results arepresented in this section Source codes for MOEAD-DE aretaken from [31]

62 Determining the Best Compromise Solution When facinga set of nondominated solutions a decision-makerrsquos judg-ment may have fuzzy or imprecise goals for each objectivefunction so useful quantitative measures for assessing thequality of each nondominated solution are needed In presentliteratures an effective method based on the concept of fuzzysets is proposed [25] and successfully used in some MOPs[32ndash34] in which a membership function 120583

119894is defined for

each objective 119891119894by taking account of the minimum and

International Journal of Antennas and Propagation 7

Table 3 The best compromise solutions for the first instance by MOEAD-IWO and MOEAD-DE (DRR = 4)

1198911(flat-top beam) 119891

2(pencil beam) (120582

1 1205822) Cost = 120582

11198911+ 12058221198912

MOEAD-DE with TE 00069 00059 (05350 04650) 00064MOEAD-DE with WS 00086 00084 (05100 04900) 00085MOEAD-IWO with TE 00037 00056 (05100 04900) 00046MOEAD-IWO with WS 00062 00060 (05450 04550) 00061

Table 4 The best compromise solutions for the second instance by MOEAD-IWO and MOEAD-DE (DRR = 5)

1198913(flat-top beam) 119891

4(cosecant beam) (120582

3 1205824) Cost = 120582

31198913+ 12058241198914

MOEAD-DE with TE 00043 00578 (05200 04800) 00300MOEAD-DE with WS 00762 01702 (04400 05600) 01288MOEAD-IWO with TE 00015 00511 (04750 05250) 00275MOEAD-IWO with WS 00016 00516 (04650 05350) 00284

maximum values of each objective function together with therate of increase of membership satisfaction As in [33 34]we assume that 120583

119894is a strictly monotonic decreasing and

continuous function defined as

120583119894=

1 if 119891119894le 119891

min119894

119891max119894

minus 119891119894

119891max119894

minus 119891min119894

if 119891min119894

le 119891119894le 119891

max119894

0 if 119891119894ge 119891

max119894

(25)

where 119891max119894

and 119891min119894

are the maximum and minimumvalues of the 119894th objective solutions among all nondominatedsolutions respectively Value 120583

119894indicates how much a non-

dominated solution satisfies the 119894th objective so the sum of120583119894(119894 = 1 2 119898) for all the objectives can be calculated

for measuring the ldquoaccomplishmentrdquo of each solution insatisfying the 119872 objectives and then a membership function119877119906(119906 = 1 2 119873

119904) can be defined for nondominated

solutions as follows

119877119906=

sum119872

119894=1120583119894(119906)

sum119873119904

V=1sum119872

119894=1120583119894(V)

(26)

where119873119904represents the number of nondominated solutions

The function 119877119906can be treated as a membership function

for the nondominated solutions in a fuzzy set Value 119877119906

represents the quality of a solution among all nondominatedsolutions and the one with the maximum value of 120583119902 is thebest compromise solution

63 Results Effectiveness of the MOEAD-IWO algorithmis verified through the design of two examples In theoptimization process excitation amplitudes range from 0 to1 and phases are restricted from minus180 to 180 degrees

631 Problem 1 Consider a 20-element 05120582-equispacedsymmetrical array with EF(120579) = sin(120579) for generating a flat-top beam and a pencil beamThe flat-top beam is realized bythe common excitation amplitudes and unknown phases andthe phases for forming pencil beam are set to zeroTherefore

there are 10 common excitation amplitude variables and 10phase variables for forming flat-top beam to be optimized inobjective function (17) The maximum dynamic range ratio(DRR) of excitation amplitudes is controlled with 4

The best compromise solutions obtained by MOEAD-IWO and MOEAD-DE are shown in Figure 1 and Table 1 Itcan be seen that for flat-top beam the peak SLL obtainedby MOEAD-IWO is 067 dB lower than that obtained byMOEAD-DE The BW value obtained by MOEAD-DE is16∘ wider than the objective value while that obtained byMOEAD-IWO is just 02∘ wider than the design objectiveFor the pencil beam the pattern obtained byMOEAD-IWOhas 094 dB lower peak SLL value when other indexes meetthe design objectives well The excitation amplitudes and thephases for the synthesized radiation patterns obtained byMOEAD-IWO are shown in Figure 2

632 Problem 2 Consider a 05120582-equispaced linear arraywith 25 isotropic elements to generate a flat-top beam and acosecant-squared beam The dimension of vector 119909 in objec-tive function (20) is 75 including 25 common amplitudesand 50 unknown phases The maximum DRR of excitationamplitudes is controlled with DRR = 5

Radiation patterns of the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE are shownin Figure 4 and Table 2 and the excitation amplitudes andthe phases for the synthesized radiation patterns obtainedby MOEAD-IWO are shown in Figure 5 It can be seenfrom Table 2 that for the flat-top beam the ripples obtainedby these two algorithms meet the design objectives wellthe peak SLL value obtained by MOEAD-IWO is 09553 dBlower than that obtained by MOEAD-DE and the BWvalues obtained by MOEAD-DE and MOEAD-IWO are088 and 086 respectively For the cosecant beam the BWobtained by MOEAD-IWO is smaller than that obtainedby MOEAD-DE and the ripple and peak SLL valuesobtained by MOEAD-IWO are 03247 dB and 08325 dBlower than those obtained by MOEAD-DE respectively

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 3

single-objective optimization subproblems and MOEADoptimizes these subproblems collectively and simultaneouslyThe objective function of each single-objective subproblemis an aggregation of all the objective functions in MOPAt each generation the population is composed of thebest solutions found so far for each subproblem which isoptimized by using information only from its neighboringsubproblems Neighborhood relations among these single-objective subproblems are defined based on the Euclidiandistances between their weight vectors MOEAD exploitsinformation sharing among subproblems which are neigh-bors to accomplish the optimization task effectively andefficiently MOEAD-IWOmaintains the basic framework ofMOEAD [11] and the IWO algorithm is used for producingnew solutions When MOEAD-IWO is used to optimizeMOP the following issues need to be considered

31 Produce theWeightVectors As in [11] we use the simplex-lattice design method [22] to produce the weight vectorsin which each individual weight takes value from 0119867

1119867 119867119867 where119867 is an integer

32 Decompose an MOP into a Number of Single-Objective Subproblems In theory weighted sum Tcheby-cheff approach boundary intersection and any otherdecomposition approach can be used to convert anMOP intoa number of single-objective subproblems [11] Tchebycheffapproach is mainly employed in our experimental studyLet 1205821 1205822 120582119875 be a set of even spread 119875 weight vectorswhere 120582

119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 is the weight vector for the 119894th

subproblem 120582119894119895ge 0 119895 = 1 2 119872 and sum

119872

119895=1120582119894

119895= 1 119872 is

the number of objectives in (1) 119894 = 1 2 119875 119911lowast representsthe reference point The 119894th single-objective optimizationsubproblem obtained by decomposing the given MOP canbe represented as [23]

Minimize 119892119905119890(119909 | 120582

119894 119911lowast) = max1le119895le119872

120582119894

119895

10038161003816100381610038161003816119891119895(119909) minus 119911

lowast

119895

10038161003816100381610038161003816

subject to 119909 isin Ω

(6)

The Tchebycheff approach searches solutions through min-imizing 119892

119905119890 toward 119911lowast which has an advantage that both

convex and concave Pareto front can be approximated[24] All these 119875 single-objective functions are optimizedsimultaneously in each run If we use a large 119875 and selectthe weight vectors properly all the optimal solutions ofthe single-objective problems from decomposition will wellapproximate the Pareto front The optimization process isaccomplished by the collaboration of the neighbors for eachsubproblem

33 Confirm the Neighbors of Each Single-Objective Sub-problem In MOEAD each subproblem is optimized byusing informationmainly from its neighboring subproblemsThe neighborhood relations among these subproblems aredefined based on the distances between their weight vectorsCompute the Euclidean distances between any two weightvectors and thenwork out the119879 closest weight vectors to each

weight vector For the 119894th subproblem its neighbor is set to119861(119894) = 119894

1 1198942 119894

119879 119894 = 1 2 119875 where 1205821198941 1205821198942 120582119894119879 are

119879 weight vectors close to 120582119894 Single-objective subproblems

which are considered neighbors will have similar fitnesslandscapes and their optimal solutions should be close inthe decision space MOEAD exploits information sharingamong subproblems which are neighbors to accomplish theoptimization task effectively and efficiently

34 The Selection of Evolution Strategy The IWO operatoris integrated into MOEAD-IWO to produce new solutionsIn IWO new solutions (seeds) produced are randomly dis-tributed in119863-dimensional space around their parents (weed)in normal distribution 119873(0 120590

2

iter) the change rule for 120590iter isexpressed as

120590iter = 120590final + (itermax minus iter

itermax)

pow(120590initial minus 120590final) (7)

where 120590initial and 120590final are initial and final standard deviationsand pow is the nonlinear regulatory factor It can be seenfrom (7) that 120590iter decreases with the increase of evolutiongenerations while the 120590iter value of each subproblem inone generation is the same This is not conducive to thealgorithm convergence Thus we modify IWO and presentan adaptive standard deviation in which the value of 120590iterin one generation changes not only with the cost functionvalue of each subproblem but also with the maximum andminimum cost function values in the generation as shown inthe following equation

120590iter119894 =

(1 + 120574FV119894minus FV119886

FVmax minus FV119886

)120590iter FV119894ge FV119886

(1 minus 120574FV119886minus FV119894

FV119886minus FVmin

)120590iter FV119894lt FV119886

(8)

where FV119894is the cost function value of the 119894th subproblem

FVmax FVmin andFV119886 denote themaximumminimum andaverage cost function values among all these subproblemsrespectively and 120574 is the zoom factor controlling the variationrange of the standard deviation which takes value from 0 to1 In this paper we take 120574 = 05

It can be seen that for the minimization problem thesubproblems with lower cost function values have smallerstandard deviation in one generation which enables theseeds distributed around the better subproblems Moreoverthe range of the adaptive standard deviation 120590iter119894 is [1 minus

120574 1 + 120574]120590iter which makes the standard deviation of onesubproblem in the younger generation likely to be larger thanthat in the older generation This will help the new producedseeds escape from local optimum improve the convergencerate and balance the global and local search capabilitieseffectively at the same time

In the present generation the number of seeds producedby the 119894th subproblem is calculated by

119904num = floor(FVmax minus FV

119894

FVmax minus FVmin(119904max minus 119904min) + 119904min) (9)

4 International Journal of Antennas and Propagation

where 119904max and 119904min are the largest and smallest numbersof seeds produced respectively and floor() represents theround-down function It is clear that better populationmembers produce more seeds

Suppose 119909119894= (119909119894

1 119909119894

2 119909

119894

119873)119879 119894 = 1 2 119875 is the

present optimal solution for the 119894th subproblem and eachnew solution produced by 119909

119894 is 119910 = (1199101 1199102 119910

119873)119879 in

which each element 119910119896is generated as follows

119910119896= 119909119894

119896+ 119873(0 120590

2

iter119894) 119896 = 1 2 119873 (10)

