Stidsen, Thomas Riis

Sørensen, Matias

A general model for the timetabling problem of high schools in Denmark is introduced, as seen from the perspective of the commercial system Lectio1, and an Adaptive Large Neighborhood Search (ALNS) algorithm is proposed for producing solutions. Lectio is a general-purpose cloud-based system for high school administration (available only for Danish high schools), which includes an embedded application for creating a weekly timetable. Currently, 230 high schools are customers of Lectio, and 191 have bought access to the timetabling software. This constitutes the majority of high schools in Denmark

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 20, 2017High School Timetabling: Modeling and solving a large number of cases in DenmarkSørensen, Matias; Stidsen, Thomas Jacob RiisPublished in:Proceedings of the Ninth International Conference on the Practice and Theory of Automated Timetabling(PATAT 2012)Publication date:2012Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Sørensen, M., & Stidsen, T. R. (2012). High School Timetabling: Modeling and solving a large number of casesin Denmark. In Proceedings of the Ninth International Conference on the Practice and Theory of AutomatedTimetabling (PATAT 2012) (pp. 359-364)High School Timetabling: Modeling and solving alarge number of cases in DenmarkMatias Sørensen · Thomas R. StidsenKeywords High School Timetabling · Modeling · Adaptive Large Neighbor-hood Search1 IntroductionA general model for the timetabling problem of high schools in Denmark isintroduced, as seen from the perspective of the commercial system Lectio1, andan Adaptive Large Neighborhood Search (ALNS) algorithm is proposed forproducing solutions. Lectio is a general-purpose cloud-based system for highschool administration (available only for Danish high schools), which includesan embedded application for creating a weekly timetable. Currently, 230 highschools are customers of Lectio, and 191 have bought access to the timetablingsoftware. This constitutes the majority of high schools in Denmark.This large customer base entails a need for a model of the problem which isgeneral enough to suit many diﬀerent requirements, while still remain tractableby computer aided solution methods. This supports the recent trend of devel-oping general models for timetabling problems (see Burke et al (1998); As-ratian and de Werra (2002); Özcan (2005); Causmaecker and Berghe (2010);Bonutti et al (2010); Post et al (2011, 2012)). Furthermore, the timetablingproblem of Danish high schools has not been formally described in the litera-ture before. Some recent formulations of related problems from other countriesinclude Wright (1996); Wood and Whitaker (1998); Bufé et al (2001); Melí-cio et al (2005); Avella et al (2007); Nurmi and Kyngas (2007); Boland et al(2008); Santos et al (2010); Minh et al (2010).Matias SørensenMaCom A/S, DKTel.: +45 3379 7900E-mail: ms@macom.dkThomas R. StidsenDepartment of Management EngineeringTechnical University Of Denmark1 http://www.lectio.dk [lectio@macom.dk], developed by MaCom A/S, Vesterbrogade 481., DK-1620 Copenhagen V.2 Matias Sørensen, Thomas R. Stidsen2 Model formulationThe following sets are given:• Timeslots, usually made up of 5 days and 4-8 daily timeslots.• Entities, the combined set of students and teachers.• Classes, a subset of entities which is taught/teaching a speciﬁc topic.• Rooms• Events (corresponding to lectures), the basic timetabling-unit which mustbe assigned exactly one timeslot in the timetable.For each class, a number of events is given (usually between 2 and 5). Thebasic timetabling problem concerns the assigning of each event to a timeslot,and a room to each event, such that no clashes among entities or rooms occur.Furthermore we introduce the concept of EventChains, which separates thisproblem from related problems described in the literature. An EventChainconsists of a subset of events, and each of these events are assigned an oﬀ-set. At least one event must have oﬀset 0, corresponding to the start of theEventChain. All events of an EventChain with the same oﬀset must be placedin the