Artificial Bee Colony (ABC), Harmony Search and Bees Algorithms on Numerical Optimization

Transcription

1 Atificial Bee Coloy (ABC), Hamoy Seach ad Bees Algoithms o Numeical Optimizatio. Kaaboga, B. Akay Eciyes Uivesity, The ept. of Compute Egieeig, 38039, Melikgazi, Kaysei, Tukiye Abstact I this pape, pefomaces of basic Atificial Bee Coloy, Hamoy Seach ad Bees algoithms ad Impoved Bees algoithm wee compaed o uimodal ad multimodal well-kow bechmak poblems. Effect of poblem dimesioality o the pefomace of the algoithms was ivestigated ad the algoithms wee compaed to each othe i tems of the umbe of cotol paametes to be tued. Keywods: Atificial Bee Coloy, Hamoy Seach. Itoductio Natue is ispiig eseaches to develop models fo solvig thei poblems. Optimizatio is a istace field i which these models ae fequetly developed ad applied. Geetic algoithm simulatig atual selectio ad geetic opeatos, Paticle Swam Optimizatio algoithm simulatig flock of bids ad school of fishes, Atificial Immue System simulatig the cell masses of immue system, At Coloy Optimizatio algoithm simulatig foagig behaviou of ats ad Atificial Bee Coloy algoithm simulatig foagig behaviou of hoeybees ae typical eamples of atue ispied optimizatio algoithms. Aothe algoithm simulatig the foagig behaviou of bees by a diffeet algoithmic model is Bees Algoithm (BA) Hamoy Seach is also a ecet optimizatio algoithm simulatig impovisatio pocess of musicias. Although thee ae may algoithms, most of these algoithms eed fie tuig of cotol paametes, ae successful oly a class of poblems. Futhemoe, o a kid of poblem, although it shows supeio pefomace o low dimesios, the algoithm ca ot peseve its supeio pefomace o high dimesioal case. I this wok, the pefomaces of Atificial Bee Coloy, Hamoy Seach ad Bees algoithms wee compaed to each othe o vayig dimesios fom low to high. Pape is ogaized as follows, i cosequet sectios Atificial Bee Coloy Algoithm, Hamoy Seach ad Bees algoithm ae aticulated, i Sectio 5, epeimetal esults ae peseted ad fially the pape is cocluded... Atificial Bee Coloy Algoithm Atificial Bee Coloy(ABC) algoithm, poposed by Kaaboga i 005 fo eal paamete optimizatio, is a ecetly itoduced optimizatio algoithm ad simulates the foagig behaviou of bee coloy [] fo ucostaied optimizatio poblems [] [5]. Fo solvig costaied optimizatio poblems, a costait hadlig method was icopoated with the algoithm [6]. I a eal bee coloy, thee ae some tasks pefomed by specialized idividuals. These specialized bees ty to maimize the ecta amout stoed i the hive by pefomig efficiet divisio of labou ad self-ogaizatio. The miimal model of swam-itelliget foage selectio i a hoey bee coloy, that ABC algoithm adopts, cosists of thee kids of bees: employed bees, olooke bees, ad scout bees. Half of the coloy compises employed bees ad the othe half icludes the olooke bees. Employed bees ae esposible fom eploitig the ecta souces eploed befoe ad givig ifomatio to the othe waitig bees (olooke bees) i the hive about the quality of the food souce site which they ae eploitig. Olooke bees wait i the hive ad decide a food souce to eploit depedig o the ifomatio shaed by the employed bees. Scouts adomly seach the eviomet i ode to fid a ew food souce depedig o a iteal motivatio o possible eteal clues o adomly. Mai steps of the ABC algoithm simulatig these behaviou ae give below:

