A COPMARISON OF PARTICLE SWARM OPTIMIZATION AND THE GENETIC ALGORITHM

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1 A COPMARISON OF PARTICLE SWARM OPTIMIZATION AND THE GENETIC ALGORITHM Rana Hassan * Babak Cohanm Olver de Weck Massachusetts Insttute of Technology, Cambrdge, MA, 39 Gerhard Venter Vanderplaats Research and Development, Inc., Colorado Sprngs, CO, 896 Partcle Swarm Optmzaton (PSO) s a relatvely recent heurstc search method whose mechancs are nspred by the swarmng or collaboratve behavor of bologcal populatons. PSO s smlar to the Genetc Algorthm (GA) n the sense that these two evolutonary heurstcs are populaton-based search methods. In other words, PSO and the GA move from a set of ponts (populaton) to another set of ponts n a sngle teraton wth lkely mprovement usng a combnaton of determnstc and probablstc rules. The GA and ts many versons have been popular n academa and the ndustry manly because of ts ntutveness, ease of mplementaton, and the ablty to effectvely solve hghly nonlnear, med nteger optmzaton problems that are typcal of comple engneerng systems. The drawback of the GA s ts epensve computatonal cost. Ths paper attempts to eamne the clam that PSO has the same effectveness (fndng the true global optmal soluton) as the GA but wth sgnfcantly better computatonal effcency (less functon evaluatons) by mplementng statstcal analyss and formal hypothess testng. The performance comparson of the GA and PSO s mplemented usng a set of benchmark test problems as well as two space systems desgn optmzaton problems, namely, telescope array confguraton and spacecraft relablty-based desgn. c c f g = self confdence factor = swarm confdence factor = ftness functon = constrant functon H o = null hypothess H a = alternatve hypothess n = sample sze N feval = number of functon evalutons Nomenclature g p k = poston of the partcle wth best global ftness at current move k p = best poston of partcle n current and all prevous moves r = penalty multpler rand = unformly dstrbuted rand varable between and * Postdoctoral Assocate, Aeronautcs & Astronautcs and Engneerng Systems, [email protected], AIAA Member. Formerly, Graduate Student, Center for Space Research, Aeronautcs & Astronautcs. Currently, Systems Engneer, The Jet Propulson Laboratory, [email protected], AIAA Member. Assstant Professor, Aeronautcs & Astronautcs and Engneerng Systems, [email protected], AIAA Member. Senor R&D Engneer, [email protected], AIAA Member. Copyrght 4 by Rana Hassan, publshed by AIAA, wth permsson.

2 Q sol = soluton qualty s () = estmate of the standard devaton of a populaton from a sample s () = the standard devaton of a sample t = t-statstc uv = poston n the ultra volet plane v k = poston of partcle n the desgn space at tme k w = nerta factor = sample mean = desgn vector k = poston of partcle n the desgn space at tme k = sgnfcance level of a test α β = power of the test µ = mean t φ = tme ncrement = objectve functon Introducton artcle Swarm Optmzaton (PSO) was nvented by Kennedy and Eberhart n the md 99s whle attemptng P to smulate the choreographed, graceful moton of swarms of brds as part of a sococogntve study nvestgatng the noton of collectve ntellgence n bologcal populatons. In PSO, a set of randomly generated solutons (ntal swarm) propagates n the desgn space towards the optmal soluton over a number of teratons (moves) based on large amount of nformaton about the desgn space that s assmlated and shared by all members of the swarm. PSO s nspred by the ablty of flocks of brds, schools of fsh, and herds of anmals to adapt to ther envronment, fnd rch sources of food, and avod predators by mplementng an nformaton sharng approach, hence, developng an evolutonary advantage. References and descrbe a complete chroncle of the development of the PSO algorthm form merely a moton smulator to a heurstc optmzaton approach. The Genetc Algorthm (GA) was ntroduced n the md 97s by John Holland and hs colleagues and students at the Unversty of Mchgan. 