A hybrid harmony search for university course timetabling

Transcription

1 MISTA 2009 A hybrid harmony earch for univerity coure timetabling Mohammed Azmi Al-Betar Ahamad Tajudin Khader Abtract. Combination of metaheuritic to olve highly contrained combinatorial optimiation problem have recently become a promiing reearch area. Some combination emerge from integrating evolutionary algorithm and local baed algorithm to trike the right balance between global eploration of the earch region and local eploitation of promiing region. Thi tudy preent hybridization between harmony earch algorithm and hill climbing optimier to tackle the univerity coure timetabling problem. The olution i eentially a harmony earch algorithm. The hill climbing optimier i reponible for improving the new harmony vector, obtained from harmony earch, with a certain probability. Furthermore, inpired by the practical warm optimiation, we modified the memory conideration operator to let it improvie rather than alway working in the ame way. Our algorithm converge ignificantly fater than doe the claical harmony earch algorithm. We evaluate our hybrid algorithm with a tandard benchmark introduced for UCTP. Our reult how that our algorithm can find higher quality olution than can previou work. Introduction Timetabling problem are comple combinatorial optimiation problem claified a NPhard []. The general requirement for timetabling problem i to aign a et of entitie (uch a tak, public event, vehicle, or people) to a limited number of reource over time, in uch a way a to meet a et of pre-defined chedule requirement [2]. Such problem appear in many form uch a chool, eamination, or coure timetabling [3, 4, 2, 5]; employee timetabling [6]; tranport timetabling [7]; and nure rotering [8]. Thi paper addree the univerity coure timetabling problem (UCTP), which include aigning et of event to particular room and time lot on a weekly bai according to contraint. There are two kind of timetabling contraint: hard and oft [3]. A timetabling Mohammed Azmi Al-Betar School of Computer Science, Univeriti Sain Malayia, 800 USM, Penang, Malayia mohbetar@c.um.my Ahamad Tajudin Khader School of Computer Science, Univeriti Sain Malayia, 800 USM, Penang, Malayia tajudin@c.um.my 57

2 olution mut atify hard contraint in order to be feaible, but need not necearily atify all oft contraint. In other word, oft contraint may be violated, and the quality of a timetable olution i optimied by atifying oft contraint. The mot common oft contraint reflect the preference of tudent and teacher. The UCTP ha attracted attention from reearcher in artificial intelligence and operational reearch for quite a long time. Many approache have been propoed to tackle thi problem. Becaue timetabling i imilar to graph colouring problem (graph vertice correpond to timetable event, and colour correpond to time period) [9, 0], the earliet approache depended on graph colouring heuritic (that i, larget degree, larget enrolment, larget weighted degree, or larget aturating degree); thee heuritic build a timetable by aigning event to valid time period and room one by one according to a particular order. Modern approache often ue thee heuritic to find a feaible timetable [, 2], and alo ue them a low-level heuritic in hyper heuritic [3, 4]. Furthermore, they erve a a fuzzy inference rule in fuzzy multiple ordering [2]. Reearcher have uccefully applied metaheuritic in both local earch and populationbaed approache to the UCTP. Local earch approache (iterated local earch [5], tabu earch [6], imulated annealing [7], and other) make equence of local change to the initial (or current) olution guided by an objective function until they reach a locally optimal olution. The key to the ucce of local approache i the definition of neighbourhood tructure, which make more of the earch pace reachable. Population-baed approache, alo called evolutionary algorithm, have alo been uccefully applied to the UCTP. Eample of uch approache include genetic algorithm [8], ant colony optimization [5], and harmony earch [9]. Population-baed approache typically ue recombination and randomiation rule, which mi different component of olution (or individual) with the hope of finding a good miture in the new olution(). The following article urvey previou metaheuritic for the UCTP [3, 4, 2, 20, 5]. Hybridization of metaheuritic i an efficient olution for the UCTP. In a recent comprehenive urvey for timetabling problem, Qu et al. [2] ugget that There are many reearch direction generated by conidering the hybridization of meta-heuritic method particularly between population-baed method and other approache. Hybridization between genetic algorithm and local algorithm or the o-called memetic algorithm i a ucceful hybrid approach that tackled univerity timetabling problem [22]. Recently, Abdullah et al.[] have tackled the UCTP uing the hybrid evolutionary algorithm. The ytem of Abdullah et al. combine variable neighbourhood algorithm and genetic algorithm, and it obtained the bet known reult for UCTP problem intance at the time of publihing. The harmony earch algorithm (HSA), developed by Geem et al. [23] i a new metaheuritic population-baed algorithm that imitate muical performance. HSA combine the key component of population-baed and local baed algorithm in a imple optimiation model [23, 24]. The HSA ha trong component (operator) that can efficiently and effectively eplore the earch pace. It combine ome component of population-baed algorithm (that i, memory conideration correponding to recombination and random conideration which correponding to mutation), which are reponible for global improvement, and ome component of local earch algorithm (that i, pitch adjutment correponding to neighbourhood tructure), which are reponible for local improvement. Our previou work adapted the HSA to the UCTP [9]. The pitch adjutment operator i deigned to work imilarly to neighbourhood tructure. At each run, thi operator make a 58

