Core Levels Algorithm for Optimization: Case of Microwave Models

Transcription

1 (IJACSA) Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 8, No. 7, 07 Core Levels Algorithm for Optimizatio: Case of Microwave Models Ali Haydar Departmet of Computer Egieerig Gire America Uiversity Mersi-0, Turkey Kamil Dimililer Departmet of Electrical ad Electroics Egieerig Gire America Uiversity Mersi-0, Turkey Ezgi Deiz Ülker Departmet of Computer Egieerig Europea Uiversity of Lefke Mersi-0, Turkey Sadık Ülker Departmet of Electrical ad Electroics Egieerig Europea Uiversity of Lefke Mersi-0, Turkey Abstract Metaheuristic algorithms are ivestigated ad used by may researchers i differet areas. It is crucial to fid optimal solutios for all problems uder study especially for the oes which require sesitive optimizatio. Especially, for real case problems, solutio quality ad covergece speed of the algorithms are highly desired characteristics. I this paper, a ew optimizatio algorithm called Core Levels Algorithm (COLA) based o the use of metaheuristics is proposed ad aalyzed. I the algorithm, two core levels are applied recursively to create ew offsprigs from the paret vectors which provides a desired balace o the eploratio ad eploitatio characteristics. The algorithm s performace is first studied o some well-kow bechmark fuctios ad the compared with previously proposed efficiet evolutioary algorithms. The eperimetal results showed that eve at the early stages of optimizatio, obtaied values are very close or eactly the same as the optimum values of the aalyzed fuctios. The, the performace of COLA is ivestigated o real case problems such as some selected microwave circuit desigs. The results deoted that COLA produces stable results ad provides high accuracy of optimizatio without high parameter depedecy eve for the real case problems. Keywords Metaheuristic algorithms; evolutioary algorithms; microwave circuits, optimizatio I. INTRODUCTION Solutio of optimizatio problems is a iterestig field of study for various areas such as array atea sythesis []-[3], fiacial aalysis [4], [5], error miimizatio ad game programmig [6], [7], microwave desig [8]-[0] ad data miig [], []. Most of the related algorithms are motivated from the ature ad are aimed to fid ear optimal solutios of give problems [3]-[7]. The performaces of proposed algorithms are usually represeted with their solutio quality ad covergece speeds. Each algorithm has several cotrol parameters which are eeded to be well tued depedig o the optimizatio problem i order to achieve better performace. This ca be cosidered as a vital step i most of the cases ad affects the eploratio ad eploitatio characteristics of the algorithms [8]-[0]. The umber of cotrol parameters ad their adustmet are quite determiistic for the performace of the algorithms. Typically, a algorithm that eeds a few cotrol parameters is assumed as a good choice i solvig a give problem. However, i some cases, eve the fie adustmet of cotrol parameters is ot sufficiet to fid the optimal poits of the problems. I such problems, the eploratio strategies of the algorithms may ot be adequate to coverge to the global optimum of the give problem. Therefore, some modificatios of the algorithms are proposed to solve these specific problems []-[4]. Itroductio of ew optimizers is still a ope area of research, because of the lack of a optimizatio algorithm that performs well i all fields. Some optimizatio algorithms perform well for some problems, while perform iadequate for other problems [5]. I the literature, may differet optimizatio algorithms have bee proposed to icrease the solutio quality for comple optimizatio problems with as little effort as possible. I may microwave desig problems, it is required to deal with some highly oliear obective fuctios with a large umber of variables. I additio, gradiet based algorithms caot yield sufficiet solutios i most of the cases sice the optimizatio parameters