3 The GA s nspred by the prncples of genetcs and evoluton, and mmcs the reproducton behavor observed n bologcal populatons. The GA employs the prncpal of survval of the fttest n ts search process to select and generate ndvduals (desgn solutons) that are adapted to ther envronment (desgn objectves/constrants). Therefore, over a number of generatons (teratons), desrable trats (desgn characterstcs) wll evolve and reman n the genome composton of the populaton (set of desgn solutons generated each teraton) over trats wth weaker undesrable characterstcs. The GA s well suted to and has been etensvely appled to solve comple desgn optmzaton problems because t can handle both dscrete and contnuous varables, and nonlnear objectve and constran functons wthout requrng gradent nformaton. The major objectve of ths paper s to compare the computatonal effectveness and effcency of the GA and PSO usng a formal hypothess testng approach. The motvaton s to valdate or refute the wdely speculated hypothess that PSO has the same effectveness as the GA (same rate of success n fndng true global optmal solutons) but wth better computatonal effcency. The results of ths test could prove to be sgnfcant for the future development of PSO. Ths remander of ths paper s organzed n fve major sectons. Frst, the PSO and GA versons that are mplemented n ths comparatve study are summarzed and the hypothess test procedure s formulated. Second, three well-known benchmark problems that are used to compare the performance of the GA and PSO are presented. Thrd, two space systems optmzaton problems that are used to test the performance of both algorthms wth respect to real lfe applcatons are presented. Results and conclusons are presented n the last two sectons. PSO versus GA Partcle Swarm Optmzaton In ths study, the basc PSO algorthm that s descrbed n Reference 4 s mplemented. The basc algorthm s frst descrbed, followed by a dscusson on sde and functonal constrant handlng, and fnally, a dscrete verson of the algorthm s presented. It should be noted that whle the GA s nherently dscrete,.e. t encodes the desgn

3 varables nto bts of s and s, therefore t easly handles dscrete desgn varables, PSO s nherently contnuous and must be modfed to handle dscrete desgn varables. The basc PSO algorthm conssts of three steps, namely, generatng partcles postons and veloctes, velocty update, and fnally, poston update. Here, a partcle refers to a pont n the desgn space that changes ts poston from one move (teraton) to another based on velocty updates. Frst, the postons, k, and veloctes, v k, of the ntal swarm of partcles are randomly generated usng upper and lower bounds on the desgn varables values, mn and ma, as epressed n Equatons and. The postons and veloctes are gven n a vector format wth the superscrpt and subscrpt denotng the th partcle at tme k. In Equatons and, rand s a unformly dstrbuted random varable that can take any value between and. Ths ntalzaton process allows the swarm partcles to be randomly dstrbuted across the desgn space. ( ) = mn + rand ma mn () ( ) mn + rand ma mn poston v = = () t tme The second step s to update the veloctes of all partcles at tme k + usng the partcles objectve or ftness values whch are functons of the partcles current postons n the desgn space at tme k. The ftness functon value g of a partcle determnes whch partcle has the best global value n the current swarm, p k, and also determnes the best poston of each partcle over tme, p,.e. n current and all prevous moves. The velocty update formula uses these two peces of nformaton for each partcle n the swarm along wth the effect of current moton, v k, to provde a search drecton, v k +, for the net teraton. The velocty update formula ncludes some random parameters, represented by the unformly dstrbuted varables, rand, to ensure good coverage of the desgn space and avod entrapment n local optma. The three values that effect the new search drecton, namely, current moton, partcle own memory, and swarm nfluence, are ncorporated va a summaton approach as shown n Equaton 3 wth three weght factors, namely, nerta factor, w, self confdence factor, c, and swarm confdence factor, c, respectvely. velocty of partcle at tme k+ g ( p k ) ( p k k ) v k + = w v k + c rand + c rand (3) t t current moton partcle memory nfluence swarm nfluence nerta factor range:.4 to.4 self confdence range:.5 to swarm confdence range: to.5 The orgnal PSO algorthm uses the values of, and for w, c, and c respectvely, and suggests upper and lower bounds on these values as shown n Equaton 3 above. However, the research presented n ths paper found out that settng the three weght factors w, c, and c at.5,.5, and.5 respectvely provdes the best convergence rate for all test problems consdered. Other combnatons of values usually lead to much slower convergence or sometmes non-convergence at all. The tunng of the PSO algorthm weght factors s a topc that warrants proper nvestgaton but s outsde the scope of ths work. For all the problems nvestgated n ths work, the weght factors use the values of.5,.5 and.5 for w, c, and c respectvely. Poston update s the last step n each teraton. The Poston of each partcle s updated usng ts velocty vector as shown n Equaton 4 and depcted n Fgure. k + = + v t (4) k k+ 3

