School Timetabling using Genetic Search

Transcription

1 School Timetabling using Genetic Seach Caldeia JP, Rosa AC Laseeb - ISR IST acosa@is.ist.utl.pt Abstact In the pape we discuss the implementation of a genetic based algoithm that is used to poduce timetables fo a small school. A poblem specific chomosome epesentation and the use of a epai algoithm afte the genetic opeatos avoid seaching though illegal timetables. We also tested the use of diffeent fitness functions and pesent esults obtained with ou pototype timetabling system implemented in C.. Intoduction The school-timetabling poblem is basically the assignment of weely lessons to time peiods. Existing solutions ae eithe difficult to use o lead to inadequate solutions. Ou objective is to develop a pogam that can easily be used in a typical school and allowing use inteaction and modification of paamete settings. In section we discuss the class-teachetimetabling poblem in moe detail. The genetic algoithm (G.A.), chomosome epesentation and the cost and fitness functions ae explained in section. Section 4 pesents the genetic opeatos and epai function. We finish off in section 5 with computational esults and conclusions.. The Class-Teache Timetabling poblem The Class-Teache Timetabling [] poblem basically consides set of classes, and a set of teaches. Each class is a set of students having a common cuiculum and studying togethe. Fo each pai <class, teache> the equied lessons and thei numbes ae defined. Data coesponding to the peiods of unavailability of each teache must also be pesent. Each day of the wee is divided into 0 60-minute peiods (time slots), which esults in a total of 50 peiods numbeed fom 0-49, as can be seen in fig. Time Mon. Tues. Wed. Thus. Fi Fig.. Division of wee into peiods (time slots) Each lesson must be assigned to a time peiod in such a way that a numbe of equiements ae met. These equiements can be divided into categoies: - had constaints - second-ode (o soft) constaints A timetable is feasible if and only if the following had constaints ae satisfied: Each lesson is scheduled to exactly one peiod Thee ae no clashes at all: Neithe a class no a teache is assigned to moe than one lesson in the same peiod Teache unavailabilities ae consideed In addition to the above conditions, good timetables satisfy as many of the following soft constaints as possible, such as: Lesson continuity students and teaches do not lie timetables with gaps between lessons. A lunch bea must be scheduled (one hou between -5 must be fee). The numbe of lesson pe day fo a teache o class can not exceed a specified limit. Lessons of the same subject must be distibuted unifomly ove the wee As fa as possible, classes should eithe have lessons in the moning o aftenoon. Classes and teaches pefe to have moe lessons on some days, in ode to have a day without lessons (fee day).

2 . The Algoithm.. Chomosome Repesentation The chomosome is constituted by genes, which depending on thei position, contain data elative to the time slot associated to each lesson in the class and teache timetables. As seen in fig., the time slot data is duplicated in the two genes that epesent a cetain lesson of the class-teache pai. One in the class timetable and the othe in the teache timetable. The numbe of genes of each chomosome is, theefoe, two times the total numbe of lessons of all classes. The heade stoes all infomation common to all chomosomes, i.e. the infomation as to whee in each chomosome you can find the genes that epesent the lessons of a cetain subject taught to a cetain class. This chomosome epesentation was chosen because it allows efficient chomosome initialisation and epai. Obseving Fig., we can see that when any coesponding genes of any two chomosomes ae switched (by a cossove o mutation opeato, fo example), the esulting chomosomes automatically satisfy constaints elative to the equied lessons and thei numbes fo each <class, teache> pai. A timetable is only id, howeve, afte possible esulting clashes have been coected by the epai function. If a time-slot wee andomly associated with a lesson, the ode of the seach space would be dim nº_of_classes*50 total_nº_of_lessons. Fo a timetable to be feasible, howeve, all had constaints (as defined in pevious section) must be met. We can geatly educe the seach space if we wo within these estictions, maing it much faste to each good solutions. This is vey impotant in a highly constained poblem lie this, othewise, one uns the is of ceating a genetic algoithm that spends most of its time euating illegal individuals []... Initialisation The G.A. stats by checing a numbe of conditions necessay fo feasible timetable to exist. The Initialisation pocedue andomly ceates a population of feasible solutions (solutions that satisfy all had constaints). To detemine which peiods will be assigned to the n i lessons of subject s i lectued to class c i, the initialisation pocedue: - Calculates the set F I of the common peiods between the set of fee peiods of the class (set F C ) and set of fee peiods of the teache (set F T ). F I F C F - Randomly selects n I peiods of F I and emoves them F C and F T which ae late used when these steps ae epeated fo the othe subjects and classes. - If the numbe of peiods in F I is less than n I the chomosome is einitialised. The lesson allocation ode is detemined by scacity of its esouces. Lesson with a moe limited numbe of possible allocations ae consideed fist. This is also the case in the epai algoithm, discussed late on. If esouces ae vey scace, the initialisation and T Class Class 4 Teache of M Teache of I Heade M M M F F F Q Q Q I I I M M M F F F Q Q Q I I I T T T T T T T T T T4 T4 T4 T T T T T T T T T T4 T4 T4 LAB Mon Tues Wed Thus Fi Mon Tues Wed Thus Fi F I M 4 0- T T4 T 4 - I 4 - T T T 4 4 T M 5 Q T 5 T T M F T 6 T F 7 Q 7 7 I Q Timetable of Class Timetable of Teache of subject M Fig. shows chomosome epesentation and how time slot (peiod) data duplicated in genes. Fo example, both fist genes in class and teache of M timetables efe to the same lesson and theefoe coesponding gene data efe to the same peiod (6).