i most of the problems are highly coupled with each other. Evolutioary algorithms are widely used whe the aalytical methods are isufficiet to obtai appropriate solutios [6]-[8]. Although Geetic Algorithm (GA) is the first domiat evolutioary algorithm, which was applied o microwave ad electromagetic based problems, Differetial Evolutio (DE) ad Particle Swarm Optimizatio (PSO) algorithms icludig their variats domiate the other evolutioary algorithms i this field [3], [9], [30]. I this paper, a ew evolutioary algorithm called Core Levels Algorithm (COLA) is proposed to solve comple optimizatio problems icludig the desig problems for microwave circuits. COLA uses the similar steps of evolutioary algorithms such as selectio of cadidates, geeratio of offsprigs ad replacemet of the parets with the ew offsprigs which have better fitess values. I additio to that, COLA focuses o two core levels to obtai better offsprig cadidates by their paret vectors. A balace 57 P a g e

2 (IJACSA) Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 8, No. 7, 07 is achieved betwee the core levels to perform iterative eploitatio i ear optimal regios ad to have better eploratio i the search regio. Moreover, the secod core level of the algorithm is desiged i such a way that it performs a eploratio scheme which is cetered o the selected vector ad covers the whole domai of iterest i order to solve differet kid of optimizatio problems without modifyig the algorithm. Optimizatio characteristics of COLA were studied with the use of bechmark fuctios. I this study, the correlatio betwee the umber of selected paret vectors ad the optimizatio capability was observed. Also, a comparative aalysis of COLA with DE, PSO ad Harmoy Search (HS) algorithms was doe for multimodal fuctios to observe the beefits of COLA. Aother goal of the paper was to verify that COLA is applicable to the real case problems. Therefore, two microwave models were selected ad optimized to achieve this goal. The rest of the paper is orgaized as follows: Sectio II itroduces the mai cocept of COLA i detail. Sectio III focuses o bechmark fuctio results, real case microwave problems ad discussios. Lastly, Sectio IV summarizes the obtaied results ad cocludes the paper. II. CORE LEVELS ALGORITHM (COLA) The Core Levels Algorithm implemets a ew method to fid the global optimum of a give fuctio which is maily based o the use of the balace betwee two core levels. This provides good eploratio ad eploitatio characteristics of the algorithm. The pseudo code of COLA is show i Fig. ad the detailed steps of COLA are eplaied i the followig paragraphs: Iitialize the populatio radomly usig uiform distributio i the related domai of the optimizatio problem. The algorithm has a few cotrol parameters to be set which are p; the umber of elemets i the populatio ad k; the umber of elemets to be combied. COLA starts by iitializig radomly the solutio set of the optimizatio problem. The iitializatio ca be defied as follows: i, mi, rad 0,, () ma, mi, Where, i,,,p ad,,, D represets the dimesio of give problem,. Here, D mi, ad ma, represet the lower ad upper bouds for the th variable respectively ad i, is the th compoet of the ith solutio vector. Evaluate the fitess of each elemet i the populatio. Select k umber of solutios accordig to their fitess values. Usig roulette wheel selectio, k umber of solutios are selected radomly accordig to their fitess values.. start. Buildig iitial Populatio (Q) 3. Evaluatio of fitess values (FV) 4. i=0 5. while i < NFE (Number of fuctio evaluatios) do 6. if i mod ==0 7. for z= to k do 8. Select a arbitrary solutio from Q. 9. Multiply the selected solutio by its Core Level fitess ad t the calculate their summatio to obtai the New Cadidate (NC). 0. ed for. NC is obtaied by dividig the summatio value with the total fitess values ad multiplied by t.. else 3. Geerate ew radom solutios. 