4 k+ g p k k v k+ current moton nfluence p swarm nfluence partcle memory nfluence v k Fgure. Depcton of the velocty and poston updates n Partcle Swarm Optmzaton. The three steps of velocty update, poston update, and ftness calculatons are repeated untl a desred convergence crteron s met. In the PSO algorthm mplemented n ths study, the stoppng crtera s that the mamum change n best ftness should be smaller than specfed tolerance for a specfed number of moves, S, as shown n Equaton 5. In ths work, S s specfed as ten moves and ε s specfed as -5 for all test problems. g g ( ) f ( p ) ε q =,,..S f p k k q (5) Unlke the GA wth ts bnary encodng, n PSO, the desgn varables can take any values, even outsde ther sde constrants, based on ther current poston n the desgn space and the calculated velocty vector. Ths means that the desgn varables can go outsde ther lower or upper lmts, mn or ma, whch usually happens when the velocty vector grows very rapdly; Ths phenomenon can lead to dvergence. To avod ths problem, n ths study, whenever the desgn varables volate ther upper or lower desgn bounds, they are artfcally brought back to ther nearest sde constrant. Ths approach of handlng sde constrants s recommended by Reference 4 and s beleved to avod velocty eploson. Functonal constrants on the other hand are handled n the same way they are handled n the GA. In ths comparatve study, functonal constrants are handled usng a lnear eteror penalty functon approach as shown n Equaton 6. f N con r,g = ( ) = ( ) + ma[ ( ) ] φ (6) In addton to applyng penalty functons to handle desgns wth volated functonal constrants, n PSO, t s recommended that the velocty vector of a partcle wth volated constrants be reset to zero n the velocty update formula as shown n Equaton 7. 4 Ths s because f a partcle s nfeasble, there s a bg chance that the last search drecton (velocty) was not feasble. Ths approach s mplemented n ths study. g ( p k ) ( p k k ) v k + = c rand + c rand (7) t t 4

5 In many desgn applcatons, especally comple systems, the optmzaton problem statement usually ncludes dscrete desgn varables, such as technology choces. Reference 4 suggests a smple but effectve way to mplement dscrete desgn varables wth PSO, that s to round partcle poston coordnates (desgn varable values) to the nearest ntegers when the desgn varables beng nvestgated are dscrete. Ths s the strategy that s mplemented n ths study. There has been no recommendaton n the lterature regardng swarm sze n PSO. Most researchers use a swarm sze of to 5 but there s no well establshed gudelne. For the purpose of comparng PSO and GA, the swarm sze that wll be used for the PSO runs n all test problems wll be the same as the populaton sze n ther equvalent GA runs and s fed at 4 partcles n the PSO swarm and 4 chromosomes n GA populaton. The Genetc Algorthm The lterature ncludes many versons of the Genetc Algorthm (GA). In ths study, a basc bnary encoded GA 5 wth tournament selecton, unform crossover and low probablty mutaton rate s employed to solve the benchmark problems and the space systems desgn problems. The GA represents the desgn varables of each ndvdual desgn wth bnary strngs of s and s that are referred to as chromosomes. It s mportant to note that the GA works wth a codng for the desgn parameters that allows for a combnaton of dscrete and contnuous parameters n one problem statement. Ths encodng feature also forces the desgn varables to only take values that are wthn ther upper and lower bounds,.e., no solutons wll ever volate the sde constrants and nfeasblty can only occur because of volaton of functonal constrants. Lke PSO, the GA begns ts search from a randomly generated populaton of desgns that evolve over successve generatons (teratons), elmnatng the need for a user-suppled startng pont. To perform ts optmzaton-lke process, the GA employs three operators to propagate ts populaton from one generaton to another. The frst operator s the Selecton operator that mmcs the prncpal of Survval of the Fttest. The second operator s the Crossover operator, whch mmcs matng n bologcal populatons. The crossover operator propagates features of good survvng desgns from the current populaton nto the future populaton, whch wll have better ftness value on average. The last operator s Mutaton, whch promotes dversty n populaton characterstcs. The mutaton operator allows for global search of the desgn space and prevents the algorthm from gettng trapped n local mnma. Many references eplan the detals of the GA mechancs of evoluton; see, for eample, Reference 3. The specfcs of the GA verson mplemented n ths study are partally based on emprcal studes developed n Reference 5, whch suggests the combnaton of tournament selecton wth