same timeslot. All events of oﬀset 1 must be assigned the timeslot fol-lowing the timeslot assigned to events of oﬀset 0, and so forth. See also Fig.1. No restrictions are posed on which events can be included in the sameEventChain, and the oﬀsets of events in an EventChain are completely forthe user to decide, providing a lot of ﬂexibility. E.g. a double lecture can beset up by creating an EventChain consisting of two events for the same class,with oﬀsets 0 and 1, respectively. EventChains also allows for parallel doublelectures for several classes, triple lectures, grouping of elective classes in thesame timeslot, etc. Many of such features have been requested by the users ofLectio, and EventChains are a pretty generic way of solving them.Monday TuesdayTS1TS2TS3Fig. 1 Six EventChains placed in a partial timetable.Fig. 2 illustrates the data-model by an example with two classes, each assignedthree events, and two EventChains.Besides the no-clashes constraints, the following hard constraints are in-cluded in the model:• An event cannot be assigned to any of its forbidden timeslots• Each event must be assigned an admissible roomTitle Suppressed Due to Excessive Length 3E3E2E1 E4 E5 E6C1 C2EC1EC2Fig. 2 Two classes and their respective events, and two EventChains.• Only one event of each class per day (unless otherwise speciﬁed in the givenEventChain)• For each teacher, a required number of days-oﬀ is givenThe following weighted objectives (soft constraints) are used:• Maximize the number of events assigned to a timeslot (very high priority)• Maximize the number of events which are assigned a room (high priority)• Maximize the number of days-oﬀ for teachers (low priority)• Minimize idle slots for entities (medium priority)• Minimize the amount of diﬀerent rooms which are assigned to events ofthe same class (room stability constraint, low priority)• Minimize the number of occurrences of two events of the same class beingassigned two consecutive days (neighbor-day constraint, low priority)3 Adaptive Large Neighborhood SearchALNS is a recent extension of the Large Neighborhood Search (LNS) paradigm,often credited to Ropke and Pisinger (2006). As in the LNS framework, ﬁrst adestruct (ruin/remove) operator is applied to the solution at hand, and thena construct (recreate/insert) operator is used to repair the solution. In anALNS framework, multiple destruct and construct operators are used, andthe adaptive layer keeps track of their individual performance, and increasesthe probability of selecting operators which have previously performed 'good'.ALNS has mainly been applied to variants of the Vehicle Routing Problem(VRP) (Azi et al (2010); Hemmelmayr et al (2011); Salazar-Aguilar et al(2011); Ribeiro and Laporte (2012)), but lately also other problem-domains(Muller et al (2011); Muller (2010); Kristiansen et al (2011); Kristiansen andStidsen (2012); Sørensen et al (2012))We propose here an ALNS heuristic for solving the described timetablingproblem. The following remove- and insertion-operators are used:• InsertGreedy: Insert EventChains in greedy way based on contributionto objective. At each insertion, also attempt to assign rooms to insertedevents.4 Matias Sørensen, Thomas R. Stidsen• InsertRegretN: This is similar to the Regret-N neighborhood applied tovariants of the VRP (Tillman and Cain (1972); Martello and Toth (1981);Potvin and Rousseau (1993)). The regret-measure is speciﬁed in terms ofbest and second-best insert-move for a given EventChain.• RemoveRandom: Select N random EventChains and unassign them.• RemoveRelated: Related to Shaw operator (Shaw (1997, 1998)). SelectEventChains to remove based on their similarity between classes, feasiblerooms and entities.• RemoveTime: Select a random timeslot, and remove EventChains assignedto it. Repeat until N EventChains has been removed.