2 : Iitialize the food souce positios. : Each employed bee poduces a ew food souce i he food souce site ad eploits the bette souce. 3: Each olooke bee selects a souce depedig o the quality of he solutio, poduces a ew food souce i selected food souce site ad eploits the bette souce. 4: etemie the souce to be abadoed ad allocate its employed bee as scout fo seachig ew food souces. 5: Memoize the best food souce foud so fa. 6: Repeat steps -5 util the stoppig citeio is met. I fist step of the algoithm, i (i =,..., SN) solutios ae adomly poduced i the age of paametes whee SN is the umbe of the food souces. I the secod step of the algoithm, fo each employed bee, whose total umbe equals to the half of the umbe of food souces, a ew souce is poduced by (): υ = φ ( - ) () i i + i i k whee φ i is a uifomly distibuted eal adom umbe withi the age [-,], k is the ide of the solutio chose adomly fom the coloy (k = it(ad SN) + ), =,..., ad is the dimesio of the poblem. Afte poducig υi, this ew solutio is compaed to i solutio ad the employed bee eploits the bette souce. I the thid step of the algoithm, a olooke bee chooses a food souce with the pobability () ad poduces a ew souce i selected food souce site by (). As fo employed bee, the bette souce is decided to be eploited. fiti pi = SN () fit = whee fit i is the fitess of the solutio i. Afte all olookes ae distibuted to the souces, souces ae checked whethe they ae to be abadoed. If the umbe of cycles that a souce ca ot be impoved is geate tha a pedetemied limit, the souce is cosideed to be ehausted. The employed bee associated with the ehausted souce becomes a scout ad makes a adom seach i poblem domai by (3). mi ma mi = + ( ) ad (3) i * 3. Hamoy Seach Hamoy seach (HS) is a ecet metaheuistic that simulates the impovisatio pocess of musicias. Solutios of the optimizatio pocess coespod to musicias ad the hamoy of the otes geeated by a musicia coespods to the fitess of the solutio [7]. Mai steps of the algoithm ae give below: : Iitialize the poblem ad algoithm paametes. : Iitialize the hamoy memoy. 3: Impovise a ew hamoy. 4: Update the hamoy memoy. 5: Repeat steps 3-4 util the stoppig citeio is met I HS algoithm, i each impovisatio, a ew hamoy ( ) is poduced ad this ew hamoy competes with the wost hamoy of the memoy. I impovisatio phase is a ew hamoy is poduced by (4) fo each compoet ( =,...,) p pa if ( ad { } else = = p hmc mi if ( ad { } phcm) + ( ma it (ad*sn) +, = ± mi < ppa) )* ad bw * ad is the ate of choosig fom memoy, is the pitch adustmet ate, bw is the distace badwidth which detemies the amout of maimum chage fo pitch adustmet. 4. Bees Algoithm I Bees algoithm (BA) [8], afte iitializig the populatio of bees by adom sites, bees ae aked accodig to the fitesses of thei sites. Fist e of the sites ae classified as elite sites ad the ep umbe of bees ae set to these elite sites. These bees poduce ew sites i the eighbouhood of the elite sites. Afte sedig bees to the elite sites, eighbouhood of the followig sites (selected sites, m-e) depedig o thei fitesses ae seached. The umbe of bees set to the selected sites is sp. Remaiig bees (-m) ae classified as scout ad allowed to coduct adom seach. Totally, the algoithm equies seve cotol paametes icludig the umbe of scout bees (), umbe of sites selected out of visited sites (m), umbe of best sites out of m selected sites (e), umbe of bees ecuited fo best e sites (ep), umbe of bees ecuited fo the othe (m-e) selected sites (sp), iitial size of patches (gh) ad maimum iteatio umbe. Mai steps of the algoithm ae give below [8]: (4)