a 5% unform crossover probablty. For the comparson wth PSO, the populaton sze s kept constant at 4 chromosomes for all problems under consderaton. A small, fed mutaton of.5% s mplemented. The basc GA verson mplemented n ths study uses a lnear eteror penalty functon approach to handle functonal constrants smlar to PSO as shown n Equaton 6 above. The GA also uses the same convergence crtera as PSO as shown n Equaton 5. Comparson Metrcs and Hypothess Testng The objectve of ths research s to statstcally compare the performance of the two heurstc search methods, PSO and the GA, usng a representatve set of test problems that are of dverse propertes. The t-test (hypothess testng) s used to assess and compare the effectveness and effcency of both search algorthms. In hypothess testng, a null hypothess, H o wll be correctly accepted wth a sgnfcance (or confdence) level ( α ) and falsely rejected wth a type I error probablty α. If the null hypothess s false, t wll be correctly rejected wth a power of the test ( β ) and wll be falsely accepted wth a type II error probablty β. The decson optons summary s presented n Table. H a corresponds to an alternatve hypothess that s complmentary to H o. Table. Possble decson outcomes n hypothess testng (adapted from Reference 6). The true stuaton may be Acton H o s true H o s false accept H o, reject H a ( α ) sgnfcance level β [type II error] reject H o, accept H a α [type I error] ( β ) power of test Sum 5

6 The t-test deals wth the estmaton of a true value from a sample and the establshng of confdence ranges wthn whch the true value can sad to le wth a certan probablty ( α ). In hypothess testng, ncreasng the sample sze, n, decreases type I and II error probabltes, α and β, based on the t-dstrbuton. When n s very large, the t-dstrbuton approaches the normal dstrbuton. 7 Hypothess testng nvolves fve steps. Frst, the null hypothess and the alternatve hypothess are defned. The hypotheses could be two sded or sngle sded. For eample, H o : µ = m (wth H a : µ m ) s a two sded hypothess testng whether an unknown populaton mean, µ, s equal to the value m or not. On the other hand, H o : µ > m (wth H a : µ m ) s a sngle sded hypothess. The second step s to decde on desred values for α, β, and n and to fnd out the correspondng t crtcal value for the gven α, β, and n parameters for the specfc type of test under consderaton (sngle or two sded). The t crtcal value could be obtaned from t-dstrbuton tables that are avalable n many basc statstcs tetbook such as Reference 7. The thrd step s to evaluate n random samples and evaluate the sample mean,, an estmate of the standard devaton of a populaton from the sample, s (), and fnally, the sample mean, s (), as shown n Equatons 8 through. n = = (8) n s( ) = n n = = n n (9) s( ) s( ) = () n The fourth step n the hypothess testng process s to calculate the t-value of the null hypothess as shown n Equaton. The formula for calculatng the t-value dffers accordng to the hypothess beng test. The form shown n Equaton s equvalent to the two sded test H o : µ = m (wth H a : µ m ). Reference 7 lsts the most mportant t-value calculaton formulas. m t = () s() The fnal step n the hypothess testng process s to compare the calculated t-value to the tabulated t crtcal α confdence level. value. If t, the null hypothess s then accepted wth ( ) tcrtcal In ths study, two hypotheses are tested. The frst test s related to the effectveness (fndng the true global optmum) of the algorthms and the second s related to the effcency (computatonal cost) of the algorthms. The frst test s the hgh effectveness test. Effectveness s defned as the ablty of the algorthm to repeatedly fnd the known global soluton or arrve at suffcently close solutons when the algorthm s started from many random dfferent ponts n the desgn space. In other words, effectveness s defned as a hgh probablty of fndng a hgh qualty soluton. Here, the qualty of a soluton s measured by how close the soluton s to the known global soluton as shown n Equaton (). soluton - known soluton Q sol = % () known soluton 6