3 epai of a chomosome will tae much longe because these algoithms will fequently fail to etun id timetables and have theefoe, to be epeated. Although all membes of the initial population ae feasible, they geneally suffe fom vey poo fitness. This is because the Initialisation outine does not conside any soft-constaints. Handling of the soft-constaints is left to the evolutionay pocess of the genetic algoithm... The Cost Function The cost function is an vey impotant pat of any G.A. It is esponsible fo detemining the ue of each chomosome, allowing us to distinguish the good fom the bad and in this way guide the G.A. to bette solutions. Ou Cost function has the fom of: Cost( Chomosome) pofscale teaches Cost classes Cost ( Timetable) ( Timetable), pofscale [ 0,] Although the costs of class and teache timetables ae detemined in the same way (by function Cost(Timetable) ), it is geneally moe impotant fo classes to have good timetables, than fo teaches. The weight, pofscale, allows us to tell the G.A. how much moe impotant the class timetables ae. The function Cost(Timetable) is composed of a weighed sum of the penalty ues imposed to the violation of each of the soft-constaints in section. It can be witten as: Cost ( Timetable) ns s ndij nlsd + nld j i + gap. n g + na MoA+ ne LH + ( no nmin ) FO s i + ni 7 Some of the penalty factos ae linea and some ae exponential and the use can change the weights of the diffeent penalties, while the pogam is unning. The default ues wee chosen assuming a weight ue w i of a cetain penalty facto and choosing the othes elative to w i in ode fo esulting timetables to have desied chaacteistics. The fist pat of the timetable cost function nd ij nlsd penalises timetables with moe than one hou of a cetain subject pe day and theefoe, contibutes to distibute a subject s lessons thoughout the wee, instead of concentating them on one day. The default ue fo nlsd is 0 and fo this ue the penalties ae: nd ji - numbe of lessons Penalty of same subject j /day i Tab.: Evolution of the facto that penalises timetables with moe than two lessons of the same subject pe day. The default ue allows the possibility of lessons of the same subject pe day, but maes lessons difficult and 4 and moe vitually impossible. These esults can only be accomplished though a nonlinea tem. s The second tem nld n i 7 penalises a timetable i with moe than seven lessons pe day. Fo the default ue of 40 we have: n i - numbe of lessons Penalty in day i Tab.: Costs esulting fom the facto that penalises timetables with moe than seven lessons pe day. These penalties allow seven lessons pe day, penalise 8 and mae moe than 8 vitually impossible. Geat cae must be taen on which tems ae made exponential. We must bea in mind that if the exponent is lage, an exponential facto will outweigh all othes. It is fo this eason that we can only use exponential factos when the maximum ue of the exponent is small. This was the case in the fist two tems of the timetable cost function. In the next two tems howeve, n g and n a may have lage ues. Theefoe, if we wee to use an exponential tems, not only will the othe tems of the timetable cost function become ielevant but the convegence of the G.A. will be vey slow. The thid tem gap. n penalises timetables with g spaces between lessons. The default ue fo gap is 50 and n g is the total amount of spaces between lessons in the timetable. The fouth tem n a MoA penalises timetables with moning and aftenoon classes. The default

4 ue fo MoA is 50 and n a is the numbe of lessons in the moning, if thee ae moe lessons in the moning than in the aftenoon othewise n a is the numbe of lessons in the aftenoon. n n a m nm nm na na nm < na numbe of moning classes n numbe of aftenoon classes a The tem ne LH (n e - numbe of days without lunch hou) penalises timetables with days without lunch hou. The default ue fo LH is 500. This lage ue guaantees maes it difficult fo timetables with lunch hou to evolve to timetables without. Finally, the last tem (n o -n min ).OD in which, n o is the numbe of days with lessons and n min is the minimum numbe of days in which the class could have all its lessons, penalises timetables with moe days occupied with lessons than necessay. The weight of OD should be chosen with the weight of MoA in mind. The lage OD is elative to MoA, the moe days will have lessons in the moning and aftenoon in ode fo some days to be without lessons. The default ue fo OD is The Fitness Function The fitness function also plays a vey impotant ole in G.A. because it is esponsible fo detemining the S.P. (Suvi Pobability) of a cetain chomosome. F S. P. i in which : i Cost F ( ) F Fitness Function i ( chomosomei ) ( ) i i Since wose solutions have geate costs (ae moe penalised), in ode fo bette solutions to have a geate S.P. F( ) < 0. We expeimented with diffeent fitness functions []: S.P. (Suvi Pobability ) Pobability of chomosome being used to spawn next geneation F ( ) F ( ). wost,. wost ], + [ best ( ln ). best F ( ) e,, ] 0 [ in which: Cost of a cetain chomosome wost - Cost of wost chomosome in population best - Value of best chomosome in population - Aveage ue of population The majo diffeence between F and the othe functions is that the latte adapt themselves fom geneation to geneation while F emains constant. In F, the Relative Suvi Pobability (R.S.P. ) of best chomosome elative to aveage chomosomes is ept constant. R. S. P. p p( BestChomosome) ( AveageChomosome) F ( best), F ( ) ] 0,[ With 0.5 the suvi pobability of the best chomosome is twice that of aveage chomosomes. As we can see in fig. vaies the slope of F with lowe ues of leading a moe elitist fitness function. A fitness function is said to be moe elitist when it is moe selective i.e. slight diffeences in chomosome cost lead to significant diffeences in S.P. SPa( ) SPb( ) SPc( ) Fig.. shows how changes in (of F ) affect the S.P. of chomosomes with diffeent costs. On the othe hand F, adjusts itself in way that the suvi pobability of wost chomosome elative to aveage chomosomes is: R.S.P. (Relative Suvi Pobability) Ratio between the S.P. s of chomosomes

5 p( WostLAB) F wost RSP ( ) p( AveageLAB) F ( ) + wost ] 0, [ ], [ This indicates that the ue of RSP (Relative Suvi Pobability) depends on elative ue between and wost. This dependence can be seen in fig.4 which show that when: In fig.5 we can see how vaies the elitism of F fo a fixed ue of. The close is to the moe elitist F becomes. SPf ( ) SPf ( ) SPf ( ) RSPa( ) RSPb( ) Fig.6 shows a compaison between all fitness functions RSPc( ) Fig.4 shows how changes in affect R.S.P. Finally, in fig. 6, we have a compaison S.P. s esulting fom all thee fitness functions with the following paamete ues: 40000,. and 0.5. Obseving fig.6 we can easily attibute F s poo pefomance to its almost non-existing slope that esults in all chomosomes having appoximately equal copy pobability. SPa ( ) SPb ( ) SPc ( ) RSP RSP 0 0 This means that when wost becomes close to ( ), which is the case when the G.A. is stalled at local maximum, maes the fitness function less elitist to allow the G.A. to find a way aound the local maximum Fig. 5 shows how vaies the elitism of F When is much smalle than wost ( 0), maes F moe elitist maing the G.A. convege faste. 4. Genetic Opeations In this application the canonical G.A. opeatos have been edesigned so that they always poduce feasible timetables. 4.. Repoduction The epoduction opeato consists of the coping of chomosomes without the changing thei chaacteistics. The epoduction method used is Roulette-wheel selection Goldbeg[6], whee the pobability that an element has of being copied, is popotional to its fitness. 4.. Cossove The cossove opeato is esponsible fo exchanging the genetic mateial of the chomosomes that wee epoduced. This is done, andomly foming pais of chomosomes with all the elements of the population. The unifom cossove opeato is applied to each pai. In this opeato, whethe o not two coesponding chomosome genes ae exchanged, depends on the ues of andomly geneated mas. If mas ue

6 is one genes ae exchanged othewise they emain in the same chomosome. This pocedue is easily undestood obseving fig.7. Mas LAB LAB Fig.7 demonstates the unifom cossove opeato 4.. Mutation Each independent gene in evey chomosome has a use defined mutation pobability. Mutation consists of changing a gene ue to a andom position The Repai Function Fo esulting chomosomes to be id they have to be epaied. Repaiing chomosomes consists of the changing of gene ues to id ues closest to the oiginal ones. This is done by fist finding the fee positions (positions unoccupied by othe subjects o classes) common to the timetable of both the class and the teache teaching uncetain subject. The fee position closest to the oiginal gene ue is then chosen and emoved fom the list of fee positions of the class and the teache. individuals ae always peseved esulting in monotonous convegence. In the second method, we stat by euating the fitness of the n ch chomosomes of the population and also of n BP best paent of the pevious geneation. Now, using Roulette-wheel selection we ceate a new population of n ch chomosomes. These chomosomes ae soted by descending ode of ue and the n BP best chomosomes ae stoed in ode to incopoate the next geneation. 5. Results and Conclusions 5.. Euation of Results Fo the simple example poblem of 4 classes and 4 teaches, the G.A. leads to good esults afte less than 0000 euations that too about 50 seconds on a pentium Mhz PC. An example of esults obtained fo this poblem ae shown in fig.9. These esults wee obtained using the fist ulta-elitism method and fitness function F on a population of 60 chomosomes, with n BP at 0 and othe paametes at thei default ues. Common Fee Positions distance 5 Oiginal Value 0 Resulting Value 8 Fig.8 shows how gene clashes ae eliminated by the epai function 4.5. Ulta-Elitism Ulta-Elitism was implemented using two diffeent methods. The fist consists simply of soting chomosomes by descending ode of ue and then eplacing the n BP (numbe of best paents) wost chomosomes with the n BP best chomosomes of the pevious geneation. The chomosomes ae now esoted and the n BP best chomosomes ae stoed in ode to incopoate the next geneation. This ulta-elitism stategy assues that the n best Fig.9 Scool timetable obtained fo a poble with 4 classes (top 4 timetables) and 4 teaches (bottom 4 timetables) Afte many expeiences, we have come to the conclusion that the tuning of the many paametes of this poblem is vey delicate matte. Fist the poblem paametes should be adjusted in ode to obtain a timetable with specific desied chaacteistics. Finally, the convegence paametes have to be adjusted. To do so we have to eep in mind that faste convegence ates nomally lead to a geate numbe of populations being stalled at a local maximum. In othes wods faste convegence

7 implies a naowe seach and a wide seach leads to a slowe convegence. Rate of convegence can be seen gaphically obseving how the fitness of the best chomosome evolves fom geneation to geneation (fig. 0). Fig.. Evolution of aveage chomosome cost with a n BP 40 population size (little vaiation naow seach) Fig.0. Convegence ate of G.A. i.e. Evolution ate of the cost of best chomosome The width of the seach can be seen obseving the ange of ues of the aveage fitness of the population (fig. and ). The paametes that a use can change ae: Population size: This ue should be lage because a bigge population size leads to a betteepesented population and a wide seach. The downside of inceasing this ue is that the calculation time inceases popotionally. Numbe of Best Paents (n BP ): This paamete and the mutation pobability stongly influence both the ate of convegence and the width of the seach. High ues of n BP lead to high convegence ates and naow seaches as can be seen in fig. in which n BP population size. _. On the othe hand, low ues of n BP lead to lowe convegence ates and wide seaches as shown in fig. 0 in which n BP. population_ size. 5 Empiical esults suggest that the ideal ue is somewhee in the inte population_ size, population_ size. Fig.. Evolution of aveage chomosome cost with a n BP 4 5 population size (geat vaiation wide seach) P( mut) total numbe of lessons which leads to appoximately mutation/chomosome. Highe mutation ates lead to moe than mutation/chomosome that esults in bad mutations destoying the effect of good mutations. Less than mutation/chomosome means that the G.A. will be stalled longe at local maximum. Mutation Pobability: Results suggest that the ideal ue fo the mutation pobability is

8 5.. Conclusions: To compae ulta-elitism methods and fitness functions the G.A. pogam was un 0 times fo each vaiation and esults ae shown in table. Ulta- Elitism Fitness Function best st method nd method F F F F G.A.P Enum Tab. Compaison between esults obtained fom diffeent fitness functions and ulta-elitism methods. The table shows that the fist ulta-elitism method is clealy bette than the second fo both fitness functions. This is due to the fact that the wost esults ae nomally vey bad, and theefoe have a geat influence on the aveage ue, esulting in an almost equal suvi pobability fo all the elements of the population. When these elements ae emoved, the G.A. becomes moe elitist and the algoithm conveges much faste. As fa as the fitness functions ae concened, F conveges much faste than F. This can be concluded obseving that gen num. F, on the othe hand, is bette at getting past local maximums as can be seen compaing G.A.Ps. 5.. Futue Diections: We thin it would be inteesting to ty unning a G.A. using as initial population, the n best chomosomes obtained afte unning the G.A. independently m times fo moe geneations. Dynamic opeato selection as suggested in [7] also seems inteesting. The extension to schools with Aveage of the ue of the best chomosomes esulting fom euations. (Empiical ue afte which most uns each goal) 4 Goal Achievement Pecentage - pecentage of populations that have best chomosome cost less than 700 afte euations. (700 is an empiical ue coesponding to the aveage cost of good timetables i.e. that satisfy all soft constaints) 5 Aveage of numbe of euations afte which the goal is achieved. (Only uns that achieve goal befoe euations contibute) aound students and diffeent couses ae undeway. Acnowledgements This eseach was patly suppoted by: Poject AGHoa, Gant /./CEG/684/95 of the Paxis XXI pogam. Refeences []. Badadym, V. A., Compute Aided School and Univesity Timetabling: The New Wave. In Lectue Notes in Compute Science, Vol.5, p-45, Spinge []. Davis, L; Steenstup, M. (987): Genetic Algoithms and Simulated Annealing: An Oveview. In: Davis l. (ed.), Genetic Algoithms and Simulated Annealing. Mogan Kaufmann Publishes Inc., Los Altos, CA:- []. Eben W. and Kepple K., A Genetic Algoithm Solving a Weely Couse- Timetabling Poblem. In Lectue Notes in Compute Science, Vol.5, p98-, Spinge [4]. Coloni A,Doigo M, Maniezzo. Genetic Algoithms: A new appoach to the TimeTable Poblem. In Combinatoial Optimisation (ed. M.Aggul et al.) Lectues Notes in Compute Science - NATO ASI Seies, Vol.F 8, p 5-9, Spinge-Velag. [5]. Allen Lima J, Gacias N, Peeia H, Rosa AC. Fitness Function Design fo Genetic Algoithms in Cost Euation Based Poblems, Poc. IEEE - Int. Conf. Evolutionay Computation, ICEC 96, pp 07-, 996 [6]. Goldbeg, D.E. In Genetic Algoithms in seach, optimization and machine Leaning, Addison-Wesley [7]. Rich, D. C. A Smat Genetic Algoithm fo Univesity Timetabling,. In Lectue Notes in Compute Science, Vol.5, p8-97, Spinge