Then 119904num new solutions are used to update the 119894th subprob-lem and its neighborhoods

35 Metrics for Comparing Nondominated Sets Several met-rics have been proposed to compare the performance ofdifferent multiobjective evolutionary algorithms quantita-tively such as generational distances (GD) [25] invertedgenerational distance (IGD) [26] and spread [27] Whenusing the above three metrics we need to know the truePareto front of the optimization problem first However inactual engineering problem we cannot obtain the true Paretofront [28] Then in this section three metrics are adoptedto make a quantitative assessment of the performance ofMOEAD-DE and MOEAD-IWO

Before giving the performance metrics we first presentnormalization of vectors in objective space Assume we have119901nondominated sets119885

1 1198852 119885

119901 First we set119885 = 119885

119894 119894 =

1 2 119901Then we define 119891min119894

= min119891119894(119909) 119865(119909) = 119911 isin 119885

and 119891max119894

= max119891119894(119909) 119865(119909) = 119911 isin 119885 which correspond to

the minimum and maximum values for the 119894th coordinate ofthe objective space Then for all coordinates 119894 = 1 2 119899we calculate all points according to

119891119894(119909) =

119891119894(119909) minus 119891

min119894

119891max119894

minus 119891min119894

(11)

The normalized vector of a vector 119911 is denoted by 119911 the set ofnormalized vectors is shown as 119885

351 119863-Metric for Accuracy [29] Suppose we have twonormalized nondominated sets 119885

1and 119885

2 forall119886 isin 119885

1 we

search for exist119887 isin 11988521

sube 1198852such that 119887 lt 119886 Then we calculate

Euclidean distances from 119886 to all points 119887 isin 11988521 So the 119863-

metric is defined as

119863(1198851 1198852) = sum

119886isin1198851

10038171003817100381710038171003817119886 minus 119887

10038171003817100381710038171003817max

radic119899100381610038161003816100381610038161198851

10038161003816100381610038161003816

(12)

where 119886 minus 119887max = max119886 minus 119887 119886 isin 1198851 119887 isin 119885

21 and 119899 is

the dimension of the objective space The 119863-metric returnsa value among 0 and 1 where the smaller the better Themaximum distance from a dominating point is taken as abasis for accuracy

352 Δ-Metric for Uniformity [29] For a given normalizednondominated set119885 we define119889

119894to be the Euclidean distance

between two consecutive vectors 119894 = 1 2 (|119885| minus 1) Let119889 = sum

|119885|minus1

119894=1119889119894(|119885| minus 1)Then we define

Δ (119885) =

|119885|minus1

sum

119894=1

10038161003816100381610038161003816119889119894minus 119889

10038161003816100381610038161003816

radic119899 (1003816100381610038161003816100381611988510038161003816100381610038161003816minus 1)

(13)

Note that 0 le Δ(119885) le 1 where 0 is the best

353 nabla-Metric for Extent [29] Given a nondominated set 119885we define

nabla (119885) =

119899

prod

119894=1

10038161003816100381610038161003816119891max119894

minus 119891min119894

10038161003816100381610038161003816 (14)

where 119899 is the dimension of the objective space A bigger valuespans a larger portion of the hypervolume and therefore isalways better

4 Basic Principle of Pattern ReconfigurableArray Antennas

Design of phase-only reconfigurable antenna array aimsat finding a common amplitude distribution and differentphase distributions such that the array can produce multipledifferent patterns

Consider a linear equispaced array with119873 elements If119872different patterns need to be produced only by changing theexcitation phases of the array under the common excitationamplitude distribution the optimization variable 119909 is a vectorwith 119872119873 + 119873 elements where 119909

119899(119899 = 1 2 119873) is the

excitation amplitude for the 119899th antenna element denotedby 119868119899and 119909

119898119873+119899(119898 = 1 2 119872 119899 = 1 2 119873) is the

excitation phase for the 119899th antenna element and the 119898thpattern denoted by 120593

119898119899 Then the complex excitation of the

119899th element in the119898th pattern is

119894119898119899

= 119868119899sdot 119890119895120593119898119899 = 119909

119899119890119895119909119899+119898119873 (15)

It can be seen from (15) that in the process of optimiza-tion the common excitation amplitude is used for119872 patternsall the time and only phases of the excitation are differentThe119898th pattern produced by the antenna array for far field isgiven by

119865119898(119906) =

EF (119906)

119865max

119873

sum

119899=1

119894119898119899

sdot 1198901198952120587119899119889119906120582

(16)

where 119898 = 1 2 119872 119889 is the spacing between arrayelements 120582 is the wavelength in free space 119906 = sin 120579 120579 isthe angle from ray direction to normal of array axis EF(120579) iselement factor (EF(120579) = 1 for isotropic source) and 119865max ispeak value of far field pattern

In this paper patterns we need to reconfigure by thephase-only reconfigurable linear arrays are presented below

(1) A flat-top beam and a pencil beam the designproblem is expressed as

minimize 119865 (119909) = (1198911(119909) 119891

2(119909))119879

(17)

International Journal of Antennas and Propagation 5

Table 1 Design objectives and simulated results for Problem 1

Design parametersFlat-top beam Pencil beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus2546 minus2613 minus3000 minus2956 minus3050

Half-power beam width(HPBW in 120579-space) 24∘ 242∘ 24∘ 68∘ 68∘ 68∘

Beam width at SLL(BW in 120579-space) 40∘ 416∘ 402∘ 20∘ 20∘ 20∘

Ripple(in dB 78∘ le 120579 le 102

∘) 050 050 049 NA NA NA

Table 2 Design objectives and simulated results for Problem 2

Design parametersFlat-top beam Cosecant-squared beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus240106 minus249659 minus2500 minus240735 minus249060

Beam width at SLL(BW in 120579-space) 084 088 086 084 088 086

Ripple(in dB)

060|119906| le 032

05998 05994 0600 le 119906 le 064

08588 05341

The cost functions of each single pattern in (17) are

1198911(119909) = Cost

119891=

4

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(18)

1198912(119909) = Cost

119901=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(19)

(2) A flat-top beam and a cosecant beam the designproblem is expressed as

minimize 119865 (119909) = (1198913(119909) 119891

4(119909))119879

(20)

The cost functions of each single pattern in (20) are

1198913(119909) = Cost

119891=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(21)

1198914(119909) = Cost

119888=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(22)

where Cost119891 Cost

119888 and Cost

119901represent the cost functions of

the flat-top beam cosecant-squared beam and pencil beamrespectively and119876

119904119889and119876

119904(119909) are the desired and calculated

values for each design specification we use Desired andcalculated values of each design index in (18) (19) (21) and(22) are summarized in Tables 1 and 2 The lower the costvalue is the closer the calculated value approaches the desiredvalue When the calculated values of all the indexes are lessthan the corresponding desired values the cost value is set tozero

5 MOEAD-IWO for Synthesis of Phase-OnlyReconfigurable Linear Array

Suppose 119875 is the population size that is to say we need tooptimize 119875 subproblems in a single run 1205821 1205822 120582119875 isa set of weight vectors The objective functions of all thepatterns are summed with assigned weights For each single-objective subproblem the objective function is described asfollows

119892119905119890(119909 | 120582

119894 119911lowast) = max1le119898le119872

120582119894

119898

1003816100381610038161003816119891119898 (119909) minus 119911lowast

119898

1003816100381610038161003816 (23)

where 120582119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 119894 = 1 2 119875 MOEAD-IWO

is used tominimize119875 objective functions (23) simultaneouslyin a single run and the neighborhood of 120582119894 is the severalweight vectors close to 120582

119894At each generation MOEAD-IWOmaintains the follow-

ing information

(i) a population of 119875 points 1199091 1199092 119909

119875isin Ω where

Ω is the feasible solution space of the excitationamplitude and phases

(ii) FV1 FV2 FV119875 where FV119894 = 119892119905119890(119909119894

| 120582119894) 119894 =

1 2 119875

(iii) Gen the current generation number

(iv) 119911 = (1199111 1199112 119911

119872)119879 where 119911

119898is the best value found

so far for objective 119891119898

6 International Journal of Antennas and Propagation

The algorithm is outlined as follows

Input

(i) The multiobjective problem (17) or (20)(ii) A stopping criterion(iii) 119875 the population size (the number of subproblems)(iv) 119879 the number of the neighbors for each subproblem(v) A uniform set of 119875 weight vectors 1205821 1205822 120582119875

Output

(i) Approximation to the PS 1199091 1199092 119909119875(ii) Approximation to the PF 119865(1199091) 119865(1199092) 119865(119909119875)

Step 1 (initialization) Consider the following

Step 11 Generate the weight vectors 1205821 1205822 120582119875

Step 12 Compute the Euclidian distances between any twoweight vectors to each weigh vector For each 119894 = 1 2 119875set 119861(119894) = 119894

1 1198942 119894

119879 where 120582

1198941 1205821198942 120582

119894119879 are 119879 weight

vectors close to 120582119894

Step 13 Generate an initial population 1199091 1199092 119909

119875 byuniformly random sampling from the search spaceΩ

Set FV119894 = 119892119905119890(119909119894| 120582119894)

Step 14 Initialize 119911 = (1199111 1199112 119911

119872)119879 by setting 119911

119898=

min1le119894le119875

119891119898(119909119894)

Step 2 (update) For 119894 = 1 2 119875 do the following

Step 21 (selection of matingupdate range) Generate a num-ber rand from [0 1] randomly Then set

119878 = 119861 (119894) if rand lt 120575

1 2 119875 otherwise(24)

120575 is the probability that parent individuals are selected fromthe neighborhood

Step 22 (reproduction) Calculate the number of new solu-tions 119904num using (9) and produce new solutions 119910

119896(119896 =

1 2 119904num) using (8)

Step 23 (repair) If an element in 119910119896 is out of the boundary of

Ω its value is reset to the boundary value

Step 24 (update of 119911) For each119898 = 1 2 119872 if119891119898(119910

min) lt

119911119898 in which 119910

min is the minimum solution among 119910119896 119896 isin

1 2 119904num then set 119911119898

= 119891119898(119910

min)

Step 25 (update of solutions) Set 119888 = 0 and then do thefollowing

(1) If 119888 = 120578119903or 119878 is empty go to Step 3 Otherwise

randomly pick an index 119895 from 119878

(2) If min119892119905119890(1199101 | 120582119895) 119892119905119890(1199102 | 120582119895) 119892119905119890(119910119904num | 120582119895) le

119892119905119890(119909119895| 120582119895) then set 119909119895 = 119910

min FV119895 = 119892119905119890(119910

min|

120582119895) in which 119910

min is the minimum solution among119910119896 119896 isin 1 2 119904num Set 119888 = 119888 + 1

(3) Delete 119895 from 119878 and go to (1)

Step 3 (stopping criteria) If stopping criteria are satisfiedthen stop and output 119909

1 1199092 119909

119875 and 119865(119909

1) 119865(119909

2)