7 The soluton qualty metrc descrbed n Equaton could then be used to synthesze a meanngful hypothess to test the effectveness of the search algorthms, PSO and the GA as shown below. It must be notes that the effectveness test must be carred out for each of the algorthms separately. In other words, ths test measures the effectveness of each algorthm wth respect to known solutons for the test problem rather than comparng the effectveness of the two algorthms to each others. Ths effectveness test wll be carred out for each algorthm wth respect to each of the test problems separately. Effectveness Test Objectve to test whether H a : µ Q > 99% H o : Q t = µ Q sol sol s( Q 99% 99% ) sol sol α, β = %, and n = takng = % Ths s a one sded test of sgnfcance of a mean (table 6., Reference 7) t =. The second hypothess that s tested n ths study, whch s the mportant test from the authors pont of vew, s the computatonal effcency test. Ths test drectly compares the computatonal effort requred by PSO and the GA to solve each of the test problems. The objectve s to support or refute the wdely speculated clam that PSO s sgnfcantly more effcent than the GA. Ths drect comparson requres a t-test called comparson of two means. For a test wth the null hypothess H o : µ µ, the t-value can be calculated as shown n Equaton (3). crtcal t = (3) s( ) n + n where ( n ) s ( ) + ( n ) s ( ) s ( ) = (4) n + n For the computatonal effcency test, the metrc that s mplemented s the number of functon evaluatons, N feval, the algorthm performed untl the convergence crtera (descrbed n Equaton 5) s met. The lower N feval s, the more effcent the algorthm. The effcency test s summarzed as follows. Effcency Test Objectve to test whether H a : PSO µ N eval < GA µ N eval H o : PSO µ N eval GA µ N eval GA µ N feval PSO µ N feval t = where s( ) nga + npso s ( ) = ( nga ) sga + ( npso ) spso nga + npso takng α = %, β = %, and n PSO = nga = Ths s a one sded test of sgnfcance of comparson of two means (table 6., Reference 7) t crtcal =. 5 Benchmark Test Problems In ths secton, a set of well-known optmzaton benchmark test problems are presented. These problems wll be solved by both PSO and the GA, ten tmes each, to carry out the hypothess testng procedure as dscussed n the prevous secton. Ths set of problems ncludes two smple, unconstraned contnuous functons, namely, the Banana 7

8 (Rosenbrock) functon and the Eggcrate functon; each has only two desgn varables. A more comple and hghly constraned benchmark problem, namely, Golnsk s Speed Reducer, s also presented; t has seven desgn varables and functonal constrants. The solutons for these test problems are known and wll be used to evaluate the effectveness of the two algorthms, PSO and the GA. The Banana (Rosenbrock) Functon Ths functon s known as the banana functon because of ts shape; t s descrbed mathematcally n Equaton 5. In ths problem, there are two desgn varables wth lower and upper lmts of [-5, 5]. The Rosenbrock functon has a known global mnmum at [, ] wth an optmal functon value of zero. + Mnmze ( ) ( ) ( ) f = (5) The Eggcrate Functon Ths functon s descrbed mathematcally n Equaton 6. In ths problem, there are two desgn varables wth lower and upper bounds of [-π, π]. The Eggcrate functon has a known global mnmum at [, ] wth an optmal functon value of zero. Mnmze f ( ) = + + 5( sn + ) sn (6) Golnsk s Speed Reducer Ths problem represents the desgn of a smple gearbo such as mght be used n a lght arplane between the engne and propeller to allow each to rotate at ts most effcent speed. The gearbo s depcted n Fgure and ts seven desgn varables are labeled. The objectve s to mnmze the speed reducer s weght whle satsfyng the constrants mposed by gear and shaft desgn practces. Full problem descrpton can be found n Reference 8. A known feasble soluton obtaned by a sequental quadratc programmng (SQP) approach (MATLAB s fmncon) s a kg gearbo wth the followng optmal values for the seven desgn varables: [ ]. Ths s a feasble soluton wth four actve constrants. It s mportant to note that the publshed soluton n Reference 8 s not a feasble soluton as s wdely beleved because t slghtly volates the 5 th, 6 th and th functonal constrants. 9 Fgure. Golnsk s Speed Reducer. Space Systems Problems Ths secton presents two space system desgn problems; the frst problem has a scentfc applcaton whle the second focuses on commercal servces. The two problems presented n ths secton, namely, ground-based rado telescope array desgn, and communcaton satellte desgn, are used as test problems to compare the performance of the two algorthms, PSO and GA, wth respect to real lfe engneerng problems. Array Confguraton Ground-based rado telescope arrays use several dstrbuted small telescopes to acheve the resoluton of a costly sngle large telescope by correlatng data to create a syntheszed beam. A measure of performance for correlaton 8

