An Adaptive Multi-population Artificial Bee Colony Algorithm for Dynamic Optimisation Problems

Transcription

1 *Revsed Manuscrpt (changes marked) Clck here to vew lnked References An Adaptve Mult-populaton Artfcal Bee Colony Algorthm for Dynamc Optmsaton Problems Shams K. Nseef 1, Salwan Abdullah 1, Ayad Turky 2 and Graham Kendall 3,4 1 Data Mnng and Optmsaton Research Group, Centre for Artfcal Intellgence Technology, Unverst Kebangsaan Malaysa, Bang, Selangor, Malaysa. E-mal: shams.shamosa91@gmal.com; salwan@ukm.edu.my 2 Swnburne Unversty of Technology, Melbourne, Vctora, Australa E-mal: aturky@swn.edu.au 3 Unversty of Nottngham Malaysa Campus, Semenyh, Malaysa 4 ASAP Research Group, Unversty of Nottngham, Nottngham, Unted Kngdom Emal: Graham.Kendall@nottngham.edu.my; Graham.Kendall@nottngham.ac.uk Abstract Recently, nterest n solvng real-world problems that change over the tme, so called dynamc optmsaton problems (DOPs), has grown due to ther practcal applcatons. A DOP requres an optmsaton algorthm that can dynamcally adapt to changes and several methodologes have been ntegrated wth populaton-based algorthms to address these problems. Mult-populaton algorthms have been wdely used, but t s hard to determne the number of populatons to be used for a gven problem. Ths paper proposes an adaptve multpopulaton artfcal bee colony (ABC) algorthm for DOPs. ABC s a smple, yet effcent, nature nspred algorthm for addressng numercal optmsaton, whch has been successfully used for tacklng other optmsaton problems. The proposed ABC algorthm has the followng features. Frstly t uses mult-populatons to cope wth dynamc changes, and a clearng scheme to mantan the dversty and enhance the exploraton process. Secondly, the number of sub-populatons changes over tme, to adapt to changes n the search space. The movng peaks benchmark DOP s used to verfy the performance of the proposed ABC. Expermental results show that the proposed ABC s superor to the ABC on all tested nstances. Compared to state of the art methodologes, our proposed ABC algorthm produces very good results. Keywords: dynamc optmsaton, artfcal bee colony algorthm, adaptve mult-populaton method, meta-heurstcs 1. Introducton Many real-world optmsaton problems have the characterstc of changng over tme n terms of decson varables, constrants and the objectve functon [1], [2]. These problems

2 are referred to as dynamc optmsaton problems (DOPs) n the scentfc lterature. A DOP requres an optmsaton algorthm that can dynamcally adapt to the changes and track the optmum soluton durng the executon of the algorthm [2]. Gven ther practcal applcatons and complexty, DOPs have attracted a lot of research attenton. Populaton-based algorthms, whch are a set of methodologes that utlse a populaton of solutons dstrbuted over the search space, have attracted partcular attenton, due to ther good performance [1], [2]. A key challenge n developng an optmsaton algorthm for DOPs s how to mantan populaton dversty durng the search process n order to keep track of landscape changes [3]. Several nterestng dversty schemes have been developed n order to mprove the search capablty of populaton-based algorthms so that they can adapt effectvely to the problem as t changes. Predcton based methods s one of the dversty schemes whch has been wdely ntegrated wth other algorthms to mantan the dversty. Ths methodology uses an algorthm to learn patterns from prevous searches, whch are then used to predct future changes. It should be noted that memory methods can be categorsed as a specal case of the predcton method as they store a set of solutons to be used when a problem changes [2]. The predcton method s sutable for problems wth cyclc changes. Hatzaks and Wallace [4] proposed a hybrd algorthm that combnes an evolutonary algorthm and a forecastng methodology for DOPs. Forecastng s used to predct the movement of the optmum based on prevous movements. The results demonstrate that ths method s sutable for problems whch change quckly f the movement of the optmum soluton s predcted correctly. Sm et al. [5] used a predcton based method to predct how the envronment would change and the tme of the next change. The authors utlsed a Markov chan that uses the prevous movement of the search n order to predct future changes and a lnear regresson to predct when the change wll occur. The results demonstrate that the hybrd algorthm performs wth predcton, than wthout. Branke and Mattfeld [6] proposed an antcpaton-based algorthm for DOPs. Ths algorthm attempts to smultaneously search for a good qualty soluton and move the search nto a dfferent area based on the prevous changes. Ths proposed algorthm was tested on a dynamc job-shop schedulng problem and t was shown to produce very good results compared to other algorthms. The advantage of the predcton method s that t can be effectve n detectng the global optma quckly, f the predctons are accurate [2]. The man drawback wth ths method s that t depends on the tranng model and n many cases the data used durng the

3 tranng process does not capture real world scenaros and there s a possblty of tranng errors due to lack of tranng data [7], [2]. Memory based methodologes am to mantan dversty. They use a memory wth a fxed sze to store some of promsng solutons that are captured durng the search process. When a change s detected, the stored solutons wll be renserted nto the current populaton and the populaton wll be fltered to nclude only the best solutons. Examples of memory based methodologes can be found n [8], [9], [10], [11]. These methodologes have worked well when the dynamc problems are perodcal or cyclc. The drawback s that they have parameter senstvtes that need to be determned n advance, and most real world problems are not cyclc n nature. Self-adaptve algorthms attempt to adaptvely mprove the dversfcaton of populatonbased algorthms based on envronmental changes. They use mechansms to adapt the algorthm to the changes n the search space [2]. Adaptve mechansms can mprove algorthm search behavour and also reduce the need for manual parameter tunng. The dea s to apply dfferent operators or parameter values for dfferent problems by adaptvely changng them durng the search process [7], [2]. Grefenstette [12] proposed a self-adaptve genetc algorthm for DOPs. The proposed algorthm adaptvely selects dfferent crossover/mutaton operators at each generaton. The author uses an agent based concept to control the selecton process, and each agent represents a crossover or mutaton operator. All agents are executed smultaneously and the one that generates the best soluton s selected for the current nstance. Promsng results were acheved when compared to other algorthms. Grefenstette [12] also proposes an dea called a genetc mutaton rate for the DOP. The dea s to set the value of the mutaton rate based on the ftness of the populaton. Ths dea was shown to generate better results compared to the basc genetc algorthm. Ursem [13] proposed a multnatonal genetc algorthm for the DOP. The man parameters are encoded wth the decson varables and are evolved durng the soluton process. The results show that ths algorthm s very good for smple nstances n whch the velocty of the movng peaks s constant. It s also able to adapt by changng the algorthm parameters durng the search. However, encodng the parameters wth the soluton decson varables requres specalst evolutonary operators. In addton, t s also very dffcult to determne the values of the parameters [2].