119865(119909119875) Otherwise go to Step 2

In the population optimization variables for each indi-vidual are the common excitation amplitudes for all patternsand different phases for forming different patterns Theoutput of the algorithm is a Pareto set in which each Paretooptimal solution corresponds to a phase-only reconfigurablearray In the optimization results different weight coefficientscorresponding to various patterns form a weight vector eachPareto optimal solution corresponds to a weight vector anddifferent Pareto optimal solutions have different weight vec-tors In actual applications decision-makers select a desiredsolution from the approximated PF or obey some standardsfor choosing the best compromise solution

6 Experiments Results

To demonstrate whether the proposed MOEAD-IWO out-performs MOEAD-DE in the design of phase-only patternreconfigurable linear arrays we carry out two experiments

61 Parameter Setting In MOEAD the population size andweight vectors are controlled by an integer 119867 1205821 1205822 120582119875are the weight vectors in which each individual weight takesa value from 0119867 1119867 119867119867 therefore the populationsize is 119875 = 119862

119872minus1

119867+119872minus1 where 119872 is the number of objectives In

the experiments population size 119875 is set to 201 (119867 = 200) for2-objective instances For the other control parameters we set119879 = 01119875 120578

119903= 001119875 and 120575 = 07 [30] which are the same

for MOEAD-DE and MOEAD-IWO For DE operator weset the crossover rate CR = 10 the scalar factor 119865 = 05120578 = 20 and 119901

119898= 1119899 in mutation operators [12 30] For

IWO operator we set 120590initial = 01 120590final = 00002 pow =

3 119904max = 3 and 119904min = 1 To be fair in the comparisonboth algorithms stop after reaching the maximum functionevaluations 500000 for all the tests In order to restrict theinfluence of random effects both algorithms run 20 timesindependently of each problem and the best results arepresented in this section Source codes for MOEAD-DE aretaken from [31]

62 Determining the Best Compromise Solution When facinga set of nondominated solutions a decision-makerrsquos judg-ment may have fuzzy or imprecise goals for each objectivefunction so useful quantitative measures for assessing thequality of each nondominated solution are needed In presentliteratures an effective method based on the concept of fuzzysets is proposed [25] and successfully used in some MOPs[32ndash34] in which a membership function 120583

119894is defined for

each objective 119891119894by taking account of the minimum and

International Journal of Antennas and Propagation 7

Table 3 The best compromise solutions for the first instance by MOEAD-IWO and MOEAD-DE (DRR = 4)

1198911(flat-top beam) 119891

2(pencil beam) (120582

1 1205822) Cost = 120582

11198911+ 12058221198912

MOEAD-DE with TE 00069 00059 (05350 04650) 00064MOEAD-DE with WS 00086 00084 (05100 04900) 00085MOEAD-IWO with TE 00037 00056 (05100 04900) 00046MOEAD-IWO with WS 00062 00060 (05450 04550) 00061

Table 4 The best compromise solutions for the second instance by MOEAD-IWO and MOEAD-DE (DRR = 5)

1198913(flat-top beam) 119891

4(cosecant beam) (120582

3 1205824) Cost = 120582

31198913+ 12058241198914

MOEAD-DE with TE 00043 00578 (05200 04800) 00300MOEAD-DE with WS 00762 01702 (04400 05600) 01288MOEAD-IWO with TE 00015 00511 (04750 05250) 00275MOEAD-IWO with WS 00016 00516 (04650 05350) 00284

maximum values of each objective function together with therate of increase of membership satisfaction As in [33 34]we assume that 120583

119894is a strictly monotonic decreasing and

continuous function defined as

120583119894=

1 if 119891119894le 119891

min119894

119891max119894

minus 119891119894

119891max119894

minus 119891min119894

if 119891min119894

le 119891119894le 119891

max119894

0 if 119891119894ge 119891

max119894

(25)

where 119891max119894

and 119891min119894

are the maximum and minimumvalues of the 119894th objective solutions among all nondominatedsolutions respectively Value 120583

119894indicates how much a non-

dominated solution satisfies the 119894th objective so the sum of120583119894(119894 = 1 2 119898) for all the objectives can be calculated

for measuring the ldquoaccomplishmentrdquo of each solution insatisfying the 119872 objectives and then a membership function119877119906(119906 = 1 2 119873

119904) can be defined for nondominated

solutions as follows

119877119906=

sum119872

119894=1120583119894(119906)

sum119873119904

V=1sum119872

119894=1120583119894(V)

(26)

where119873119904represents the number of nondominated solutions

The function 119877119906can be treated as a membership function

for the nondominated solutions in a fuzzy set Value 119877119906

represents the quality of a solution among all nondominatedsolutions and the one with the maximum value of 120583119902 is thebest compromise solution

63 Results Effectiveness of the MOEAD-IWO algorithmis verified through the design of two examples In theoptimization process excitation amplitudes range from 0 to1 and phases are restricted from minus180 to 180 degrees

631 Problem 1 Consider a 20-element 05120582-equispacedsymmetrical array with EF(120579) = sin(120579) for generating a flat-top beam and a pencil beamThe flat-top beam is realized bythe common excitation amplitudes and unknown phases andthe phases for forming pencil beam are set to zeroTherefore

there are 10 common excitation amplitude variables and 10phase variables for forming flat-top beam to be optimized inobjective function (17) The maximum dynamic range ratio(DRR) of excitation amplitudes is controlled with 4

The best compromise solutions obtained by MOEAD-IWO and MOEAD-DE are shown in Figure 1 and Table 1 Itcan be seen that for flat-top beam the peak SLL obtainedby MOEAD-IWO is 067 dB lower than that obtained byMOEAD-DE The BW value obtained by MOEAD-DE is16∘ wider than the objective value while that obtained byMOEAD-IWO is just 02∘ wider than the design objectiveFor the pencil beam the pattern obtained byMOEAD-IWOhas 094 dB lower peak SLL value when other indexes meetthe design objectives well The excitation amplitudes and thephases for the synthesized radiation patterns obtained byMOEAD-IWO are shown in Figure 2

632 Problem 2 Consider a 05120582-equispaced linear arraywith 25 isotropic elements to generate a flat-top beam and acosecant-squared beam The dimension of vector 119909 in objec-tive function (20) is 75 including 25 common amplitudesand 50 unknown phases The maximum DRR of excitationamplitudes is controlled with DRR = 5

Radiation patterns of the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE are shownin Figure 4 and Table 2 and the excitation amplitudes andthe phases for the synthesized radiation patterns obtainedby MOEAD-IWO are shown in Figure 5 It can be seenfrom Table 2 that for the flat-top beam the ripples obtainedby these two algorithms meet the design objectives wellthe peak SLL value obtained by MOEAD-IWO is 09553 dBlower than that obtained by MOEAD-DE and the BWvalues obtained by MOEAD-DE and MOEAD-IWO are088 and 086 respectively For the cosecant beam the BWobtained by MOEAD-IWO is smaller than that obtainedby MOEAD-DE and the ripple and peak SLL valuesobtained by MOEAD-IWO are 03247 dB and 08325 dBlower than those obtained by MOEAD-DE respectively

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

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4 International Journal of Antennas and Propagation

where 119904max and 119904min are the largest and smallest numbersof seeds produced respectively and floor() represents theround-down function It is clear that better populationmembers produce more seeds

Suppose 119909119894= (119909119894

1 119909119894

2 119909

119894

119873)119879 119894 = 1 2 119875 is the

present optimal solution for the 119894th subproblem and eachnew solution produced by 119909

119894 is 119910 = (1199101 1199102 119910

119873)119879 in

which each element 119910119896is generated as follows

119910119896= 119909119894

119896+ 119873(0 120590

2

iter119894) 119896 = 1 2 119873 (10)

Then 119904num new solutions are used to update the 119894th subprob-lem and its neighborhoods

35 Metrics for Comparing Nondominated Sets Several met-rics have been proposed to compare the performance ofdifferent multiobjective evolutionary algorithms quantita-tively such as generational distances (GD) [25] invertedgenerational distance (IGD) [26] and spread [27] Whenusing the above three metrics we need to know the truePareto front of the optimization problem first However inactual engineering problem we cannot obtain the true Paretofront [28] Then in this section three metrics are adoptedto make a quantitative assessment of the performance ofMOEAD-DE and MOEAD-IWO

Before giving the performance metrics we first presentnormalization of vectors in objective space Assume we have119901nondominated sets119885

1 1198852 119885

119901 First we set119885 = 119885

119894 119894 =

1 2 119901Then we define 119891min119894

= min119891119894(119909) 119865(119909) = 119911 isin 119885

and 119891max119894

= max119891119894(119909) 119865(119909) = 119911 isin 119885 which correspond to

the minimum and maximum values for the 119894th coordinate ofthe objective space Then for all coordinates 119894 = 1 2 119899we calculate all points according to

119891119894(119909) =

119891119894(119909) minus 119891

min119894

119891max119894

minus 119891min119894

(11)

The normalized vector of a vector 119911 is denoted by 119911 the set ofnormalized vectors is shown as 119885

351 119863-Metric for Accuracy [29] Suppose we have twonormalized nondominated sets 119885

1and 119885

2 forall119886 isin 119885

1 we

search for exist119887 isin 11988521

sube 1198852such that 119887 lt 119886 Then we calculate

Euclidean distances from 119886 to all points 119887 isin 11988521 So the 119863-

metric is defined as

119863(1198851 1198852) = sum

119886isin1198851

10038171003817100381710038171003817119886 minus 119887

10038171003817100381710038171003817max

radic119899100381610038161003816100381610038161198851

10038161003816100381610038161003816

(12)

where 119886 minus 119887max = max119886 minus 119887 119886 isin 1198851 119887 isin 119885

21 and 119899 is

the dimension of the objective space The 119863-metric returnsa value among 0 and 1 where the smaller the better Themaximum distance from a dominating point is taken as abasis for accuracy

352 Δ-Metric for Uniformity [29] For a given normalizednondominated set119885 we define119889

119894to be the Euclidean distance

between two consecutive vectors 119894 = 1 2 (|119885| minus 1) Let119889 = sum

|119885|minus1

119894=1119889119894(|119885| minus 1)Then we define

Δ (119885) =

|119885|minus1

sum

119894=1

10038161003816100381610038161003816119889119894minus 119889

10038161003816100381610038161003816

radic119899 (1003816100381610038161003816100381611988510038161003816100381610038161003816minus 1)

(13)

Note that 0 le Δ(119885) le 1 where 0 is the best

353 nabla-Metric for Extent [29] Given a nondominated set 119885we define

nabla (119885) =

119899

prod

119894=1

10038161003816100381610038161003816119891max119894

minus 119891min119894

10038161003816100381610038161003816 (14)

where 119899 is the dimension of the objective space A bigger valuespans a larger portion of the hypervolume and therefore isalways better

4 Basic Principle of Pattern ReconfigurableArray Antennas

Design of phase-only reconfigurable antenna array aimsat finding a common amplitude distribution and differentphase distributions such that the array can produce multipledifferent patterns

