Bees algorithm for multimodal function optimisation Zhou, Z.; Xie, Y.; Pham, Duc; Kamsani, S.; Castellani, Marco

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1 Bees algorthm for multmodal functon optmsaton Zhou, Z.; Xe, Y.; Pham, Duc; Kamsan, S.; Castellan, Marco DOI:.77/ Lcense: None: All rghts reserved Document Verson Peer revewed verson Ctaton for publshed verson (Harvard): Zhou, Z, Xe, Y, Pham, D, Kamsan, S & Castellan, M 5, 'Bees algorthm for multmodal functon optmsaton' Insttuton of Mechancal Engneers. Proceedngs. Part C: Journal of Mechancal Engneerng Scence. DOI:.77/ Lnk to publcaton on Research at Brmngham portal General rghts Unless a lcence s specfed above, all rghts (ncludng copyrght and moral rghts) n ths document are retaned by the authors and/or the copyrght holders. The express permsson of the copyrght holder must be obtaned for any use of ths materal other than for purposes permtted by law. Users may freely dstrbute the URL that s used to dentfy ths publcaton. Users may download and/or prnt one copy of the publcaton from the Unversty of Brmngham research portal for the purpose of prvate study or non-commercal research. User may use extracts from the document n lne wth the concept of far dealng under the Copyrght, Desgns and Patents Act 988 (?) Users may not further dstrbute the materal nor use t for the purposes of commercal gan. Where a lcence s dsplayed above, please note the terms and condtons of the lcence govern your use of ths document. When ctng, please reference the publshed verson. Take down polcy Whle the Unversty of Brmngham exercses care and attenton n makng tems avalable there are rare occasons when an tem has been uploaded n error or has been deemed to be commercally or otherwse senstve. If you beleve that ths s the case for ths document, please contact UBIRA@lsts.bham.ac.uk provdng detals and we wll remove access to the work mmedately and nvestgate. Download date:. Dec. 8

2 Bees Algorthm for Multmodal Functon Optmsaton Z. D. Zhou, Y. Q. Xe, D. T. Pham, S. Kamsan, M. Castellan School of Informaton Engneerng, Wuhan Unversty of Technology, Chna School of Mechancal Engneerng, Unversty of Brmngham, U.K Abstract: The am of multmodal optmsaton (MMO) s to fnd sgnfcant optma of a multmodal objectve functon ncludng ts global optmum. Many real-world applcatons are MMO problems requrng multple optmal solutons. The Bees Algorthm (BA) s a global optmsaton procedure nspred by the foragng behavour of honeybees. In ths paper, several procedures are ntroduced to enhance the algorthm s capablty to fnd multple optma n MMO problems. In the proposed Bees Algorthm for MMO, dynamc colony sze s permtted to automatcally adapt the search effort to dfferent objectve functons. A local search approach called balanced search technque (BST) s also proposed to speed up the algorthm. In addton, two procedures of radus estmaton and optma eltsm are added, to respectvely enhance the Bees Algorthm s ablty to locate unevenly dstrbuted optma, and elmnate nsgnfcant local optma. The performance of the modfed Bees Algorthm s evaluated on well-known benchmark problems, and the results are compared wth those obtaned by several other state-of-the-art algorthms. The results ndcate that the proposed algorthm nherts excellent propertes from the standard Bees Algorthm, obtanng notable effcency for solvng MMO problems due to the ntroduced modfcatons. Keywords: Swarm-based algorthms, multmodal optmsaton, Bees Algorthm, balanced search, hll valley Correspondng author: xyqwhut78@6.com

3 Nomenclature MMO: multmodal optmsaton BA: Bees Algorthm BST: balanced search technque SOA: swarm-based optmsaton algorthm EA: Evolutonary Algorthm GA: Genetc Algorthm PSO: Partcle Swarm Optmsaton IWO: Invasve Weed Optmsaton HV: hll valley DE: Dfferental Evoluton SDE: Speces-based Dfferental Evoluton. Introducton Swarm-based optmsaton algorthms (SOAs) are usually employed to fnd the global soluton to an optmsaton problem, dscardng any alternatve soluton of equal or comparable ftness. However, n a multmodal optmsaton (MMO) task, the man purpose s to fnd multple optmal solutons []. MMO problems are ganng ncreasng attenton due to ther frequent occurrence n scentfc and engneerng applcatons, such as object detecton n machne vson, parameter tunng n vared-lne-spacng holographc gratng desgn, and proten structure predcton. MMO apples to those problems whch have more than one global optmum n the feasble soluton space, or one global optmum and several

4 local optma. As they represent alternatve solutons, t s sometmes desrable to locate all the sgnfcant optma of a gven ftness landscape. In addton, the knowledge of multple optmal solutons may provde useful nsght nto the problem doman. A smlarty analyss of multple optmal solutons may brng about helpful nnovatve and hdden prncples, smlar to what s often observed n Pareto-optmal solutons n a mult-objectve problem solvng task []. Investgaton of the performance of SOAs for MMO problems has been recevng growng nterest n the SOA communty. Evolutonary algorthms (EAs) are wdely used to solve MMO problems due to ther populaton-based searchng ablty. Nchng, clusterng, and specaton methods have been used to dstrbute the EA populaton on dfferent peaks n the search regon. Smlarly, several modfed versons of Partcle Swarm Optmsaton (PSO) and Invasve Weed Optmsaton (IWO) have been used to search multple optmal solutons. More detals about methods for solvng MMO problems are presented n the second secton. Frst proposed by Pham [3], the Bees Algorthm s a SOA that mmcs the foragng behavour of honey bees, a speces whch has been successfully survvng for hundreds of thousands of years n varous knds of natural envronments. Ths paper wll ntroduce several modfcatons to the basc Bees Algorthm wth the am to fnd multple optmal solutons smultaneously n a sngle run. The modfcatons are necessary snce the basc Bees Algorthm s desgned to fnd only one optmum. Frst of all, unlke some SOAs that use a predefned clusterng radus (or parameter wth the smlar functon) for the peaks n the ftness landscapes, the proposed algorthm estmates the radus usng an amended hll valley (HV) method. The second modfcaton ntroduces a local search operator named balanced