3 : Iitialise populatio with adom solutios. : Evaluate fitess of the populatio. 3: Repeat 4: Select sites fo eighbouhood seach. 5: Recuit bees fo selected sites (moe bees fo best e sites) ad evaluate fitesses. 6: Select the fittest bee fom each patch. 7: Assig emaiig bees to seach adomly ad evaluate thei fitesses. 8: Util stoppig citeio is met. BA was impoved by usig shikig pocess. I this pape, this model is called Impoved Bees Algoithm (IBA). I IBA, the patch size (gh) is gadually educed by dividig it by a shikig costat at each iteatio. Theefoe, a ew cotol paamete is itoduced to the algoithm, called shikig costat. 5. Epeimets I the epeimets, esults of basic HS, BA ad ABC algoithms ae compaed to each othe o uimodal (Sphee, Rosebock) ad multimodal (Giewak,Rastigi, Ackley) fuctios with diffeet dimesios. Beside, the esults of IBA ae povided. Fo HS algoithm, paametes ae set to the values ecommeded i [9]. Size of the hamoy memoy was 5, p hmc = 0.9, p pa = 0.3, bw = 0.0 ad the maimum impovisatio umbe was fo all test poblems. Fo BA, =40, e=5, m=5, ep=5, sp=5, gh={0.5,3} ad maimum iteatio umbe was 5. BA was implemeted based o the MATLAB code give i the site dedicated to the BA. Theefoe, maimum evaluatio umbe was fo all fuctios. Fo IBA, =40, e=, m=3, ep=50, sp=0, maimum iteatio umbe was 468, iitial gh was 6, ad shikig costat (sc) was.05. Fo ABC algoithm, umbe of food souces was 0 ad maimum cycle umbe was 500 i ode to equalize the maimum evaluatio umbe to as fo HS ad BA. Limit cotol paamete was 00 fo all poblems. Each of the epeimets was epeated 30 times with diffeet adom seeds, ad the mea fuctio values of the best solutios ad the stadad deviatios thoughout the optimizatio us was ecoded ude diffeet dimesios. I the epeimets, mea ad stadad deviatios of 30 us with diffeet adom seeds fo 5, 0, 30, 50 ad 00 dimesios wee epoted o Table II fo Sphee fuctio, o Table III fo Rosebock fuctio, o Table IV fo Rastigi fuctio, o Table V fo Giewak fuctio ad o Table VI fo Ackley fuctio. Fom the tables, it ca be see that ABC algoithm outpefoms HS, BA ad IBA o all fuctios ad o all dimesios. Oly o uimodal Sphee fuctio ABC algoithm ad IBA poduced simila esults. Results epoted o tables ae plotted o Figues (a)-(e). Both fom tables ad figues, it ca be said that the pefomace of HS, BA ad IBA algoithms ae moe sesitive to the icemet i poblem dimesios as compaed to ABC algoithm. ABC algoithm poduced easoable esults eve fo high dimesios fo both uimodal ad multimodal fuctios. Aothe issue that is a dawback of HS, BA ad IBA is that they employ much moe cotol paametes tha ABC algoithm does. Ecept fo commo cotol paametes (populatio size ad maimum iteatio umbe), ABC algoithm has oly oe cotol paamete, limit, while HS algoithm has thee cotol paametes: p hmc, p pa, bw ; BA has five cotol paametes: e, m, ep, sp, gh ad IBA has si cotol paametes icludig the paametes of BA ad a shikig costat. The pefomaces of BA ad IBA ae highly depedet to the cotol paametes. We have tested the BA fo gh=0.5 ad 3. The esults wee sigificatly diffeet fom each othe as ca be see fom the tables ad figues. Especially o Sphee ad Rosebock fuctios, diffeet gh values ifluece the supeioess of the BA ove HS. IBA has bette pefomace with espect to basic BA. 6. Coclusio I this wok, we ivestigated the pefomaces of Hamoy Seach, Bees algoithms ad Atificial Bee Coloy algoithm. Fom the esults, Atificial Bee Coloy algoithm is supeio ove Bees algoithms ad Hamoy Seach. Atificial Bee Coloy algoithm peseves poducig easoable esults eve fo high dimesios. Atificial Bee Coloy algoithm employs less cotol paamete to be tued with espect to Bees Algoithms ad Hamoy Seach. Refeeces []. Kaaboga. A idea based o hoey bee swam fo umeical optimizatio. Techical Repot TR06, Eciyes Uivesity, Egieeig Faculty, Compute Egieeig epatmet, 005. [] B. Bastuk ad. Kaaboga. A atificial bee coloy (abc) algoithm fo umeic fuctio optimizatio. I IEEE Swam Itelligece Symposium 006, Idiaapolis, Idiaa, USA, May 006. [3]. Kaaboga ad B. Bastuk. A poweful ad efficiet algoithm fo umeical fuctio optimizatio: Atificial bee coloy (abc) algoithm. Joual of Global Optimizatio, 39(3):459 47, 007. [4]. Kaaboga ad B. Bastuk. O the pefomace of atificial bee coloy (abc) algoithm. Applied Soft Computig, 8(): , 008. [5]. Kaaboga ad B. Akay. A compaative study of atificial bee coloy algoithm. Applied Mathematics ad Computatio, I Pess, 009. [6]. Kaaboga ad B. Bastuk. Advaces i Soft Computig: Foudatios of Fuzzy Logic ad Soft Computig, volume 459/007 of LNCS, chapte Atificial Bee Coloy (ABC) Optimizatio Algoithm fo Solvig Costaied Optimizatio Poblems, pages Spige-Velag, 007. [7] K.S. Lee ad Z.W. Geem. A ew meta-heuistic