4. Core Level Combie them with the selected solutio from Q to fid the NC. 5. ed if 6. if FV (NC) > FV (Worst solutio) 7. Replace worst solutio with NC. 8. ed if 9. i++ 0. ed while. Select the solutio that has the best FV value. ed Fig.. Pseudo code of COLA. Obtai a ew cadidate solutio from the selected solutio ad the radomly geerated solutio by usig two core levels. The ew cadidate solutio vector is formed which is a weighted sum of the selected k umber of solutio vectors by core level. New cadidate solutio vector (NC) is calculated usig the followig epressio: ew t fits, t fit S, tk fit k Sk,, () k t fit ew Where, is the th compoet of the ew cadidate t, ad is selected as - or with equal vector, probability, vector ad i fit is the fitess of the pth selected solutio p S p, is the th compoet of the pth selected vector. The ew cadidate formatio usig () is performed oce i every two geeratios of the cadidate vector. Core level is implemeted for the et geeratio of the ew cadidate by usig the followig epressio: S rad ew,, (3) ew where, is the th compoet of the ew cadidate vector, α is a radom umber i the uiformly distributed iterval 58 P a g e

3 (IJACSA) Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 8, No. 7, 07 [0, ], S, is the th compoet of the radomly selected S vector from the populatio ad rad is a radomly obtaied umber i the related domai which is calculated usig (). I the Core level, ew vector ca be obtaied by applyig i three differet methods. ew ) Oly oe of the radomly selected compoets of S is updated. ) The k umber of compoets i the rage of < k < D are updated where the umber of k is selected radomly. 3) All of the compoets of S are updated by their liear combiatio with radom umbers. For each core level applicatio, oly oe of these methods is performed ad this selectio is doe recursively. Replacemet of the ew cadidate solutio with the paret vector, if the fitess value of the ew cadidate is better. The fitess of ew vector is compared with the fitess of the worst paret solutio vector i the populatio. If the fitess of ew vector is better tha fitess of the worst paret solutio vector, the the ew vector replaces with the worst paret solutio vector i order to advace to the et geeratio. This ca be epressed as follows: worst ew if fitess ( ew ) fitess ( worst ). (4) worst otherwise. The steps will be repeated util the stoppig criterio is met. III. EXPERIMENTAL RESULTS AND DISCUSSIONS A. Testig with bechmark fuctios I this sectio, COLA is applied to well-kow bechmark fuctios to demostrate its performace. These fuctios are take from literature ad they have bee widely used for testig of optimizatio problems [3]. The selected bechmark fuctios are show i Table. Amog these fuctios, the first seve fuctios are uimodal fuctios ad the followig five fuctios are multimodal fuctios. For uimodal fuctios, covergece rates are the distiguishig characterictics of the optimizatio algorithms rather the fial results. However, for multimodal fuctios, due to the may optimum poits of problems, the fial result obtaied by algorithm is sigificat. The preseted eperimetal results are average, stadard deviatio ad the best value of the fuctios. All values are gathered over 40 idepedet rus. Average value idicates the solutio quality, stadard deviatio value specifies the stability of the algorithm for udergoig radom operatios ad the best value simply epresses the closest result to the optimal solutio out of 40 idepedet rus. Fuctio ame Sphere TABLE I. THE SELECTED BENCHMARK FUNCTIONS USED IN EXPERIMENTS Fuctio epressio Domai f mi i i. i i.. Rosebrock i i i ma{ i i [-00, 00] 0 i [-0, 0] 0 [-00, 00] 0, i } [-00, 00] 0 00 i i i Step ( i 0.5) i 4 Quartic i i radom, i [-30, 30] 0 [-00, 00] 0 0 [-.8,.8] 0 i si i i [-500, 500] Rastrigi i 0 cos i i 0 [-5., 5.] 0 Ackley 0ep 0. i i ep cos( i ) 0 e i [-3, 3] 0 Griewak Pealized 4000 i 0si i i cos i i y yi i y yi ui,0,00,4 i yi i 4 m k( i a), i a ui, a, k, m 0, a i a m k( i a), i a [-600, 600] 0 0si [-50, 50] 0 Populatio size is fied to 00, dimesio is set to 30 ad the umber of fuctio evaluatios is set to 0000 for all bechmark fuctios. The algorithm cotiues util the stoppig coditio is met. The results obtaied for the listed fuctios above are give i Table. I this table, it is aimed to observe the performace of COLA for differet k parameter values i the set {3, 4, 5} ad a radom selectio of k from the same set for each iteratio P a g e