5 search technque (BST) to search for the solutons of hghest ftness n a ftness peak. Some algorthms calculate the gradent of the objectve functon. Ths can be very helpful n smulaton but unfortunately many real world problems are non-dfferentable. Compared to the purely random local search n the basc Bees Algorthm, ths modfcaton ams at mprovng the algorthm s search speed. Furthermore, the algorthm allows for varable colony sze. That s, the populaton sze n each generaton s allowed to ncrease f more optma are detected, or decrease f only a small number of optma exst n the objectve functon. Ths s bologcally plausble, snce bologcal bees optmse the number of harvestng bees accordng to the abundance of food sources (.e. nectar). The remander of ths paper s organsed as follows. In Secton, a revew s provded of related SOAs for solvng MMO problems. Secton 3 outlnes the basc Bees Algorthm. The modfcatons ntroduced to perform MMO search are explaned n Secton 4. Thereafter, the ndvdual mpact of each of the new features on the search capablty of the algorthm s hghlghted n Secton 5. In Secton 6, the expermental results of the proposed algorthm and comparsons are presented. Fnally, Secton 7 concludes the paper and suggests topcs for future work.. Related work When applyng SOAs to MMO problems, t s very mportant to consder two apparently contradctory requrements: preservng promsng ndvduals from one generaton to the next and mantanng the dversty of the populaton [4]. Ths secton brefly revews some recently developed technques to address the above trade-off.

6 De Jong tred to solve the MMO problem usng an EA for the frst tme n 975 [5]. He used populaton crowdng. Crowdng encourages populaton dversty by elmnatng from the parent populaton those ndvduals whch are most smlar to the offsprng. Ftness sharng was proposed by Goldberg and Rchardson n 987 [6] to ncrease the chance of locatng multple optma. Instead of usng an absolute ftness functon, they desgned a shared functon whch takes nto account the genotypc or phenotypc smlarty of the ndvduals. Snce then, an ncreasng number of researchers explored dfferent ways to deal wth the populaton dversty problem. These methods nclude speces conservaton, pre-selecton, eltsm, and clearng. The adaptve eltst-populaton search method was used n a Genetc Algorthm (GA) for MMO [7-8]. Vtela and Castanos proposed a sequental nchng algorthm for MMO. They combned hll-clmbng, a deratng functon, nchng and clearng technques wthn a GA for a multple optma search [9]. In the lterature [-], authors dscussed a clusterng genetc algorthm based on dynamc nchng wth nche mgraton. They studed the nchng method ntensvely and clamed very lttle pror knowledge s requred to determne the radus and the number of nches. Other technques as the dstance measurng method [3] and memetc algorthms [4] were used wthn a GA for MMO purpose. Partcle Swarm Optmsaton (PSO) s another promnent member of the SOA famly, and s now recevng a great deal of nterests for MMO purpose. A memetc algorthm, along wth a local search operator, was hybrdsed wth PSO by Wang [5] for MMO. Ths hybrd PSO obtaned excellent performance. Lkewse, other practtoners [6-] hybrdsed PSO wth nchng and clusterng technques to desgn dfferent populaton topologes or ftness

7 evaluaton methods to obtan several multple optmal solutons. Some latest studes related to PSO for MMO are summarsed n [3]. L [4] utlsed a SOA for determnng speces n conjuncton wth a basc Dfferental Evoluton (DE) procedure named Speces-based Dfferental Evoluton (SDE). In [5], the prncple of localty, a wdely used concept n computng, was ncorporated wth dfferental evoluton for MMO. Spatal localty and temporal localty were adopted n the proposed methods. Other learnng algorthms such as artfcal weed colony optmsaton are contnuously nvestgated for MMO [6-8]. 3. The basc Bees Algorthm The basc Bees Algorthm s nspred by the foragng behavour of honeybees n nature, and was desgned to search for the best soluton to a gven optmsaton problem. A soluton n the search space s thought of as a nectar source. Scout bees randomly sample the soluton space and apprase the qualty of the vsted locatons through the ftness functon. Foragers are recruted to explot the most promsng m locatons found by the scout bees. Each scout drects a number of foragers to the neghbourhood of the solutons found. The scouts that found the e top-rated locatons recrut nre foragers, the scouts that found the remanng m-e most promsng solutons recrut nrb< nre foragers. The neghbourhood of a soluton s regarded as a flower patch. Overall, the orgnal Bees Algorthm employs a combnaton of local explotatve and global exploratory search technques. For the global search, scout bees are sent to random ponts of the search space to look for potental solutons. For the local search, foragers are sent to the neghbourhood of the most favourable solutons.