4 Mult-populaton methods mprove dversty by dvdng the populaton of solutons nto several sub-populatons and dstrbutng them throughout the search landscape so that they can more effectvely capture the problem changes. The dea s to mantan populaton dversty by assgnng a dfferent sub-populaton to a dfferent area, where each one s responsble for ether ntensfyng or dversfyng the search process [7], [2]. These subpopulatons nteract wth each other va a merge and dvde process when a change n the envronment s detected. The mult-populaton method has been shown to be effectve n dealng wth varous problem changes, whether they are cyclc or non-cyclc, and t has outperformed other methods on varous problem szes. Branke et al. [14] proposed a selforgansng scouts mult-populaton evolutonary algorthm for the DOP. The populaton of solutons s dvded nto two groups; small and large. The small populaton group s responsble for trackng promsng solutons found so far, whle the large populaton group tres to fnd a new regon of the search space that has a new peak. The proposed algorthm was tested on the movng peaks benchmark (MPB), obtanng very good results. Blackwell and Branke [15] proposed a mult-swarm optmsaton algorthm for the DOP. The swarm s dvded nto subsets of swarms. These mult-swarms nteract wth each other locally, through algorthm parameters, and globally by usng an ant-convergence mechansm. The antconvergence mechansm searches for new peaks by removng the worst ones and rentalsng them nto a dfferent area n the search space. The proposed algorthm obtaned very good results when tested on MPB problems. Mendes and Mohas [16] presented a multpopulaton dfferental evoluton algorthm for the DOP. The populaton of solutons s dvded nto several sub-populatons. Each sub-populaton s assgned to a dfferent area of the search space. The expermental results show that ths algorthm obtans very good results for MPB problems. L and Yang [17] proposed a fast mult-swarm Partcle Swarm Optmsaton (PSO) algorthm for the DOP. The swarm populaton s dvded nto two types of swarms; parents and chldren. The parent swarm explores the entre search space to seek the global optma, whle the chld swarm s responsble for montorng the search behavour around the best soluton obtaned by the parent swarm. The poston of the chld swarm s dynamcally updated durng the process. The algorthm was tested on the MPB problems and produced good results when compared to other methods. Yang and L [18] presented a clusterng-based partcle swarm optmser for the DOP. The swarm s dvded based on a herarchcal clusterng method to locate and track multple peaks. The algorthm acheved

5 very good results when tested on the MPB. Turky and Abdullah [19] proposed a multpopulaton electromagnetc algorthm for DOPs. The proposed algorthm dvdes the populaton nto several sub-populatons to smultaneously explore and explot the search process. The algorthm was tested on MPB problems and obtaned very good results when compared to other populaton dversty mechansms. The same authors [20] also presented a mult-populaton harmony search algorthm for the DOP. The populaton s dvded nto subpopulatons. Each sub-populaton s responsble for ether explorng or explotng the search space. An external archve s utlsed to track the best solutons found so far, whch are used to replace the worst ones when a change s detected. The results show that ths algorthm produces good results when compared to other methods. Sharf et al. [21] proposed a hybrd PSO and local search algorthm for DOPs. The algorthm utlses a fuzzy socal-only model to locate the peaks. The results show that ths algorthm can produce very good results for MPB problems. In L et al. [22] comprehensve expermental analyss was reported on the performance of a mult-populaton method wth varous algorthms n relaton to DOPs. The authors concluded that the mult-populaton method s able to deal effectvely wth varous DOPs and has the ablty to mantan populaton dversty. It s also able to help the search n locatng a new area through a dvde and merge process and nformaton exchange. The authors also hghlghted several weaknesses of ther method that relate to the number of subpopulatons, the dstrbuton of solutons and the reacton to problem changes. Exstng works on DOPs demonstrate that employng mult-populaton methods are the most effectve method n mantanng populaton dversty. The features that make the multpopulaton methodologes popular are [3]: ) t dvdes the populaton nto sub-populatons, where the overall populaton dversty can be mantaned snce dfferent populatons can be located n dfferent areas of the problem landscape, ) t has the ablty to search dfferent areas smultaneously, enablng t to track the movement of the optmum, and ) varous sngle populaton-based algorthms can be ntegrated wthn mult-populaton methods. Although mult-populaton methods have shown success when appled to DOPs, most of them use a number of sub-populatons and the populaton dversty s mantaned only through the sub-populaton dstrbuton [3]. The number of sub-populatons has a crucal mpact on algorthm performance as t relates to the dffculty of the problem, whch s not known n advance, and changes durng the search. In addton, the solutons n the subpopulatons may not be dverse enough as some methods are only concerned wth how to