4 Stadard Deviatio Value Average Value (IJACSA) Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 8, No. 7, 07 TABLE II. THE DESCRIPTIVE VALUES OF BENCHMARK FUNCTIONS BY SELECTED NUMBER OF K PARAMETER AMONG THE PARENT VECTORS 0000 fuctio evaluatios Sphere... Rosebrock Step Quartic Rastrigi Ackley Griewak Pealized k = 3 k = 4 k = 5 k = rad[3-5] Avg E-4 Best.9E-7.9E-7.9E-7.9E-7 Stdev Avg Best Avg -.4E-6 -.4E-6 -.4E-6 -.4E-6 Best -.4E-6 -.4E-6 -.4E-6 -.4E-6 Avg 3.67E-6.58E-6.8E-6.74E-6 Best.36E-.36E-.36E-.36E- Stdev 3.E-06.38E-6.09E E-7 It ca be observed from the results i Table, ecept Rosebrock, Ackley ad Pealized fuctios, all other fuctios are coverged to their optimal values regardless of selectio of k. Moreover, whe the results of Rosebrock ad Pealized fuctios are ivestigated, istead of selectig k parameter as a fied umber, the selectio of it radomly i give iterval for each iteratio decreases the average ad stadard deviatio values. This is the idicatio of better solutio quality ad faster covergece speed of the algorithm uder the radom selectio of k parameter i the give iterval. For these two fuctios, the average ad stadard deviatio are show for differet values of k i Fig. ad 3, respectively. The figures illustrate that selectig parameter k radomly i a give iterval for each iteratio presets better results. It is obvious that COLA performs effectively to reach to the optimum poits of Rosebrock ad fuctios. Sice the cotrol parameter k is radomized i the give iterval, COLA performs efficietly by usig oly a sigle cotrol parameter which is p; the umber of elemets i the populatio Fig.. Average fuctio values obtaied for Rosebrock ad Pealized fuctios for differet values of k parameter Rosebrock Avg Pealized Avg rad[3-5] parameter k Rosebrock Stdev Pealized Stdev rad[3-5] parameter k Fig. 3. Stadard deviatio of fuctio values obtaied for Rosebrock ad Pealized fuctios for differet values of k parameter. I order to verify the advatage of COLA over some effective evolutioary algorithms which are Differetial Evolutio, Particle Swarm Optimizatio ad Harmoy Search algorithms, it is compared for the multimodal bechmark fuctios uder the same coditios. These comparative results are preseted i Table 3. The results for DE, PSO ad HS algorithms are take from the previously reported work [3]. Sice the global optimum poit is located i may local optimum poits, it is quite challegig to reach the global optimum of give multimodal fuctios. It is also kow that the solutio quality of the last value achieved by a algorithm is a distiguishig characteristic especially for multimodal fuctios. Therefore, a compariso aalysis is performed oly for the multimodal fuctios. It ca be see from Table 3 that COLA optimizes the multimodal fuctios with a good covergece rate, while the others are away from the global poits of the fuctios eve after 0000 umber of fuctio evaluatios. B. Microwave Models Importace of optimizatio ad computer desig has bee realized for may years. Oe of the earliest papers o the area of optimizatio methods for microwave circuits was Badler ad Macdoald s work [33], [34]. A classical paper o the aalysis part of microwave circuits i computer aided desig was itroduced by Moaco ad Tiberio [35] P a g e