8 The parameters need to be set for the basc algorthm are: the number of scout bees (n); the number of patches selected for the local search (m); the number of top-rated patches (elte) n selected patches (e); the number of foragers recruted for the top patches (nre); the number of foragers to be recruted for the other selected patches (nrb); the ntal sze of each patch (ngh); and fnally the stoppng crtera. Snce ts orgnal formulaton, the Bees Algorthm has undergone many varatons [9-34], and for ther applcatons reader can refer to [35-39]. 4. The proposed modfed Bees Algorthm The proposed algorthm ncludes a number of modfcatons to the basc Bees Algorthm to fnd multple satsfactory solutons to an objectve problem n a sngle run. Ths secton detals the proposed algorthm and presents the modfcatons. Wthout loss generalty, t wll be assumed n the rest of ths paper that the optmsaton problem requres the maxmsatons of a gven ftness functon. 4. Man procedures of the proposed algorthm Before detalng the proposed algorthm, a number of terms wll frst be defned. Defnton. Feld: a feld defnes an area n the search space that may potentally cover a ftness peak. It helps to dfferentate one peak from another. Each feld should cover one and only one ftness peak deally. The sze of a feld s determned by ts radus (refer to Defnton. 3). Each feld contans at least one scout. Defnton. Feld centre: the feld centre s the locaton of hghest ftness found so far by a scout bee n the feld. If there s only one scout n that feld, ts locaton automatcally becomes the feld centre. Defnton 3. Feld radus: the feld radus s the Eucldean dstance from the feld centre

9 to the border. It determnes the sze of feld. Defnton 4. Neghbourhood: the neghbourhood s used to constran the range for a refned local search wthn a feld to locate the local optma. Only wthn the neghbourhood scouts and foragers are allowed to land. To enhance the search accuracy, the sze of the neghbourhood shrnks f the search stagnates n a feld. The man body of the proposed algorthm s desgned based on the framework of the basc verson. Fgure summarses the proposed algorthm, and Table lsts the parameters to be ntalsed n Step. In Step 3, felds are allowed to merge or splt accordng to the dstance between them or the dstrbuton of scouts. The felds rad are updated through an estmaton procedure. Local search s performed n Step 4 to look for the optmal solutons on detected ftness peaks. A balanced search strategy s employed n ths step to enhance the search speed. Ths s followed by global search n Step 5. The global search tres to detect potental ftness peaks that have not been dentfed n the search space. The hll valley method s mplemented n global search to enhance the possblty of fndng undentfed peaks. Input: Objectve functon Step : Intalsaton Step : Whle (stoppng crtera not met) Step 3: Felds update Step 4: Local search (neghbourhood search) Step 5: Global search Step 6: End whle

10 Output: Optmal solutons Fgure. Pseudo code of man body of the proposed algorthm Table. Parameters of the Bees Algorthm for MMO ns nr nre nrb ngh rad ntal number of scouts n Felds ntal number of random scouts number of foragers recruted by the scout at the feld centre number of foragers recruted by other scouts n a feld (nre> nrb) ntal neghbourhood sze ntal feld radus stlm lmt of cycles to determne the stagnaton of a feld 4. Feld update Felds are allowed to merge and splt when the certan condtons are met, and the radus of each feld s made adaptve to the objectve problem through an estmaton procedure. 4.. Dstance between felds Let P and P j be two felds wth radus R and R j, and C and C j be the respectve centre. The dstance between P and P j s the Eucldean dstance between C and C j calculated as Equaton (): Ds P, P ) = pos( C ) pos( C ) () ( j j where pos ( ) returns the poston of a specfed pont n the search space. 4.. Rule for mergng felds P and P j are allowed to merge when Relaton () s satsfed: Ds P, P ) <.7 ( R + R ) () ( j j

11 Let P k, C k, R k denote respectvely the newly formed feld, ts centre and radus, then C k s the one selected from C and C j whch has hgher ftness, and R k s the larger one of R and R j. All the scouts of P and P j are moved nto the common feld P k. A restrcton on the number of scouts n a feld s placed snce a feld cannot sustan an overly dense populaton. If the number of scouts n a feld exceeds a predefned upper lmt (3 n ths paper), only those at the fttest postons wll be kept for the next generaton, whlst the others are regarded as redundant and transferred to global search Rule for splttng a feld A feld P s allowed to splt when Relaton (3) s satsfed: Ds ( C, S ) >. 4 R (3) where S scouts n P S s not at C } s one of the scouts n P. Each feld gets only one { chance to splt n an evolvng teraton. Denotng the two chldren felds as P m and P n, C m, C n, R m, R n are updated as C m = C, n S C = and R m = Rn =. 7 R, and P m nherts all the scouts from P except S Feld radus estmaton The feld radus s updated through a radus estmaton algorthm, where an amended hll valley (HV) [4] s appled. Fgure helps to explan how the amended HV works on a one dmensonal functon. Nevertheless, the valdty of the procedure extends to any dmensonalty of the search space.

12 ' C C B ' B M ' B B M ' C C (a) (b) ' B B M B ' B ' C C M ' C C (c) (d) Fgure. (a) (b) and (c) demonstrates the condtons (), (), and (3)respectvely, (d) demonstrates a valley beng omtted. Fgure llustrates four possble cases n feld radus estmaton. Let C and R be the centre and estmated radus of the current feld P. A partcular bee B s created and sent to the poston obtaned by Equaton (4). pos( B ) = pos( C ) + D R (4) r where D r symbolses a normalsed random drecton, Then three sample ponts M are created accordng to Equatons (5). ' pos( C ) = pos( C ) + δ [ pos( B ) pos( C )] ' pos( B ) = pos( B ) + δ [ pos( C ) pos( B )] pos( M ) = pos( C ) + [ pos( B ) pos( C )] ' C, (5) ' B and where δ s an absolute small postve value. The basc motvaton underneath the radus estmaton s: f a valley s detected between B and C, the current radus ought to be reduced,