9 arrays s the degree of coverage n the nstantaneous uv plane. The uv plane s defned by Equaton 7 where and y are the ground postons of the statons measured n a convenent set of unts such as klometers. u j = j, vj = y j y (7) If autocorrelaton ponts are removed (the correlaton of data from one telescope staton wth tself), the number of vsblty ponts, N uv s gven by Equaton 8. ( N ) N uv = N statons statons (8) Each uv pont samples the Fourer transform of the sky brghtness. The more unform the uv coverage s, the lower the undesrable sde lobes n the syntheszed beam of the array., Cornwell proposed a metrc for the uv coverage as the dstance between all uv ponts as shown n Equaton 9. Ths metrc s mplemented as the an objectve functon n the Mamze uv = ( uj ukl ) + ( vj vkl ) coverage (9), j, k, l Another objectve functon that s usually consdered s the mnmzaton of the dstance between the dfferent statons as these dstances represent the cost of the fber optc cable needed to connect the statons. These are two opposng objectves because mamzng the uv coverage requres the statons to be placed as far as possble from each other. In ths case study, only the uv coverage mamzaton metrc s consdered. Regardless of the number of statons, the soluton of ths sngle objectve, unconstraned problem s known to place all avalable statons on a permeter of a crcle. The desgn varables then become the angles that descrbe the locaton the staton on the crcle. Consequently, the number of desgn varables s equal to the number of statons n the telescope array to be desgned. In ths research, four cases are nvestgated; these are: fve, s, seven and eght staton telescope arrays. The solutons for these cases are gven n Reference, and were verfed usng an SQP approach (MATLAB s fmncon) for use n the PSO and GA effectveness tests. These known solutons are shown n Fgure 3 below. 4 N statons = 5, uv coverage = 48.9 N statons = 6, uv coverage = y v y v u u N statons = 7, uv coverage = 337. N statons = 8, uv coverage = y v y v u u Fgure 3. Optmal placement of telescope array statons and the coresspondng uv coverage. 9

10 Communcaton Satellte Relablty-Based Desgn Ths problem nvolves desgnng the payload and bus subsystems of a commercal communcaton Geosynchronous satellte wth gven payload requrements. The desgn objectve s to mnmze the spacecraft overall launch mass, whch s a surrogate for cost, gven desgn constrants on payload as well as overall system relablty. The problem also nvolves geometrcal constrants mposed by the choce of the launch vehcle. The problem ncludes s functonal constrants and 7 dscrete desgn varables representng the technology choces and redundancy levels of the satellte payload and bus subsystems. For a complete descrpton of the problem, see References 3, 4, and 5. The best soluton found for ths dscrete problem s satellte wth a an optmal mass of kg. The above references descrbe the technology choces and redundancy levels of the dfferent subsystems n ths optmal spacecraft desgn. Results and Dscussons Two performance tests were carred out for both algorthms under consderaton, namely, PSO and the GA. The performance tests were carred out for eght sample problems: the Banana functon, the Eggcrate functon, Golnsk s Speed Reducer, Relablty-based Satellte Desgn, 5-staton, 6-staton, 7-staton and 8-staton Rado Array Confguraton. The frst test s the effectveness test, whch measures the qualty of the solutons found by the heurstc algorthm wth respect to known solutons for the test problems. Ths test nvestgates whether the qualty of the solutons obtaned s greater than 99%. The second test s the effcency test, whch nvestgates whether the computatonal effort of PSO s less than that of the GA for the sample problem set usng the same convergence crtera. These statements of the two tests were set as the alternatve hypotheses whle ther complmentary statements were set as the null hypotheses, whch s a tradtonal hypothess testng approach. Both tests were conducted usng acceptable Type I and Type II errors of % each. The t-statstc values that were calculated for the PSO runs and GA runs for all eght sample problems are summarzed n Table below. The sample problems below are organzed n order of ncreasng number of desgn varables, a measure of complety. Table : Calculated t-values for the effectveness and effcency hypotheses tests Effectveness Test, t crtcal =. Calculated t-value PSO GA Effcency Test, t crtcal =.5 Calculated t-value - Banana Functon Eggcrate Functon Array (5 Statons) Array (6 Statons) Array (7 Statons) Array (8 Statons) Golnsk s Speed Reducer Satellte Desgn The effectveness tests for PSO and the GA show that t > tcrtcal n most sample test problems. Ths leads to the rejecton of the null hypothess and the acceptance of the alternatve hypothess, that s the qualty of the solutons of both approaches s equal to or greater than 99% n most of the test problems. Ths alternatve hypothess s accepted wth a confdence level of 99%. The nfnty t-values n the telescope array problems are obtaned because n each of the runs, both PSO and the GA consstently found the known solutons for the 5, 6, 7, and 8 statons telescope array problems. Therefore, the qualty of the ten solutons n each case s % and the standard devaton s zero. In only one case, the Banana functon, the null hypothess s accepted for the GA runs, therefore t cannot be proved that the GA solutons have hgh qualty for ths problem. By further nvestgatng the data, t s found that mean qualty of the GA solutons for ths problem s 9% wth a standard devaton of 9.5%. Ths mean s much lower than the 99% qualty level mplemented n the test. It s surprsng that the GA could fnd hgh qualty solutons for the rest of the test problems wth contnuous desgn varables and wth larger number of desgn

11 varables (problems to 7) but were not able to obtan the same qualty for the Banana problem. A larger sample sze mght lead to mprovng the mean and standard devaton of GA soluton qualty for the Banana problem. PSO solutons for all test problems eceeded the 99% qualty lmts ecept for the last two problems. Further nvestgaton of the data reveals that the mean qualty of the PSO solutons for Golnsk s speed reducer s 98.8% wth a standard devaton of.7%. The scatter n the data of the sample does not allow for the qualty lmt of 99% to be met wth suffcent confdence. As for the satellte desgn problem, t s epected that PSO does not perform well on ths problem gven the fact that all the desgn varables are dscrete and that PSO s nherently contnuous. Fgure 4 below summarzed the values mplemented n the effectveness test for both PSO and the GA. Soluton Qualty % 95% 9% 85% PSO mean +/- standard devaton GA mean +/- standard devaton 8% Banana Functon Eggcrate Functon Array (5) Array (6) Array (7) Array (8) Golnsk's Speed Reducer Satellte Desgn Fgure 4. The mean qualty of the solutons obtaned by PSO and the GA usng run samples for eght test problems. The effcency test shows t > tcrtcal for all test problems ecept for the Banana functon. These results lead to the rejecton of the null hypothess and the acceptance of the alternatve hypothess wth a confdence level of 99%. The nterpretaton of these results s that for all sample problems, ecept for the Banana functon, the computatonal effort requred by PSO to converge to a soluton s less than that of the GA 99% of the tme usng the same convergence crtera. From Fgure 5, t appears that the scatter of the GA runs for the Banana functon s the reason behnd the acceptance of the null hypothess that PSO does not outperform the GA for the Banana functon as t dd n the rest of the test problems. No. of Functon Evaluatons Banana Functon PSO mean +/- standard devaton GA mean +/- standard devaton Eggcrate Functon Array (5) Array (6) Array (7) Array (8) Golnsk's Speed Reducer Satellte Desgn Fgure 5. The mean computatonal effort of the solutons obtaned by PSO and the GA usng run samples for eght test problems.

12 Fgure 6 below couples the results of the effectveness and effcency tests. It shows the ratos of the mean values of the soluton qualtes and computatonal effort of the GA and PSO. The nsght learned from Fgure 6 s that for a range of test problems wth varable complety, PSO and the GA were able to arrve at solutons wth the same qualty, whch s ndcated by the flat lne of a rato of for the means of the soluton qualtes. However, the varyng ratos of the computatonal effort provdes two conclusons: frst, PSO does offer a less epensve approach than the GA n general, second, the epected computatonal savngs offered by PSO over the GA s problem dependent. It appears that PSO offers more computatonal savngs for unconstraned nonlnear problems wth contnuous desgn varables whereas the computatonal savngs s lower for constraned and med nteger nonnear problems. 7 Performance Rato (GA/PSO) Rato of means of No. of functon evalutons Rato of means of qualty of solutons Banana Functon Eggcrate Functon Array (5) Array (6) Array (7) Array (8) Golnsk's Speed Reducer Satellte Desgn Fgure 6. Comparson of ratos of soluton qualty and computatonal effort of PSO and the GA. Conclusons Partcle Swarm Optmzaton (PSO) s a relatvely recent heurstc search method that s based on the dea of collaboratve behavor and swarmng n bologcal populatons. PSO s smlar to the Genetc Algorthm (GA) n the sense that they are both populaton-based search approaches and that they both depend on nformaton sharng among ther populaton members to enhance ther search processes usng a combnaton of determnstc and probablstc rules. Conversely, the GA s a well establshed algorthm wth many versons and many applcatons. The objectve of ths research s to test the hypothess that states