5 (IJACSA) Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 8, No. 7, 07 TABLE III. COMPARATIVE VALUES OF ALGORITHMS FOR MULTIMODAL FUNCTIONS. Fuctio ame COLA DE PSO HS Rastrigi Ackley Griewak Pealized Avg Best Stdev Avg Best Stdev Avg -.4E Best -.4E Stdev Avg Best Stdev Avg.74E Best.36E Stdev 8.43E I this paper, differet methods used i the aalysis programs of liear circuits i frequecy domai were described. Also, determiatio of sesitivity ad coveiece of usig oe or the other method i relatio to the umber of parameters ad differet aalysis methods were eplaied. Differet methodology which used the combiatio of eperimetal desig ad computer-aided desig was demostrated i [36]. I 00, computer-aided desig summary of works to date is icluded as a survey paper [37]. It is also idicated that there are three essetial reasos for simulatio of radio frequecy ad microwave circuits; to uderstad the physics of a comple system of iteractig elemets, to test ew cocepts ad to optimize the desigs. Over the years may papers for the computer aided desig or optimizatio of microwave circuits ca be foud. Recetly a ew techique for rapid multi-obective optimizatio of the compact microwave passive compoets was preseted [38], [39]. C. Microwave Tapered Matchig Circuit Desig Microwave matchig etworks are importat i the desig of may differet types of microwave circuitry. Oly with proper matchig such a circuit ca attai maimum power trasfer ad elimiate the reflectio. I microwave matchig circuit desig, especially whe oe eeds to match real load impedaces, oe of the most useful etwork is a tapered microwave matchig etwork which ca be cosidered as a series of cascaded quarter wavelegth trasmissio lies. The desig for tapered lies is usually doe by usig computer algorithms for cotiuous sectios [40], [4]. For this kid of structure desig optimizatio usig ature-ispired metaheuristic methods, amely, particle swarm optimizatio, was doe [0]. A eample circuit is show i Fig. 4. Fig. 4. Microwave Tapered matchig etwork. As ca be see from Fig. 4, the tapered lie i this model of study cosists of a series of λ/4 trasmissio lies. This type of cofiguratio ca be used to achieve a match for real loads. I the model,00ω load is matched to a 50Ω lie usig a series of cascaded three trasmissio lies of a quarter wavelegth log. Startig from the load higher impedace trasmissio lies eist ad as movig alog the tapered lie, lower impedace trasmissio lies are obtaied. The obective fuctio of this circuit is give as below: 3 (5) f (,, 3) 50, 00 Where, the values of parameters, which are the characteristic impedaces,, ad 3 must to be foud also with the coditio that 3 < < ad the values are restricted to be i the rage [0-00]. I order to observe the performace of COLA o this microwave circuit, 40 idepedet rus were performed. For each ru, 000 fuctio evaluatios were eecuted to obtai values of, ad 3 that provide optimal value of fuctio (5). From the performed 40 idepedet trials, a sample of five impedece values (Ω) are show i Table 4. The reflectio coefficiet obtaied by usig first ad the secod trial values are plotted i Fig. 5 for the desig frequecy of 5 GHz to demostrate the impedace values. As it ca be see i Fig. 5, reflectio coefficiet values are aroud 35 db ad below at 5 GHz, which idicates a good matchig. This idicates that the values obtaied by COLA are all correct yieldig proper desigs at the ed. TABLE IV. IMPEDENCE VALUES FOR FIVE INDEPENDENT TRIALS (Ω) (Ω) 3 (Ω) Trial Trial Trial Trial Trial P a g e

6 Reflectio Coefficiet (db) (IJACSA) Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 8, No. 7, Frequecy (GHz) Trial Trial Fig. 5. Reflectio coefficiet values for two differet trial sets. Aother aspect that eeds to be aalyzed is the sesitivity aalysis of the obtaied results usig the algorithm. Simple 5% error is itroduced to the obtaied values to see how the desig is affected. I this case it is observed that there is ot a chage i desig operatig frequecy sice the legths are ot differet. The reflectio value is worse, however still lower tha -5 db which yields a reasoable match sice the reflectio coefficiet value is very low. D. Microwave Amplifier Desig Amplificatio is ecessary for most of the electroic circuits ad for microwave circuit systems. Nowadays, with the developmet