13 otherwse t should be ncreased. A valley s sad to exst f at least one of the followng condtons s met: ' () ftness ( B ) < mn{ ftness( B ), ftness( C )}, as shown n Fgure (a); ' () ftness ( C ) < mn{ ftness( B ), ftness( C )}, as shown n Fgure (b); (3) ftness M ) < mn{ ftness( B ), ftness( C )}, as shown n Fgure (c). ( where the functon ftness( ) returns the ftness value of a sampled poston. Otherwse, t s assumed there s no valley between B and C. Fgure (d) shows a case n whch the condtons (), () and (3) are not satsfed but actually a valley does exst between B and C. The HV method omts a valley between two end ponts wth a probablty nversely proportonal to the number of adopted sample ponts. To reduce the chance of such an event, the amended HV s repeated k (k > ) tmes (k equals to the dmenson of the objectve functon n ths paper) before determnng the radus. Fgure 3 shows the pseudo code of feld radus estmaton algorthm where α ( < α < ) s ntroduced to control the radus alterablty. Throughout ths paper t s kept constant to α =.. Input: current radus R (t), k Step : Send B accordng to Equaton (4), k = k-; Step : Produce sample ponts accordng to Equatons (5); Step 3: Determne whether a valley exsts accordng to condtons (), () and (3), f a valley exsts, go to Step 4, f a valley does not exst and k =, go to Step 5, otherwse go to Step ;

14 Step 4: R (t+) ( α) R (t), return; Step 5: R (t+) ( + α) R (t). Output: updated radus R (t+) 4.3 Local search Fgure 3. The pseudo code of feld radus estmaton The basc Bees Algorthm and most of ts varants mplement local search n a neghbourhood usng a random operator, as expressed by Equatons (6). j pos( F ( t)) = pos( S( t)) + ngh( t) r ftness( S( t + )) = max { ftness( S( t)), j=,,..., nf ftness( F j ( t))} (6) where F j (t) stands for the jth forager recruted by the scout S(t) n generaton t, n F denotes the number of foragers recruted by S, r s a unformly dstrbuted random value n (,). The functon max{ } embodes the greedy selecton strategy adopted n the local search, that s the scout s replaced by the recruted forager f the forager s landng at a poston of hgher ftness than the scout. The balanced search technque (BST) s developed to speed up the algorthm as descrbed below Obtanng the gude A gradent-lke vector s obtaned as a search gude for the recruted foragers. It does not requre the functon to be dfferentable. The gude s calculated as follows: G ( t) = pos( C ( t)) pos( C ( t )) (7) where G (t) denote the gude of the feld P n generaton t. Equaton (7) ndcates that the gude for the current teraton depends on the poston of the feld centre n the last two teratons. The gude s thereafter normalsed by dvdng t by the norm ( t) as n G Equaton (8). A search conducted by ths gude s called guded search.

15 G ( t) G ( t) =,f G ( t) (8) G ( t) 4.3. Formulatng the balanced search The basc prncple of BST s to keep a balance between random and guded search, as shown n Equatons (9). j pos( F ( t)) = pos( C ( t)) + ( µ ( t) G ( t) + ( µ ( t)) D ) ngh ( t) r r j ftness( C ( t + )) = max { ftness( C ( t)), ftness( F ( t))} j=,,..., nf (9) n whch D r symbolses a normalsed random drecton, r s a unformly dstrbuted random value n (,), and µ (t) s ntroduced as an adaptve weght to balance the nfluence of the two local search operators. The larger µ (t) s, the more local search depends on guded search. Conversely, the local search wll rely more on randomness Updatng the balance weght The weght µ (t) s updated accordng to Equatons ().. µ ( t), angle( G ( t), G ( t )) θ µ ( t + ) =.8 µ ( t), angle( G ( t), G ( t )) θ () µ ( t), other. subject to < µ ( t). θ and θ are two thresholds that determne the sze of µ (t), and the functon angle( ) returns the angle between two vectors. Equatons () ndcate that f mprovements n ftness are obtaned n consecutve teratons wthout alterng substantally the drecton of the gude, µ wll keep growng untl t reaches ts upper lmt. Accordng to Equatons (9), an ncrease of µ wll result n the domnance of the guded search. On the contrary, f the angle between two successve gudes exceeds the threshold θ, µ wll gradually decrease and then random search wll take domnance.

16 4.3.4 Preservng stagnant felds A feld s consdered stagnant f no mprovement can be obtaned after a predefned number of evolutonary cycles (stlm). Instead of abandonng the stagnated feld, the proposed algorthm records the nformaton of the feld ncludng ts centre and radus. The poston of the feld centre s one of the optmal solutons located by the algorthm. All the scouts n the stagnated feld except the one at the centre are released and become random scouts, so the search n ths feld s termnated. 4.4 Global search Global search focuses on yet unknown areas of the soluton space. It s ntally carred out by a predefned number of scouts, called random scouts. To guarantee that a ftness peak dscovered by a random scout has never been searched before, the regonal evoluton and the standard hll valley are combned. The regonal evoluton s used to prevent that a random scout whch just began to clmb a ftness peak s beng neglected due to ts current low ftness. A random scout s allowed to evolve a few tmes (equals to the dmenson of the objectve functon n ths paper) before competng wth those scouts at the feld centre. Normally, the HV would ncur a fast ncrease of functon evaluatons, snce t has to evaluate the ftness values of a number of sample ponts. In the proposed algorthm, ths ncrease s restraned by restrctng HV only to the felds n the vcnty of the peak under consderaton. When a random scout dscovers a new promsng regon n the ftness landscape, a new feld s generated around ths random scout. Ths random scout therefore become a scout n felds and s nvolved n the local search n the next teraton. The global search process conssts essentally of the followng steps n Fgure 4:

17 Input: current felds, random scouts Step : send a scouts randomly to the search space; Step : mplement the regonal evoluton; Step 3: mplement the HV to determne whether the random scout has found a new peak n ftness landscape; Step 4: f a new ftness peak s dentfed, a new feld coverng ths peak s formed and nserted nto current felds Step 5: go to Step untl all the random scouts are sent out Output: updated felds Fgure 4. The pseudo code of global search 4.5 The mechansm of varable colony sze In the proposed algorthm, the varable colony sze s acheved by transferrng part of the scouts between local search and global search, and settng a range for the number of random scouts. The rule governng how felds are merged has set a restrcton on the number of scouts n a feld. If the number of scouts grows above a specfed level n a feld, some scouts landng at low-ftness postons wll be released and added to the random scouts. The number of random scouts s hence ncreased. However t cannot exceed an upper boundary, otherwse some surplus random scouts wll be removed from the colony. In the global search process, the random scouts that have dscovered a potental ftness peak are becomng the scouts n felds and wll be nvolved n the local search process n the followng evolutonary teraton. In ths stuaton the number of scouts n felds ncreases

18 whle the random scouts decrease. A few random scouts wll be created by the algorthm f the number of current random scouts falls below the lower boundary. The colony sze s therefore ncreased. Through ths way the algorthm ensures the number of random scouts falls wthn the allowed boundares. Fgure 5 helps explan the mechansm of the varable populaton sze n a colony. Fgure 5. Varable populaton sze n a colony 5. Effects of new features used 5. Evoluton of the colony sze Two functons are used here to demonstrate how the colony sze evolves durng the optmsaton process. () Deb s functon: 6 f ( x) = sn (5π ) () x where x and the fve global maxmal at x =., x =. 3, x =. 5, x =. 7 x =.9 respectvely. () Two dmensonal multmodal functon:, and

19 f x) = x sn(4πx ) x sn(4πx + π ) () ( + where x, x. There are totally optmal solutons (ncludng local optma). Evoluton of colony sze Evoluton of colony sze Entre scouts Scouts n felds Random scouts Entre scouts Scouts n felds Random scouts 6 4 s t u o c s 5 4 s t u o c s f o r e b m u N 3 f o r e b m u N Cycle of teratons Cycle of teratons (a) (b) Evoluton of colony sze Evoluton of colony sze Entre colony Scouts n felds Random scouts Entre colony Scouts n felds Random scouts 5 4 s t u o c s f o 4 3 s n o t a r e t 8 r e b m u N f o e l c y C Cycle of teratons Number of scouts (c) (d) Fgure 6. Evoluton of the colony sze durng optmsaton: n (a) and (b) the algorthm starts wth a large and small populaton sze respectvely to search the ftness landscape of functon (); n (c) and (d) the algorthm starts wth a large and small populaton sze respectvely to search the ftness landscape of functon (). In the case of functon (), the algorthm starts wth 5 feld scouts and 5 random scouts, and then wth feld scouts and 3 random scouts. As can be seen n Fgure 6(a), the entre populaton drops dramatcally when the sze of the ntal colony s unnecessarly large to fnd the fve optma n the soluton space. The number of random scout grows at the begnnng

20 because the redundant scouts n felds are transferred to global search, and then declnes due to upper boundary of the number of random scouts. On the contrary, when the colony sze s nsuffcent for fndng many optma at the begnnng of the search (Fgure 6(b)), new members are gradually added to the colony so the colony sze keep growng untl enough for explorng the ftness peaks. The developmental pattern n Fgure 6(c) bascally matches that of 6(a), and the pattern n Fgure 6(d) wth 6(b). The two cases show that the populaton n an evolvng colony s able to adapt to the task, and fnally run n a relatvely stable state. 5. Effects of BST Also, a set of well-known functons are used to demonstrate the effects of BST, as gven n Table. Table. Functons used for evaluatng the effects of BST () Deb s functon (5 optma): 6 f ( x) = sn (5π ), x [, ] ; x () Deb s decreasng functon (5 optma): (( x.) /.9) 6 f ( x) = sn (5πx), x [, ]; (3) Roots functon (6 optma): f ( 3 x) =, x C, x = x 6 + x [, ] ; + x (4) Two dmensonal multmodal functon ( optma): f x) = x sn(4πx ) x sn(4πx + π ), x x [, ] ; 4 ( +, (5) Eght dmensonal multmodal functon (56 optma): f ( x, x 5,..., x 8 ) = 8 = sn(π ( 8 x ) 3 5 ), x [, ], =,,..., 8. For each functon, the algorthm starts wth the same parameter confguratons. For problem () and (), the colony starts wth feld scouts and 5 random scouts. The search