3 limited number of local change to the new improvied olution, called new harmony, baed on the pitch adjutment rate (PAR) and harmony memory conideration rate (HMCR). Thee local change move the new harmony olution to the neighbouring olution a a random walk acceptance criterion without conidering the objective function value. Therefore, the HSA cannot be fine-tune the earch pace region of new harmony to find a local optimal olution at thi region. In thi tudy, we introduce a Hybrid Harmony Search Algorithm (HHSA) to be applied to the UCTP. The hill climbing optimizer (HCO) i incorporated a a new operator of the HSA to eplore the earch pace region of new harmony thoroughly to fine-tune thi region. In addition, we propoe an idea temming from practical warm algorithm, modified by a memory conideration operator, to lower the election preure of thi operator. Our method dramatically improve the convergence of HSA. In the HHSA, the eential application of the HSA, the HCO erve a a new operator, controlled by a parameter we call Hill Climbing Rate (HCR). Furthermore, inpired by practical warm optimiation, we modified the functionality of the memory conideration operator to increae it improviational ability. Thi paper i organized a follow: Section 2 dicue the univerity coure timetabling problem, and Section 3 eplain the HSA. Section 4 preent the HSA for the UCTP and our hybridization trategy. Section 5 dicue our eperimental reult and compare them to the previou literature. In the final ection, we preent concluion and future direction for our algorithm. 2 Univerity coure timetabling problem 2. Problem Decription Napier Univerity coure timetabling in Edinburgh wa conidered a an application problem by a Metaheuritic Network (MN). The MN decribed the contet of thi problem a aigning certain event to uitable room and timelot according to hard and oft contraint; a olution mut atify hard contraint, while it hould minimie violation of oft contraint. The UCTP benchmark we conider wa contructed by Socha et al. [5] who propoed eleven data intance uing Paechter' generator 2 to mimic the ame Napier Univerity coure timetabling. Thee data intance are grouped into three clae: five mall, five medium, and one large, and they include three hard contraint: H. Student mut not be double-booked for event. H2. Room ize and feature mut be uitable for aigned event. H3. Room mut not be double booked for event. And three oft contraint: S. A tudent hall not have a cla in the lat lot of the day. S2. A tudent hall not have more than two clae in a row. S3. A tudent hall not have a ingle cla on one day. The main objective of thi contet i to atify the hard contraint and to minimie the violation of oft contraint. Metaheuritic Network official webite (04 Feb 2008). 2 Ben Paechter i a Profeor in the School of Computing at Napier Univerity, UK and a member of Metaheuritic Network. Hi official home page i (04 Feb 2008). 59

4 2.2 Problem formulation In order to illutrate the bridge between the UCTP and harmony earch algorithm, it i ueful to dicu UCTP mathematically to overview the et of parameter and deciion variable. Indeed, the formulation, preented here, could not be olved by eact approache ince the number of deciion variable would be large. In the following ubection, the et, parameter and deciion variable are epreed before the problem formulation i introduced Set and Parameter decription A et E = {... e} of event, each of which contain certain tudent and need certain feature. A et R = {... r} of room, each of which ha a eat capacity and it own feature. A et S = {... } of tudent, each of whom enrol in ome event. A et F = {... f } of feature, uch a overhead projector or pecial whiteboard. A et P = {... p} of timelot where p=45(5 day with 9 period on each day). ~ ~5 A et D ~ d = { D... D} of day, where each day ha nine period. ordered ubet P ~ d of P. correponding to a period in a day d. where ~ ~2 P = { p, p2,..., p9}, P = { p0, p,..., p8}... etc. ~ An ordered ubet L d = p, p, p, p, p } that contain the lat period of each day. { ~ L d ~d P. d D. e, r,, f, p are the number of event, room, tudent, feature, and timelot repectively. = the ize of room r. r R. R r E = the number of tudent enrolled in event e. e e E. w f, = if event e require feature f, 0 otherwie. e E and f F. e y f, = if room r contain feature f, 0 otherwie. r R and f F. r t, = if tudent i enrolled in event e, 0 otherwie. S and e E. e Deciion variable are binary deciion variable indeed by event, room and timelot. Their value e, r, p i if and only if event e occurred in room r and time period p. e E, r R and p P. 60

5 ldp C (Lat period of day) are deciion variable indeed by tudent; their value indicating the number of violation of oft contraint S by tudent. S. R C 2 (More than two event in a row) are deciion variable indeed by tudent; their value indicate the number of violation of oft contraint S2 by tudent. S. d C (Single cla in a day) are deciion variable indeed by tudent; their value indicate to the number of violation of oft contraint S3 by tudent. S. z, are binary deciion variable indeed by tudent and day; their value indicate d that the tudent ha a ingle cla in a day d. S and d D. ~d Formulation The objective function of the obtained olution can be decribed a follow: Minimize S ldp 2R d ( C C + C ) The objective function i decribed in Eq. (), C, + () ldp C 2 R, and d C conecutively decribe the violation of the oft contraint S,S2,S3 made againt the will of each tudent. When each violation occur in the olution, it will be penalized by. The hard contraint are decribed by Eq. (2-6); thee contraint are mandatory for any feaible olution. e E r R p P p P. S r R. p P e E r R e E f F. r R. p P e E r R. p P e E (2) e, r, p = t (3), e e, r, p, (4) E e e r, p R r w y (5) = f, e f, r e, r, p (6) e, r, p Eq. (2) decribe the implicit contraint, not mentioned in Section 2., which meant that timetable olution mut be complete and each event mut be preented once. Eq. (3) decribe H, which mean that no tudent may be double-booked for event. Eq. (4, 5) decribe H2, which reflect room allocation with uitable capacity and feature for event. Eq. (6) reflect H3, which eplain that no room may be double-booked for event. ldp S C = t, e e, r, q (7) e E r R ~ q L 2R S C = t t, t, i, r, p j, r, q k, r, m (8) ~ d S. d D z, d, i j k i, j, k E r R p, q, m P i j k q= p+ m= q+ = d S C = z ~ d q D t =, e e, r, i e E r R ~ d i P 0 otherwie, q (9) (0) 6