Consider a linear equispaced array with119873 elements If119872different patterns need to be produced only by changing theexcitation phases of the array under the common excitationamplitude distribution the optimization variable 119909 is a vectorwith 119872119873 + 119873 elements where 119909

119899(119899 = 1 2 119873) is the

excitation amplitude for the 119899th antenna element denotedby 119868119899and 119909

119898119873+119899(119898 = 1 2 119872 119899 = 1 2 119873) is the

excitation phase for the 119899th antenna element and the 119898thpattern denoted by 120593

119898119899 Then the complex excitation of the

119899th element in the119898th pattern is

119894119898119899

= 119868119899sdot 119890119895120593119898119899 = 119909

119899119890119895119909119899+119898119873 (15)

It can be seen from (15) that in the process of optimiza-tion the common excitation amplitude is used for119872 patternsall the time and only phases of the excitation are differentThe119898th pattern produced by the antenna array for far field isgiven by

119865119898(119906) =

EF (119906)

119865max

119873

sum

119899=1

119894119898119899

sdot 1198901198952120587119899119889119906120582

(16)

where 119898 = 1 2 119872 119889 is the spacing between arrayelements 120582 is the wavelength in free space 119906 = sin 120579 120579 isthe angle from ray direction to normal of array axis EF(120579) iselement factor (EF(120579) = 1 for isotropic source) and 119865max ispeak value of far field pattern

In this paper patterns we need to reconfigure by thephase-only reconfigurable linear arrays are presented below

(1) A flat-top beam and a pencil beam the designproblem is expressed as

minimize 119865 (119909) = (1198911(119909) 119891

2(119909))119879

(17)

International Journal of Antennas and Propagation 5

Table 1 Design objectives and simulated results for Problem 1

Design parametersFlat-top beam Pencil beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus2546 minus2613 minus3000 minus2956 minus3050

Half-power beam width(HPBW in 120579-space) 24∘ 242∘ 24∘ 68∘ 68∘ 68∘

Beam width at SLL(BW in 120579-space) 40∘ 416∘ 402∘ 20∘ 20∘ 20∘

Ripple(in dB 78∘ le 120579 le 102

∘) 050 050 049 NA NA NA

Table 2 Design objectives and simulated results for Problem 2

Design parametersFlat-top beam Cosecant-squared beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus240106 minus249659 minus2500 minus240735 minus249060

Beam width at SLL(BW in 120579-space) 084 088 086 084 088 086

Ripple(in dB)

060|119906| le 032

05998 05994 0600 le 119906 le 064

08588 05341

The cost functions of each single pattern in (17) are

1198911(119909) = Cost

119891=

4

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(18)

1198912(119909) = Cost

119901=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(19)

(2) A flat-top beam and a cosecant beam the designproblem is expressed as

minimize 119865 (119909) = (1198913(119909) 119891

4(119909))119879

(20)

The cost functions of each single pattern in (20) are

1198913(119909) = Cost

119891=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(21)

1198914(119909) = Cost

119888=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(22)

where Cost119891 Cost

119888 and Cost

119901represent the cost functions of

the flat-top beam cosecant-squared beam and pencil beamrespectively and119876

119904119889and119876

119904(119909) are the desired and calculated

values for each design specification we use Desired andcalculated values of each design index in (18) (19) (21) and(22) are summarized in Tables 1 and 2 The lower the costvalue is the closer the calculated value approaches the desiredvalue When the calculated values of all the indexes are lessthan the corresponding desired values the cost value is set tozero

5 MOEAD-IWO for Synthesis of Phase-OnlyReconfigurable Linear Array

Suppose 119875 is the population size that is to say we need tooptimize 119875 subproblems in a single run 1205821 1205822 120582119875 isa set of weight vectors The objective functions of all thepatterns are summed with assigned weights For each single-objective subproblem the objective function is described asfollows

119892119905119890(119909 | 120582

119894 119911lowast) = max1le119898le119872

120582119894

119898

1003816100381610038161003816119891119898 (119909) minus 119911lowast

119898

1003816100381610038161003816 (23)

where 120582119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 119894 = 1 2 119875 MOEAD-IWO

is used tominimize119875 objective functions (23) simultaneouslyin a single run and the neighborhood of 120582119894 is the severalweight vectors close to 120582

119894At each generation MOEAD-IWOmaintains the follow-

ing information

(i) a population of 119875 points 1199091 1199092 119909

119875isin Ω where

Ω is the feasible solution space of the excitationamplitude and phases

(ii) FV1 FV2 FV119875 where FV119894 = 119892119905119890(119909119894

| 120582119894) 119894 =

1 2 119875

(iii) Gen the current generation number

(iv) 119911 = (1199111 1199112 119911

119872)119879 where 119911

119898is the best value found

so far for objective 119891119898

6 International Journal of Antennas and Propagation

The algorithm is outlined as follows

Input

(i) The multiobjective problem (17) or (20)(ii) A stopping criterion(iii) 119875 the population size (the number of subproblems)(iv) 119879 the number of the neighbors for each subproblem(v) A uniform set of 119875 weight vectors 1205821 1205822 120582119875

Output

(i) Approximation to the PS 1199091 1199092 119909119875(ii) Approximation to the PF 119865(1199091) 119865(1199092) 119865(119909119875)

Step 1 (initialization) Consider the following

Step 11 Generate the weight vectors 1205821 1205822 120582119875

Step 12 Compute the Euclidian distances between any twoweight vectors to each weigh vector For each 119894 = 1 2 119875set 119861(119894) = 119894

1 1198942 119894

119879 where 120582

1198941 1205821198942 120582

119894119879 are 119879 weight

vectors close to 120582119894

Step 13 Generate an initial population 1199091 1199092 119909

119875 byuniformly random sampling from the search spaceΩ

Set FV119894 = 119892119905119890(119909119894| 120582119894)

Step 14 Initialize 119911 = (1199111 1199112 119911

119872)119879 by setting 119911

119898=

min1le119894le119875

119891119898(119909119894)

Step 2 (update) For 119894 = 1 2 119875 do the following

Step 21 (selection of matingupdate range) Generate a num-ber rand from [0 1] randomly Then set

119878 = 119861 (119894) if rand lt 120575

1 2 119875 otherwise(24)

120575 is the probability that parent individuals are selected fromthe neighborhood

Step 22 (reproduction) Calculate the number of new solu-tions 119904num using (9) and produce new solutions 119910

119896(119896 =

1 2 119904num) using (8)

Step 23 (repair) If an element in 119910119896 is out of the boundary of

Ω its value is reset to the boundary value

Step 24 (update of 119911) For each119898 = 1 2 119872 if119891119898(119910

min) lt

119911119898 in which 119910

min is the minimum solution among 119910119896 119896 isin

1 2 119904num then set 119911119898

= 119891119898(119910

min)

Step 25 (update of solutions) Set 119888 = 0 and then do thefollowing

(1) If 119888 = 120578119903or 119878 is empty go to Step 3 Otherwise

randomly pick an index 119895 from 119878

(2) If min119892119905119890(1199101 | 120582119895) 119892119905119890(1199102 | 120582119895) 119892119905119890(119910119904num | 120582119895) le

119892119905119890(119909119895| 120582119895) then set 119909119895 = 119910

min FV119895 = 119892119905119890(119910

min|

120582119895) in which 119910

min is the minimum solution among119910119896 119896 isin 1 2 119904num Set 119888 = 119888 + 1

(3) Delete 119895 from 119878 and go to (1)

Step 3 (stopping criteria) If stopping criteria are satisfiedthen stop and output 119909

1 1199092 119909

119875 and 119865(119909

1) 119865(119909

2)

119865(119909119875) Otherwise go to Step 2

In the population optimization variables for each indi-vidual are the common excitation amplitudes for all patternsand different phases for forming different patterns Theoutput of the algorithm is a Pareto set in which each Paretooptimal solution corresponds to a phase-only reconfigurablearray In the optimization results different weight coefficientscorresponding to various patterns form a weight vector eachPareto optimal solution corresponds to a weight vector anddifferent Pareto optimal solutions have different weight vec-tors In actual applications decision-makers select a desiredsolution from the approximated PF or obey some standardsfor choosing the best compromise solution

6 Experiments Results

To demonstrate whether the proposed MOEAD-IWO out-performs MOEAD-DE in the design of phase-only patternreconfigurable linear arrays we carry out two experiments

61 Parameter Setting In MOEAD the population size andweight vectors are controlled by an integer 119867 1205821 1205822 120582119875are the weight vectors in which each individual weight takesa value from 0119867 1119867 119867119867 therefore the populationsize is 119875 = 119862

119872minus1

119867+119872minus1 where 119872 is the number of objectives In

the experiments population size 119875 is set to 201 (119867 = 200) for2-objective instances For the other control parameters we set119879 = 01119875 120578

119903= 001119875 and 120575 = 07 [30] which are the same

for MOEAD-DE and MOEAD-IWO For DE operator weset the crossover rate CR = 10 the scalar factor 119865 = 05120578 = 20 and 119901

119898= 1119899 in mutation operators [12 30] For

IWO operator we set 120590initial = 01 120590final = 00002 pow =

3 119904max = 3 and 119904min = 1 To be fair in the comparisonboth algorithms stop after reaching the maximum functionevaluations 500000 for all the tests In order to restrict theinfluence of random effects both algorithms run 20 timesindependently of each problem and the best results arepresented in this section Source codes for MOEAD-DE aretaken from [31]

62 Determining the Best Compromise Solution When facinga set of nondominated solutions a decision-makerrsquos judg-ment may have fuzzy or imprecise goals for each objectivefunction so useful quantitative measures for assessing thequality of each nondominated solution are needed In presentliteratures an effective method based on the concept of fuzzysets is proposed [25] and successfully used in some MOPs[32ndash34] in which a membership function 120583

119894is defined for

each objective 119891119894by taking account of the minimum and

International Journal of Antennas and Propagation 7

Table 3 The best compromise solutions for the first instance by MOEAD-IWO and MOEAD-DE (DRR = 4)

1198911(flat-top beam) 119891

2(pencil beam) (120582

1 1205822) Cost = 120582

11198911+ 12058221198912

MOEAD-DE with TE 00069 00059 (05350 04650) 00064MOEAD-DE with WS 00086 00084 (05100 04900) 00085MOEAD-IWO with TE 00037 00056 (05100 04900) 00046MOEAD-IWO with WS 00062 00060 (05450 04550) 00061

Table 4 The best compromise solutions for the second instance by MOEAD-IWO and MOEAD-DE (DRR = 5)

1198913(flat-top beam) 119891

4(cosecant beam) (120582

3 1205824) Cost = 120582

31198913+ 12058241198914

MOEAD-DE with TE 00043 00578 (05200 04800) 00300MOEAD-DE with WS 00762 01702 (04400 05600) 01288MOEAD-IWO with TE 00015 00511 (04750 05250) 00275MOEAD-IWO with WS 00016 00516 (04650 05350) 00284

maximum values of each objective function together with therate of increase of membership satisfaction As in [33 34]we assume that 120583

119894is a strictly monotonic decreasing and

continuous function defined as

120583119894=

1 if 119891119894le 119891

min119894

119891max119894

minus 119891119894

119891max119894

minus 119891min119894

if 119891min119894

le 119891119894le 119891

max119894

0 if 119891119894ge 119891

max119894

(25)

where 119891max119894

and 119891min119894

are the maximum and minimumvalues of the 119894th objective solutions among all nondominatedsolutions respectively Value 120583

119894indicates how much a non-

dominated solution satisfies the 119894th objective so the sum of120583119894(119894 = 1 2 119898) for all the objectives can be calculated

for measuring the ldquoaccomplishmentrdquo of each solution insatisfying the 119872 objectives and then a membership function119877119906(119906 = 1 2 119873

119904) can be defined for nondominated

solutions as follows

119877119906=

sum119872

119894=1120583119894(119906)

sum119873119904

V=1sum119872

119894=1120583119894(V)

(26)

where119873119904represents the number of nondominated solutions

The function 119877119906can be treated as a membership function

for the nondominated solutions in a fuzzy set Value 119877119906

represents the quality of a solution among all nondominatedsolutions and the one with the maximum value of 120583119902 is thebest compromise solution

63 Results Effectiveness of the MOEAD-IWO algorithmis verified through the design of two examples In theoptimization process excitation amplitudes range from 0 to1 and phases are restricted from minus180 to 180 degrees

631 Problem 1 Consider a 20-element 05120582-equispacedsymmetrical array with EF(120579) = sin(120579) for generating a flat-top beam and a pencil beamThe flat-top beam is realized bythe common excitation amplitudes and unknown phases andthe phases for forming pencil beam are set to zeroTherefore

there are 10 common excitation amplitude variables and 10phase variables for forming flat-top beam to be optimized inobjective function (17) The maximum dynamic range ratio(DRR) of excitation amplitudes is controlled with 4

The best compromise solutions obtained by MOEAD-IWO and MOEAD-DE are shown in Figure 1 and Table 1 Itcan be seen that for flat-top beam the peak SLL obtainedby MOEAD-IWO is 067 dB lower than that obtained byMOEAD-DE The BW value obtained by MOEAD-DE is16∘ wider than the objective value while that obtained byMOEAD-IWO is just 02∘ wider than the design objectiveFor the pencil beam the pattern obtained byMOEAD-IWOhas 094 dB lower peak SLL value when other indexes meetthe design objectives well The excitation amplitudes and thephases for the synthesized radiation patterns obtained byMOEAD-IWO are shown in Figure 2

632 Problem 2 Consider a 05120582-equispaced linear arraywith 25 isotropic elements to generate a flat-top beam and acosecant-squared beam The dimension of vector 119909 in objec-tive function (20) is 75 including 25 common amplitudesand 50 unknown phases The maximum DRR of excitationamplitudes is controlled with DRR = 5

Radiation patterns of the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE are shownin Figure 4 and Table 2 and the excitation amplitudes andthe phases for the synthesized radiation patterns obtainedby MOEAD-IWO are shown in Figure 5 It can be seenfrom Table 2 that for the flat-top beam the ripples obtainedby these two algorithms meet the design objectives wellthe peak SLL value obtained by MOEAD-IWO is 09553 dBlower than that obtained by MOEAD-DE and the BWvalues obtained by MOEAD-DE and MOEAD-IWO are088 and 086 respectively For the cosecant beam the BWobtained by MOEAD-IWO is smaller than that obtainedby MOEAD-DE and the ripple and peak SLL valuesobtained by MOEAD-IWO are 03247 dB and 08325 dBlower than those obtained by MOEAD-DE respectively

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

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International Journal of

International Journal of Antennas and Propagation 5

Table 1 Design objectives and simulated results for Problem 1

Design parametersFlat-top beam Pencil beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus2546 minus2613 minus3000 minus2956 minus3050

Half-power beam width(HPBW in 120579-space) 24∘ 242∘ 24∘ 68∘ 68∘ 68∘

Beam width at SLL(BW in 120579-space) 40∘ 416∘ 402∘ 20∘ 20∘ 20∘

Ripple(in dB 78∘ le 120579 le 102

∘) 050 050 049 NA NA NA

Table 2 Design objectives and simulated results for Problem 2

Design parametersFlat-top beam Cosecant-squared beam

Desired Obtained byMOEAD-DE

Obtained byMOEAD-IWO Desired Obtained by

MOEAD-DEObtained by

MOEAD-IWOSide lobe level(SLL in dB) minus2500 minus240106 minus249659 minus2500 minus240735 minus249060

Beam width at SLL(BW in 120579-space) 084 088 086 084 088 086

Ripple(in dB)

060|119906| le 032

05998 05994 0600 le 119906 le 064

08588 05341

The cost functions of each single pattern in (17) are

1198911(119909) = Cost

119891=

4

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(18)

1198912(119909) = Cost

119901=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(19)

(2) A flat-top beam and a cosecant beam the designproblem is expressed as

minimize 119865 (119909) = (1198913(119909) 119891

4(119909))119879

(20)

The cost functions of each single pattern in (20) are

1198913(119909) = Cost

119891=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(21)

1198914(119909) = Cost

119888=

3

sum

119904=1

(max 119876119904119889

minus 119876119904(119909) 0)

2

(22)

where Cost119891 Cost

119888 and Cost

119901represent the cost functions of

the flat-top beam cosecant-squared beam and pencil beamrespectively and119876

119904119889and119876

119904(119909) are the desired and calculated

values for each design specification we use Desired andcalculated values of each design index in (18) (19) (21) and(22) are summarized in Tables 1 and 2 The lower the costvalue is the closer the calculated value approaches the desiredvalue When the calculated values of all the indexes are lessthan the corresponding desired values the cost value is set tozero

5 MOEAD-IWO for Synthesis of Phase-OnlyReconfigurable Linear Array

Suppose 119875 is the population size that is to say we need tooptimize 119875 subproblems in a single run 1205821 1205822 120582119875 isa set of weight vectors The objective functions of all thepatterns are summed with assigned weights For each single-objective subproblem the objective function is described asfollows

119892119905119890(119909 | 120582

119894 119911lowast) = max1le119898le119872

120582119894

119898

1003816100381610038161003816119891119898 (119909) minus 119911lowast

119898

1003816100381610038161003816 (23)

where 120582119894= (120582119894

1 120582119894

2 120582

119894

119872)119879 119894 = 1 2 119875 MOEAD-IWO

is used tominimize119875 objective functions (23) simultaneouslyin a single run and the neighborhood of 120582119894 is the severalweight vectors close to 120582

119894At each generation MOEAD-IWOmaintains the follow-

ing information

(i) a population of 119875 points 1199091 1199092 119909

119875isin Ω where

Ω is the feasible solution space of the excitationamplitude and phases

(ii) FV1 FV2 FV119875 where FV119894 = 119892119905119890(119909119894

| 120582119894) 119894 =

1 2 119875

(iii) Gen the current generation number

(iv) 119911 = (1199111 1199112 119911

119872)119879 where 119911

119898is the best value found

so far for objective 119891119898

6 International Journal of Antennas and Propagation

The algorithm is outlined as follows

Input

(i) The multiobjective problem (17) or (20)(ii) A stopping criterion(iii) 119875 the population size (the number of subproblems)(iv) 119879 the number of the neighbors for each subproblem(v) A uniform set of 119875 weight vectors 1205821 1205822 120582119875

Output

(i) Approximation to the PS 1199091 1199092 119909119875(ii) Approximation to the PF 119865(1199091) 119865(1199092) 119865(119909119875)

Step 1 (initialization) Consider the following

Step 11 Generate the weight vectors 1205821 1205822 120582119875

Step 12 Compute the Euclidian distances between any twoweight vectors to each weigh vector For each 119894 = 1 2 119875set 119861(119894) = 119894

1 1198942 119894

119879 where 120582

1198941 1205821198942 120582

119894119879 are 119879 weight

vectors close to 120582119894

Step 13 Generate an initial population 1199091 1199092 119909

119875 byuniformly random sampling from the search spaceΩ

Set FV119894 = 119892119905119890(119909119894| 120582119894)

Step 14 Initialize 119911 = (1199111 1199112 119911

119872)119879 by setting 119911

119898=

min1le119894le119875

119891119898(119909119894)

Step 2 (update) For 119894 = 1 2 119875 do the following

Step 21 (selection of matingupdate range) Generate a num-ber rand from [0 1] randomly Then set

119878 = 119861 (119894) if rand lt 120575

1 2 119875 otherwise(24)

120575 is the probability that parent individuals are selected fromthe neighborhood

Step 22 (reproduction) Calculate the number of new solu-tions 119904num using (9) and produce new solutions 119910

119896(119896 =

1 2 119904num) using (8)

Step 23 (repair) If an element in 119910119896 is out of the boundary of

Ω its value is reset to the boundary value

Step 24 (update of 119911) For each119898 = 1 2 119872 if119891119898(119910

min) lt

119911119898 in which 119910

min is the minimum solution among 119910119896 119896 isin

1 2 119904num then set 119911119898

= 119891119898(119910

min)

Step 25 (update of solutions) Set 119888 = 0 and then do thefollowing

(1) If 119888 = 120578119903or 119878 is empty go to Step 3 Otherwise

randomly pick an index 119895 from 119878

(2) If min119892119905119890(1199101 | 120582119895) 119892119905119890(1199102 | 120582119895) 119892119905119890(119910119904num | 120582119895) le

119892119905119890(119909119895| 120582119895) then set 119909119895 = 119910

min FV119895 = 119892119905119890(119910

min|

120582119895) in which 119910

min is the minimum solution among119910119896 119896 isin 1 2 119904num Set 119888 = 119888 + 1

(3) Delete 119895 from 119878 and go to (1)

Step 3 (stopping criteria) If stopping criteria are satisfiedthen stop and output 119909

1 1199092 119909

119875 and 119865(119909

1) 119865(119909

2)

119865(119909119875) Otherwise go to Step 2

In the population optimization variables for each indi-vidual are the common excitation amplitudes for all patternsand different phases for forming different patterns Theoutput of the algorithm is a Pareto set in which each Paretooptimal solution corresponds to a phase-only reconfigurablearray In the optimization results different weight coefficientscorresponding to various patterns form a weight vector eachPareto optimal solution corresponds to a weight vector anddifferent Pareto optimal solutions have different weight vec-tors In actual applications decision-makers select a desiredsolution from the approximated PF or obey some standardsfor choosing the best compromise solution

6 Experiments Results

To demonstrate whether the proposed MOEAD-IWO out-performs MOEAD-DE in the design of phase-only patternreconfigurable linear arrays we carry out two experiments

61 Parameter Setting In MOEAD the population size andweight vectors are controlled by an integer 119867 1205821 1205822 120582119875are the weight vectors in which each individual weight takesa value from 0119867 1119867 119867119867 therefore the populationsize is 119875 = 119862

119872minus1

119867+119872minus1 where 119872 is the number of objectives In

the experiments population size 119875 is set to 201 (119867 = 200) for2-objective instances For the other control parameters we set119879 = 01119875 120578

119903= 001119875 and 120575 = 07 [30] which are the same

for MOEAD-DE and MOEAD-IWO For DE operator weset the crossover rate CR = 10 the scalar factor 119865 = 05120578 = 20 and 119901

119898= 1119899 in mutation operators [12 30] For

IWO operator we set 120590initial = 01 120590final = 00002 pow =

3 119904max = 3 and 119904min = 1 To be fair in the comparisonboth algorithms stop after reaching the maximum functionevaluations 500000 for all the tests In order to restrict theinfluence of random effects both algorithms run 20 timesindependently of each problem and the best results arepresented in this section Source codes for MOEAD-DE aretaken from [31]

62 Determining the Best Compromise Solution When facinga set of nondominated solutions a decision-makerrsquos judg-ment may have fuzzy or imprecise goals for each objectivefunction so useful quantitative measures for assessing thequality of each nondominated solution are needed In presentliteratures an effective method based on the concept of fuzzysets is proposed [25] and successfully used in some MOPs[32ndash34] in which a membership function 120583

119894is defined for

each objective 119891119894by taking account of the minimum and

International Journal of Antennas and Propagation 7

Table 3 The best compromise solutions for the first instance by MOEAD-IWO and MOEAD-DE (DRR = 4)

1198911(flat-top beam) 119891

2(pencil beam) (120582

1 1205822) Cost = 120582

11198911+ 12058221198912

MOEAD-DE with TE 00069 00059 (05350 04650) 00064MOEAD-DE with WS 00086 00084 (05100 04900) 00085MOEAD-IWO with TE 00037 00056 (05100 04900) 00046MOEAD-IWO with WS 00062 00060 (05450 04550) 00061

Table 4 The best compromise solutions for the second instance by MOEAD-IWO and MOEAD-DE (DRR = 5)

1198913(flat-top beam) 119891

4(cosecant beam) (120582

3 1205824) Cost = 120582

31198913+ 12058241198914

MOEAD-DE with TE 00043 00578 (05200 04800) 00300MOEAD-DE with WS 00762 01702 (04400 05600) 01288MOEAD-IWO with TE 00015 00511 (04750 05250) 00275MOEAD-IWO with WS 00016 00516 (04650 05350) 00284

maximum values of each objective function together with therate of increase of membership satisfaction As in [33 34]we assume that 120583

119894is a strictly monotonic decreasing and

continuous function defined as

120583119894=

1 if 119891119894le 119891

min119894

119891max119894

minus 119891119894

119891max119894

minus 119891min119894

if 119891min119894

le 119891119894le 119891

max119894

0 if 119891119894ge 119891

max119894

(25)

where 119891max119894

and 119891min119894

are the maximum and minimumvalues of the 119894th objective solutions among all nondominatedsolutions respectively Value 120583

119894indicates how much a non-

dominated solution satisfies the 119894th objective so the sum of120583119894(119894 = 1 2 119898) for all the objectives can be calculated

for measuring the ldquoaccomplishmentrdquo of each solution insatisfying the 119872 objectives and then a membership function119877119906(119906 = 1 2 119873

119904) can be defined for nondominated

solutions as follows

119877119906=

sum119872

119894=1120583119894(119906)

sum119873119904

V=1sum119872

119894=1120583119894(V)

(26)

where119873119904represents the number of nondominated solutions

The function 119877119906can be treated as a membership function

for the nondominated solutions in a fuzzy set Value 119877119906

represents the quality of a solution among all nondominatedsolutions and the one with the maximum value of 120583119902 is thebest compromise solution

63 Results Effectiveness of the MOEAD-IWO algorithmis verified through the design of two examples In theoptimization process excitation amplitudes range from 0 to1 and phases are restricted from minus180 to 180 degrees

631 Problem 1 Consider a 20-element 05120582-equispacedsymmetrical array with EF(120579) = sin(120579) for generating a flat-top beam and a pencil beamThe flat-top beam is realized bythe common excitation amplitudes and unknown phases andthe phases for forming pencil beam are set to zeroTherefore

there are 10 common excitation amplitude variables and 10phase variables for forming flat-top beam to be optimized inobjective function (17) The maximum dynamic range ratio(DRR) of excitation amplitudes is controlled with 4

The best compromise solutions obtained by MOEAD-IWO and MOEAD-DE are shown in Figure 1 and Table 1 Itcan be seen that for flat-top beam the peak SLL obtainedby MOEAD-IWO is 067 dB lower than that obtained byMOEAD-DE The BW value obtained by MOEAD-DE is16∘ wider than the objective value while that obtained byMOEAD-IWO is just 02∘ wider than the design objectiveFor the pencil beam the pattern obtained byMOEAD-IWOhas 094 dB lower peak SLL value when other indexes meetthe design objectives well The excitation amplitudes and thephases for the synthesized radiation patterns obtained byMOEAD-IWO are shown in Figure 2

632 Problem 2 Consider a 05120582-equispaced linear arraywith 25 isotropic elements to generate a flat-top beam and acosecant-squared beam The dimension of vector 119909 in objec-tive function (20) is 75 including 25 common amplitudesand 50 unknown phases The maximum DRR of excitationamplitudes is controlled with DRR = 5

Radiation patterns of the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE are shownin Figure 4 and Table 2 and the excitation amplitudes andthe phases for the synthesized radiation patterns obtainedby MOEAD-IWO are shown in Figure 5 It can be seenfrom Table 2 that for the flat-top beam the ripples obtainedby these two algorithms meet the design objectives wellthe peak SLL value obtained by MOEAD-IWO is 09553 dBlower than that obtained by MOEAD-DE and the BWvalues obtained by MOEAD-DE and MOEAD-IWO are088 and 086 respectively For the cosecant beam the BWobtained by MOEAD-IWO is smaller than that obtainedby MOEAD-DE and the ripple and peak SLL valuesobtained by MOEAD-IWO are 03247 dB and 08325 dBlower than those obtained by MOEAD-DE respectively

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

6 International Journal of Antennas and Propagation

The algorithm is outlined as follows

Input

(i) The multiobjective problem (17) or (20)(ii) A stopping criterion(iii) 119875 the population size (the number of subproblems)(iv) 119879 the number of the neighbors for each subproblem(v) A uniform set of 119875 weight vectors 1205821 1205822 120582119875

Output

(i) Approximation to the PS 1199091 1199092 119909119875(ii) Approximation to the PF 119865(1199091) 119865(1199092) 119865(119909119875)

Step 1 (initialization) Consider the following

Step 11 Generate the weight vectors 1205821 1205822 120582119875

Step 12 Compute the Euclidian distances between any twoweight vectors to each weigh vector For each 119894 = 1 2 119875set 119861(119894) = 119894

1 1198942 119894

119879 where 120582

1198941 1205821198942 120582

119894119879 are 119879 weight

vectors close to 120582119894

Step 13 Generate an initial population 1199091 1199092 119909

119875 byuniformly random sampling from the search spaceΩ

Set FV119894 = 119892119905119890(119909119894| 120582119894)

Step 14 Initialize 119911 = (1199111 1199112 119911

119872)119879 by setting 119911

119898=

min1le119894le119875

119891119898(119909119894)

Step 2 (update) For 119894 = 1 2 119875 do the following

Step 21 (selection of matingupdate range) Generate a num-ber rand from [0 1] randomly Then set

119878 = 119861 (119894) if rand lt 120575

1 2 119875 otherwise(24)

120575 is the probability that parent individuals are selected fromthe neighborhood

Step 22 (reproduction) Calculate the number of new solu-tions 119904num using (9) and produce new solutions 119910

119896(119896 =

1 2 119904num) using (8)

Step 23 (repair) If an element in 119910119896 is out of the boundary of

Ω its value is reset to the boundary value

Step 24 (update of 119911) For each119898 = 1 2 119872 if119891119898(119910

min) lt

119911119898 in which 119910

min is the minimum solution among 119910119896 119896 isin

1 2 119904num then set 119911119898

= 119891119898(119910

min)

Step 25 (update of solutions) Set 119888 = 0 and then do thefollowing

(1) If 119888 = 120578119903or 119878 is empty go to Step 3 Otherwise

randomly pick an index 119895 from 119878

(2) If min119892119905119890(1199101 | 120582119895) 119892119905119890(1199102 | 120582119895) 119892119905119890(119910119904num | 120582119895) le

119892119905119890(119909119895| 120582119895) then set 119909119895 = 119910

min FV119895 = 119892119905119890(119910

min|

120582119895) in which 119910

min is the minimum solution among119910119896 119896 isin 1 2 119904num Set 119888 = 119888 + 1

(3) Delete 119895 from 119878 and go to (1)

Step 3 (stopping criteria) If stopping criteria are satisfiedthen stop and output 119909

1 1199092 119909

119875 and 119865(119909

1) 119865(119909

2)

119865(119909119875) Otherwise go to Step 2

In the population optimization variables for each indi-vidual are the common excitation amplitudes for all patternsand different phases for forming different patterns Theoutput of the algorithm is a Pareto set in which each Paretooptimal solution corresponds to a phase-only reconfigurablearray In the optimization results different weight coefficientscorresponding to various patterns form a weight vector eachPareto optimal solution corresponds to a weight vector anddifferent Pareto optimal solutions have different weight vec-tors In actual applications decision-makers select a desiredsolution from the approximated PF or obey some standardsfor choosing the best compromise solution

6 Experiments Results

To demonstrate whether the proposed MOEAD-IWO out-performs MOEAD-DE in the design of phase-only patternreconfigurable linear arrays we carry out two experiments

61 Parameter Setting In MOEAD the population size andweight vectors are controlled by an integer 119867 1205821 1205822 120582119875are the weight vectors in which each individual weight takesa value from 0119867 1119867 119867119867 therefore the populationsize is 119875 = 119862

119872minus1

119867+119872minus1 where 119872 is the number of objectives In

the experiments population size 119875 is set to 201 (119867 = 200) for2-objective instances For the other control parameters we set119879 = 01119875 120578

119903= 001119875 and 120575 = 07 [30] which are the same

for MOEAD-DE and MOEAD-IWO For DE operator weset the crossover rate CR = 10 the scalar factor 119865 = 05120578 = 20 and 119901

119898= 1119899 in mutation operators [12 30] For

IWO operator we set 120590initial = 01 120590final = 00002 pow =

3 119904max = 3 and 119904min = 1 To be fair in the comparisonboth algorithms stop after reaching the maximum functionevaluations 500000 for all the tests In order to restrict theinfluence of random effects both algorithms run 20 timesindependently of each problem and the best results arepresented in this section Source codes for MOEAD-DE aretaken from [31]

62 Determining the Best Compromise Solution When facinga set of nondominated solutions a decision-makerrsquos judg-ment may have fuzzy or imprecise goals for each objectivefunction so useful quantitative measures for assessing thequality of each nondominated solution are needed In presentliteratures an effective method based on the concept of fuzzysets is proposed [25] and successfully used in some MOPs[32ndash34] in which a membership function 120583

119894is defined for

each objective 119891119894by taking account of the minimum and

International Journal of Antennas and Propagation 7

Table 3 The best compromise solutions for the first instance by MOEAD-IWO and MOEAD-DE (DRR = 4)