• RemoveClass: Select a random class, and remove EventChains which con-tains it. Repeat until N EventChains has been removed.Furthermore we apply a special set of operators, RoomRemove/Insert, whichare coupled. In a coupled set of operators, the choice of remove operator impliesthe choice of repair operator. In RoomRemove, N random room-assignmentare removed from the solution. In RoomInsert, rooms are assigned to eventsin a greedy way.4 Preliminary resultsThe Lectio database contains about 4000 datasets from 94 diﬀerent highschools. Grouping these by school and year entails about 200 'unique' datasets.The authors plan to make at least some of these datasets public, most likelyusing the XHSTT format (Post et al (2012)).Table 1 shows statistics and preliminary computational results for sixdatasets. These results show that the heuristic ﬁnds solutions where manyevents are unassigned, but if they are assigned a timeslot, a suitable roomis also assigned. In the full paper, comprehensive computational studies willbe made. This will include comparison of the found solutions with a boundprovided by an integer programming model.Name #E. #E.C. #R. #C. #Ent. #T. #E./w pos. #E./w roomSorø2011 424 424 70 132 230 50 360.6 (13.6) 360.6 (13.6)Skive2010 2955 1749 58 331 304 90 2358.2 (13.9) 2358.2 (13.9)Fjerritslev2009 530 387 51 152 122 40 487.9 (2.9) 487.9 (2.9)ViborgT2010 536 447 21 116 85 50 515.2 (1.8) 515.2 (1.8)Herning2010 1671 1616 86 355 213 60 1545.2 (2.5) 1545.2 (2.5)Hasseris2011 1343 1080 79 404 578 50 1331.8 (3.2) 1331.8 (3.2)Table 1 Preliminary results for 10 runs of ALNS heuristic on 6 datasets. Columns 2-9shows the number of Events, EventChains, Rooms, Classes, Entities, Timeslots, Avg. eventsassigned to a timeslot, Avg. events assigned to a room, respectively. For columns 8 and 9,also the standard deviation is shown.Title Suppressed Due to Excessive Length 5ReferencesAsratian A, de Werra D (2002) A generalized class-teacher model for some timetabling prob-lems. European Journal of Operational Research 143(3):531  542, DOI 10.1016/S0377-2217(01)00342-3Avella P, DAuria B, Salerno S, Vasilâev I (2007) A computational study of localsearch algorithms for italian high-school timetabling. Journal of Heuristics 13:543556,10.1007/s10732-007-9025-3Azi N, Gendreau M, Potvin JY (2010) An Adaptive Large Neighborhood Search for a VehicleRouting Problem with Multiple Trips. CIRRELTBoland N, Hughes B, Merlot L, Stuckey P (2008) New integer linear programming ap-proaches for course timetabling. Computers & Operations Research 35(7):2209  2233,DOI DOI: 10.1016/j.cor.2006.10.016, part Special Issue: Includes selected papers pre-sented at the ECCO'04 European Conference on combinatorial OptimizationBonutti A, De Cesco F, Di Gaspero L, Schaerf A (2010) Benchmarking curriculum-basedcourse timetabling: formulations, data formats, instances, validation, visualization, andresults. Annals of Operations Research pp 112, 10.1007/s10479-010-0707-0Bufé M, Fischer T, Gubbels H, Häcker C, Hasprich O, Scheibel C, Weicker K, WeickerN, Wenig M, Wolfangel C (2001) Automated solution of a highly constrained schooltimetabling problem - preliminary results. In: Boers E (ed) Applications of EvolutionaryComputing, Lecture Notes in Computer Science, vol 2037, Springer Berlin / Heidelberg,pp 431440Burke E, Kingston J, Pepper P (1998) A standard data format for timetabling instances.In: Burke E, Carter M (eds) Practice and Theory of Automated Timetabling II, Lec-ture Notes in Computer Science, vol 1408, Springer Berlin / Heidelberg, pp 213222,10.1007/BFb0055891Causmaecker PD, Berghe G (2010) Towards a reference model for timetabling and rostering.Annals of Operations Research pp 110, 10.1007/s10479-010-0721-2Hemmelmayr VC, Cordeau JF, Crainic TG (2011) An adaptive large neighborhood searchheuristic for two-echelon vehicle routing problems arising in city logistics. Tech. Rep.CIRRELT-2011-42, Interuniversity Research Centre on Enterprise Networks, Logisticsand TransportationKristiansen S, Stidsen TR (2012) Adaptive large neighborhood search for student sectioningat danish high schools. In: Proceedings of the Ninth International Conference on thePractice and Theory of Automated Timetabling (PATAT 2012)Kristiansen S, Sørensen M, Stidsen T, Herold M (2011) Adaptive large neighborhood searchfor the consultation timetabling problem, to appearMartello S, Toth P (1981) An algorithm for the generalized assignment problem. Operationalresearch 