6 The deciion variable C, C 2, C reflect the violation of oft contraint S,S2,and S3, ldp R d conecutively. Thee oft contraint are formalied in Eq. (7), which decribe the oft contraint S, Eq. (8) decribe the oft contraint S2, and Eq. (0) decribe the oft contraint S3. Eq. (9) i neceary for decribing S3, which penalie tudent who have only attended a ingle event in a day by, while Eq. (0) calculate all violation of any tudent for all day. 3 Harmony earch algorithm Current metaheuritic originate from natural phenomena and mimic real ituation. Eample include phyical annealing in imulated annealing, ant behaviour in ant ytem, vertebrate immune ytem in artificial immune ytem, human neural ytem in artificial neural network, human memory in tabu earch, and evolution in genetic algorithm. The HSA i a new metaheuritic algorithm developed by Geem et al.[23] to imitate the natural phenomenon of muical performance, in which muician jointly tune the pitche of their intrument to find a euphoniou harmony. Thi intene and prolonged muical proce lead them to the perfect (or Nirvana) tate. The HSA i a calable tochatic earch mechanim, imple in it concept, and ha few parameter, needing no derivational information at the initial tage [24]. It can find a trade-off between global improvement of earch pace and local improvement of the new harmony. Furthermore, HSA work at the component level rather than the individual level, which conider component (or deciion variable) of the tored olution (or harmony memory vector) a eential element to generate a new olution called the new harmony. The HSA ha been uccefully applied to everal optimiation problem [25]. Algorithm decribe the baic HSA, which ha five tep, a follow [23]: Step. Initialize the HSA and optimiation problem parameter. Step 2. Initialize the Harmony Memory (HM). Step 3. Improvie a new harmony. Step 4. Update the harmony memory. Step 5. Check the topping criterion. Thee tep will be eplained imilarly to Lee and Geem [24] in the net five ubection: 3. Initialize the HSA and optimiation problem parameter In tep, the optimiation problem can be pecified a follow: min Subject to 0 and 0 () Where i the objective function; i the et of each deciion variable, that i,, i the poible range of value for each deciion variable, that i,, 2,,. i the number of deciion variable, and K i the number of poible value of deciion variable. i inequality contraint function and i equality contraint function. Furthermore, the parameter of the HSA required to olve the optimiation problem are alo pecified in thi tep: Harmony Memory Conideration Rate (HMCR), Harmony Memory Size (HMS) (that i, equivalent to population ize), Pitch Adjutment Rate (PAR), and Number of Improviation (NI) (that i, the maimum number of generation). The HSA parameter will be eplained in more detail in the net tep. 62

7 Algorithm The framework of a baic HSA : Input : the data intance P of the optimiation problem and the parameter for HSA(HMCR,PAR,NI,HMS). HMS 2: Initialize-HM {,..., } wort HMS 3: Recognize (,..., ). 4: while not termination criterion pecified by NI do 5: // new harmony vector 6: for j=,..., N do // N i the number of deciion variable. 7: if U(0,) HMCR then 8: begin HMS,..., // memory conideration 9: ( ) j j j 0: if U(0,) PAR then // pitch adjutment : N( ) // N( ) i the neighbouring value of variable j 2: end 3: ele //random conideration 4: j X j 5: end if 6: end for 7: if f( ) f( wort ) then 8: wort = new 9: end if 20: end while 2:output: the bet olution obtained o far. j j j 3.2 Initialize the harmony memory The harmony memory (HM) i a vector of olution with ize HMS a hown in Eq. (2). In thi tep, thoe olution are randomly contructed and decreaingly filled to HM baed on the value of the objective function. HM 2 =. HMS HMS N 2 N. HMS N ( ) 2 ( ) f f. ( ) HMS f (2) 3.3 Improvie a new harmony In thi tep, the HSA generate (improvie) a new harmony vector, = (,,,..., ) 2 3 N, baed on three operator:() memory conideration,(2) random conideration, and (3) pitch adjutment. In the memory conideration, the deciion variable of the new harmony randomly inherited the hitorical value tored in the HM with probability HMCR. Retating, the value of 2 3 HMS deciion variable ( ) i choen from {,,,.., } that tored in HM, the net deciion 2 3 HMS variable ( 2 ) i choen from {,,,.., } , the other deciion variable,,..., are choen conecutively in the ame way with probability HMCR where ( ) 3,

8 ( 0 HMCR ). Other value that are not choen depending on memory conideration with probability (-HMCR) are elected according to their poible range by random conideration a in Eq. (3). 2 3 HMS i { i, i, i,.., i } with probability HMCR i = (3) i X i with probability HMCR The HMCR parameter i the probability of electing one value of the deciion variable, i, baed on hitorical value tored in the HM. For intance, if (HMCR =0.90), that indicate that the probability of electing the value of deciion variable from hitoric value in the HM with the probability i 90%, and the value of deciion variable i elected from it poible range with a probability of 0%. All deciion variable, = (,,,..., ) 2 3 N, choen by memory conideration are eamined to be pitch-adjuted with the probability PAR. Pitch adjutment make a equence of local change (pitch adjutment) on the new harmony with probability PAR where (0 PAR ). The ize of thee local change i elected a in Eq. (4). The pitch adjuting deciion for i i: Ye with probability PAR i = (4) No with probability PAR The value of deciion variable not obtained by memory conideration with probability of ( PAR) are not changed. The deciion variable, uppoe it i ( k) element (or value) in i, that i, the kth, that eamined to be eamined for pitch adjuting, i choen a a neighbouring value with probability (PAR HMCR). For eample, if HMCR=90% and PAR=20%, the probability of electing the neighbouring value of any deciion variable i 8%. If the pitch adjuting deciion for i Ye, the pitch-adjuted value of ( k) i = ( k m) where m i the neighbouring inde, {..., 2,,0,,2,... } i i + i i: m (5) The HMCR parameter help the HSA to find globally improved olution, and the PAR parameter help it to find locally improved olution [26]. 3.4 Update the harmony memory If the new harmony vector, (,,..., ), 2 3 =, ha a better objective function value than the wort harmony tored in the HM, then it i included in the HM and the wort harmony vector i ecluded from the HM. N 3.5 Check the topping criterion Step 3 and 4, preented in ubection 3.3 and 3.4, are repeated until the top criterion (maimum number of improviation) i met. Thi i determined by the NI parameter. 64

9 4 A hybrid harmony earch algorithm for the UCTP Thi ection decribe our HHSA for UCTP. Firt, we briefly decribe HSA a preented in our previou work on UCTP; then, we illutrate the combination of HSA, HCO, and PSO. Figure give peudo-code for each tep of HHSA 4. The HSA for UCTP The HSA repreent the UCTP a a deciion variable indeed by room and time lot, where r,p ay whether a room contain event e in room r and timelot p or - if it i empty. r R and p P ; See Eq. (6). The value of parameter e indicate the number of room, and p i the number of time lot. 0,0,0 = 2,0... r,0 0,, 2,... r, Step pecifie the UCTP parameter and poible range for each deciion variable (that i, event). The objective function, f(), which decribe the violation of oft contraint S, S2, and S3, i defined in thi tep. The parameter of HSA (HMCR, PAR, HMS, and NI) are alo elected in the ame tep. 2 HMS In tep 2, the random feaible timetable, {,,... }, are generated according to the HMS. Thee timetable will be decreaingly tored to the HM according to the objective function. See Eq. (7). The feaibility for all HM member i maintained with a method that combine larget weighted degree (LWD), backtracking algorithm, and the MultiSwap algorithm (for more detail, ee the peudo-code in tep 2 of Figure ). HMS HMS HMS 0,0 0, 0, p 0,0 0, 0, p HMS HMS HMS,0,, p,0,, p = HMS HMS HMS HM ~ 2,0... r,0 2,... r, 2, p... r, p 2,0... HMS r,0 0, p, p 2, p... r, p 2,... HMS r, 2, p... HMS r, p In tep 3, the new harmony (ee Eq. (8)) i improvied baed on memory conideration, random conideration, and pitch adjutment operator; if the complete new harmony i not obtained, then the repair trategy ha to take over. 0,0,0 = 2,0... r,0 0,, 2,... r, In memory conideration, the value of the deciion variable 2 HMS elected from {, },0 0,0 0,0 0, 0 0, p, p 2, p... r, p, 0 (6) (7) (8) 0 hould be randomly 0,..., the value of other deciion variable, 65

10 (,...,,..., ) 0, 0,2 0,3,0 2,0,.., r, p,, according to the baic HSA, hould be elected equentially in the ame way [23, 24, 27]. In the UCTP cae, thi equential proce i not ueful becaue, in our cae, the feaibility for new harmony hould be maintained (that i, we worked in feaible earch pace region). The main difficulty of the UCTP lie in the attempt to find valid location for each event. Indeed, each event ha a pecific number of valid location in the HM member to be cheduled in the new harmony. For thi reaon, the election of the new event,, to be cheduled in new harmony hould be baed on available location for thi event in the HM member. New For UCTP, we propoe the mallet poition algorithm to elect the appreciate event, i, j, to be cheduled in a new harmony with probability HMCR. In thi algorithm, the event e E i ordered iteratively baed on the minimum valid location available in the HM member. Let 2 et Q = { Q, Q,..., Q } K e e e e be the HM member that have a valid location for event e to be cheduled in new harmony. The event e with the minimal number of valid place will be cheduled firt. The valid location for event e will be randomly choen from Q uch that New 2 { } K 2 2 k k,,..., e e e e with probability HMCR, where ( Q, Q,..., Q ) e e e and K i the total number of HM member that have a valid location for event e.. In random conideration, the event e elected according to the mallet poition algorithm i cheduled to the new harmony baed on the poible available location in the new harmony rather than in HM member with probability (-HMCR). Indeed, in certain ituation the two above operator cannot find a feaible timetable (thi happen in ome medium and large data intance). In thi cae, the algorithm initiate a repair proce through one-level backtracking. In the pitch adjutment operator, every deciion variable (event),, obtained by the memory conideration i eamined to determine whether it hould be pitch-adjuted with probability PAR. Moreover, for the UTCP, we deign two pitch adjuting procedure, each one of which reflect a neighbourhood tructure. Each pitch adjuting procedure i controlled by a particular PAR value a follow: - PAR- PAR denote the rate of local change in the new harmony by moving an eamined event e to a different location while preerving feaibility. More formally, let an event e be aigned to feaible timetable in location k, e = k baed on memory conideration. Let location k be randomly elected. Location k i free and uitable for event e. The event e i moved to another location k uch that ( k ) e =. 2- PAR2- It denote the rate of local change in the new harmony by wapping the location of the eamined event e and another event e while preerving feaibility. More formally, let an eamined event e be aigned to feaible timetable in the location k, e = k, baed on memory conideration. Let an event e aigned to location k, e = k, be randomly elected. Note that both location k and k are uitable for both event e and e. The location of event e and event e are wapped uch that = k and k e e =. Thee local change of new harmony do a random walk without adopting any acceptance criteria like firt improvement or bet improvement. A uch, thi operator emphaie eploration rather than eploitation, which hould be et to a mall value. new i, j i, j 66