21 accuracy s set to be.. For functon (3), (4) and (5), these confguratons are,,.; 5, 3. and 3, 85,. respectvely. Table 3 compares the results of the algorthms usng random and balanced local search n terms of functon evaluaton, teraton cycles and the number of optma found. The functon evaluaton s the prmary crteron for comparng the performance of the varous algorthms. The statstcal sgnfcance of the dfference between the results s evaluated through student s t-tests. The t-tests are run wth a confdence level of 95% and the p-values are lsted n Table 4. The p-value below the sgnfcance level sgnal (.5) ndcates a statstcally sgnfcant dfference between the results obtaned by the two algorthms.. Table 3 shows that the BST allows the Bees Algorthm to fnd the optmal solutons usng about 3.8%, 3.4%, 43%, 3.% less functon evaluatons than the random local search based Bees Algorthm on functons (), (3), (4), (5) respectvely, and the correspondng p-values of.4,.5,.. n Table 4 are below the acceptance value.5, suggestng that the mprovement s statstcally sgnfcant. For functon (), The Bees Algorthm usng BST needs 8.% less functon evaluatons but the p-value ndcates the mprovement s not sgnfcant. The data shows both algorthms are successful n fndng all the optma. In these experments, the BST based Bees Algorthm requres less functon evaluatons and less teraton cycles on the majorty of functons than the basc Bees Algorthm. Therefore BST can be consdered as a promsng local search approach to speed up the algorthm. Table 3. Comparson betweem random local search and BST Test functon Local search method Functon evaluatons(std.) Iteraton cycles(std.) Optma found(std.)

22 () Deb s functon () Deb s decreasng functon (3) Roots functon (4) Multmodal functon (D) (5) Multmodal functon (8D) Random,96(7) 9.3(5.3) 5() BST,3(8) 6.(4.7) 5() Random,6(3) 9.5(6.8) 5() BST,67(54) 7.6(5.) 5() Random,3(69) 36.4(8.3) 5.8(.4) BST 9,96(63) 3.(8.) 5.8(.4) Random 77,86(,4) 59.9(7.3) 94.9(.75) BST 44,37(6,5) 3.6(5.3) 96.7(.7) Random 836,83(68,345) 49.(4.38) 47.(7.8) BST 77,777(66,564) 47.6(5.94) 5.8(6.7) Table 4. p-values showng statstcal sgnfcance Test functon Functon evaluatons Iteraton cycles Optma found () Deb s functon () Deb s decreasng functon (3) Roots functon.5. - (4) Multmodal functon (D) <. <. <. (5) Multmodal functon (8D) < Evoluton of feld radus Three benchmark functons are selected to demonstrate how the feld radus evolves as the algorthm proceeds. For vsualsaton purposes, two benchmarks are chosen to be two dmensonal (as plotted n Fgure 7) whlst the other one s four dmensonal, (Table 5). Table 5. Functons used for demonstraton of feld radus evolvement () Inverted Rastrgn (9 optma) ( x) = x cos( x ) = f π,.5 x. 5 ; () Fve hlls (Ursem functon, 5 optma) x 3 x x f ( x) = sn(.πx +.5π ) + sn(.5πx +.5π ) 3 x,

23 .5 x.5, x ; (3) Four dmensonal multmodal functon (6 optma) f ( x, x 3,. x 3 ) = 3 = sn(π ( 3 x ) 3 5 ), x [, ], =,, 3. For each functon, the algorthm s executed several tmes usng dfferent ntal feld rad, whlst all the other parameters are kept the same. For functon (), the radus s ntalzed at.5 and.. For functon () and (3), t s set to respectvely. and., and.3 and.3. Fgure 8 shows how the radus evolves as the search proceeds. The plots n Fgure 8 show two trends, whch can be summarsed as follows: the feld radus self-adapts at the begnnng, then reaches a relatvely steady value, and fnally fluctuates around ths value. Fgures 8 (e) and (f) show that the feld radus estmaton method s also applcable to mult-dmensonal problems. From ths group of tests, t can be concluded that the radus estmaton method makes the radus adaptve and less dependent on the preset ntal value. However, the fnal phase of fluctuatons ndcates there s stll room for enhancng the radus estmaton accuracy (a) (b) Fgure 7. functons for estmatng feld radus (a) functon ; (b)functon ()

24 Evoluton of feld radus Evoluton of feld radus s u d a R s u d a R Cycle of teratons Cycle of teratons (a) (b) Evoluton of feld radus Evoluton of feld radus s u d a R.6 s u d a R Cycles of teratons Cycles of teratons (c) (d) Evoluton of feld radus Evoluton of feld radus s u d a R..5 s u d a R Cycle of teratons (e) Cycle of teratons (f) Fgure 8. Evoluton of the feld radus durng the optmsaton process, (a)(c)(e) The algorthm starts wth a large feld radus for each functon; (b)(d)(f) The algorthm starts wth a small feld radus for each functon. 6. Evaluaton of the Bees Algorthm for MMO

25 Ths secton tests the proposed performance on optmsng MMO problems. The purpose s to underlne the Bees Algorthm s effcency for MMO, rather than focusng on ts superorty to other algorthms. 6. Expermental setup To evaluate the performance of the proposed algorthm, some commonly used multmodal functons of varous characterstcs, such as rregular landscape, symmetrc or equal dstrbuton of optma, unevenly spaced optma, multple global optma n the presence of multple local optma, are employed as gven n Table 6. Table 6. Benchmark functons for evaluatng the performance of the algorthm f : Two-peak trap ( global optma/ local optma) 6 (5 x), x < 5 5 f ( x) =, x. ( x 5), 5 x < 5 f : Central two-peak trap (/) 6 x, x < 6 f ( x) = ( x 5), x < 5, x 5 ( x 5), 5 x 5 f 3 : Fve-uneven-peak-trap (/3) 8(.5 x), x <.5 64( x.5),.5 x < 5 64(7.5 x), 5 x < 7.5 8( x 7.5), 7.5 x <.5 f ( x) =, x 3 8(7.5 x),.5 x < 7.5 3( x 7.5), 7.5 x <.5 3(7.5 x),.5 x < 7.5 8( x 7.5), 7.5 x < 3 f 4 : Equal maxma (5/)

