Kristiansen, Simon

Mason, Andrew

Stidsen, Thomas Riis

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Student Sectioning at High Schools in DenmarkKristiansen, Simon; Mason, Andrew; Stidsen, Thomas Jacob RiisPublished in:Proceedings of the Multidisciplinary International Conference on Scheduling: Theory and ApplicationsPublication date:2013Document VersionEarly version, also known as pre-printLink back to DTU OrbitCitation (APA):Kristiansen, S., Mason, A., & Stidsen, T. R. (2013). Student Sectioning at High Schools in Denmark. InProceedings of the Multidisciplinary International Conference on Scheduling: Theory and Applications MISTA.(Proceedings of the Multidisciplinary International Conference on Scheduling: Theory and Applications).MISTA 2013 manuscript No.(will be inserted by the editor)Student Sectioning at High Schools in DenmarkSimon Kristiansen · Andrew Mason ·Thomas R. Stidsenthe date of receipt and acceptance should be inserted later1 IntroductionHigh School Student Sectioning (HSSS) is a planning problem each Danish highschool faces every year. There has been an increase of scientific research withinEducation Timetabling and High School Timetabling the last decade due to thethree International Timetabling Competitions. However, many of the articles areconcerned with the actual timetabling problem and not the sectioning of students.The HSSS is closely related to the Elective Course Student Sectioning (ECSS)(Kristiansen et al (2011)). Both problems are yearly recurrent planning problemsat the Danish high schools. But where ECSS is concentrated on assigning 2nd and3rd year students to elective courses given their requests, the HSSS is concernedwith the partitioning of the 1st year students to form cohort. The overall goal of theHSSS is to create a solution method which can be implemented in the cloud basedhigh school administration software Lectio and hence available to the majority ofDanish high schools. The purpose of this abstract is to create an IP model for theproblem and try to solve it using the state-of-art MIP solver Gurobi.2 High School Student SectioningThe first problem we consider is the partitioning of students to form cohorts,where a cohort of students will be timetabled as one entity for most of theirclasses. Students are distinguished by their choice of specialization, where somepossible specializations for linguistic students are listed in Table 1. Each studentS. KristiansenMaCom A/S, Vesterbrogade 48, 1., DK-1620 Copenhagen V, DenmarkS. Kristiansen (E-mail: sikr@dtu.dk) · T.R. StidsenSection of Operations Research, Department of Management Engineering, Technical Universityof DenmarkBuilding 426, Produktionstorvet, DK-2800 Kgs. Lyngby, DenmarkA. MasonDept. of Engineering Science, University of Auckland, Auckland, New Zealand2 Simon Kristiansen et al.chooses exactly one specialization. In this example, the school has specified thatwe need to form 5 cohorts each containing no more than 30 students where eachcohort contains students from one or more specializations. We wish to minimizethe number of specializations in each cohort. Table 1 shows a possible solution.Table 1 An example of forming cohorts for linguistic students who have chosen one of 4linguistic specializations. The upper limit on cohort size is 30 studentsSpecialization # Students Cohort 1 Cohort 2 Cohort 3 Cohort 4 Cohort 5English-German 18 18English-Spanish 40 28 12English-Music 9 9English-SocialSci 77 20 30 27The example above considered students each of whom has chosen a special-ization in the arts area. A similar independent problem needs to be solved forstudents selecting a specialization within science.As well as choosing a specialization, each student chooses one Linguistic sub-ject, such as French or Italian, and one Arts subject such as Drama or Media. Afterthe partitioning of students to cohorts, we wish to assign the cohorts to time-slots.One for each type of subject. There will typically be many cohorts assigned to thesame time-slot. These assignments then determine the number of students tak-ing each subject, such as French or Italian, in each time-slot, and elective courseclasses can then be created based on these numbers.The primary objective of the HSSS is to assign all the students to a cohort,while minimizing the number of specializations represented in each cohort. Assecond objective, we want to minimize the amount of classes needed to fulfill allcourse requests and to minimize the number of courses represented in each time-slot.2.1 IP modelFor the partitioning of students to cohorts at a high school we have a set of stu-dents, a set of specializations S, a set of cohorts C. To avoid some of the symmetrywhich the problem contains, we introduce the set of student groups, G. A studentgroup consists of students which have requested exactly the same specializationand elective courses. Lc ∈ N and Uc ∈ N are the lower and upper limit on thecohort size, respectively. Let Fg ∈ N be the size of the student group g, and hencethe flow demand of student group g. And let Ag,g′ be 1 if the student groups gand g’ are not allowed to be in the same cohort. The decision variable xg,c ∈ {0, 1}takes value 1 if there exists a flow from student group g to cohort c, zero otherwise.The variable x¯g,c ∈ N is the amount of flow from student group g to cohort c.min α ·∑g,cxg,c + β ·∑gx¯g,c∅ (1)s.t.