11 Step 4 calculate the objective function for new harmony; if the objective function value of new harmony i better than the objective function of wort harmony in HM, then the new harmony i included in HM and the wort olution i ecluded from the HM. In tep 5, tep 3 and 4 are repeated until the top criterion determined by NI i atified. 4.2 Hybridization Thi ection dicue the hybridiation trategy between HSA and hill climbing optimier (HCO), called the hybrid harmony earch algorithm (HHSA). We then eplain the need to hybridie the HHSA with Practical Swarm Optimiation (PSO) concept to increae the convergence rate. Section 4.2. dicue the HHSA and ection dicue hybridiation between HHSA and PSO Hybridization with HCO The HSA ha three key component that generate a new harmony in every run. Memory and random conideration are the two component reponible for global improvement: memory conideration i a ource of eploitation, and random conideration are a ource of eploration [28]. The pitch adjutment component i the third component of ha, which i reponible for local improvement. The ize of the local improvement thi component make i determined mainly by the number of deciion variable of each problem. The ize i pecified by the probability of HMCR and PAR: each deciion variable that i elected according to memory conideration with probability HMCR i eamined for potential pitch adjutment with probability PAR. Thu, the probability that each variable will be pitchadjuted i HMCR PAR. A uch, the amount of local improvement in the new harmony doe not guarantee that the new harmony converge to the local optimal olution in each run. Furthermore, thee local improvement are done randomly, which mean that not all local change might be reponible for the improvement in the new harmony. Recently, Feanghary et al. [29] have propoed a new hybridiation trategy for the harmony earch algorithm to olve engineering optimization problem. The main motivation of thi hybrid algorithm are to improve the olution quality and lower the computational cot of HSA. Their work incorporate equential quadratic programming into the tep 4 of the HSA to improve the quality of the new harmony vector and other vector in HM. Thi incorporation proce, a Feanghary et al. eplain, need careful attention when the HMS i choen for the problem, epecially conidering that the local optimier i computationally epenive. Finally, Feanghary et al. ugget that the applicability of hybrid harmony earch can be found by uing a few harmony memory vector. In fact, the HHSA preented in thi tudy i imilar to the memetic algorithm (MA), which i ued motly for timetabling problem by incorporating hill-climbing into genetic algorithm (GA) operator[22, 30]. One poible way to incorporate hill-climbing i Elaborate Encoding and Operator [22], in which GA and the local earch operator work in a feaible earch pace region. The main retriction to thi trategy i that the eploratory ability of MA may be decreaed. In thi tudy, we combine the hill climbing optimier (HCO) with the HSA a a new operator (ee Figure, tep 3) which i controlled by a new parameter called hill climbing rate (HCR). The HCR i the rate of uing the HCO to improve the new harmony vector relative to the number of improviation (NI). The HCO tart with a feaible new harmony generated by the original HSA operator. In each run, the HCO eplore the neighbourhood tructure of the new harmony and move to another neighbouring olution that ha a better or equivalent objective function value. Indeed, we ue the bet improvement and ide walk acceptance criteria to guide the HCO, which mean that the HCO accept only the bet local change (the change with the bet objective function value) among all poible local change. Side 67

12 walk mean the current olution walk to it neighbouring olution without changing the objective function. We conider five different neighbourhood tructure, a follow: N- Move one event from a timelot to another empty one; N2- Swap two event in two different timelot; N3- Echange three event in three eparate timelot; N4- Move one event from a room to another empty one in the ame timelot; N5- Swap two event in two different room in the ame timelot. The firt three neighbourhood tructure directly affect the olution cot, becaue they are directly linked to time lot and event. Thi mean that the local change that are linked to the interaction between time lot and event affect the oft contraint S, S2, and S3, epecially by penaliing the event with relation to the time lot. The other two remaining neighbourhood tructure do not affect the olution cot ince they eecute the ame time lot proce which i mainly ueful in the ide walk. Thee two neighbourhood tructure very uefully reveal more option to the HCO in the coming iteration, becaue event may acquire a room with a better ize and feature Hybridization with PSO Memory conideration i the mot vital operator in HSA. Mot of the deciion variable in the new harmony are elected from the other vector tored in HM. In the timetabling cae, the deciion variable (or event) are, by nature, highly correlated, which mean that the random election of the value of the event from any HM vector might influence other event in the new harmony, and thi in turn might violate ome oft contraint. Therefore, the HSA ha a low probability of generating a good-quality new harmony when the value of each ingle event i different in the HM vector, which may get tuck in a bottleneck. Actually, in HSA, thi limitation ha little or no effect on the convergence behaviour of HSA while, in the propoed HHSA, uch a limitation doe practically impede the effect of the peed of convergence. Here, the quetion below hould be eplicitly anwered: Why doe memory conideration work well in HSA while it doe not work well in HHSA? To the bet of our knowledge, the vector tored in HM on HSA moothly converge to the ame earch pace region during the generation proce, in which any combination (or miture) done by memory conideration to generate a new harmony lead to the ame earch pace region, and the functionality of thi operator i thu efficient. Keep in mind that random conideration and pitch adjutment are the ource of eploration to other earch pace region. In contrat, the vector tored in HM in the propoed HHSA belong to different earch pace region that have a different tructure than that ued by HCO. Thu, the functionality of the memory conideration i lot in mot cae where diverity i too high. To addre thi limitation ariing from the memory conideration of HHSA, we need another election criterion that reduce the election preure of memory conideration. Therefore, the new harmony cot in mot iteration can be improved. In thi tudy, we modify the memory conideration to ue idea from Particle Swarm Optimization (PSO)[3]a an auiliary rule to elect a promiing value of the deciion variable from the vector tored in HM. Thi auiliary concept mimic the bet harmony among the HM vector to contruct the new harmony. Figure, tep 3, preent the peudocode for the PSO concept. The memory conideration elect the value(or location) of the deciion variable primarily from the bet harmony tored in harmony memory, if it i not feaible; the value of the deciion variable are elected from any valid value tored in any HM vector. Similar to our work preented in thi ection, Omran and Mahdavi [32] propoed a new variation of HSA called global bet harmony earch. In their work, the pitch adjutment, which wa modified to mimic the bet o-far olution obtained in HM, i imilar a the PSO concept. 68