of trasistor techology, all microwave amplifiers use trasistor devices which are more reliable ad rugged. The mai advatage of usig trasistor devices is that they ca easily be itegrated ito moolithic circuits. Desig of amplifiers i geeral requires the matchig etwork desig for iput ad output parts of the etwork. If the work is doe by had, first the stability of the trasistor is checked ad drawig the Gai Circles ad selectig optimum poits, oe ca perform the desig operatio usig Smith Chart. This process ca be performed by usig metaheuristic algorithms especially the ature-ispired metaheuristic algorithms. Similar works, usig metaheuristic algorithms to solve amplifier desig problems, were performed by the followig researchers for the give specific problems [9], [4]-[46]. Simple two-port microwave etwork which produces amplificatio with a proper desig is show i Fig. 6. I the Fig. 6, there are two matchig ework desigs that should be doe simultaeously to achieve the desired gai. The overall desig also eeds a compromise i gai ad i retur loss at the same time. The desig i this case is the desig of two impedace matchig etworks to achieve the desired gai goal. I other words, the desig requires fidig the proper legths of trasmissio lies d, d, l ad l at the cetral operatio frequecy. The power fuctio that eeds to be optimized is give by the followig epressio: S L S G 6. (6) T S S S S S L S L Fig. 6. Microwave amplifier desig illustratio. This epressio depeds o trasistor s-parameters: S, S, S ad S. The reflectio parameters are Γ S ad Γ L for source ad load respectively. Epressios for Γ S ad Γ L ad,, 3 ad 4 were derived i [9] ad are as follows: S L, 4 4, Re( ) ( ota( d) Im( )), (9) o Im( ) ta( d ) Re( ) ta( d ) o ta( l), (0) ta( l ) Re( 3) ( ota( d) Im( 3)) 4, () o Im( ) ta( d ) Re( ) ta( d ) o ta( l ), () ta( l ) Where, o is the characteristic impedace of the trasmissio lie. I our desig, trasistor FHX35X, maufactured by Fuitsu Cooperatio was used. With the chose s-parameters, the desig was cetered at frequecy of 0 GHz. The desig was optimized to get a gai of 6, which i decibels is db. The characteristics were gathered over 40 idepedet rus. Each ru had 000 fuctio evaluatios. The values for d, d, l ad l were restricted to be rage [0, π]. These values for d, d, l ad l were obtaied i terms of radias ad five of them were tabulated i Table 5. I additio, a microwave simulator was used to obtai characteristics. Fig. 7 demostrates results of Trial for gai ad reflectio values. The plot shows S ad S which are the reflectios i the ports ad, respectively ad S which is the trasmissio from port to. I this case, gai value is S value. I a desig, especially at the desig frequecy S ad S values should be miimized ad if possible S ad S values should be kept below 0 db at all times to avoid oscillatios. It is see that this is roughly happeig all throughout the bad of observatio from 7 GHz to 3 GHz. (7) (8) 6 P a g e

7 Gai ad Reflectio Coefficiet (db) (IJACSA) Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 8, No. 7, 07 TABLE V. A SAMPLE SOLUTION SET FOR DIFFERENT NUMBER OF TRIALS d d l l Trial Trial Trial Trial Trial S S S Fig. 7. Gai ad reflectio coefficiet values of Trial for the amplifier desig. At the desig frequecy S value is -8 db ad S value is - db. The gai value is also at db. A sesitivity test is also performed to see if the values produced by the algorithm are sesitive to errors. 