that although PSO and the GA on average yeld the same effectveness (soluton qualty), PSO s more computatonally effcent (uses less number of functon evaluatons) than the GA. To nvestgate ths clam, two statstcal tests were set to eamne the two elements of ths clam, equal effectveness but superor effcency for PSO over the GA. To carry out the t-tests, eght sample test problems were solved usng both PSO and the GA over multple runs. The test problems ncludes three well-known benchmark test problems; these are: the Banana (Rosenbrock functon, the Eggcrate Functon, and Golnsk s Speed Reducer. Two space systems desgn problems were also nvestgated to test the algorthms on real-lfe engneerng problems. The frst problem s the confguraton of a ground-based multstaton rado telescope array. Four versons of ths problem were nvestgated ncludng 5, 6, 7 and 8-staton arrays. The second test problem nvolves the relablty-based desgn of a commercal communcaton satellte. All test problems nvolve contnuous desgn varables only ecept for the satellte desgn problem whch contans dscrete desgn varables representng subsystems technology choces and redundancy levels. The eght test problems represent a wde range of complety, nonlnearty, and constrant levels. Two metrcs were dentfed for the two t-tests. The effectveness test for both PSO and the GA uses a qualty of soluton metrc that measures the normalzed dfference between the solutons obtaned by the heurstc approaches and known solutons of the test problems. The effcency test uses the number of functon evaluatons needed by the heurstc approaches to reach convergence. The same convergence crteron s enforced on PSO and the GA. The results of the t-tests support the hypothess that whle both PSO and the GA obtan hgh qualty solutons, wth qualty ndces of 99% or more wth a 99% confdence level for most test problems, the computatonal effort

13 requred by PSO to arrve to such hgh qualty solutons s less than the effort requred to arrve at the same hgh qualty solutons by the GA. The results further show the computatonal effcency superorty of PSO over the GA s statcally proven wth a 99% confdence level n 7 out of the 8 test problems nvestgated. Further analyss shows that the dfference n computatonal effort between PSO and the GA s problem dependent. It appears that PSO outperforms the GA wth a larger dfferental n computatonal effcency when used to solve unconstraned nonlnear problems wth contnuous desgn varables and less effcency dfferental when appled to constraned nonlnear problems wth contnuous or dscrete desgn varables. The authors encourage the readers to nvestgate the results presented n ths paper to help set the agenda for further PSO development efforts. References Kennedy, J. and Eberhart, R., Partcle Swarm Optmzaton, Proceedngs of the IEEE Internatonal Conference on Neural Networks, Perth, Australa 995, pp Kennedy, J. and Eberhart, R., Swarm Intellgence, Academc Press, st ed., San Dego, CA,. 3 Goldberg, D., Genetc Algorthms n Search, Optmzaton and Machne Learnng, Addson-Wesley, Readng, MA, 989, pp Venter, G. and Sobesk, J., Partcle Swarm Optmzaton, AIAA -35, 43rd AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamcs, and Materals Conference, Denver, CO., Aprl. 5 Wllams, E. A., and Crossley, W. A., Emprcally-Derved Populaton Sze and Mutaton Rate Gudelnes for a Genetc Algorthm wth Unform Crossover, Soft Computng n Engneerng Desgn and Manufacturng, P. K. Chawdhry, R. Roy and R. K. Pant (edtors), Sprnger-Verlag, 998, pp Harnett, D., Statstcal Methods, 3 rd ed., Addson-Wesley, 98, pp Volk, W., Appled Statstcs for Engneers, nd ed., McGraw-Hll, 969, pp Web ste cted August 6th, 4. 9 Ray, T., Golnsk s Speed Reducer Problem Revsted, the AIAA Journal, Vol. 4, No. 3, 3, pp Thompson, A., Moran, J., and Swenson, G., Interferometery and Synthess n Rado Astronomy, Wley Interscence, New York, 986. Cornwell, T., Novel Prncple for Optmzaton of the Instantaneous Fourer Plane Coverage of Correlaton Arrays, IEEE Transactons on Antennas and Propagaton, AP, Vol. 36, 988, pp. 65. Keto, E., Shapes of Cross-Correlaton Interferometers, APJ, Vol. 475, 997, pp Hassan, R., Genetc Algorthm Approaches for Conceptual Desgn of Spacecraft Systems Includng Multobjectve Optmzaton and Desgn under Uncertanty, doctoral thess, Purdue Unversty, May 4. 4 Hassan, R., and Crossley, W., Multobjectve Optmzaton of Communcaton Satelltes wth a Two-Branch Tournament Genetc Algorthm, Journal of Spacecraft & Rockets, Vol. 4, No., 3, pp Hassan, R., and Crossley, W., Varable Populaton-Based Samplng for Probablstc Desgn Optmzaton wth a Genetc Algorthm, AIAA-4-45, 4nd Aerospace Scences Meetng, Reno, NV, January 4. 3