13 4.2.3 Rationale for adopting the propoed hybrid mechanim. To better undertand our motivation for introducing our hybrid algorithm, recall two term from metaheuritic algorithm: eploration and eploitation. Any effective and robut metaheuritic algorithm mut be baed on two earch technique to find a global optimal olution: eploration to reach not-yet-viited region in the earch pace when it i neceary, and eploitation to make ue of the previou viited earch pace region to yield a high quality olution. Thee term contradict each other. The uitable balance between both eploration and eploitation mut be achieved in any metaheuritic to find high quality olution. Any ucceful metaheuritic (local or population-baed) ha component reponible for eploring the earch pace efficiently and effectively and eploitation to fine-tune the region of earch pace that were already viited. Thi i achieved by the objective function conideration. The more the metaheuritic component oberve th 3 e objective function of previou tate already viited during the earch, the more the earch drift toward eploitation. In contrat, the more the ame component oberve randomne, the more the earch i drawn to eploration [33]. Briefly, the pure hill climbing algorithm ha an eploitation component with no eploration power [34]. The convergence rate of any metaheuritic relie on the balance between eploration and eploitation during the earch. In general, local earch metaheuritic are very effective in eploring the ingle earch pace region (eploitation) and finding the local optimal olution in thi region (eploitation), yet they fail to eplore the entire earch pace and may get tuck in a locally optimal olution (lack of eploration). On the other hand, the component power of population-baed algorithm i very powerful in viiting multiple earch pace region at the ame time (eploration), yet they cannot fine-tune promiing region to find a globally optimal olution (eploitation) in each region. A pointed out earlier, recent urvey of metaheuritic direct the new reearcher to turn to hybridization between population-baed and local earch algorithm. For eample, Blum and Roli [33] write that In ummary, population-baed method are better in identifying promiing area in the earch pace, wherea trajectory method are better in eploring promiing area in the earch pace. Thu, metaheuritic hybrid that in ome way manage to combine the advantage of population-baed method with the trength of trajectory method are often very ucceful.. 3 A an eample for the metaheuritic component [33]:In a population baed algorithm like a genetic algorithm or ant colony ytem, the recombination operator work from the argument that a better offpring i obtained by a collection of good piece from it parent. Thi mean that the higher the recombination rate (that i, the croover rate) i increaed, the more the offpring i like it parent. A uch, the recombination operator i ueful in eploitation. The oppoite cae take place if it decreae. Another eample of a metaheuritic i the acceptance criterion in imulated annealing, which depend on the value of temperature and the objective function. The high temperature that mean the chance of accepting the uphill move increae (the amount of eploration increae).the oppoite happen if it decreae. A final eample of metaheuritic i the tabu lit in tabu earch, in which the more the ize of the tabu lit (that i, tabu tenure) increae, the more the diverification power increae. 69

14 Step. Initialize parameter : Initialize the problem parameter - initialize the et and parameter (ection 2.2) - f() : objective function of timetable. - : et of Deciion variable (event). - X : et of poible range for each deciion variable (in our tudy, all valid place of each event) 2: Initialize HHSA parameter -HMCR : Harmony memory conideration rate. -PAR : Pitch Adjuting Rate. -NI : Number of Improviation. -HMS : Harmony Memory Size -HCR : Hill Climbing Rate Step 2. Initialize the harmony memory : for k = to HMS do 2: begin //Generate timetable k 3: Repeat 4: Larget-weight-degree( k ) 5: while( k i incomplete && predefined Iteration are not met) do 6: begin 7: backtracking ( k ) 8: MultiSwap ( k ) 9: end // while 0: Until ( k i complete) or (retart after N iteration) : Calculate f( k ) 2: Store k to HM according to f( k ) 3: end // for k Step 3. Improvie a new harmony : for i= to n e do // ne i the number of event 2: begin 3: e = mallet poition() 4: If (U(0,)<HMCR) //Memory Conideration 5 begin Bet 6: if ( e i valid to 7: e = Bet e e ) then // PSO concept 8: ele 2 K 2 K 9: e { e, e,..., e }// where { e, e,..., e } i the olution in the HM that have a valid place for event e 0: If (U(0,)<PAR) Pitch_ Adjutment _Move( e ) // Pitch Adjutment : If (U(0,)<PAR2) Pitch_ Adjutment _Swap( e ) 2: end 3: ele // Random Conideration 4: e M // M i the et of valid place for event e 5: end :If (U(0,)<HCR) then // hill climbing Optimizer 2: begin 3: calculate f( New ) 4: bet_sol = New ; 5: for k= to maimum number of iteration do 6: begin 7: for a = to L do // L i the number of neighbourhood tructure 8: begin 9: Temp_Sol(a)= Apply neighbourhood tructure (a) to ; 0: Calculate f(temp_sol(a)) : end // for a 2: Sol* = bet olution among Temp_Sol(a); 3: f(sol*)= f(bet olution among Temp_Sol(a)); 4: if (f(sol*) f(bet_sol)) 5: begin 6: bet_sol = Sol*; 7: end // if 8: end // for k 9 : New = Sol*; 20:end // if Step 5. Check the topping criterion Step 4. Update the harmony memory : Repeat 2: tep 3 and tep 4 3: Until(NI i atified) 4: Done Figure. The tep and peudo-code of our hybrid algorithm. : calculate f( New ) 2: t = wort timetable in HM 3: if ( f( New ) i better than the wort f( t ) ) 4: begin 5: eclude t from HM 6: include t to HM 7: end 8: endif 70