3% error is itroduced to all of the legth values, the overall gai dropped to 0.77 db; however the simulatio idicated that the trasistor is stable. Whe a 5% error is itroduced to all of the legth values, the maimum gai dropped from db to 9.8 db which is ot a very desirable feature. However this kid of result is epected, sice the gai performace i the model relies o correct trasmissio lie legths ad itroducig a 5% error i all legths is actually a sigificat chage i the desig. These results overall idicate that COLA performed efficietly ad it ca be a good cadidate of optimizatio algorithm for desigig a stable amplifer. IV. Frequecy (GHz) CONCLUSIONS I this paper, a ew optimizatio algorithm COLA is proposed to fid the global optimum poits of give problems by providig good solutio quality with robust solutios to the radom operatios i the algorithm. Two core levels are applied simultaeously to reach these goals by providig a balace betwee eploratio ad eploitatio characteristics. The algorithm is tested with differet characteristics of bechmark fuctios, compared with powerful evolutioary algorithms ad the applied to two real microwave circuit desig problems. The results obtaied for bechmark fuctios idicate that COLA provides better solutio quality tha the aalyzed algorithms ad its covergece speed is fairly good eve for the first stages of optimizatio such as 0000 fuctio evaluatios. The results for microwave circuits obtaied by COLA are verified by microwave simulators ad it is see that it produces accurate results. It is studied that the cotrol parameter k is radomized i give iterval produces better results. Therefore, COLA uses oly oe cotrol parameter p which is ot ecessary to be well tued for the problems studied i this paper. The umber of populatio p is fied to 00 for all the problems i this study. I other words, it ca be cosidered that the algorithm is almost parameter free which ca be used for ay problem without tuig ay cotrol parameter. This advatage of the algorithm makes it very practical to be used for ay real case problems. Accordig to these results, it ca be cocluded that the solutio quality of the algorithm is better tha the aalyzed algorithms ad also is quite robust eve though there is o parameter to be tued. The algorithm COLA ca be suggested as a cadidate for optimizatio problems icludig real case problems from differet fields. As future work, a ew research ca be doe to compare COLA with recet hybrid optimizatio algorithms. REFERENCES [] Y. Che, S. Yag ad. Nie, The Applicatio of a modified differetial evolutio strategy to some array patter sythesis problems, IEEE Tras. Ateas ad Propagatio, 56, pp , 008. [] B. Choudhury, S. Maickam ad R.M. Jha, Particle Swarm Optimizatio for Multibad Metamaterial Fractal Atea, J. Optim. 03, pp. -8, 03. [3] E.D. Ulker, A. Haydar ad K. Dimililer, Applicatio of Hybrid Optimizatio Algorithm i the Sythesis of Liear Atea Array, Math. Prob. Eg. 04, pp. -7, 04. [4] P. We-Tsao, A ew Fruitfly Optimizatio Algorithm: Takig a Fiacial Distress model as a Eample, Kowledge-Based Systems, 6, pp , 0. [5] E. Hadavadi, H. Shavadi, A. Ghabari, Itegratio of Geetic Fuzzy Systems ad Artificial Neural Networks for Stock Price Forecastig, Kowledge-based Systems. 3, pp , 00. [6] R. Köker, Geetic Algorithm Approach to a Neural-Network based iverse kiematics solutio of robotic maipulators based o error miimizatio, Spec. Sect. o New Treds i Ambiet Itelligece ad Bio-ispired Syst., pp , 03. [7].W. Geem, Harmoy Search Algorithm for Solvig Sudoku, Kowledge-based Itelliget Iformatio ad Egieerig Systems, Lect. Notes Comput. Sci. 469, pp , 007. [8] E.D. Ulker ad S. Ulker, Applicatio of Particle Swarm Optimizatio to Microwave Tapered Microstrip Lies, It. J. Comput. Sci. Eg (CSEIJ) 4, pp.59-64, 04. [9] E. Deiz, S.Ulker, Cloal selectio algorithm applicatio to simple microwave matchig etwork, Microw. Opt. Techol. Lett. 53, pp , 0. [0] S. Ulker, Broadbad Microwave Amplifier Desig Usig Particle Swarm Optimizatio, J. Comput. 6, pp. 7-76, 0. [] A.A. Freitas, Data miig ad kowledge discovery with evolutioary algorithms. Spriger Sciee ad Busiess Media, 03. [] A. Mukhopadhyay, U. Maulik, S. Badyopadhyay, C.A.C Coello, A survery of multiobective evolutioary algorithms for data miig: Part I, IEEE Tras. o Evol. Comput.. 8, pp. 4-9, 04. [3].W. Geem, J.H. Kim ad G.V. Logaatha, Harmoy Search Optimizatio: Applicatio to Pipe Network Desig, J. Model. Optim., pp , P a g e

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