26 6 f ( x) = sn (5πx), x f 5 : Decreasng maxma (/4) x. 6 f ( x) = exp log() sn (5πx), x.8 f 6 : Uneven maxma (5/) f ( x) = sn [5π ( x.5)], x f 7 : Uneven decreasng maxma (/4) x f ( x) = exp log() sn [5π ( x.5)], x.854 f 8 : Hmmelblau s functon (4/) f ( x) = ( x x, 6 x 6 + x ) ( x + 7) f 9 : Sx-hump camel back (/) r 4 f ( x) = 4[(4.x + x ) x + xx + ( 4 + 4x ) 3.9 x.9,. x. x ], f : Shekel s foxholes (/4) r f ( x) = b( )] 6 = + + [ x a( )] + [ x b ( ) = 6( ) ; x f : Inverted Shutert (8/many) r f ( x) = 5 = j= j cos[( j + ) x + j], x 6,where a ( ) = 6( mod5 ), f : D nverted Vncent functon (6/) f 3 : D nverted Vncent functon (36/) f 4 : 3D nverted Vncent functon (6/) n f ( x r ) = sn[ log( x )],.5 x n = The performance of Bees Algorthm for MMO s compared wth that of rpsols, r3psols, rpsolhcls, r3psolhcls and IWO-δ-GSO. In the experments, the parameter settngs are matched as closely as possble wth those used by the other algorthms under comparson.

27 However, due to the dfferent mechansms of the algorthms, some of the parameters are dmensonalty depended, that s, nre= (D+), nrb=d+ and stlm=5 D. The ntal ngh s correlated wth the search range: ngh= range/ns. Table 7 lsts the other parameter settngs for the experments. 6. Results and dscusson Table 7. Intal colony sze, ntal feld radus and search resoluton No. ns nr rad resoluton f -f f 4 -f 7, f f f 3. f 5..5 f 4.. f f The success rate and average number of optma found are recorded and presented n Table 8 and Table 9 respectvely, and agan p-values are gven n Table to show the statstcal sgnfcance of the performance dfferences between the compared algorthms and the proposed Bees Algorthm (wth confdence level of 95%) n the number of optma. Some p-values cannot be obtaned because the compared and the proposed algorthm can both reach % success rate n fndng the optma. The best performance s reported n boldface. As can be seen from Table 8, the proposed Bees Algorthm and the IWO-δ-GSO generally perform better than others. The functons f, f, f 7 and f are the easest, and all the algorthms can fnd the optma. The r3psolhcls s the only algorthm that cannot reach % success rate n fndng the 5 optma of the functon f 4, and the correspondng p-value of. (below.5) ndcates the proposed Bees Algorthm sgnfcantly dffers from r3psolhcls. For functon f 6, all the algorthms except rpsols can fnd the 5 optma wth % success rate,

28 and the p-value of.346 suggests the dfference between the results s sgnfcant. For functons f 3, f 5 and f, the proposed algorthm and IWO-δ-GSO show a dstnctve advantage over the other algorthms, snce they are capable of locatng all the optma wth % success rate whle the other algorthms cannot. Furthermore, the p-values ndcate that the dfferences n results are statstcally sgnfcant, except for the rpsolhcls on functon f. The success rate and the number of optma found show that the proposed Bees Algorthm outperforms the other algorthms on functon f 8, and the correspondng p-values mply the results dfference s sgnfcant except for the IWO-δ-GSO. However, the data from the three tables dsplay the nferor performances of the IWO-δ-GSO, r3psols and r3psolhcls to the Bees Algorthm, rpsols and rpsolhcls on the functon f 9. For f, even though rpsolhcls generates the hghest success rate, the correspondng p-values suggest all the algorthms except the r3psols produce statstcally smlar results. On functons f 3 and f 4 all algorthms fal to acheve a non-zero success rate. However, the average number of optma found, and the respectve p-values, mply that the proposed Bees Algorthm and IWO-δ-GSO produce statstcally resemblng results, and that they outperform the other algorthms except for r3psols on f 3. Table 8. Success rate No. BA (%) rpso ls(%) r3pso ls(%) rpso lhcls(%) r3pso lhcls(%) IWO-δ-GSO (%) f f f f 4 96 f f 6 96 f 7 f f f

29 f f f 3 f 4 Table 9. Average number of optma found No. BA rpsols r3psols rpsolhcls r3psolhcls IWO-δ-GSO f f f f f f f f f f f f f f Table. p-values showng statstcal sgnfcance No. rpsols r3psols rpsolhcls r3psolhcls IWO-δ-GSO f f f 3 <. <..5 <. - f f 5 <. <..57 <. - f f f f <. f f f f f Assocated dagrams showng bees dstrbutons n search space Functon f has two peaks, of whch one s global and the other s local. Fgure 9 shows a smulaton run of the proposed Bees Algorthm for MMO on f startng from 5 bees. The

30 ntal populaton s much larger than the necessary for fndng only two optmal solutons. Most of the members n the colony become redundant and are removed. After 3 functon evaluatons, the bees get attracted towards the global optmum at x=, and the local optmum at x=. Fnally both peaks are located by the bees. 5 5 ftness ftness x 5 5 x (a) ntalsaton (b) at 3 evaluatons Fgure 9. Dstrbuton of ndvduals n the search space durng the evoluton process for f 5 5 ftness ftness x 5 5 x (a) ntalsaton (b) at evaluatons Fgure. Dstrbuton of ndvduals n the search space durng the evoluton process for f 5 5 ftness ftness x x (a) ntalsaton (b) at 36 evaluatons Fgure. Dstrbuton of ndvduals n the search space durng the evoluton process for f 3

31 Also functon f has two peaks: one s the global peak and the other s the local one. Fgure shows two snapshots from a sample smulaton run of the Bees Algorthm wth an ntal colony sze of 5 bees, the frst snapshot taken at ntalsaton and the other at functon evaluatons. f 3 has fve peaks, of whch two are global and three are local. Fgure shows two snapshots of a sample run of the Bees Algorthm on f3 at ntalsaton and after 36 functon evaluatons..8.8 ftness.6.4 ftness x x (a) at 5 evaluatons (b) at 57 evaluatons Fgure. Dstrbuton of ndvduals n the search space durng the evoluton process for f ftness.6.4 ftness x x (a) at 8 evaluatons (b) at 575 evaluatons Fgure 3. Dstrbuton of ndvduals n the search space durng the evoluton process for f 6

32 .8.8 ftness.6.4 ftness x x (a) at 8 evaluatons (b) at 558 evaluatons Fgure 4. Dstrbuton of ndvduals n the search space durng the evoluton process for f 5 Functons f 4 and f 6 have fve global peaks. The peaks of f 4 are evenly spaced, whereas the peaks of f 6 are unevenly spaced. Fgure and Fgure 3 show two stages n a sample run on f 4 and f 6 respectvely. It shows that the Bees Algorthm s able to fnd all the global and local peaks of f 4 and f 6 usng very few functon evaluatons. Functons f 5 and f 7 have one global peak and four local peaks. The peaks n f 5 are evenly spaced, whereas n f 7 they are unevenly spaced. Fgure 4 and Fgure 5 show a sample run of the Bees Algorthm on f5 and f7 respectvely. The above fgures show the colony dstrbutons of the Bees Algorthm and llustrate the algorthm s ablty to detect all the peaks. Functon f 8 has four global peaks. Fgure 6 shows four stages of the search process of a smulaton run of the Bees Algorthm on f8, usng ntally 5 bees, and sampled at: ntalsaton, after 495 evaluatons, 36 evaluatons and 736 evaluatons. Functon f has 5 evenly spaced peaks of unequal heghts, of whch one s the global peak whlst the others are local peaks. Fgure 7 plots the evoluton of the search process on functon f at the ntalsaton stage, after 9 functon evaluatons, 5735 functon evaluatons and 853 functon evaluatons. Also on ths functon the algorthm locates all the

33 peaks successfully..8.8 ftness.6.4 ftness x x (a) at 9 evaluatons (b) at 63 evaluatons Fgure 5. Dstrbuton of ndvduals n the search space durng the evoluton process for f 7 ftness - ftness x -5-5 x 5-5 x -5-5 x 5 (a) ntalsaton (b) after 495 evaluatons ftness - ftness x -5-5 x 5-5 x -5-5 x 5 (c) after 36 evaluatons (d) after 736 evaluatons Fgure 6. Dstrbuton of ndvduals n the search space durng the evoluton process for f 8

34 x -5-5 x x -5-5 x (a) ntalsaton (b) after 9 evaluatons x -5-5 x x -5-5 x (a) after 5735 evaluatons (b) after 853 evaluatons Fgure 7. Dstrbuton of ndvduals n the search space durng the evoluton process for f For vsualsaton reasons only a part of the benchmark functons showng how the bees dstrbute n the optmsng process are llustrated. The results clearly ndcate the hgh effcency of the Bees Algorthm n solvng MMO problems. The Bees Algorthm exhbts good exploratve and explotatve abltes of locatng global and local peaks. At the same tme, the Bees Algorthm needs very few functon evaluatons to locate the optma. The speed of the Bees Algorthm s lkely due to ts adaptve colony sze, n whch surplus bees are removed. 7. Concluson Ths paper has proposed an evolutonary optmsaton technque based on the basc Bees Algorthm for solvng MMO problems. Some fundamental concepts of the basc Bees Algorthm are retaned, such as the mechansm of combnng exploraton and explotaton,

35 and the waggle dance whch allocates foragers accordng to the ftness of the dscovered flower patches. The basc Bees Algorthm has been mproved by addng several new procedures, whch are also partly nspred by the honeybees natural behavour. The experments proved the ablty of the proposed algorthm to solve MMO problems. Ths stems from the capablty of the Bees Algorthm to adjust the number of ndvduals accordng to the complexty of the search space. Ths ablty s also enhanced by the proposed local search method called balanced search technque (BST), whch gudes the foragers towards the ftness gradent whlst stll retanng some randomness for exploratve purposes. The HV s executed n the global search stage to detect good solutons n yet uncharted regons of the search space. The expermental results show that the proposed Bees Algorthm s compettve for the MMO problems compared wth other algorthms. A frst step to extend the current work may be to stablse the feld radus. Currently, ts fluctuaton around an estmated value may degrade the accuracy of locatng optmal solutons n the search space. Secondly, the BST can be further studed or modfed for other optmsaton problems lke fnd one global optmum or trackng a dynamc optmum. Fnally, a very mportant future research ssue would be to develop a tool for estmatng the rato of the number of scouts n felds to the number of random scouts. The partton n scouts drectly determnes how the algorthm balances between explotatve and exploratve search. The former decdes the accuracy and speed of convergence when searchng around an dentfed soluton and the latter affects the ablty of explorng potental spaces. Acknowledgement

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