15 5 Eperimental reult and analyi We now turn our attention to evaluate our hybrid method. Section 5. dicue the problem intance, and Section 5.2 preent the comparion reult of our hybrid method and other preented in the previou tudy that ue the ame problem intance. Section 5.3 analye different individual and hybrid algorithm propoed in our tudy. Finally, we illutrate an empirical tudy of the performance of our hybrid method on a different HCR in Section 5.4. We ran the eperiment on an Intel 2 GHz Core 2 Quad proceor with 2 GB of RAM. We implemented our method in Microoft Viual C++ verion 6.0 under Window XP. 5. The problem intance The UCTP data ued in the eperiment in thi tudy are freely available at prepared by Socha et al.[5]. For the purpoe of our tudy, we call them the Socha benchmark. The problem intance, which are grouped into five mall problem intance, five medium problem intance and one large problem intance, have different level of compleity and variou ize, a hown in Table. The olution to all problem intance mut atify the hard contraint tated in Section 2.. Furthermore, the olution cot i meaured by the defined oft contraint violation. Table The characteritic of each cla of Socha benchmark [5]. Cla Small Medium Large number of event number of room number of feature number of tudent number of timelot approimate feature per room percentage of feature ue maimum number of event per tudent maimum number of tudent per event Comparion with previou work Thi ection compare the bet reult obtained by HHSA to thoe applied by other method in the literature to the ame Socha benchmark which are lited the Table 2 a follow: RRLS Random Retart Local earch [5]. MMAS MAX-MIN Ant Sytem [5]. THH Tabu-earch Hyper Heuritic [3]. VNS Variable Neighbourhood earch [35]. FMHO Fuzzy Multiple Heuritic Ordering [2]. GHH Graph-baed Hyper-Heuritic [4]. RII Randomied Iterative Improvement [36]. HEA Hybrid Evolutionary Approach []. HSA Harmony Search Algorithm [9]. GD Great Deluge [37]. GDNLDR Great Deluge with Non-Linear Decay Rate [37]. HHSA The propoed Hybrid Harmony Search Algorithm. 7

16 We compare the obtained reult of HHSA with thoe of previou method in Table 2. The bet reult reported among all method are highlighted. The comparion are meant to how the ability of HHSA to find high quality olution to the UCTP. In hort, the HHSA outperformed the previou method in four out of five medium problem intance. The reult of HHSA alo hared the ame bet known reult with RII, HEA, and ome reult introduced by MMSA, THH, VNS, and GDNLDR for mall problem intance. In addition, HHSA report the econd-bet reult in the large problem intance. Table 2 A comparion of reult on the mall / medium/large data intance with previou method. Small Small 2 Small 3 Small 4 Small 5 Medium Medium 2 Medium 3 Medium 4 Medium 5 Large RRLS Avg MMAS Avg THH Bet VNS Bet FMHO Bet GHH Bet RII Bet Avg HEA Bet Avg HSA Bet GD Bet GDNLDR Bet HHSA Bet We are particularly intereted in comparing our reult with HEA, which i the cloet approach to our HHSA. The HEA hybridied variable neighbourhood tructure with genetic algorithm (memetic algorithm), and our propoed algorithm alo combined HCO with the harmony earch algorithm. Both method incorporate local earch into population-baed algorithm. Clearly, combining local earch algorithm with population-baed algorithm produced better reult than other that were olely baed either on local earch or populationbaed algorithm eparately in mot cae. Note that any method can obtain good reult by conidering the right balance between global and local improvement incorporating both population and local earch algorithm. 5.3 Performance of the propoed hybrid method Thi ection dicue the performance of five different method propoed by u on the Socha benchmark. Specifically, thee are the HHSA with PSO, HHSA without PSO, HSA with PSO, HSA without PSO, and HCO. The ame program i ued for all of thee method (in the econd method and fourth one, the PSO code are deactivated in the program), and the parameter configuration for each method i preented in Table 3. The firt two method are teted in the ame parameter configuration to meaure the effectivene of the PSO on HHSA. Furthermore, the effectivene of PSO upon the HSA i alo tudied by the eperiment done in the third and fourth method, which run in the ame parameter configuration. Finally, we eperiment with the HCO with more relaed computational time than that ued in HHSA. Table 4 how the comparion reult obtained by the five propoed method. The bet reult obtained are highlighted. Each method run ten time for each Socha benchmark. The 72