1198911(flat-top beam) 119891

2(pencil beam) (120582

1 1205822) Cost = 120582

11198911+ 12058221198912

MOEAD-DE with TE 00069 00059 (05350 04650) 00064MOEAD-DE with WS 00086 00084 (05100 04900) 00085MOEAD-IWO with TE 00037 00056 (05100 04900) 00046MOEAD-IWO with WS 00062 00060 (05450 04550) 00061

Table 4 The best compromise solutions for the second instance by MOEAD-IWO and MOEAD-DE (DRR = 5)

1198913(flat-top beam) 119891

4(cosecant beam) (120582

3 1205824) Cost = 120582

31198913+ 12058241198914

MOEAD-DE with TE 00043 00578 (05200 04800) 00300MOEAD-DE with WS 00762 01702 (04400 05600) 01288MOEAD-IWO with TE 00015 00511 (04750 05250) 00275MOEAD-IWO with WS 00016 00516 (04650 05350) 00284

maximum values of each objective function together with therate of increase of membership satisfaction As in [33 34]we assume that 120583

119894is a strictly monotonic decreasing and

continuous function defined as

120583119894=

1 if 119891119894le 119891

min119894

119891max119894

minus 119891119894

119891max119894

minus 119891min119894

if 119891min119894

le 119891119894le 119891

max119894

0 if 119891119894ge 119891

max119894

(25)

where 119891max119894

and 119891min119894

are the maximum and minimumvalues of the 119894th objective solutions among all nondominatedsolutions respectively Value 120583

119894indicates how much a non-

dominated solution satisfies the 119894th objective so the sum of120583119894(119894 = 1 2 119898) for all the objectives can be calculated

for measuring the ldquoaccomplishmentrdquo of each solution insatisfying the 119872 objectives and then a membership function119877119906(119906 = 1 2 119873

119904) can be defined for nondominated

solutions as follows

119877119906=

sum119872

119894=1120583119894(119906)

sum119873119904

V=1sum119872

119894=1120583119894(V)

(26)

where119873119904represents the number of nondominated solutions

The function 119877119906can be treated as a membership function

for the nondominated solutions in a fuzzy set Value 119877119906

represents the quality of a solution among all nondominatedsolutions and the one with the maximum value of 120583119902 is thebest compromise solution

63 Results Effectiveness of the MOEAD-IWO algorithmis verified through the design of two examples In theoptimization process excitation amplitudes range from 0 to1 and phases are restricted from minus180 to 180 degrees

631 Problem 1 Consider a 20-element 05120582-equispacedsymmetrical array with EF(120579) = sin(120579) for generating a flat-top beam and a pencil beamThe flat-top beam is realized bythe common excitation amplitudes and unknown phases andthe phases for forming pencil beam are set to zeroTherefore

there are 10 common excitation amplitude variables and 10phase variables for forming flat-top beam to be optimized inobjective function (17) The maximum dynamic range ratio(DRR) of excitation amplitudes is controlled with 4

The best compromise solutions obtained by MOEAD-IWO and MOEAD-DE are shown in Figure 1 and Table 1 Itcan be seen that for flat-top beam the peak SLL obtainedby MOEAD-IWO is 067 dB lower than that obtained byMOEAD-DE The BW value obtained by MOEAD-DE is16∘ wider than the objective value while that obtained byMOEAD-IWO is just 02∘ wider than the design objectiveFor the pencil beam the pattern obtained byMOEAD-IWOhas 094 dB lower peak SLL value when other indexes meetthe design objectives well The excitation amplitudes and thephases for the synthesized radiation patterns obtained byMOEAD-IWO are shown in Figure 2

632 Problem 2 Consider a 05120582-equispaced linear arraywith 25 isotropic elements to generate a flat-top beam and acosecant-squared beam The dimension of vector 119909 in objec-tive function (20) is 75 including 25 common amplitudesand 50 unknown phases The maximum DRR of excitationamplitudes is controlled with DRR = 5

Radiation patterns of the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE are shownin Figure 4 and Table 2 and the excitation amplitudes andthe phases for the synthesized radiation patterns obtainedby MOEAD-IWO are shown in Figure 5 It can be seenfrom Table 2 that for the flat-top beam the ripples obtainedby these two algorithms meet the design objectives wellthe peak SLL value obtained by MOEAD-IWO is 09553 dBlower than that obtained by MOEAD-DE and the BWvalues obtained by MOEAD-DE and MOEAD-IWO are088 and 086 respectively For the cosecant beam the BWobtained by MOEAD-IWO is smaller than that obtainedby MOEAD-DE and the ripple and peak SLL valuesobtained by MOEAD-IWO are 03247 dB and 08325 dBlower than those obtained by MOEAD-DE respectively

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Advances inOptoElectronics

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

Table 3 The best compromise solutions for the first instance by MOEAD-IWO and MOEAD-DE (DRR = 4)

1198911(flat-top beam) 119891

2(pencil beam) (120582

1 1205822) Cost = 120582

11198911+ 12058221198912

MOEAD-DE with TE 00069 00059 (05350 04650) 00064MOEAD-DE with WS 00086 00084 (05100 04900) 00085MOEAD-IWO with TE 00037 00056 (05100 04900) 00046MOEAD-IWO with WS 00062 00060 (05450 04550) 00061

Table 4 The best compromise solutions for the second instance by MOEAD-IWO and MOEAD-DE (DRR = 5)

1198913(flat-top beam) 119891

4(cosecant beam) (120582

3 1205824) Cost = 120582

31198913+ 12058241198914

MOEAD-DE with TE 00043 00578 (05200 04800) 00300MOEAD-DE with WS 00762 01702 (04400 05600) 01288MOEAD-IWO with TE 00015 00511 (04750 05250) 00275MOEAD-IWO with WS 00016 00516 (04650 05350) 00284

maximum values of each objective function together with therate of increase of membership satisfaction As in [33 34]we assume that 120583

119894is a strictly monotonic decreasing and

continuous function defined as

120583119894=

1 if 119891119894le 119891

min119894

119891max119894

minus 119891119894

119891max119894

minus 119891min119894

if 119891min119894

le 119891119894le 119891

max119894

0 if 119891119894ge 119891

max119894

(25)

where 119891max119894

and 119891min119894

are the maximum and minimumvalues of the 119894th objective solutions among all nondominatedsolutions respectively Value 120583

119894indicates how much a non-

dominated solution satisfies the 119894th objective so the sum of120583119894(119894 = 1 2 119898) for all the objectives can be calculated

for measuring the ldquoaccomplishmentrdquo of each solution insatisfying the 119872 objectives and then a membership function119877119906(119906 = 1 2 119873

119904) can be defined for nondominated

solutions as follows

119877119906=

sum119872

119894=1120583119894(119906)

sum119873119904

V=1sum119872

119894=1120583119894(V)

(26)

where119873119904represents the number of nondominated solutions

The function 119877119906can be treated as a membership function

for the nondominated solutions in a fuzzy set Value 119877119906

represents the quality of a solution among all nondominatedsolutions and the one with the maximum value of 120583119902 is thebest compromise solution

63 Results Effectiveness of the MOEAD-IWO algorithmis verified through the design of two examples In theoptimization process excitation amplitudes range from 0 to1 and phases are restricted from minus180 to 180 degrees

631 Problem 1 Consider a 20-element 05120582-equispacedsymmetrical array with EF(120579) = sin(120579) for generating a flat-top beam and a pencil beamThe flat-top beam is realized bythe common excitation amplitudes and unknown phases andthe phases for forming pencil beam are set to zeroTherefore

there are 10 common excitation amplitude variables and 10phase variables for forming flat-top beam to be optimized inobjective function (17) The maximum dynamic range ratio(DRR) of excitation amplitudes is controlled with 4

The best compromise solutions obtained by MOEAD-IWO and MOEAD-DE are shown in Figure 1 and Table 1 Itcan be seen that for flat-top beam the peak SLL obtainedby MOEAD-IWO is 067 dB lower than that obtained byMOEAD-DE The BW value obtained by MOEAD-DE is16∘ wider than the objective value while that obtained byMOEAD-IWO is just 02∘ wider than the design objectiveFor the pencil beam the pattern obtained byMOEAD-IWOhas 094 dB lower peak SLL value when other indexes meetthe design objectives well The excitation amplitudes and thephases for the synthesized radiation patterns obtained byMOEAD-IWO are shown in Figure 2

632 Problem 2 Consider a 05120582-equispaced linear arraywith 25 isotropic elements to generate a flat-top beam and acosecant-squared beam The dimension of vector 119909 in objec-tive function (20) is 75 including 25 common amplitudesand 50 unknown phases The maximum DRR of excitationamplitudes is controlled with DRR = 5

Radiation patterns of the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE are shownin Figure 4 and Table 2 and the excitation amplitudes andthe phases for the synthesized radiation patterns obtainedby MOEAD-IWO are shown in Figure 5 It can be seenfrom Table 2 that for the flat-top beam the ripples obtainedby these two algorithms meet the design objectives wellthe peak SLL value obtained by MOEAD-IWO is 09553 dBlower than that obtained by MOEAD-DE and the BWvalues obtained by MOEAD-DE and MOEAD-IWO are088 and 086 respectively For the cosecant beam the BWobtained by MOEAD-IWO is smaller than that obtainedby MOEAD-DE and the ripple and peak SLL valuesobtained by MOEAD-IWO are 03247 dB and 08325 dBlower than those obtained by MOEAD-DE respectively

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

0 18016014012010080604020minus50

minus40

minus30

minus20

minus10

0

F (d

B)

MOEAD-DEMOEAD-IWO

120579 (deg)

(a)

minus50

minus40

minus30

minus20

minus10

0

F (d

B)

0 18016014012010080604020

MOEAD-DEMOEAD-IWO

120579 (deg)

(b)

Figure 1 Synthesized radiation patterns and their masks (dashed black lines) for Problem 1 (a) flat-top beam and (b) pencil beam

2 20181614121086402

04

06

08

10

Exci

tatio

n am

plitu

de

Element number

(a)

2 201816141210864minus150minus100minus50

050100150

Exci

tatio

n ph

ase (

deg)

Element number

Pencil beam Flat-top beam

(b)

Figure 2 Excitation coefficients for the synthesized radiation patterns obtained by MOEAD-IWO for Problem 1 (a) common amplitudesand (b) different phases

0000 00120006000400020000

0002

0004

0006

0008

0010

0012

MOEAD-IWOMOEAD-DE

f2

(cos

t val

ue o

f pen

cil b

eam

)

f1 (cost value of flat-top beam)00100008

Figure 3 Approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 1

Thus we conclude that the patterns obtained by MOEAD-IWO have better performance The advantage of MOEAD-IWO over MOEAD-DE can also be seen from the obtainedapproximate PFs shown in Figures 3 and 6

633 Comparisons with Different Decomposition ApproachesIn order to make further comparisons between MOEAD-IWO and MOEAD-DE weighted sum method [23] is alsoused in the two algorithms to optimize Problem 1 and Prob-lem 2 Related parameters are set the same as in Section 61All the algorithms run 20 times independently and the bestresults are compared