6 dvde the populaton nto sub-populatons, rather than focussng on dversfcaton. To address these ssues, ths work proposes an adaptve mult-populaton artfcal bee colony (ABC) algorthm for the DOP. The proposed ABC utlses a clearng scheme to remove redundant solutons n order to mantan dversty and enhance the exploraton process. To effcently track the landscape changes, the proposed ABC algorthm adaptvely updates the number of sub-populatons based on the problem change strength. In ths paper, the key objectves are:. To propose an artfcal bee colony algorthm that utlses a mult-populaton and a populaton clearng scheme to effcently solve the dynamc optmsaton problem.. To propose an adaptve mult-populaton algorthm that updates the number of the sub-populatons based on the problem change strength.. To test the performance of the proposed algorthm on dynamc optmsaton problems usng dfferent scenaros and compare the results wth other methodologes. We used the movng peaks benchmark DOP wth a dfferent number of peaks to evaluate the effectveness of the proposed ABC. Results demonstrate that the proposed ABC performs better than a basc ABC on all tested scenaros. Compared to the state of the art method, the proposed ABC produces very good results for many nstances. 2. The proposed algorthm Ths secton presents the basc artfcal bee colony algorthm, as well as our proposed adaptve mult-populaton algorthm. 2.1 Basc artfcal bee colony algorthm The Artfcal Bee Colony (ABC) algorthm s a smple, yet effcent, nature nspred algorthm for addressng numercal optmzaton problems. It was proposed n [23] as a nature nspred swarm ntellgence algorthm based on the observaton of bee foragng behavour. In ABC, there are a set of food sources and a set of bees. The qualty of the food sources s based on the amount of nectar they contan. Bees search and collaborate wth each other, seekng better food sources. To address an optmzaton problem usng ABC, food sources represent the populaton of solutons for a gven problem and bees are categorsed nto three types: scout, employee and onlooker bees. The amount of nectar corresponds to the

7 qualty (objectve functon) of the problem beng addressed. The three types of bees work together n an teratve manner to mprove the qualty of the populaton of solutons (food sources). The pseudo-code of a basc ABC s shown n Algorthm 1 [24]. ABC frst sets the man parameters, ntalzes the populaton of solutons and then evaluates them. Next, the man loop s executed n an attempt to solve the gven optmsaton problem by callng the employee bees, onlooker bees and scout bees untl the stoppng condton s satsfed. Algorthm 1: The pseudo-code of basc ABC Step 1: Set the parameter values Step 2: Intalze the populaton of solutons Step 3: Evaluate the populaton of solutons whle termnaton condton s not met do Step 4: Employed Bees step Step 5: Onlooker Bees step Step 6: Scout Bees step end whle The basc ABC has the followng steps: Step 1- Set ABC parameters. In ths step the man parameters of ABC are ntalzed. These nclude: the maxmum number of teratons (MaxIt) whch represents the stoppng condton of ABC, the number of solutons or populaton sze (Ps) whch represent how many solutons wll be generated, the total number of bees (Sbees) whch s set to be twce the sze of Ps, where half of them are employee bees and the other half are onlooker bees, the lmt parameter (Lt), whch s used to determne f the soluton should be replaced by a new one. Step 2- Intalse the populaton of solutons. A set of solutons wth sze equal to Ps are randomly generated as follows: x mn max mn x Rand [0,1]( x x ) (1), j, j, j, j where s the ndex of the soluton, j s the current decson varable, Rand [0,1] generates a random number between zero and one and upper bonds for the j th decson varable. mn x, j and max x, j are the lower and

8 Step 3- Evaluate the populaton of solutons. The ftness (qualty) of the generated solutons are calculated usng the objectve functon. The objectve functon s problem dependent. The objectve functon used n ths work s shown n Secton 3.2. Step 4- Employed bees. Each employee bee s sent to one food source (soluton). Its man role s to explore the neghbourhood of the current soluton, seekng an mprovng soluton. A neghbourhood soluton, v, s created by modfyng the th soluton, x, as follows: v, j, j, j (, j k, j x x x ) (2) where k s a randomly selected soluton from Ps and Φ s a random number between [-1, 1]. The generated neghbourhood soluton wll be replaced wth current soluton f t has better ftness. Step 5- Onlooker bees. Onlooker bees seek to mprove the current populaton of solutons by explorng ther neghbourhood usng Equaton (2), the same as the employee bee. The dfference s that onlooker bees select the solutons probablstcally based on ther ftness values as follows: p Ps j ftness ftnessj 1 (3) That s, the soluton wth the hgher ftness has a hgher chance of beng selected (.e. roulette wheel selecton). Onlooker bees use a greedy selecton mechansm, where the better soluton n terms of ftness s selected. Step 6- Scout bees. Ths step s actvated f both employed and onlooker bees cannot mprove the current soluton for a number of consecutve teratons defned by the lmt parameter, Lt. Ths ndcates that the current soluton s not good enough to search ts neghbourhood and t should be dscarded. In ths case, the scout bee wll generate a new

9 soluton usng Equaton (1) to replace the dscarded one. Ths can help ABC to escape from a local optmum and explore a dfferent area of the search space. 2.2 The proposed artfcal bee colony algorthm Exstng works on DOPs have demonstrated that mult-populaton methods are state of the art, n that they outperform other methods on many scenaros. However, although multpopulaton methods have acheved success n solvng DOPs, most of them use a fxed number of sub-populatons and the populaton dversty s mantaned through the subpopulaton dstrbuton. To address these ssues, ths work proposes an adaptve populaton ABC (denoted as Mult-pop-ABC). In Mult-pop-ABC, three major modfcatons are added to the basc ABC. These are:. Mult-populaton method. To deal wth DOP, the proposed ABC uses a mult-populaton method to dvde the populaton nto several sub-populatons. By usng a mult-populaton method, the solutons are scattered over the search space nstead of focusng on a specfc area. Thus the algorthm can generate hgh qualty solutons and track the problem changes.. Adaptve scheme. To track the landscape changes that occur durng the search process, the proposed Mult-pop-ABC updates the number of sub-populatons based on the strength of the problem change. That s the number of sub-populatons s ether decreased or ncreased durng the search process. By usng the proposed adaptve method, the number of subpopulatons can be changed adaptvely based on the strength of the envronment changes, whch helps the search track the optmum soluton and also mproves the dversfcaton and exploraton processes.. Populaton clearng scheme. To ensure that the solutons are dverse enough, a populaton clearng scheme s called when a change s detected to delete redundant solutons and replace them wth new solutons. Ths scheme removes redundant solutons n order to mantan dversty and enhance the exploraton process. The flowchart of the proposed Mult-pop-ABC for DOPs s shown n Fgure 1. It starts by settng the parameter values. It creates the populaton of solutons and then evaluates them. Next, the populaton of solutons s dvded nto m sub-populatons. Each sub-