81:589603Melício F, Caldeira P, Rosa A (2005) Solving real school timetabling problems with meta-heuristics. In: Proceedings of the 4th WSEAS International Conference on AppliedMathematics and Computer Science, World Scientiﬁc and Engineering Academy andSociety (WSEAS), Stevens Point, Wisconsin, USA, pp 4:14:8Minh K, Thanh N, Trang K, Hue N (2010) Using tabu search for solving a high schooltimetabling problem. In: Nguyen N, Katarzyniak R, Chen SM (eds) Advances in In-telligent Information and Database Systems, Studies in Computational Intelligence, vol283, Springer Berlin / Heidelberg, pp 305313Muller L (2010) An adaptive large neighborhood search algorithm for the multi-moderesource-constrained project scheduling problem. l, Department of Management En-gineering, Technical University of Denmark Produktionstorvet, Building 426, DK-2800Kgs. Lyngby, DenmarkMuller L, Spoorendonk S, Pisinger D (2011) A hybrid adaptive large neighborhood searchheuristic for lot-sizing with setup times. European Journal of Operational ResearchVolume 218(Issue 3):614623Nurmi K, Kyngas J (2007) A framework for school timetabling problem. In: Proceedings ofthe 3rd multidisciplinary international scheduling conference: theory and applications,pp 3863936 Matias Sørensen, Thomas R. StidsenPost G, Kingston JH, di Gaspero L, McCollum B, Schaerf A (2011) The third internationaltimetabling competition (itc2011), http://www.utwente.nl/ctit/hstt/itc2011/.Post G, Ahmadi S, Daskalaki S, Kingston J, Kyngas J, Nurmi C, Ranson D (2012) Anxml format for benchmarks in high school timetabling. Annals of Operations Research194:385397Potvin JY, Rousseau JM (1993) A parallel route building algorithm for the vehicle routingand scheduling problem with time windows. European Journal of Operational Research66(3):331  340Ribeiro GM, Laporte G (2012) An adaptive large neighborhood search heuristic for thecumulative capacitated vehicle routing problem. Computers &amp; Operations Research39(3):728  735, DOI 10.1016/j.cor.2011.05.005Ropke S, Pisinger D (2006) An adaptive large neighborhood search heuristic for the pickupand delivery problem with time windows. Transportation Science 40:455472Salazar-Aguilar M, Langevin A, Laporte G (2011) An adaptive large neighborhood searchheuristic for a snow plowing problem with synchronized routes. In: Pahl J, ReinersT, Voss S (eds) Network Optimization, Lecture Notes in Computer Science, vol 6701,Springer Berlin / Heidelberg, pp 406411Santos H, Uchoa E, Ochi L, Maculan N (2010) Strong bounds with cut and columngeneration for class-teacher timetabling. Annals of Operations Research pp 114,10.1007/s10479-010-0709-yShaw P (1997) A new local search algorithm providing high quality solutions to vehiclerouting problemsShaw P (1998) Using constraint programming and local search methods to solve vehiclerouting problems. In: Maher M, Puget JF (eds) Principles and Practice of ConstraintProgramming  CP98, Lecture Notes in Computer Science, vol 1520, Springer Berlin/ Heidelberg, pp 417431Sørensen M, Kristiansen S, Stidsen TR (2012) International timetabling competition 2011:An adaptive large neighborhood search algorithm. In: Proceedings of the Ninth Inter-national Conference on the Practice and Theory of Automated Timetabling (PATAT2012)Tillman FA, Cain TM (1972) An upperbound algorithm for the single and multiple terminaldelivery problem. Management Science 18(11):664  682Wood J, Whitaker D (1998) Student centred school timetabling. The Journal of the Oper-ational Research Society 49(11):11461152Wright M (1996) School timetabling using heuristic search. The Journal of the OperationalResearch Society 47(3):347357Özcan E (2005) Towards an xml-based standard for timetabling problems: Ttml. In: KendallG, Burke EK, Petrovic S, Gendreau M (eds) Multidisciplinary Scheduling: Theory andApplications, Springer US, pp 163185

High School Timetabling: Modeling and solving a large number of cases in Denmark

http://orbit.dtu.dk/ws/files/52417410/High_School_Timetabling.pdf

High School Timetabling: Modeling and solving a large number of cases in Denmark

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology