A New Artificial Fish Swarm Algorithm for Dynamic Optimization Problems

Transcription

1 WCCI 2012 IEEE World Congress on Computatonal Intellgence June, 10-15, Brsbane, Australa IEEE CEC A New Artfcal Fsh Swarm Algorthm for Dynamc Optmzaton Problems 1 Danal Yazdan Department of Electrcal, Computer and I Engneerng, Qazvn Branch, Islamc Azad Unversty, Iran d_yazdan@qau.ac.r 3 Babak Nasr Department of Electrcal, Computer and I Engneerng, Qazvn Branch, Islamc Azad Unversty, Iran nasr.babak@qau.ac.r 2 Mohammad Reza Akbarzadeh- otonch Center of Excellence on Soft Computng and Intellgent Informaton Processng, Ferdows Unversty of Mashhad, Iran akbarzadeh@eee.org 4 Mohammad Reza Meybod Department of Computer Engneerng and Informaton echnology, Amrkabr Unversty of echnology, ehran, Iran. mmeybod@aut.ac.r Abstract Artfcal fsh swarm algorthm s one of the swarm ntellgence algorthms whch performs based on populaton and stochastc search contrbuted to solve optmzaton problems. hs algorthm has been appled n varous applcatons e.g. data clusterng, neural networks learnng, nonlnear functon optmzaton, etc. Several problems n real world are dynamc and uncertan, whch could not be solved n a smlar manner of statc problems. In ths paper, for the frst tme, a modfed artfcal fsh swarm algorthm s proposed n consderaton of dynamc envronments optmzaton. he results of the proposed approach were evaluated usng movng peak benchmarks, whch are known as the best metrc for evaluatng dynamc envronments, and also were compared wth results of several state-of-the-art approaches. he expermental results show that the performance of the proposed method outperforms that of other algorthms n ths doman. Keywords- dynamc optmzaton problems; artfcal fsh swarm algorthm; movng peaks benchmark; dynamc envronments. I. INRODUCION Artfcal fsh swarm algorthm (AFSA) s one of the algorthms nspred both from the nature and swarm ntellgence algorthms. AFSA was presented by L ao Le n 2002[1]. hs algorthm s an approach based on swarm behavors that was nspred from socal behavors of fsh swarm n the nature. hs algorthm has some characterstcs such as hgh convergence rate, nsensblty to ntal values, flexblty and hgh fault tolerance. AFSA has been utlzed n optmzaton applcatons e.g. PID controller parameters settng[2] mult-objectve optmzaton[3], global optmzaton[4], neural network learnng[5], data clusterng[6], color quantzaton[7] and etc. In many real world problems, uncertanty s obvous and clear. Swarm ntellgence and evolutonary algorthms could be contrbuted to solve these knds of problems. Uncertan problems could be dvded nto four categores: exstence of nose n evaluaton functon, dsturbance n desgn varables, approxmaton n ftness functon and tme-dependent ftness functon. In ths paper, tme-dependent ftness functon has been consdered, whch belongs to the most general types of uncertantes. Up to now, dfferent methods have been presented for solvng dynamc problems by means of evolutonary processng[8] and swarm ntellgence methods[9]. hree man challenges of evolutonary and swarm ntellgence algorthms that cause nablty n drect use of these methods for optmzaton n dynamc envronments are memory lmtatons, losng dversty and exstng several potental optmums n the problem space. When the envronment s changed, current obtaned solutons n memory are not vald anymore. hus t should be ether forgotten completely or should be evaluated agan. In addton, snce most evolutonary algorthms and swarm ntellgence methods converge to a pont due to ther characterstcs, group dversty s lost n the envronment and n case of changes n the envronment, convergence to a new optmal pont would be mpossble or tmeconsumng and slow. here are varous methods for generatng or mantanng group dversty n the envronment that wll be dscussed as follows. U.S. Government work not protected by U.S. copyrght Downloaded from

2 he smplest way for reactng aganst envronment change s consderng any change as an entrance of a new optmzaton problem whch must be solved agan. In case of lack of tme lmtatons, ths soluton s an approprate opton. However, frequently ths tme s rather short for re-optmzaton. A natural attempt to accelerate optmzaton process after a change s usng related nformaton of the search space before change n order to mprove the search after change. For nstance, f t s supposed that a new optmal pont s near the prevous optmal one, t can lmt the search space to the prevous optmal neghbor pont. he case whether reusng prevous nformaton s sutable depends on change characterstcs. If the change s broad and there s lttle smlarty between the envronments before and after change, restartng the optmzaton algorthm can be the only opton and any new use of collected nformaton based on the problem space before change wll be msleadng. In most real world problems, changes are hoped to be rather smooth, whch means there should be a relaton between the problem envronment before change and the problem envronment after change, n order to be able to use prevous nformaton for mprovng optmzaton process. he fundamental queston dscussed here s that what nformaton should be kept and how should ths nformaton be used to accelerate the search process after changes n the envronment. Another challenge s that n most optmzaton algorthms, after the algorthm converges to a pont, t loses ts dversty whch leads to decreased compatblty wth changes n the envronment. In fact, when algorthms agents converge to a pont, they lose ther ablty of followng ths pont after ts dsplacement. herefore, besde transferrng nformaton among optmzaton algorthms agents before and after the changes, we should search ways to ncrease dversty and compatblty of the algorthm after envronment change so that these agents could follow the goal pont after dsplacement and converge fast to the new poston of the goal pont. Another challenge n solvng dynamc problems s that there are several potental optmums n these problems. In fact, there are some peaks n dynamc envronments whose wdth, heght and locaton may be vared after envronment change n dfferent problems. herefore, n such problems, each of these peaks can be modfed to the global optmum after envronment change. So, the algorthms whch are desgned for solvng these problems should have the ablty to cover all peaks so that whenever one of them s transformed to a global optmum, they can fnd t fast. Multswarm s an approprate way to resolve ths problem. In [9], there were some predetermned groups that consst of some agents and every group has to cover one peak. he problem wth ths method was that f the number of peaks s not equal to the number of groups, algorthm s effcency was decreased. In[10], the number of groups was determned adaptvely accordng to the number of found peaks n the problem space. Generally, algorthms n whch the number of groups s determned wth respect to the number of found peaks n the problem space have more effcency than those wth a constant number of groups. he best condton s that each peak s covered by one group of agents, but sometmes more than one group may converge to a peak. o solve ths problem, excluson was used n [8]. In ths method, when two swarms get close more than a specfc dstance called r excl, the group wth the worse poston would be rentalzed. herefore, n each peak, one group can resde at most. In ths paper, a modfed AFSA s proposed for optmzaton n dynamc envronments. All requrements of dynamc envronments were satsfed n proposed algorthm. In ths algorthm, basc behavors mechansms of standard AFSA have been changed whch are: Prey, Follow, Swarm and Free_move. Also procedure of basc algorthm s completely changed. he proposed algorthm has been appled on dfferent confguraton of movng peaks benchmark (MPB) [11] that s one of the most famous benchmarks of dynamc envronments. Performance of proposed algorthm s compared wth ten other algorthms whch were presented for optmzng MPB. Comparng crteron s offlne_error whch s the man comparng crtera of desgned algorthms for dynamc envronments [11]. Expermental results show that the proposed algorthm has a sutable effcency. In followng, ths paper s organzed as: n secton two, AFSA algorthm wll be descrbed brefly. he proposed algorthm wll be dscussed n secton three. In secton four, experments and ther results wll be dscussed and last secton wll present the concluson. II. ARIFICIAL FISH SWARM ALGORIHM In water world, fshes can fnd areas that have more foods, whch s done wth ndvdual or swarm search by fshes. Accordng to ths characterstc, artfcal fsh (AF) model s represented by prey, free_move, swarm and follow behavors. AFs search the problem space by those behavors. he envronment, whch AF lves n, substantally s soluton space and other AF s doman. Food consstence degree n water area s AFSA objectve functon. Fnally, AFs reach to a pont whch ts food consstence degree s maxma (global optmum). As t s observed n fgure 1, AF perceves external concepts wth sense of sght. Current poston of AF s shown by vector =(x 1, x 2,, x n ). he vsual s equal to sght feld of AF and v s a poston n vsual where the AF wants to go. hen f v has better food consstence than current poston of AF, t goes one step toward v whch causes change n AF poston from to next, but f the current poston of AF s better than v, t contnues searchng n ts vsual area. Food consstence n poston s ftness value of ths poston and s shown wth f(). he step s equal to maxmum length of the movement.

3 he dstance between two AFs whch are n and j postons s shown by Ds j = - j (Eucldean dstance). AF model conssts of two parts of varables and functons. Varables nclude (current AF poston), step (maxmum length step), vsual (sght feld), try-number (the maxmum test nteractons and tres) and crowd factor δ (0<δ<1). Also functons consst of prey behavor, free move behavor, swarm behavor and follow behavor. In each step of optmzaton process, AF looks for locatons wth better ftness values n problem search space by performng these four behavors based on algorthm procedure[13]. Fgure 1. Artfcal Fsh and the Envronment Around It. III. HE PROPOSED ALGORIHM In ths secton, a new algorthm based on artfcal fsh swarm algorthm s presented for optmzaton n dynamc envronment. In proposed algorthm, parameters, behavors and AFSA procedure are modfed to be approprate for optmzaton n dynamc envronments. In the proposed algorthm, prey, follow and swarm behavors are done for AFs whch have man dfferences wth prey, follow and swarm behavors n Standard AFSA[5]. he reason to perform these changes s to adapt AFSA for workng n dynamc envronments. In followng, after descrpton of modfed artfcal fsh swarm algorthm (MAFSA), a new algorthm based on MAFSA wll be presented for dynamc envronments. A. Modfed AFSA algorthm Frst, we dscuss about MAFSA behavors. 1) Prey behavor hs behavor s an ndvdual behavor and each AF performs t wthout consderng other swarm members. Along performng ths behavor, each AF does a local search around tself. By performng ths behavor each AF attempts try_number tmes to replace to a new poston wth better ftness. Let suppose AF s n poston and wants to execute prey behavor. Followng steps are done n prey behavor: AF consders a target poston n vsual by equaton (1), then evaluates ts ftness value. d s dmenson number and Rand generates a random number wth unform dstrbuton n [-1, 1]: + Vsual Rand ( 1,1) (1), d =, d d If ftness value of poston s better than current poston of AF, poston wll be updated by equaton (2): = (2) Steps a and b are performed try_number tmes. By executng above steps, n the best case, an AF can update ts poston at most try_number tmes and move toward better postons. In the worst case, none of AF s attempts to fnd better poston wll be succeed. hen n ths stuaton after performng prey behavor, there wll be no replacement at all. 2) Follow behavor AFs n Standard AFSA n case of not fndng better postons n performng standard prey behavor, move one step randomly [11, 12] and lose ther prevous poston. But, n MAFSA as t was dscussed n 3-1-1, n prey behavor f an AF s not able to move to better postons, t won t move at all and wll keep ts prevous poston. hs causes that the best AF (accordng to ftness value) of swarm locates n the best found poston by swarm member so far. he reason s that n prey behavor n MAFSA, an AF dsplace f only moves to a better poston. In follow behavor, each of AFs moves one step toward the best AF of swarm by usng equaton 3: Best ( ) ( ) () t t + 1 = t + [ Vsual Rand( 0,1) ] (3) Ds, Best s poston vector of AF whch performs follow behavor and x best s poston vector of the best AF n swarm. herefore, AF moves at most to Vsual extent n each dmenson toward the best AF of swarm. In fact, after fndng more food by a fsh, other swarm members follow after t to reach more foods. Followng the best AF of swarm causes convergence rate ncrease and helps to keep ntegrty of AFs n a swarm. hs behavor s a group behavor and nteractons between swarm members are done globally. 3) Swarm behavor hs functon s a group behavor too and s performed globally among members of swarm. In swarm behavor, frst of all, central poston of swarm s calculated n terms of arthmetc average of all swarm members poston n every dmenson. Central poston of swarm center s obtaned by: 1 N Center, d = = 1 (4), d N

4 As t s observed, component d of vector center s the arthmetc mean of component d of all AFs of swarm. For AF, move condton toward central poston s checked,.e. f( Center ) f( ) and f ths condton s satsfed, next poston of AF s obtaned by: Center ( ) ( ) () t t + 1 = t + [ Vsual Rand( 0,1) ] (5) Ds, Center Equaton (5) s used for all AFs that have worse postons than central poston so they move toward Center. But for the best AF locatng n Best, f ftness value of Center s better than Best, next poston of the best AF s obtaned from: Best = Center (6) he reason of usng equaton (6) for the best AF s that t may be located n worse poston than ts current poston by movng toward Center by usng equaton (5), because t s possble to have worse postons n the way endng to Center from Best. herefore, t may cause to lose the best poston found by all members of swarm so far. hs problem s removed by usng of (6) for the best AF. he reason of not usng (6) for all AFs s that changng poston of swarm fshes to a smlar poston leads to extreme decrease n dversty of swarm and consderable decrease of convergence rate. IV. MAFSA PROCEDURE In MAFSA, for every AF, prey, follow and swarm behavors are performed n each teraton. In Standard AFSA executng one of the standard swarm and standard follow behavors ddn t affect on AFs movement and huge amount of computatons were wasted. Unlke standard AFSA, n MAFSA, all three behavors nfluence on movement of AFs and swarm move toward better postons. In MAFSA, frst, all AFs perform prey behavor and ther poston s updated based on ths behavor procedure. By executng ths behavor, every AF can dsplace up to try_number tmes. hen, all of them wth respect to ther new poston and other AFs poston whch performed prey behavor, execute follow behavor and all members except the best AF of swarm move to a new locaton n drecton of movng toward the best found poston by swarm. hen each AF performs swarm behavor. By performng swarm behavor, AFs whch are apart from swarm and locate n worse postons than other swarm members, can jon to the swarm faster. In fact, ths behavor causes faster movement of AFs whch are n worse postons. As a result of Performng ths behavor, convergence rate would ncrease and unlke follow behavor t can causes poston mprovement of best AF. At the end of each teraton of performng MAFSA algorthm, Vsual value s updated for AFs. Accordng to varous experments done on AFSA, t s concluded that bg value of Vsual parameter results n convergence rate ncrease snce AFs move wth larger steps. But when swarm converges, AFs are not able to do an acceptable local search because after convergence, space where has to be searched for better postons becomes smaller and AFs wth large Vsual by performng prey behavor would have less chance to reach better postons. On the other hand, f Vsual s small, local search ablty ncreases but because of small move steps, convergence rate decreases so much. hus, t can conclude that the best condton s that at frst Vsual to be large so AFs to converge fast to ther goals. Along wth convergng of swarm to goal, Vsual value has to decrease gradually so at last AFs wth small Vsual could reach acceptable results by dong acceptable local search. For ths reason, Vsual value has to be multpled by a number less than one n each teraton. We can use varous mechansms to determne ths number. In ths paper, a specfed random number generator functon s used as followng: Vsual ( t +1) = Vsual ( t) ( L + ( Rand ( L L ) Low (7) In equaton (5), Vsual n each teraton s obtaned randomly based on ts value at prevous teraton. L Low and L Hgh are lower lmt and upper lmt of Vsual change percentage n compare wth prevous teraton. Rand s the random number generator functon wth unform dstrbuton n [0, 1]. So, Vsual value n each teraton randomly s n [Vsual(t-1) L Low, Vsual(t-1) L Hgh ]. For ths reason, L Hgh value should be consdered less than one and L Low L Hgh. MAFSA pseudo code s represented n fgure 2. Here we suppose optmzaton problem s maxmzng problem. A. MAFSA confguraton for dynamc envronments In ths part, MAFSA algorthm wll be confgured for workng n dynamc envronments whch s called dynamc MAFSA (DMAFSA). In MPB, when there s more than one peak n problem space, each peak can modfy nto global optma after envronment change. Hence, all peaks have to be covered by AFs as much as possble so f each of them transforms nto global optma, the algorthm could fnd t fast. hus, one swarm should be located at each peak and cover t. So, mult-swarm has to be used n DMAFSA and each swarm act ndependently from other swarms and do accordng to MAFSA procedure. In ths paper, at the begnnng, there s only one swarm n problem space. If the swarm converges to a peak, another swarm s generated n problem space. A swarm has converged when the best AF poston s almost constant after some teraton n problem space. It means Eucldean dstance of current poston of the best AF from ts poston n some prevous teraton to be less than a specfed amount. In DMAFSA, whenever all swarms have converged, a new swarm generates n problem space and starts searchng. If recently generated Hgh Low

5 swarm converges to a peak whch s covered by another swarm, a race condton occurs. In ths condton, the swarm whch has better best AF accordng to ftness value contnues ts actvty and the loser wll expel and rentalze. New generated swarm would converge to a covered peak by another swarm when Eucldean dstance of best AF poston of new generated swarm form best AF poston n one of the other swarms n problem space s less than a specfed value r excl. hereafter, any new generated swarm has a chance to converge to a peak whch no other swarm has resded there yet. But f t converges to a covered peak, wth hgh probablty t wll be rentalzed. It s because the probablty of beng placed n a better poston than prevously resdng swarm s very low. If new swarm converges to an empty peak whch ddn t cover by any swarm yet, another swarm wll be generated n problem space. herefore, the number of avalable swarms n problem space s always one more than the number of found peaks n problem space. hs mechansm of generatng new swarms s as lke as [10] whch was presented for PSO but wth some modfcatons n t. MAFSA: for each Artfcal Fsh [1.. N] ntalze x repeat: //Searchng Food Behavor for each AF for counter=1 to try_number Obtan wth equaton (1) and Calculate f( ) f f( ) f( ) then = end f // Follow Behavor for each AF Apply equaton (3) // Swarm Behavor Calculate Central Poston by equaton (4) Center for each AF f f( ) f( Center ) then f s Best AF Apply equaton (6) for AF else Apply equaton (5) for AF end f end f Update Vsual accordng to equaton(7) untl stoppng crteron s met Fgure 2. MAFSA Pseudo Code o dscover change n envronment, at the begnnng of algorthm executon, a random pont called test pont s selected n problem space and ts ftness value wll be stored. After executng each teraton of algorthm for all actve swarms, ftness value of test pont s reevaluated. If obtaned ftness value s not equal to stored value based on prevous ftness evaluaton of test pont, then envronment has been changed. It should be noted that test pont poston would reman fxed untl the end of algorthm. After detectng change n envronment, frst, problem of dversty reducton has to be resolved. Indeed, when a swarm converges to a pont, the dstance of AFs decreases very much from each other and dversty n swarm decreases too much. As a result, after envronment change, AFs cannot take advantage of follow and swarm behavors because dstance of artfcal fshes from each other has decreased too much. o resolve ths ssue, after detectng change n envronment, the best artfcal fsh poston of the swarm s kept for converged swarms and other AFs of that swarm are dstrbuted randomly n a d-dmensonal ball around the best AF poston of the swarm. Radus of ths d-dmensonal ball n each dmenson s determned n terms of shft length of peaks n MPB snce t s expected that new peak poston placed n a d-dmensonal ball wth the centered of prevous peak poston before envronment change. Also, Vsual has decreased wth respect to space where the swarm has to search for better postons by usng equaton (7). hus, after detectng envronment change, Vsual value should be determned n terms of shft length. Consequently, new peak poston would be n the searchng neghborhood of AFs and they can move fast toward t. For swarm that has not converged, there s no need to change the poston of AFs because dversty has not been lost but ts Vsual value must adjust equal to ts ntal value to avod reducng the convergence