10 populaton utlses an ABC algorthm. If a change n the problem s detected, the algorthm calculates the change strength to update the sub-populaton sze and checks the stoppng condton. If the specfed stoppng condton (we set ths as a maxmum number of ftness evaluatons) has been reached, the algorthm termnates and the best soluton s returned. Otherwse, the algorthm merges all the sub-populatons, updates the populaton, runs the clearng method, re-dvdes the populaton nto m sub-populatons and starts a new teraton. The man steps are descrbed n further detal below: - Step 1: Set parameters. The man parameters of Mult-pop-ABC are ntalsed. The algorthm has fve parameters. Four of them are the same as the basc ABC. These are: the maxmum number of teratons (MaxIt), populaton sze (Ps), number of bees (Sbees), and the lmt parameter (Lt). The ffth parameter s the sub-populaton sze (m), whch represents the number of sub-populatons (Ps/m). Intally, m=2 and durng the search process, t s ether decreased or ncreased. 1- Step 2: Intalse the populaton of solutons. Same as Step 2 n the basc ABC, Secton Step 3: Evaluate the populaton of solutons. Same as Step 3 n the basc ABC, Secton Step 4: Dvde the populaton. The populaton of solutons s dvded nto m subpopulatons (Ps/m). Each sub-populaton s assgned to explore a dfferent area of the search space. These sub-populatons nteract wth each other through mergng and redvdng every tme a change n the envronment s detected. Each soluton n the populaton s randomly assgned to a sub-populaton. The number of sub-populatons m s ether ncreased or decreased based on the envronment change strength. The ntal value of m s set to two (m=2) and t s updated durng the search. 4- Step 5: Assgn ABC to each sub-populaton. Each sub-populaton has ts own ABC algorthm. Each ABC executes all the steps presented n Secton 2.1. It starts wth a

11 populaton of solutons and teratvely calls the followng untl the stoppng condton s satsfed (the algorthm stops when a change n the envronment s detected):. Employee bees. Same as Step 4 n the basc ABC, Secton Onlooker bees. Same as Step 5 n the basc ABC, Secton Scout bees. Same as Step 6 n the basc ABC, Secton Step 6: Check the change strength. Ths step s actvated when a change n the envronment s detected. Its man role s to update the number of sub-populatons based on the envronment change strength. It frst calculates the objectve functon of the best soluton before and after the envronment change as follows: Cs f ( best _ before) f ( best _ after) (4) where Cs s the change strength, f(best_before) s the qualty of the best soluton before the envronment change and f(best_after) s the qualty of the best soluton after the envronment change. If the Cs s less than the defned threshold (Tv) and m s greater than 2, the number of sub-populatons m s decreased as the algorthm needs to be more explotve than exploratve (m=m-1). Otherwse, m s ncreased by one wth the am of ncreasng the exploraton aspect of the search (m=m+1). It should be noted that when m s an odd number, the extra soluton s randomly assgned to one of the sub-populatons. 6- Step 7: Check the stoppng condton. Ths step checks the termnaton crteron of the search process. In ths work, t s set as a maxmum number of ftness evaluatons n lne wth prevous works. If the specfed stoppng condton s reached, the search process stops and returns the best soluton. Otherwse, the algorthm performs the followng processes:. Populaton clearng scheme: Ths scheme calculates the smlarty between solutons n the populaton. The smlarty s calculated by usng a matchng algorthm, whch matches each par of solutons n terms of phenotype. Two

12 solutons are smlar f they have the same values n all the cells of both solutons. If two or more solutons are smlar, these solutons are deleted and replaced wth randomly generated ones.. Populaton update: All sub-populatons are merged to form one populaton.. Re-dvde the populaton: The populaton s re-dvded nto m sub-populatons and the algorthm contnues by startng the process at step 1 wth a new generaton.

13 Fgure 1. The proposed Mult-pop-ABC 3. Expermental Setup Ths secton dscusses the Movng Peak Benchmark (MPB), evaluaton metrc and the parameter settngs.

14 3.1 The Movng Peak Benchmark The movng peak benchmark (MPB) s a maxmzaton dynamc contnuous optmzaton problem proposed by [9], [25], and has been commonly used as a testbed for the performance of optmsaton algorthms. MPB conssts of a set of peaks that move over the problem landscape. It takes the gven soluton as an nput and returns the value of the hghest peak. The returned value represents the qualty of ths soluton. MPB can be mathematcally expressed as follows: H( t) F( x, t) max (5) 1,..., p D 2 1W ( t) ( x j( t) X j( t)) j1 where F(x, t) s the qualty of soluton x at tme t, p s the number of peaks, D s the problem dmenson (number of decson varables where each varable has an upper and lower boundary (DB)), H (t) s the heght of peak, W (t) s the wdth of peak, and X j s the j th element of the locaton of peak. Note that Equaton (5) s a statonary optmzaton problem. Thus, to change t to a dynamc problem, MPB randomly shfts the poston of all peaks by vector v degree) as follows: of a dstance s (s s also known as the shft length that determnes the severty s v ( t) ((1 ) r v ( t 1)) (6) r v ( t 1) where r s a random vector, λ s the correlaton between consecutve movements of a sngle peak that takes ether 0 f the movement of peaks are completely uncorrelated or 1 f they move n the same drecton. To make a far comparson wth exstng algorthms, n ths paper, we used λ=0 [6]. The change of heght and wdth of a peak n a gven soluton can be mathematcally expressed as follows: H ( t) H ( t 1) heght _ severty (7) W ( t) W ( t 1) wdth _ severty (8) where heght_severty and wdth_severty are calculated based on the problem severty. σ s a normally dstrbuted random number between 0 and 1. Then, the change of a soluton x s gven as follows:

15 X ( t) X ( t)( t 1) v ( t) (9) The change frequency (cf) occurs every 5,000 ftness evaluatons [9]. The parameter values of all MPBs that have been used n our experments are shown n Table 1 [25]. Table 1 MPB parameter values Parameters Descrpton Value p Number of peaks cf Change frequency 5000 heght_severty Heght severty 7.0 wdth_severty Wdth severty 1.0 Peak shape Peak shape Cone s Shft length 1.0 D Number of dmensons 5 λ Correlaton coeffcent 0 DB Each dmenson boundares [0,100] H Peak heght [30.0,70.0] W Peak wdth [1,12] 3.2 Evaluaton Metrc To farly compare the proposed ABC wth exstng algorthms, we use the same evaluaton metrc known as the offlne error as suggested by [25]. Ths has also been used by other researchers. The offlne error s calculated as follows: off 1 g g 1 (10) where g s the number of generatons and Ω s the best performance snce the last change at th ftness evaluaton. 3.3 Parameter Settngs The parameter values of our Mult-pop-ABC are set by carryng out a set of ntal experments, wth the excepton of the stoppng condton whch was set to be the same as the compared algorthms (50,000 ftness evaluatons). For each parameter, we tested varous values and the best values were selected. Ths s acheved by varyng the value of one parameter whle fxng others. We have selected two scenaros of MPB for the parameter tunnng process: 50 peaks and 200 peaks. The proposed ABC has three parameters:

16 populaton sze (Ps), lmt (Lt) and the change strength threshold (Tv). Frst, we fxed Lt to 30, Tv to 0.09 and changed Ps. Table 2 shows the offlne error of varous Ps values for 50 and 200 peaks. The best result s hghlghted n bold. Next, we fxed Ps to 60, Tv to 0.09 and changed Lt as shown n Table 3. Fnally, we fxed Ps to 60, Lt to 30 and changed Tv as shown n Table 4. The parameter settngs of the proposed ABC that were used across all scenaros are presented n Table 5. Table 2 The value of Ps parameter Ps value 50 peaks 200 peaks Table 3 The value of Lt parameter Lt value 50 peaks 200 peaks Table 4 The value of Tv parameter Tv value 50 peaks 200 peaks Table 5 The parameter settngs of the proposed ABC # Parameter Value 1- Maxmum number of teratons (MaxIt) 50,000 ftness evaluatons 2- Populaton sze (Ps) Lmt parameter (Lt) Change strength threshold (Tv) Results We carred out three set of experments. In frst one, we compare the results of Mult-pop- ABC wth the basc ABC. In second one, the results obtaned by Mult-pop-ABC are

17 compared wth state of the art methods. In the thrd experment, the results of Mult-pop-ABC on well-known test functons are compared wth state of the art methods. 4.1 Results comparson of Mult-pop-ABC and the basc ABC Ths secton ams to verfy the effectveness of the addtonal components that we have added to the basc ABC. Specfcally, the objectve s to nvestgate the mpact of the proposed enhancements on the performance of the basc ABC when dealng wth DOPs. Four dfferent algorthms were derved as follows: - Mult-pop-ABC: the proposed ABC that utlses the adaptve mult-populaton and populaton clearng scheme - Mult-pop-ABC 1 : same as above but wthout the populaton clearng scheme - Mult-pop-ABC 2 : same as above but uses a fxed number of sub-populatons and wthout the populaton clearng scheme. The sub-populatons were fxed to be the same as [26] - ABC: basc ABC algorthm. The computatonal comparsons of Mult-pop-ABC, Mult-pop-ABC 1, Mult-pop-ABC 2 and basc ABC are presented n Table 6. The comparson s n terms of the offlne error, ± standard error for each number of peaks. The best results are hghlghted n bold. The results clearly show the good performance of Mult-pop-ABC when compared to Mult-pop-ABC 1, Mult-pop-ABC 2 and basc ABC. Indeed, Mult-pop-ABC outperformed Mult-pop-ABC 1, Mult-pop-ABC 2 and basc ABC on both the offlne error and the standard error on all tested scenaros. The results demonstrate that the enhancements we made to the basc ABC mprove the algorthmc performance. Table 6 Results of the Mult-pop-ABC, Mult-pop-ABC 1, Mult-pop-ABC 2 and basc ABC Number of Peaks Algorthm Mult-pop- ABC ± ± ± ± Mult-pop- ABC ± ± ± ± ± ± ± ± ± ± ±0.10 Mult-pop- ABC ± ± ± ± ± ± ± ± ± ± ±0.12 Basc ABC 5.88 ± ± ± ± ± ± ± ± ± ± ±3.44 Note: Values n bold font ndcate the best results.

18 To further verfy the results, we conducted a comparson between Mult-pop-ABC and each method separately. We used a Wlcoxon statstcal test wth a confdence level of The p- values of Mult-pop-ABC aganst Mult-pop-ABC 1, Mult-pop-ABC 2 and basc ABC for each scenaro s presented n Table 7. A value less than 0.05 ndcates Mult-pop-ABC s superor (.e. statstcally dfferent). As can be seen from Table 7, Mult-pop-ABC s superor to Multpop-ABC 1, Mult-pop-ABC 2 and basc ABC on 9 out of 11 tested scenaros (p < 0.05). The table also shows than on two scenaros (1 peak and 2 peaks) Mult-pop-ABC s not superor to Mult-pop-ABC 1 and Mult-pop-ABC 2. Ths can be attrbuted to the fact that these two scenaros are relatvely easy to solve and thus all methods produce very good solutons. The results of the statstcal test also demonstrate that the proposed enhancements have a postve mpact and mprove the search process. Table 7 p-values of the of Mult-pop-ABC aganst other methods Number of Peaks Mult-pop ABC vs. Mult-pop ABC 1 Mult-pop ABC 2 Basc ABC Note: Values less than 0.05 ndcate that Mult-pop-ABC s better than the compared methods. 4.2 Comparson wth state of the art methods There are numerous methods that use dfferent schemes to handle dversfcaton, and whch have been tested on MPB. In ths secton, we evaluate the performance of our algorthm by comparng t wth several recently proposed algorthms taken from the scentfc lterature. The algorthms are: - Multswarms, excluson, and ant-convergence n dynamc envronments (mcpso) [27]. - Multswarms, excluson, and ant-convergence n dynamc envronments (mqso) [27] - Multswarms, excluson, and ant-convergence n dynamc envronments (mcpso * ) [27]

19 - Multswarms, excluson, and ant-convergence n dynamc envronments (mqso * ) [27]. - Compettve populaton evaluaton n a dfferental evoluton algorthm for dynamc envronments (CDE) [28]. - Dfferental evoluton for dynamc envronments wth unknown numbers of optma (DynPopDE) [29]. - Dynamc functon optmzaton wth hybrdzed extremal dynamcs (EO + HJ) [30] - A compettve clusterng partcle swarm optmzer for dynamc optmzaton problems (CCPSO) [31]. - A novel hybrd adaptve collaboratve approach based on partcle swarm optmzaton and local search for dynamc optmzaton problems ( CHPSO(ES-NDS)) [32]. To ensure a far comparson, we used the same stoppng condton (50,000 ftness evaluatons), the same change frequency (every 5,000 ftness evaluatons) and the same evaluaton metrc (Offlne error). We also used 11 MPB nstances wth a dfferent number of peaks rangng between 1 to 200 peaks. The results of Mult-pop-ABC and the compared algorthms are presented n Table 8. The results n the table are n terms of offlne error, ± standard error and computatonal tmes for each number of peaks. In the table, the symbol - ndcates that the scenaro has not been tested. We ndcate n bold the best obtaned results. From Table 8, t can be seen that Multpop-ABC s superor to the other algorthms n most of the cases n terms of offlne error. In partcular, Mult-pop-ABC obtaned new best results for 9 out of 11 tested MPB nstances. Mult-pop-ABC was nferor on only two MPB nstances: 1 peak and 2 peaks. Nevertheless, the results of Mult-pop-ABC for these two scenaros are very compettve, where t obtaned the second best results. In terms of the standard error, Mult-pop-ABC produced a better standard error for 6 scenaros, beng smlar on 5 scenaros out of the 11 tested. Table 8 Results of Mult-pop-ABC compared to the state of the art methods Number of Peaks Algorthm Mult-pop- ABC mcpso 4.93 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.04

20 mqso 5.07 ± ± ± ± ±0.06 mcpso * ±0.17 ±0.26 ±0.11 ±0.11 ±0.07 mqso * ±0.17 ±0.23 ±0.07 ±0.07 ±0.06 CDE ±0.07 DynPopDE ±0.13 ±0.07 EO + HJ ±1.99 ±0.10 CCPSO ±0.01 ±0.03 ±0.06 CHPSO(ES NDS) ± 0.00 ±0.02 ±0.02 Note: Values n bold font ndcate the best results ± ± ± ± ± ± ±0.08 ±0.11 ±0.11 ±0.11 ±0.09 ± ±0.07 ±0.11 ±0.08 ±0.11 ±0.08 ± ± ±0.10 ±0.09 ±0.09 ± ±0.08 ±0.07 ±0.08 ± ±0.01 ±0.01 ±0.01 ± ± ± ± ± ± To further verfy the effectveness of the proposed Mult-pop-ABC, we statstcally compare t wth other methods. We followed the procedure descrbed n [33]. Frst, Fredman test and Iman and Davenport statstcal tests wth 0.05 confdence levels are carred out to detect f there s a dfference between the results of Mult-pop-ABC and other methods. It should be noted that only those methods that were tested on all scenaros were consdered for ths test. Both the Fredman test and Iman and Davenport tests returned p-values ( and ) less than 0.05 ndcatng the compared results are statstcally dfferent. We next conducted a Fredman test to obtan rankngs, and Holm and Hochberg post-hoc tests. The rankng value for each method obtaned by a Fredman test s presented n Table 9 (the lower the better), where Mult-pop-ABC obtaned the frst rank followed by mqso second rank, mcpso thrd rank, mqso* fourth rank and mcpso* ffth rank. Consequently, Multpop-ABC wll be the controllng method for the Holm and Hochberg post-hoc tests. The p- values of Holm and Hochberg tests are shown n Table 10. From the table, one can see that Mult-pop-ABC s statstcally better than the compared methods on both Holm and Hochberg tests n whch all the obtaned p-values are less than Table 9 The average rankng of Fredman test # Algorthm Rankng 1 Mult-pop-ABC 1 2 mqso mcpso mqso* mcpso* Table 10 The adjusted p-value of the compared methods # Algorthm Unadjusted P P Holm P Hochberg

21 1 mcpso* mqso* mcpso mqso The above results reveal that, n most of the tested scenaros, the proposed Mult-pop-ABC s better than the compared methods. These results are supported by statstcal tests. We hypothesse that several key features contrbute to the hgh performance of the proposed algorthm (Mult-pop-ABC) on the dynamc problem. These can be summarsed as follows: - Mult-populaton: Ths feature s benefcal for mantanng the dversty of solutons n the populaton durng the search process. - Adaptve number of sub-populatons: Ths feature helps the algorthm n changng the soluton dstrbuton over the search landscape to get better dversfcaton and ntensfcaton based on the problem change strength. - Populaton clearng scheme: Ths feature helps avod havng smlar solutons wthn the populaton n order to further add to the dversfcaton. 4.3 Comparson wth state-of-the-art approaches on test functons In ths secton, we evaluate our proposed algorthm based on other well-known ten test functons. The tested functons are wdely used by researchers [34-37]. These functons are: f 1 x n = 1 x 2 n n f x 2 = x 1 x 1 [-100, 100] n [-10, 10] n x n x f = j [-100, 100] n f x = max 4 { x,1 n} [-100, 100] n x n f = x x 1 2 x [-30, 30] n 1 x n 4 f = 6 x random 0,1 1 [-1.28, 1.28] n

22 x n f = x sn 7 x [-500, 500] n x 1 n 2 f = 8 x cos 2 x x [-5.12, 5.12] n 1 n 1 n n n f = 2 20exp 0.2 x exp 9 cos2 x 20 e 1 1 [-32, 32] n 1 n 2 f x 10 = 1 cos n x x [-600, 600] n For every benchmark functon, respectvely assume the dmenson as 30, 50 and 100. The results n Tables 11, 12 and 13 demonstrate that Mult-pop-ABC performs better than the compared ABC, PS-ABC and PS-ABCII algorthms [34-36] n terms of both mean and standard devaton (SD). Note that the best results are hghlghted n bold. The presented results ndcate that the Mult-pop-ABC outperforms other methods over all test functons.

23 Table 11 Mean, the standard devaton (SD) of functons wth 30 dmensons. F ABC PS-ABC PS-ABCII LWGSODE CFOA Mult-pop-ABC Dm Mean SD Mean SD Mean Mean Mean SD Mean SD SD SD f x x x x x f x x x x x f x x x x x x x x 10 2 f x f x x x f x x x x x x x x x 10-3 f x x x x 10 1 f x x x x f x x x x x x x x x x

24 Table 12 Mean, the standard devaton (SD) of functons wth 50 dmenson. F ABC PS-ABC PS-ABCII Mult-pop-ABC Dm Mean SD Mean SD Mean SD Mean SD f x x f x x f x x x x x x x x 10 3 f x x f x x x x x x x x 10 1 f x x x x x x x x 10-4 f x x x x 10 1 f f x x x x x x Table 13 Mean, the standard devaton (SD) of functons wth 100 dmenson. F ABC PS-ABC PS-ABCII Mult-pop-ABC Dm Mean SD Mean SD Mean SD Mean SD f x x x x f x x f x x x x x x x x 10 4 f x x f x x x x x x x x 10-1 f x x x x x x x 10-3 f x x x x 10 2 f x x f x x x x x x x x Concluson Ths paper has presented a modfed artfcal bee colony algorthm for dynamc optmzaton problems. The ams of our modfcatons were to enhance the capablty of the algorthm to effcently deal wth DOPs. We frst ntegrated t wth a mult-populaton method to scatter the soluton over the search process so that they can search and track the optmum soluton smultaneously. An adaptve mult-populaton was also proposed to adaptvely change the number of sub-populatons based on the problem change strength. In addton, a populaton clearng scheme was proposed to remove redundant solutons n the populaton. To evaluate the performance of the proposed algorthm, expermental tests were carred out usng the movng peaks benchmark DOP, wth a dfferent number of peaks. Comparsons were carred out between the proposed algorthm, the basc ABC and state of the art methods. The results demonstrated that the proposed algorthm outperforms basc ABC on all tested scenaros. It

25 also produced better results than the state of the art methods on many scenaros, ndcatng that the proposed algorthm s an effectve method for the DOP. Acknowledgements Ths work was supported by the Mnstry of Educaton, Malaysa (FRGS/1/2015/ICT02/UKM/01/2) and the Unverst Kebangsaan Malaysa (DIP ). References 1. Jn, Y. and J. Branke, Evolutonary optmzaton n uncertan envronments-a survey. IEEE Transactons on Evolutonary Computaton, (3): p Nguyen, T.T., S. Yang, and J. Branke, Evolutonary dynamc optmzaton: A survey of the state of the art. Swarm and Evolutonary Computaton, : p L, C., T.T. Nguyen, M. Yang, S. Yang, and S. Zeng, Mult-populaton methods n unconstraned contnuous dynamc envronments: The challenges. Informaton Scences, : p Hatzaks, I. and D. Wallace. Dynamc mult-objectve optmzaton wth evolutonary algorthms: a forward-lookng approach. n Proceedngs of the 8th annual conference on Genetc and evolutonary computaton p ACM. 5. Smões, A. and E. Costa. Improvng predcton n evolutonary algorthms for dynamc envronments. n Proceedngs of the 11th Annual conference on Genetc and evolutonary computaton p ACM. 6. Branke, J. and D.C. Mattfeld, Antcpaton and flexblty n dynamc schedulng. Internatonal Journal of Producton Research, (15): p Cruz, C., J.R. González, and D.A. Pelta, Optmzaton n dynamc envronments: a survey on problems, methods and measures. Soft Computng, (7): p Branke, J. Memory enhanced evolutonary algorthms for changng optmzaton problems. n In Congress on Evolutonary Computaton CEC : Branke, J., Evolutonary optmzaton n dynamc envronments. Vol : Sprnger Scence & Busness Meda. 10. Yang, S. On the desgn of dplod genetc algorthms for problem optmzaton n dynamc envronments. n IEEE Congress on Evolutonary Computaton CEC pp Daneshyar, M. and G.G. Yen. Dynamc optmzaton usng cultural based PSO. n IEEE Congress on Evolutonary Computaton CEC 2011 pp Grefenstette, J.J. Evolvablty n dynamc ftness landscapes: A genetc algorthm approach. n IEEE Congress on Evolutonary Computaton CEC Vol. 3, pp Ursem, R.K. Multnatonal GAs: Multmodal Optmzaton Technques n Dynamc Envronments. n In GECCO, pp Branke, J., T. Kaußler, C. Schmdt, and H. Schmeck, A mult-populaton approach to dynamc optmzaton problems. Adaptve computng n desgn and manufacturng,2000:, 2000: p Blackwell, T. and J. Branke, Mult-swarm optmzaton n dynamc envronments. Applcatons of Evolutonary Computng, 2004: p

26 16. Mendes, R. and A.S. Mohas. DynDE: a dfferental evoluton for dynamc optmzaton problems. n IEEE Congress on Evolutonary Computaton CEC 2005 vol. 3, pp L, C. and S. Yang. Fast mult-swarm optmzaton for dynamc optmzaton problems. n Fourth Internatonal Conference on Natural Computaton, ICNC'08.. vol. 7, pp IEEE. 18. Yang, S. and C. L, A clusterng partcle swarm optmzer for locatng and trackng multple optma n dynamc envronments. IEEE Transactons on Evolutonary Computaton,, (6): p Turky, A.M. and S. Abdullah, A mult-populaton electromagnetc algorthm for dynamc optmsaton problems. Appled Soft Computng, (1): p Turky, A.M. and S. Abdullah, A mult-populaton harmony search algorthm wth external archve for dynamc optmzaton problems. Informaton Scences, (1): p Sharf, A., J.K. Kordestan, M. Mahdavan, and M.R. Meybod, A novel hybrd adaptve collaboratve approach based on partcle swarm optmzaton and local search for dynamc optmzaton problems. Appled Soft Computng, (1): p L, C., T.T. Nguyen, M. Yang, S. Yang, and S. Zeng, Mult-populaton methods n unconstraned contnuous dynamc envronments: The challenges. Informaton Scences, (1): p Karaboga, D., An dea based on honey bee swarm for numercal optmzaton. 2005, Techncal report-tr06, Ercyes unversty, engneerng faculty, computer engneerng department. 24. Karaboga, D. and B. Basturk, A powerful and effcent algorthm for numercal functon optmzaton: artfcal bee colony (ABC) algorthm. Journal of global optmzaton, (3): p Branke, J. and H. Schmeck, Desgnng evolutonary algorthms for dynamc optmzaton problems, n Advances n evolutonary computng. 2003, Sprnger. p Branke, J., T. Kaußler, C. Smdt, and H. Schmeck, A mult-populaton approach to dynamc optmzaton problems, n Evolutonary Desgn and Manufacture. 2000, Sprnger. p Blackwell, T. and J. Branke, Multswarms, excluson, and ant-convergence n dynamc envronments. IEEE Transactons on Evolutonary Computaton, (4): p Du Plesss, M.C. and A.P. Engelbrecht, Usng compettve populaton evaluaton n a dfferental evoluton algorthm for dynamc envronments. European Journal of Operatonal Research, (1): p Du Plesss, M.C. and A.P. Engelbrecht, Dfferental evoluton for dynamc envronments wth unknown numbers of optma. Journal of Global Optmzaton, (1): p Moser, I. and R. Chong, Dynamc functon optmsaton wth hybrdsed extremal dynamcs. Memetc Computng, (2): p Nckabad, A., M.M. Ebadzadeh, and R. Safabakhsh, A compettve clusterng partcle swarm optmzer for dynamc optmzaton problems. Swarm Intellgence, (3): p Sharf, A., J.K. Kordestan, M. Mahdavan, and M.R. Meybod, A novel hybrd adaptve collaboratve approach based on partcle swarm optmzaton and local

27 search for dynamc optmzaton problems. Appled Soft Computng, : p García, S., A. Fernández, J. Luengo, and F. Herrera, Advanced nonparametrc tests for multple comparsons n the desgn of experments n computatonal ntellgence and data mnng: Expermental analyss of power. Informaton Scences, (10): p L, G., P. Nu, Y. Ma, H. Wang, and W. Zhang, Tunng extreme learnng machne by an mproved artfcal bee colony to model and optmze the boler effcency. Knowledge-Based Systems, : p Cu, H., J. Feng, J. Guo, and T. Wang, A novel sngle multplcatve neuron model traned by an mproved glowworm swarm optmzaton algorthm for tme seres predcton. Knowledge-Based Systems, : p Mtć, M., N. Vukovć, M. Petrovć, and Z. Mljkovć, Chaotc frut fly optmzaton algorthm. Knowledge-Based Systems, : p Yeh, W.-C., An mproved smplfed swarm optmzaton. Knowledge-Based Systems, : p

28 *Revsed Manuscrpt (Clean Verson) Clck here to vew lnked References An Adaptve Mult-populaton Artfcal Bee Colony Algorthm for Dynamc Optmsaton Problems Shams K. Nseef 1, Salwan Abdullah 1, Ayad Turky 2 and Graham Kendall 3,4 1 Data Mnng and Optmsaton Research Group, Centre for Artfcal Intellgence Technology, Unverst Kebangsaan Malaysa, Bang, Selangor, Malaysa. E-mal: shams.shamosa91@gmal.com; salwan@ukm.edu.my 2 Swnburne Unversty of Technology, Melbourne, Vctora, Australa E-mal: aturky@swn.edu.au 3 Unversty of Nottngham Malaysa Campus, Semenyh, Malaysa 4 ASAP Research Group, Unversty of Nottngham, Nottngham, Unted Kngdom Emal: Graham.Kendall@nottngham.edu.my; Graham.Kendall@nottngham.ac.uk Abstract Recently, nterest n solvng real-world problems that change over the tme, so called dynamc optmsaton problems (DOPs), has grown due to ther practcal applcatons. A DOP requres an optmsaton algorthm that can dynamcally adapt to changes and several methodologes have been ntegrated wth populaton-based algorthms to address these problems. Mult-populaton algorthms have been wdely used, but t s hard to determne the number of populatons to be used for a gven problem. Ths paper proposes an adaptve multpopulaton artfcal bee colony (ABC) algorthm for DOPs. ABC s a smple, yet effcent, nature nspred algorthm for addressng numercal optmsaton, whch has been successfully used for tacklng other optmsaton problems. The proposed ABC algorthm has the followng features. Frstly t uses mult-populatons to cope wth dynamc changes, and a clearng scheme to mantan the dversty and enhance the exploraton process. Secondly, the number of sub-populatons changes over tme, to adapt to changes n the search space. The movng peaks benchmark DOP s used to verfy the performance of the proposed ABC. Expermental results show that the proposed ABC s superor to the ABC on all tested nstances. Compared to state of the art methodologes, our proposed ABC algorthm produces very good results. Keywords: dynamc optmsaton, artfcal bee colony algorthm, adaptve mult-populaton method, meta-heurstcs 1. Introducton Many real-world optmsaton problems have the characterstc of changng over tme n terms of decson varables, constrants and the objectve functon [1], [2]. These problems