Tables 3 and 4 show the best compromise solutionsobtained by MOEAD-IWO and MOEAD-DE for Problem1 and Problem 2 respectively in which TE and WS are theabbreviations of Tchebycheff approach and weighted sum 119891

1

and 1198912are the cost values of flat-top beam and pencil beam

in Problem 1 1198913and 119891

4are the cost values of flat-top beam

and cosecant beam in Problem 2 (1205821 1205822) and (120582

3 1205824) are

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

minus50

minus40

minus30

minus20

minus10

0F

(dB)

u

minus10 100500minus05

MOEAD-DEMOEAD-IWO

(a)

minus10 100500minus05minus50

minus40

minus30

minus20

minus10

0

F (d

B)

u

MOEAD-DEMOEAD-IWO

(b)

Figure 4 Synthesized radiation patterns and their masks (dashed black lines) for Problem 2 (a) flat-top beam and (b) cosecant-squaredbeam

02

04

06

08

10

Exci

tatio

n am

plitu

de

Element number5 25201510

(a)

5 25201510minus200minus150minus100minus50

050

100150200

Exci

tatio

n ph

ase

Element number

Cosecant beam Flat-top beam

(b)

Figure 5 Excitation coefficients for the synthesized radiation pattern obtained byMOEAD-IWO for Problem 2 (a) common amplitudes and(b) different phases

0000 0020001500100005003

004

005

006

007

008

009

010

f2

(cos

t val

ue o

f cos

ecan

t bea

m)

f1 (cost value of flat-top beam)

MOEAD-IWOMOEAD-DE

Figure 6 The approximate PFs obtained by MOEAD-DE andMOEAD-IWO for Problem 2

the weight vectors for the compromise solutions The overallcosts are computed by120582

11198911+12058221198912and120582

31198913+12058241198914 Best values

are shown in bold We can conclude that MOEAD-IWOwith Tchebycheff approach can obtain better compromisesolutions than other algorithms

Tables 5 and 6 present the comparison of the threemetrics obtained in the two experiments best values areshown in bold It is obvious that when using Tcheby-cheff approach as the decomposition approach MOEAD-IWO has higher accuracy (119863-metric) better uniformity (Δ-metric) and larger extent (nabla-metric) than MOEAD-DEAmong the four algorithms MOEAD-DE with weightedsum shows the best mean maximum and minimum valuesof nabla-metric MOEAD-IWO with Tchebycheff approachperforms better than the other three algorithms in termsof 119863-metric Δ-metric and the standard deviation of nabla-metric Comparison between the four algorithms reveals thesuperiority of MOEAD-IWO and confirms its potential forarray synthesis problems

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

Table 5 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 1

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 897119890 minus 3 547119890 minus 3 927119890 minus 3 654119890 minus 3

MOEAD-DE with WS 912119890 minus 3 534119890 minus 3 964119890 minus 3 735119890 minus 3

MOEAD-IWO with TE 589e minus 3 327e minus 3 684e minus 3 431e minus 3MOEAD-IWO with WS 592119890 minus 3 335119890 minus 3 691119890 minus 3 539119890 minus 3

Δ-metricMOEAD-DE with TE 214119890 minus 3 121119890 minus 3 307119890 minus 3 165119890 minus 3

MOEAD-DE with WS 204119890 minus 3 102119890 minus 3 298119890 minus 3 132119890 minus 3

MOEAD-IWO with TE 514e minus 4 317e minus 4 612e minus 4 387e minus 4MOEAD-IWO with WS 523119890 minus 4 324119890 minus 4 635119890 minus 4 397119890 minus 4

nabla-metricMOEAD-DE with TE 905119890 minus 5 524119890 minus 5 936119890 minus 5 879119890 minus 5

MOEAD-DE with WS 289e minus 4 102119890 minus 5 302e minus 4 239e minus 4MOEAD-IWO with TE 934119890 minus 5 214e minus 6 948119890 minus 5 931119890 minus 5

MOEAD-IWO with WS 161119890 minus 4 832119890 minus 6 164119890 minus 4 151119890 minus 4

Table 6 Performance comparison of MOEAD-DE and MOEAD-IWO with different decomposition approaches for Problem 2

Algorithm Mean Std Max Min119863-metric

MOEAD-DE with TE 531119890 minus 2 239119890 minus 2 725119890 minus 2 421119890 minus 2

MOEAD-DE with WS 621119890 minus 2 452119890 minus 2 795119890 minus 2 531119890 minus 2

MOEAD-IWO with TE 358e minus 2 154e minus 2 431e minus 2 301e minus 2MOEAD-IWO with WS 491119890 minus 2 231119890 minus 2 612119890 minus 2 395119890 minus 2

Δ-metricMOEAD-DE with TE 387119890 minus 2 135119890 minus 2 459119890 minus 2 274119890 minus 2

MOEAD-DE with WS 698119890 minus 2 532119890 minus 2 725119890 minus 2 535119890 minus 2

MOEAD-IWO with TE 178e minus 2 101e minus 2 214e minus 2 121e minus 2MOEAD-IWO with WS 232119890 minus 2 129119890 minus 2 402119890 minus 2 254119890 minus 2

nabla-metricMOEAD-DE with TE 670119890 minus 4 427119890 minus 4 685119890 minus 4 587119890 minus 4

MOEAD-DE with WS 230e minus 2 329119890 minus 4 243e minus 2 201e minus 2MOEAD-IWO with TE 102119890 minus 3 324e minus 5 110119890 minus 3 102119890 minus 3

MOEAD-IWO with WS 123119890 minus 2 123119890 minus 4 134119890 minus 2 117119890 minus 2

7 Conclusion

In this paper synthesis of phase-only reconfigurable antennaarray is formulated as anMOPwhere the design objectives arethe different patterns We integrate the modified IWO algo-rithm in MOEAD and propose a new version of MOEADcalled MOEAD-IWO for solving the multiobjective opti-mization problemsThe new algorithmmakes good use of thepowerful searching and colonizing ability of invasive weedsand maintains the advantages of the original MOEAD Twophase-only reconfigurable antenna array examples are syn-thesized by the proposed algorithm and the results have beencompared with those obtained by the existing MOEAD-DE algorithm The comparing simulation results indicatethe superiority of the proposed algorithm over MOEAD-DE in synthesizing the phase-only reconfigurable antennaarray The proposed algorithm opens a new prospect inarray synthesis problems It may also be applied in otherchallenging fields of MOPs for future research

Conflict of Interests

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

Acknowledgment

This research was fully supported by the National NaturalScience Foundation of China (no 61201022)

References

[1] O M Bucci G Mazzarella and G Panariello ldquoReconfigurablearrays by phase-only controlrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 919ndash925 1991

[2] R Vescovo ldquoReconfigurability and beam scanning with phase-only control for antenna arraysrdquo IEEE Transactions on Antennasand Propagation vol 56 no 6 pp 1555ndash1565 2008

[3] A F Morabito A Massa P Rocca and T Isernia ldquoAn effectiveapproach to the synthesis of phase-only reconfigurable linear

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 11

arraysrdquo IEEETransactions onAntennas andPropagation vol 60no 8 pp 3622ndash3631 2012

[4] S Baskar A Alphones and P N Suganthan ldquoGenetic-algorithm-based design of a reconfigurable antenna array withdiscrete phase shiftersrdquo Microwave and Optical TechnologyLetters vol 45 no 6 pp 461ndash465 2005

[5] G K Mahanti A Chakrabarty and S Das ldquoPhase-only andamplitude-phase synthesis of dual-pattern linear antenna arraysusing floating-point genetic algorithmsrdquoProgress in Electromag-netics Research vol 68 pp 247ndash259 2007

[6] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[7] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[8] D Gies and Y Rahmat-Samii ldquoReconfigurable array designusing parallel particle swarmoptimizationrdquo inProceedings of theIEEE International Antennas and Propagation Symposium andUSNCCNCURSI North American Radio Science Meeting vol1 pp 177ndash180 June 2003

[9] X Li and M Yin ldquoHybrid differential evolution with biogeog-raphy-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[10] X Li andMYin ldquoDesign of a reconfigurable antenna arraywithdiscrete phase shifters using differential evolution algorithmrdquoProgress In Electromagnetics Research B vol 31 pp 29ndash43 2011

[11] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[12] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[13] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceeding of the IEEE Congress on Evolutionary Computation(CEC 09) pp 203ndash208 Trondheim Norway May 2009

[14] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[15] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[16] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[17] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[18] Z D Zaharis C Skeberis and T D Xenos ldquoImproved antennaarray adaptive beam-forming with low side lobe level using anovel adaptive invasive weed optimization methodrdquo Progress inElectromagnetics Research vol 124 pp 137ndash150 2012

[19] A R Mallahzadeh H Oraizi and Z Davoodi-Rad ldquoApplica-tion of the invasive weed optimization technique for antenna

configurationsrdquo Progress in Electromagnetics Research vol 79pp 137ndash150 2008

[20] A R Mallahzadeh S Esrsquohaghi and H R Hassani ldquoCom-pact U-array MIMO antenna designs using IWO algorithmrdquoInternational Journal of RF and Microwave Computer-AidedEngineering vol 19 no 5 pp 568ndash576 2009

[21] D Kundu K Suresh S Ghosh S Das and B K PanigrahildquoMulti-objective optimization with artificial weed coloniesrdquoInformation Sciences vol 181 no 12 pp 2441ndash2454 2011

[22] H Scheffe ldquoExperiments with mixturesrdquo Journal of the RoyalStatistical Society B Methodological vol 20 pp 344ndash360 1958

[23] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[24] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Conference on Genetic and EvolutionaryComputation pp 645ndash652 New York NY USA 2014

[25] D V VeldhuizenMultiobjective evolutionary algorithms classi-fications analyses and new innovations [PhD thesis] Depart-ment of Electrical and Computer Engineering Air Force Insti-tute of Technology Wright-Patterson AFB Ohio USA 1999

[26] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[27] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[28] K DebMulti-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons Chichester UK 2001

[29] C Erbas S Cerav-Erbas and A D Pimentel ldquoMultiobjectiveoptimization and evolutionary algorithms for the applicationmapping problem in multiprocessor system-on-chip designrdquoIEEE Transactions on Evolutionary Computation vol 10 no 3pp 358ndash374 2006

[30] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[31] httpcswwwessexacukstaffqzhang[32] C G Tapia and B AMurtagh ldquoInteractive fuzzy programming

with preference criteria in multiobjective decision-makingrdquoComputers amp Operations Research vol 18 no 3 pp 307ndash3161991

[33] J S Dhillon S C Parti and D P Kothari ldquoStochastic economicemission load dispatchrdquoElectric Power SystemsResearch vol 26no 3 pp 179ndash186 1993

[34] S Pal B Qu S Das and P N Suganthan ldquoLinear antennaarray synthesis with constrained multi-objective differentialevolutionrdquo Progress in Electromagnetics Research B vol 21 pp87ndash111 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of