Comparison on the Selection Strategies in the Artificial Bee Colony Algorithm for Examination Timetabling Problems

Transcription

1 Internatonal Journal o Sot Computng and Engneerng (IJSCE) ISSN: , Volume-1, Issue-5, November 2011 Comparson on the Selecton Strateges n the Artcal Bee Colony Algorthm or Examnaton Tmetablng Problems Malek Alzaqebah, Salwan Abdullah Abstract Ths paper presents an nvestgaton o selecton strateges upon the Artcal Bee Colony (ABC) algorthm n examnaton tmetablng problems. ABC s a global stochastc optmsaton algorthm that s based on the behavor o honey bee swarms. Onlooker bees n ABC algorthm choose ood source based on the proportonal selecton strategy. In ths paper, three selecton strateges are ntroduced (.e. dsruptve, tournament and rank selecton strateges), n order to mprove the dversty o the populaton and avod the premature convergence n the evolutonary process. Expermental results show that the moded ABC wth the three selecton strateges outperorms the ABC algorthm alone. Among the selecton strateges, the dsruptve selecton strategy shows the better perormance when tested on standard benchmark examnaton tmetablng problem. Index Terms: Artcal Bee Colony Algorthm, Examnaton Tmetablng problems, Selecton Strateges. I. INTRODUCTION Several approaches have been proposed or solvng optmzaton problems. In recent years, the research trend ocuses more on heurstc methods rather than the tradtonal methods to solve the optmsaton problems. Swarm ntellgence or example ocuses on the behavour o nsects to develop some meta-heurstcs whch can mmc the nsect s problem-solvng. Artcal bee colony (ABC) algorthm s a part o swarm ntellgence algorthms that mmcs the natural behavor o real honey bees on searchng or ood sources. It was proposed by Karaboga [12] or numercal optmsaton problem [12]. [11] Proposed an extended verson o ABC algorthm or solvng constraned optmsaton problems. In ths paper, we treat the Examnaton Tmetablng Problems (ETTP), whch can be dened as a classc combnatoral optmsaton problem. ETTP deals n assgnng a number o exams nto a lmted number o tmeslots and locatons, whlst reducng the volatons o a predened set o constrants. Usually there are two types o constrants consdered n ETTP.e. hard and sot constrants. Hard constrants cannot be volated n any crcumstances and Manuscrpt receved July 09, Malek Alzaqebah, Data Mnng and Optmzaton Research Group (DMO) Center or Artcal Intellgence Technology, Unverst Kebangsaan Malaysa, Bang, Selangor, Malaysa, , (e-mal: malek_zaqeba@tsm.ukm.my). Salwan Abdullah, Data Mnng and Optmzaton Research Group (DMO) Center or Artcal Intellgence Technology, Unverst Kebangsaan Malaysa, Bang, Selangor, Malaysa, , (e-mal: salwan@tsm.ukm.my). volaton o sot constrants s mnmsed as much as possble. Interested readers can nd more detals about ETTP research and comprehensve survey papers n [7], [8], [1], [14], [18] and [19]. Example o one o the bee amly algorthms.e. honey-bees matng optmsaton algorthm that has been appled to solve ETTP can be ound n [16]. The paper s organsed as ollow: Secton II ormally presents the ETTP and ormulaton. Secton III descrbes the orgnal ABC algorthm. The selecton strateges that have been appled n ABC are presented n Secton IV. The proposed approach s presented n Secton V. Our expermental comparson are presented, dscussed and evaluated n Secton VI. Ths s ollowed by some bre concludng comments n Secton VII. II. PROBLEM DESCRIPTION AND FORMULATION ETTP can be dened as NP-hard problem that due to the dculty o satsyng the pre-dened number o constrants. In ths paper, the problem descrpton s adapted rom the explanaton presented n [8]. ETTP consst o a number o nputs as ollow: N s the number o exams. E s an exam, {1 N}. T s the gven number o avalable tmeslots. M s the number o students. C (c ) NxN s the conlct matrx where each element denoted by c,, {1,,N} s the number o students takng exams and. t k (1 t k T) speces the assgned tmeslot or exam k (k {1,,N}). An obectve uncton s ormulated whch tres to space out students exams throughout the exam perod (Eq. (1)) whch s consdered as the sot constrant. It can be ormulated as the mnmsaton o: Where And N 1 1 F 1 ( ) (2) M N F1 ( ) c + 1. proxmty ( t, t ) (3) 158