17 bet cot, the wort cot, and the approimate computational time among all run are reported. We alo report the average and tandard deviation of the bet reult obtained for each of the ten run. The reult demontrate that the HHSA with PSO can find higher quality olution for all Socha benchmark than thoe of other preented in thi tudy. Table 3 The parameter ued for different method Parameter HHSA with PSO HHSA without PSO NI HMS HMCR PAR PAR2 HCI* HCR.5 e % 2% 4% 2 e 3 30%.5 e % 2% 4% 2 e 3 30% * HCI (the number of iteration of HCO). HSA with PSO e % 2% 4% 0 0% HSA without PSO e % 2% 4% 0 0% HCO 00% 0% 0% e 5 00% Note that the number of iteration for both hybrid method (HHSA with PSO and HHSA without PSO) are the ame, but the reult are ignificantly different, which demontrate the effect of the PSO on HHSA convergence behaviour. See Figure 2 a an eample of the effect of the PSO on HHSA, where the two trend are the average cot of HM vector in each iteration. Thi mean that the HHSA without PSO required more computational reource becaue it had to do additional iteration to be able to find the ame reult obtained by HHSA with PSO. For the large problem intance, the PSO help the HHSA to find a feaible olution in each run. Furthermore, we tudy the impact of PSO on the convergence behaviour of HSA. Here, we do ten run for each Socha benchmark uing two method: the HSA with PSO and the HSA without PSO. Our reult how that the PSO ha not affected the ability of HSA to converge to a minimal olution, while it help the HSA find a feaible olution for large problem intance. Finally, we can oberve that hybridiation between HSA a a global improvement earch trategy and HCO a a local improvement earch trategy, with auiliary concept from PSO, had a better olution cot than HSA without PSO and HCO, when each of them i individually running. Thi hybrid method tand out for it ability to trike a balance between global improvement and local improvement in a parallell optimiation environment. Figure 2. The impact of PSO on the HHSA for medium problem intance. 73

18 Table 4 The performance of different variation of the propoed method on each of the Socha benchmark. (Note that h mean hour and m mean minute). HHSA with PSO HHSA without PSO HSA with PSO HSA without PSO mall Bet Average Wort Std.dev Time 8.4 m 7.04m 5.m 4.83m.06m Small2 Bet Average Wort Std.dev Time 32.45m 28.5m 4.55m 4.6m 2.70m Small3 Bet Average Wort Std.dev Time.03 h 47.56m 20.7m 20.53m 3.20m Small4 Bet Average Wort Std.dev Time 38.7 m 40.33m 2.9m 9.59m 4.65m Small5 Bet Average Wort Std.dev Time 2.44 m 9.47m 25.0m 29.5m 7.7m Medium Bet Average Wort Std.dev Time 2.06 h.56h 25.57m 22.27m 37.50m Medium2 Bet Average Wort Std.dev Time 2.03 h 2.4h 2.4m 20.53m 32.55m Medium3 Bet Average Wort Std.dev Time.54 h.35h 22.47m 24.23m 2.54m Medium4 Bet Average Wort Std.dev Time.65 h 2.23 h 50.5m 34.73m 2.43m Medium5 Bet Average Wort Std.dev Time 4.6 h 4.5 h 35.93m 3.58m 3.9m Large Bet Average Wort Std.dev Time 2.36h - 46 m m HCO 74

19 5.4 Empirical tudy of the effect of HCR on convergence rate of HHSA In thi ection, we aim to evaluate our hybrid method with PSO on different HCR etting, bearing in mind that the ame parameter etting tated in Table 3 are ued with different HCR etting. A we eplained earlier, HCR i the probability of uing the HCO in our hybrid method at each iteration, that i, NI. For eample, if HCR=50%, thi mean that the HHSA ue the HCO in each iteration with probability of 50%. Value of HCR variation are etting of three cenario: too mall being elected 5%, medium elected 30% and too large elected 80%. Thee variation of HCR ran ten time for each Socha problem intance. The bet cot, wort cot, and the computational time are recorded in Table 5. The average and tandard deviation indicate the average and tandard deviation of the bet value among the HM vector obtained from each ingle run. The bet reult between HCR variant i highlighted. From the reult recorded in Table 5, we can ee that the bet reult of all Socha benchmark are obtained from a high etting of HCR, which mean that uing HCO with a high HCR improve our algorithm' performance. The computational time i ignificantly different in the three variant, where increaing the HCR lead to increaing computation time with a ignificant improvement in the olution quality. A further effect of the variant HCR on our hybrid method can be een in Figure 3 which demontrate tatitical evidence that the HHSA i very effective in obtaining a high quality olution and that the HCR ha a trong effect on the behaviour of HHSA. Bo-plot in Figure 2 demontrate the approimation ditribution of cot value of the HCR variant for each Socha problem intance. The bo how the cot value between firt and third quartile, while the bold line inide the bo denote the median of obtained cot. The lower and upper cot value are indicated by whiker. The aterik denote highly etreme cot value. 6 Concluion and future work In thi paper, we preented HHSA to tackle UCTP. We have incorporated HCO with harmony earch algorithm a a new operator to improve the quality of new harmony in each run with probability HCR. Thi idea tem from an analogy with MA to find a trade-off between the locally improved olution from HCO and the globally improved olution from HSA. The reult demontrate that our algorithm can find a high-quality olution with a reaonable computational time, compared to previou work. Having been inpired by PSO concept, we modified the memory conideration operator to mimic the bet harmony o far found that i tored in harmony memory. Thi modification influence the behaviour of the propoed HHSA, being able to configure a good-quality new harmony that might be further improved by HCO. In hort, our HHSA with PSO concept could find the bet reult for four out of five medium problem intance prepared by Socha et al. It alo hared the bet reult with ome pat literature reearch on mall problem intance, while the econd bet reult for large problem intance i obtained. Thi how that hybridiation between local earch algorithm and population-baed algorithm i promiing in thi reearch area and can introduce highquality olution. In fact, our previou work and the contribution in thi paper work on a feaible region of earch pace. The limitation mainly implie the difficulty of producing a feaible olution in each iteration, which alo affect running time. Our future aim i to tudy the hybrid harmony earch algorithm over the entire earch pace (feaible and infeaible region) to diverify earching even more efficiently. A a recommendation, we propoe that the hybrid harmony and harmony earch algorithm to the UCTP problem ha broken new ground and thu can be ued with great efficiency in the optimiation proce for combining global and local component in one algorithm, which can then be ued in other cheduling problem like eamination timetabling and chool timetabling. 75