4 algoithm fo cotiues egieeig optimizatio: hamoy seach theoy ad pactice. Comput. Meth. Appl. Mech. Eg, 94: , 004. [8] Pham.T., Ghabazadeh A., Koc E., Oti S., Rahim S., ad M.Zaidi "The Bees Algoithm - A Novel Tool fo Comple Optimisatio Poblems"", Poceedigs of IPROMS 006 Cofeece, pp [9] M.G. H. Oma ad M. Mahdavi. Global-best hamoy seach. Applied Mathematics ad Computatio, 98(): , 008 Table :Test fuctios used i the epeimets Fuctio Fomulae Rage Miimum Value Sphee ( f 0) = Rosebock Giewak Rastigi Ackley f ( [-00,00] 0 ) = f ( ) = i [00( i+ i ) + ( i ) ] [-30,30] f ( ) = 0 i f ( [-600,600] f ( 0) = 0 ) = i cos i [-5.,5.] f ( 0) = 0 f ( ) = [ i 0 cos(π i ) + 0] f ( ) = 0 ep i 0. i + + e ep cos(π ) 0 [-3,3] f ( 0) = 0 Table : Mea ad stadad deviatios of the esults obtaied by HS, BA, IBA ad ABC algoithms fo Sphee fuctio, :imesio of the poblem. HS Mea 3.0E E Std.89E E BA(gh=0.5) Mea Std BA (gh=3) Mea Std IBA Mea 3.95E E-7.9E E-6.45E-5 Std.4E-7.30E E-7.07E-6.63E-6 ABC Mea 4.30E E E-6.9E-5.99E-6 Std.07E E-7.07E E-6.6E-6 Table 3: Mea ad stadad deviatios of the esults obtaied by HS, BA, IBA ad ABC algoithms fo Rosebock fuctio, :imesio of the poblem. HS Mea Std BA(gh=0.5) Mea Std BA (gh=3) Mea Std IBA Mea Std ABC Mea Std Table 4: Mea ad stadad deviatios of the esults obtaied by HS, BA, IBA ad ABC algoithms fo Rastigi fuctio, :imesio of the poblem.

5 HS Mea 6.07E-8.05E Std 5.5E-8 5.3E BA(gh=0.5) Mea Std BA(gh=3) Mea Std IBA Mea Std ABC Mea 4.34E E E Std.0E-7.98E Table 5: Mea ad stadad deviatios of the esults obtaied by HS, BA, IBA ad ABC algoithms fo Giewak fuctio, :imesio of the poblem. HS Mea Std BA(gh=0.5) Mea Std BA(gh=3) Mea Std IBA Mea Std ABC Mea 4.04E E-7 5.8E Std.E E-7 3.3E Table 6: Mea ad stadad deviatios of the esults obtaied by HS, BA, IBA ad ABC algoithms fo Ackley fuctio, :imesio of the poblem. HS Mea.68E Std.4E E BA(gh=0.5) Mea Std BA(gh=3) Mea Std IBA Mea 6.35E Std 9.77E ABC Mea 9.64E-7 3.5E E E Std 5.4E-7 6.3E-7 3.6E E

6 (a) Sphee (b) Rosebock (c) Rastigi (d) Giewak (e) Ackley Fig.. Results fo fuctios fo diffeet dimesios