rate. After adjustng Vsual and determnng AFs poston, ther ftness values are reevaluated to remove outdated memory. hen, the algorthm contnues searchng n new envronment. In DMAFSA, to ncrease search ablty around the best found peak by all swarms, try_number amount s consdered for AFs of the best swarm more than other swarms. he best swarm s that one whch ts ftness value of the best AF s better than the other swarms best AF ftness value. herefore, AFs of the best swarm would have more opportunty to fnd better postons by performng prey behavor. he reason whch try_number amount for all swarms s not ncreased, s that ncreasng of local search ablty n local optmal peaks cannot affect on obtaned result. Also, by ncreasng try_number n all swarms, the number of ftness evaluatons whch AFs perform at each teraton ncrease. It causes that AFs perform less teratons untl envronment change and the algorthm adaptablty becomes slow n comparson wth envronment change perod. hereafter, by ncreasng try_number just n the best swarm, search ablty

6 ncreases around optmal peak and on the other sde, the algorthm doesn t waste ftness evaluaton. Pseudo code of the proposed algorthm DMAFSA s shown n fgure 3. Dynamc MAFSA: //Intalzng frst swarm for each AF j n frst_swarm ntalze 1,j randomly ntalze est_pont randomly repeat for each swarm //Searchng Food Behavor for each AF j n swarm for counter=1 to try_number obtan by equaton (1) and evaluate f( f f( ) f( ) then, j ), = j end f // Follow Behavor for each AF j n swarm apply equaton (3) based on best_af n swarm // Swarm Behavor Calculate accordng equaton (4) Center, for each AF j n swarm f f( ) f( Center,, ) then j f, s Best_AF n swarm then j =, j Center, else Apply equaton (5) on, j end f end f Update Vsual accordng equaton (7) f All swarm are converged then Create new_swarm and ntalze t end f //Excluson for each swarm f dstance(bestaf and bestaf new_swarm) < r excl then f f(bestaf ) f(bestaf new_swarm) then rentalze Swarm else rentalze new_swarm end f //est for Change Evaluate est_pont f new value s dfferent from last teraton then for each swarm f swarm s converged keep best_af and randomze others around t based on Shft_length end f Set Vsual based on Shft_ length end f Untl stoppng crteron s met Fgure 3. Pseudo Code of Dynamc MAFSA. I. EPERIMENAL RESULS o assess correctness and effcency of the proposed algorthm, ths algorthm was compared wth varous known algorthms on MPB whch s one of the most famous benchmarks of dynamc envronments [9,11]. Experments were done wth respect to MPB parameters whch are gven n table I. ABLE I. MPB PARAMEERS SEING Parameter Value Number of peaks, M Varable between 1 to 200 Change frequency 5000 Heght change 7.0 Wdth change 1.0 Peaks shape Cone Basc functon No Shft length, s 1.0 Number of dmensons,d 5 Correlaton Coeffcnet,λ 0 peaks locaton range [0 100] Peak heght [ ] Peak wdth [1 12] Intal value of peaks 50.0 he man metrc for evaluatng the performance of algorthms n ths doman s offlne error whch s ndcates the average of the best found poston s ftness usng algorthms durng the runnng optmzaton process[9,11]. In other word, the value of offlne error s the average of current errors. Current error at tme t s the devance of the best found poston usng algorthm at tme t n the current envronment and the poston of global optmum n the current envronment. he value of offlne error s equal or greater than zero, where zero ndcates the deal stuaton. Adjustment of parameters n the proposed algorthm s done accordng to varous executons of experments. Populaton sze n each swarm s 2. try_number s equal to 20 for the best swarm and 2 for other swarms. Vsual value for recently generated swarm s 20. After detectng envronment change, vsual value for converged swarms s consdered equal to peaks shft length and AFs of these swarms are located randomly n a d-dmensonal ball wth radus of shft_length around the best AF of swarm. L Low and L Hgh are adjusted 0.4 and 1 n equaton (7), respectvely. A swarm has converged, f Eucldean dstance between the best AF poston of ths swarm and ts poston n 3 prevous teratons s less than 0.1. Also, determnng r excl value accordng to [9] has approprate results. Experments were repeated ndependently 50 tmes and each tme they were performed by dfferent random seeds. Every experment contnued untl envronment changes 100 tmes. For example when change frequency was 5000, experments performed ftness evaluatons and durng ths tme the envronment changes 100 tmes. In table II, the proposed algorthm DMAFSA effcency s compared wth 10 other known algorthms: mqso[9], mcpso[9], Adaptve mqso(amqso) [10],

7 CellularPSO[14], FMSO[15], RPSO[16], SPSO[17], rspso[18], mpso[19], PSO-CP[20] wth change frequency 5000, shft length equal 1 and dfferent number of peaks n terms of offlne error and Standard error. Gven results n table II for some algorthms are brought from ther respectve paper drectly. As t s observed, the proposed algorthm has better effcency than other algorthms and only n the case that the problem has only one peak, the proposed algorthm s placed n second rank. In the proposed algorthm, because the number of swarms s proportonal to the number of found peaks, the algorthm effcency s acceptable both when number of peaks s low and when the number of peaks s hgh. ABLE II. COMPARISON OF OFFLINE ERROR (SANDARD ERROR) OF 11 ALGORIHMS ON MPB PROBLEM WIH DIFFEREN NUMBER OF PEAKS. ALG. mqso AmQSO CLPSO FMSO RPSO mcpso SPSO rspso mpso PSO-CP DMAFSA (0.21) 0.51(0.04) 2.55(0.12) 3.44(0.11) 0.56(0.04) 4.93(0.17) 2.64(0.10) 1.42(0.06) 0.90(0.05) 3.41(0.06) 0.55(0.06) (0.08) 1.01(0.09) 1.68(0.11) 2.94(0.07) 12.22(0.76) 2.07(0.08) 2.15(0.07) 1.04(0.03) 1.21(0.12) (0.06) (0.08) 1.51(0.10) 1.78(0.05) 3.11(0.06) 12.98(0.48) 2.08(0.07) 2.51(0.09) 1.50(0.08) 1.61(0.12) 1.31(0.06) 1.01(0.05) NUMBER OF PEAKS (0.09) 2.51(0.10) 2.00(0.15) 2.19(0.17) 2.61(0.07) 2.93(0.08) 3.36(0.06) 3.28(0.05) 12.79(0.54) 12.35(0.62) 2.64(0.07) 2.63(0.08) 3.21(0.07) 3.64(0.07) 2.20(0.07) 2.62(0.07) 2.05(0.08) 2.18(0.06) (0.07) 1.42(0.06) 1.63(0.06) (0.08) 2.43(0.13) 3.26(0.08) 3.22(0.05) 11.34(0.29) 2.65(0.06) 3.86(0.08) 2.72(0.08) 2.34(0.06) (0.07) (0.06) 2.68(0.12) 3.41(0.07) 3.06(0.04) 9.73(0.28) 2.49(0.04) 4.01(0.07) 2.93(0.06) 2.32(0.04) 2.14(0.08) 1.95(0.05) (0.05) 2.62(0.10) 3.40(0.06) 2.84(0.03) 8.90(0.19) 2.44(0.04) 3.82(0.05) 2.79(0.05) 2.34(0.03) 2.04(0.07) 1.99(0.04) Scenaro 2 Current-Err Offlne-Err 10 0 Err FEs x 10 5 Fgure 4. Offlne Error and Current Error Obtaned from the Proposed Algorthm DMAFSA. In DMAFSA algorthm, parameters are set such that swarms whch locate n local optmums perform fewer ftness evaluatons n each teraton. herefore, snce the envronment change crteron s number of ftness evaluatons, there are more chances for the best swarm to seek better around the global optmum before envronment change. In MAFSA desgnng, t tred to keep ntegrty of swarm by performng follow and swarm behavors. On the other hand, wth followng the best AF of swarm by other swarm members, the algorthm convergence rate has ncreased much more. Moreover, n prey behavor, AFs can move some steps toward better postons wth respect to local search and use all done ftness evaluatons to mprove ther poston. In addton, ncrease n local search ablty wth decrease n vsual parameter cause AF to be able to reach more accurate results. All these mprovements caused that swarms n the proposed algorthms could converge fast to goal and after reachng to goal, perform an acceptable local search. In fgure 4, offlne error and current error graphs of the proposed algorthm DMAFSA are shown on MPB for 100 tmes of envronment change wth 5000 change frequency, 10 peaks and shft length equal to 1 whch called scenaro 2 n MPB problem. hs graph s plotted wth respect to mean of 50 tmes runnng the algorthm. As t s observed, after each tme of envronment change, the proposed algorthm follows the optmum well. he reason of hgh effcency of ths algorthm n lookng after the optmums s adjustment of dversty and vsual. As a result of ths, new peak locaton s placed wthn the area of artfcal fsh swarm. Also, acceptable decrement amount of current error n any envronment shows approprate local search ablty of swarm n proposed algorthm.

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