∑cx¯g,c + x¯g,c∅ = Fg ∀ g (2)x¯g,c ≤ min[Fg, Uc] · xg,c ∀ g, c (3)Student Sectioning at High Schools in Denmark 3∑gx¯g,c ≤ Uc ∀ c (4)∑gx¯g,c ≥ Lc ∀ c (5)xg,c + xg′,c ≤ 1 ∀ c, g, g′, Ag,g′ = 1 (6)xg,c ∈ {0, 1} (7)x¯g,c ∈ N (8)Constraints (2) ensure that all students in a student group are assigned a cohort.Constraints (3) ensure that the maximum flow from student group g into a givencohort c, cannot be greater that the upper cohort size or the size of the studentgroup. Constraints (4) and (5) make sure that the total flow into a cohort is limitedby the upper and lower cohort size. Finally constraints (6) make sure that two stu-dent groups which are not allowed to be placed in the same cohort are not done so.The second part of the HSSS is the elective course assignment where we wantto assign the cohort to one time-slot, T of each course type, D. The types are thelinguistic and the artistic courses. Let Td⊆ T be the set of times used for courseof type d. Rg,e denotes whether student group g has requested elective course e.The decision variable yc,t,d ∈ {0, 1} denote whether cohort c are assigned time-slot t of course type d. And let variable v¯e,t ∈ N be the number of course classesneeded for elective course e in time slot t. The parameter Ke,t ∈ N denote themaximum number of classes of course e which is possible to create in time t.The objective of this part is to minimize the number of course classes and thefollowing constraints are imposed.min γ ·∑e,tv¯e,t (9)s.t.∑t∈Tdyc,t,d = 1 ∀ c, d (10)∑cyc,t,d ≥ 1 ∀ t, d (11)∑gRg,e · x¯g,c −M(1− yc,t,d) ≤ Ue · v¯e,t ∀ c, e, t, d (12)v¯e,t ≤ Ke,t ∀ e, t,Ke,t > 0 (13)yc,t,d ∈ {0, 1} (14)v¯e,t ∈ N (15)Constraints (10) make sure that every cohort is assigned exactly one time slot foreach course type. Constraints (11) ensure that there is at least one cohort in eachtime slot, whereas Constraints (12) sets the value for variable v¯e,t. Constraints (13)ensure that the recourse limitation for a given course at a given time is respected.By making small adjustment on how the cohorts are formed and in whichtime slots they are placed, it is possible to improve the distribution and thenease the problem of creating classes and the entire timetable problem. This isdone by minimizing the number of specializations and courses represented in eachcohort, and by minimizing the number of courses represented in each time slot. Letthe parameters Dg,s and Rg,e denote whether the student group g has requested4 Simon Kristiansen et al.specialization s and course e, respectively. The variables zs,c ∈ {0, 1} and ze,c∈ {0, 1} denote whether specialization s or course e is represented in cohort c,respectively. And let ve,t ∈ {0, 1} take value 1 if course e is presented in time slott.min∑s,cδs · zs,c +∑e,cδe · ze,c +∑e,tδ · ve,t (16)s.t. Dg,s · xg,c ≤ zs,c ∀ g, c, s (17)Rg,e · xg,c ≤ ze,c ∀ g, e, c (18)ze,c + yc,t,d − 1 ≤ ve,t ∀ e, c, d (19)zs,c ∈ {0, 1} (20)ze,c ∈ {0, 1} (21)ve,t ∈ {0, 1} (22)Constraints (17) and (18) sets the value for the variables zs,c and ze,c and hencedenote if specialization s or course e is represented in cohort c, respectively. Con-straints (19) denote whether elective course e is represented in time slot t, usingvariables ve,t.3 Solution Method and outlookCurrently the model has been tested using Gurobi and 15 real life datasets providedby MaCom A/S. MaCom A/S is the owner of Lectio and they have access todatasets from more than 200 Danish high schools. The results shows that for largeinstances Gurobi has a hard time finding good solutions within the given runningtime of 60 seconds. I.e. more work is needed before an implementation of HSSS inLectio.ReferencesKristiansen S, Srensen M, Stidsen TR (2011) Elective course planning.European Journal of Operational Research 215(3):713 – 720, DOI10.1016/j.ejor.2011.06.039

Student Sectioning at High Schools in Denmark

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Student Sectioning at High Schools in Denmark

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