2 Comparson o the Selecton Strategy n the Artcal Bee Colony Algorthm or Examnaton Tmetablng ProblemsBees Algorthm or Examnaton Tmetablng 5 2 proxmty ( t, t ) Subect to: Where λ N 1 N ( t, t ) / 2 t t 0 c λ 1 t otherwse ( t, t ) 0 1 t 0 otherwse t t 5 Eq. (2) presents the cost or an exam whch s gven by the proxmty value multpled by the number o students n conlct. Eq. (3) represents a proxmty value between two exams [10]. Eq. (4) represents a hard constrant (clash-ree requrement) so that no student can st two exams at the same tme. III. ARTIFICIAL BEE COLONY ALGORITHM (ABC) A. The Basc Artcal Bee Colony (ABC) Algorthm Artcal Bee Colony Algorthm (ABC) was ntroduced by Karaboga [12] as a global optmsaton algorthm that smulates the oragng behavor o honey bees. In ABC, the artcal agents are dened and classed nto three types.e. the employed bees, the onlooker bees, and the scout bees. Each o them plays a derent role n the process. The employed bees stay on a ood source and provde the neghborhood o the source n ts memory. The onlooker bees get the normaton o ood sources rom the employed bees n the hve, and select one o the ood sources to gather the nectar, and the scout bees s responsble or ndng new ood sources. ABC system combnes local and global search methods, where the local search method s carred out by employed bees and onlooker bees. Whle the global search method s managed by onlooker bees and scout bees. The possble solutons n the ABC algorthm represent ood sources (lowers), and the tness o the soluton s corresponded to the nectar amount o the assocated ood source, [11]. Intal ood sources are produced or all employed bees REPEAT Each employed bee les to a ood source n her memory and determnes a neghbor source, then evaluates ts nectar amount and dances n the hve. Each onlooker watches the dance o employed bees and chooses one o ther sources dependng on the dances, and then goes to that source. Ater choosng a neghbor around that, onlooker evaluates ts nectar amount. Abandoned ood sources are determned and are replaced wth the new ood sources dscovered by scout bees. The best ood source ound so ar s regstered. UNTIL (requrements are met) Fgure I. Orgnal artcal bee colony algorthm (4) (5) Fgure I shows the basc ABC algorthm as n [12]. In ABC, the number o the employed bees or onlooker bees s equal to the number o ood source (SN). At the rst step, ntal populatons (ood source postons) are generated based on a constructve heurstc algorthm. An employed bee produces an adustment on the source poston n her memory and dscovers a new ood source poston. I the nectar amount o the new ood source s hgher than the prevous one, then the bee memorses the new ood source poston, otherwse the bee keeps the old poston n her memory. Ater the employed bees complete the search process, they share the normaton about the poston o the sources wth the onlooker bees at the dance area. Each onlooker bee evaluates the nectar normaton that s collected rom all employed bees. Based on the nectar amounts o sources she chooses a ood source to produce an adustment on the source poston n her memory, and checks ts nectar amount. Scout bees determne the abandoned sources and produce new sources randomly n order to replace the abandoned ones. B. Onlooker Bees Selecton Process Onlooker bees select the soluton by a stochastc selecton scheme, whch conssts o three steps: 1. Calculates the tness value by usng the tness uncton as ollow: t Where s tness uncton and t s the tness uncton ater a transormaton. 2. Calculate the probablty value by usng the ollowng expresson: t p SN (6) t 1 Where SN s the number o ood sources, s the tness uncton o the th ood source. 3. Fnally, chose a canddate soluton based on the selecton probablty by roulette wheel selecton method. As stated n [3], they menton that, there are two problems o usng basc ABC selecton strategy as below: () A superndvdual beng too oten selected the whole populaton tends to converge towards hs poston. The dversty o the populaton s then too reduced to allow the algorthm to progress; () wth the progresson o the algorthm, the derences between tness are reduced. The best ones then get qute the same selecton probablty as the others and the algorthm stops progressng. Thus, ths selecton strategy s hard to keep the dversty and to avod the premature convergence. In order to allevate these problems, ths paper employed three derent selecton strateges to mprove the perormance o the ABC algorthm. IV. SELECTION STRATEGIES In ths work, we ncorporate three selecton strateges wth (5) 159

3 Internatonal Journal o Sot Computng and Engneerng (IJSCE) ISSN: , Volume-1, Issue-5, November 2011 ABC algorthm and tested on ETTP. The descrpton o the selecton strategy used s descrbed as below: Tournament selecton: The tournament selecton s a selecton process where a number o ndvduals (N tour ) rom the populaton are chosen at random, and then the comparson s made dependng on the tness n order to take the best ndvdual. Parameter N tour s called a tournament sze. Normally, tournaments are held only between two ndvduals (bnary tournament), but a generalsaton s possble to an arbtrary group. Tournament selecton gves more chances or the ndvduals wth hgh tness to survve, [5]. In ths work, we select two ndvduals rom the populaton and compare ther tness values, then assgn one score (coded as a) to a better ndvdual. Repeat such process or all the ndvduals n the populaton as shown n Fgure II, where s the tness value o 0...n, where n s the populaton sze (adapted rom, Bao and Zeng, (2009)). or 1:n a 0; or 1:n a a +1; end end or endor Fgure II. Tournament selecton pseudo code Ater calculatng the value o (a) or all the ndvduals, the selecton probablty or each ndvduals s calculated usng Eq. (7). P n a 0 a Rank selecton: In the rank selecton, the tness value s calculated usng Eq. (5). Indvduals are sorted n descendng order based on the tness value. The ndex k s gven to each ndvdual rom the best to the worst,.e. or the best tness k 1, and or the worst tness k n, where n s the populaton sze and N s the maxmum number o teratons. Fnally, the selecton probablty s calculated usng Eq. (8), [17]: 1 n + 1 2k P k + a( t), k (1,2,... n) n n( n + 1) (7) (8) 3t where a( t) 0.2 +, t (1,2,... N ) 4N Dsruptve selecton: Dsruptve selecton gves more chance or hgher and lower ndvduals to be selected by changng the denton o the tness uncton as n Eq. (9), [13]. t t t P n t (9) 0 Where s the tness uncton, s the average value o the tness uncton o the ndvduals n the populaton. V. THE ALGORITHM: ARTIFICIAL BEE COLONY SEARCH ALGORITHM A. Constructon Heurstc In ths work, we used the graph colourng approach (.e. largest degree heurstc) to generate the ntal soluton, where examnatons wth the largest number o conlcts are scheduled rst. For more detals about graph colourng applcatons to tmetablng see [8]. B. Improvement Algorthm: Artcal Bee Colony Search Algorthm Fgure III llustrates the pseudo code o the proposed approach. The algorthm starts wth easble ntal solutons whch are generated by a largest degree heurstc n the constructve phase. The poston o a ood source represents a possble soluton and the nectar amount o a ood source corresponds to the qualty (tness value) o the assocated soluton. The number o the employed bees s equal to the number o solutons n the populaton. The employed bees work on random solutons and apply a random neghborhood structure on each soluton. Provded that the nectar amount o the new one s hgher than that o the prevous source, the bee memorzes the new source poston and orgets the old one. Otherwse she keeps the poston o the one n her memory. Ater all the employed bees complete the search process, they share the poston normaton o the sources wth the onlooker bees on the dance area. Onlooker bees work on the selected soluton based on the selecton strategy explaned above, and enhance t by applyng a random neghborhood structure n order to reduce the volaton o the sot constrants. Fnally, scout bees determne the abandoned ood sources and replace them wth a new ood source by perormng several moves. Intalzaton: Intalse the ntal populaton and evaluate the tness; Calculate the ntal tness value, (Sol); Set best soluton, Solbest Sol; Set maxmum number o teraton, NumOIte; Set the populaton sze; //where populaton sze OnlookerBee EmployeedBee; teraton 0; t 160

4 Comparson o the Selecton Strategy n the Artcal Bee Colony Algorthm or Examnaton Tmetablng ProblemsBees Algorthm or Examnaton Tmetablng Improvement: do whle (teraton < NumOIte) or 1: EmployeedBee Sol* Select a random soluton Sol** Apply a random neghbourhood structure on Sol*; (Sol** < Solbest) SolbestSol**; end or or 1: OnlookerBee Calculate the selecton probablty P, based on the correspondng selecton strategy (mnmse o [4]: Eq.(6), Eq.(7), Eq.(8) or Eq.(9), respectvely) Sol* select the soluton dependng on P ; Sol** Apply a random neghbourhood structure on Sol*; (Sol** < Solbest) SolbestSol**; end end or ScoutBee determnes the abandoned ood source and replace t wth the new ood source. teraton++; end do Fgure III. The pseudo code or the artcal bee colony search algorthm Nbs2 Choose a sngle exam at random and move to a new random easble tmeslot. Nbs3 Select two tmeslots at random and smply swap all the exams n one tmeslot wth all the exams n the other tmeslot. Nbs4 Select 3 exams randomly and swap the tmeslots between them easbly. Nbs5 Select 4 exams randomly and swap the tmeslots between them easbly. Nbs6 Take two tmeslots at random, say t and t (where >) where tmeslots are ordered t 1,t 2,t 3,,t p. Take all exams that n t and allocate them to t, then allocate those that were n t -1 to t -2 and so on untl we allocate those that were t +1 to t and termnate the process. Nbs7 Move the hghest penalty exams rom a random 10% selecton o the exams to a random easble tmeslot. Nbs8 Carry out the same process as n Nbs7 but wth 20% o the exams. Nbs9 Move the hghest penalty exams rom a random 10% selecton o the exams to a new easble tmeslots whch can generate the lowest penalty cost. Nbs10 Carry out the same process as n Nbs9 but wth 20% o the exams. VI. EXPERIMENTAL COMPARISON Table I shows the parameters settng, whch have been used n ths work. C. Neghborhood Structure In ths paper, ten neghborhood structures are employed n order to enhance the perormance o searchng algorthms. These neghborhood structures are presented as [1]: Nbs1 Select two exams at random and swap tmeslots. TABLE I. PARAMETERS SETTING. Parameter Value Iteraton 500 populaton sze 50 scout bee 1 TABLE II. RESULTS COMPARISON. Dataset ABC DABC RABC TABC Best known Best Avg. Best Avg. Best Avg. Best Avg. car car ear83i hec92i ku lse sta83i tre uta92i ute yor83i Table II provdes the results comparson between three moded ABC algorthms (.e. ABC wth derent selecton 161

5 Internatonal Journal o Sot Computng and Engneerng (IJSCE) ISSN: , Volume-1, Issue-5, November 2011 strateges) and the basc ABC. Three derent moded ABC algorthms are called ABC algorthm based on dsruptve selecton (DABC), ABC algorthm based on rank selecton (RABC), and ABC algorthm based on tournament selecton (TABC). We run the experments or 5 tmes or each dataset. Note that, the dataset s speccaton o the examnaton tmetablng problems that were proposed by [10]. As shown n Table II the best results are presented n bold. The above comparson shows that, the ABC algorthm wth three selecton strateges perorm better than the basc ABC algorthm alone. However, n most o the tested datasets, DABC outperorms other algorthms n comparson. The comparson between ABC algorthm wth three selecton strateges and the best known results shows that even we are unable to beat any o the best known results n the lterature, but we are stll able to produce promsng solutons Penalty Cost 22 RABC TABC DABC Iteratons Fgure IV. Convergence o ku93 dataset Fgure IV shows that the behavour o ABC algorthm based on three selecton strateges over ku93 dataset. The x-axs represents the number o teraton, whle the y-axs represents the penalty cost. Ths graph shows how DABC, TABC and RABC explore the search space. We beleve that the way the algorthm behaves has a correlaton wth the complexty o the datasets (represented by the conlct densty value). Note that the detals o the conlct densty values can be ound n [15]. The hgher conlct densty sgnes that more exams are conlctng wth each other. The conlct densty value or ku93 s As shown n Fgure IV, the behavour o the three selecton strateges works smlar at the begnnng o the teratons where the mprovement o the soluton can easly be obtaned. Later, t becomes steady and hard to be mproved. However, the graph shows that DABC can explore the search space better than RABC and TABC. Ths s due to the nature o the selecton strategy, where the tournament selecton randomly selects a number o solutons (Ntour) and compares them based on a probablty. The soluton wth a hghest tness value wll be chosen. In a rank selecton, the solutons are ranked based on the tness values, so ths uncton s based to work wth the soluton at hgher rank (.e. good tness), whle the dsruptve selecton concentrates on both the worse and the hgh tness, and tres to keep the dversty o populaton by mprovng the worse tness solutons n concurrent wth the hgh tness solutons. Fgure V shows the convergence o three datasets.e. hec92i, sta83i and tre92. The x-axs represents the number o soluton, whle the y-axs represents the penalty cost. Fgure V. Convergence o three datasets.e. hec92i, sta83i and tre92 These graphs show how the DABC, TABC and RABC spray the populaton at ntal stage (represented by the 162

6 Comparson o the Selecton Strategy n the Artcal Bee Colony Algorthm or Examnaton Tmetablng ProblemsBees Algorthm or Examnaton Tmetablng trangle symbol), and then ater 500 teratons the mproved solutons are represented by the square symbol. From these gures, we can conclude that the DABC gves a chance or all the solutons n the populaton to be mproved and converged together. Ths can be seen that the plotted square symbols are concentrated (not scattered) to each other, that represents the closeness o the qualty o the solutons n the populaton. I. CONCLUSION AND FUTURE WORK The am o ths paper s to compare the perormance o the ABC algorthm when uses derent selecton strateges. Through the results obtaned, t s concluded that ABC algorthm wth a dsruptve selecton strategy s able to produce better results when compared to other selecton strateges tested n ths work. 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