Davison, Matthew

Kheiri, Ahmed

Zografos, K. G.

Lancaster E-Prints

Optimising Scheduling of Hybrid Learning usingMixed Integer ProgrammingMatthew Davison1, Ahmed Kheiri2, Konstantinos Zografos21 STOR-i Centre for Doctoral Training, Lancaster University, Lancaster, UKm.davison2@lancaster.ac.uk2 Department of Management Science, Lancaster University, Lancaster, UK{a.kheiri,k.zografos}@lancaster.ac.ukKeywords: University Timetabling · Hybrid Learning · Mixed Integer Pro-gramming1 IntroductionThe COVID-19 pandemic artificially reduced the already limited capacity ofphysical spaces at universities. It forced a greater use of hybrid learning, which isa mode of teaching that combines online and in-person elements. Previous studiesdealt with limited capacity by controlling the quantity and flow of students oncampus [5] or on the areas surrounding campus [1]. Other studies investigatedwhat policy changes allow a better use of resources [2].Online classes are one way to reduce demand for physical space. Universitiesin the future will likely continue to offer a mix of online and in-person classes be-yond the pandemic [3]. Since students and staff typically prefer to attend classesin-person, this motivates the need to limit the number of classes held online,whilst still taking advantage of their ability to reduce the demand for physicalspace. More specifically, the problem is to investigate how universities can max-imise the number of courses that they offer, whilst simultaneously limiting thenumber of online classes that are used to achieve this. This timetabling problemdiffers to the classic timetabling problem in that the input is a list of classes, butnot all classes need to be assigned. Solutions to this problem identify how manycourses could be offered, which is useful information for universities planningsemesters. The preliminary model presented in this paper illustrates one way ofsolving this particular problem.2 Model FormulationTimeslots are lengths of time that have a start and end. In this problem weassume these are five minutes long. Timesets are defined as a subset of the setof all timeslots. These are used to better model complicated arrangements. Forexample, a timeset could describe a two hour class meeting every other week.Table 1 provides the notation for the sets used within the formulation of themodel.We define the matrix A where the entry Ar1,r2 is equal to the number oftimeslots it takes to travel from room r1 to room r2. In particular, for d a non-negative integer, Ar1,r2 = d represents a travel time of 5d minutes.2 M. Davison et al.Table 1. Key notation. The first six sets are primarily used to describe elements ofthe problem, the last seven sets are primarily used in the construction of constraints.S Set of timeslotsT Set of timesets. Each t ∈ T is a subset of SR Set of roomsC Set of classesK Set of coursesLk Set of sections for course k. Each l ∈ Lk is a subset of CRur Set of timeslots when room r ∈ R is unavailableRc Set of rooms suitable for class c ∈ CTc Set of timesets suitable for class c ∈ CRG Let G ⊆ C. RG := ∩c∈GRcCG Let G ⊆ C. CG := {(c1, c2) ∈ G×G : c1 ̸= c2}Rcr Let r ∈ R. Rcr := {c ∈ C : r ∈ Rc}Os Let s ∈ S. Os := {a ∈ T : s ∈ a}The set C contains all classes regardless of if they can be held online, in-person or both. The online space is modelled as a room that is always availableand can host multiple classes at the same time. Let r∗ represent this onlinespace. A class, c ∈ C, can be held online if and only if r∗ ∈ Rc. For r∗ assumethat Ar∗,r∗ = 0 and Ar∗,r = d∗ for all r ∈ R \ r∗ where d∗ is a fixed numberof timeslots. For this paper, we assume any class can happen online and thatd∗ = 0.2.1 Definition of variablesOne set of variables used in this problem are binary variables indicating if a classis assigned to a particular room and timeset. They are defined as follows:xc,r,t ={1 Class c ∈ C is held in room r ∈ R in timeset t ∈ T ,0 Otherwise.The second set of variables are binary variables that indicate if a class usesa room, or if a class uses a timeset. They are defined as follows:yc,r ={1 Class c ∈ C is held in room r ∈ R,0 Otherwise.yc,t ={1 Class c ∈ C is held in timeset t ∈ T ,0 Otherwise.These variables are related to each other by linking constraints:yc,r =∑t∈Txc,r,t, ∀r ∈ R, c ∈ C,Optimising Scheduling of Hybrid Learning 3yc,t =∑r∈Rxc,r,t, ∀t ∈ T, c ∈ C.We also define binary variables to indicate if courses are offered:gk ={1 Course k ∈ K is offered,0 Otherwise.Parameter arrays Defined in Table 2, parameter arrays are fully determinedby a given problem instance. They are used to indicate if a group of resourcessatisfy a particular condition, thus indicating what constraints to include in themodel corresponding to that instance.Table 2. Definition of each parameter array. t, t1 and t2 are in the set T . r, r1 and r2are in the set RArray DescriptionD0 A matrix where D0[r, t] is equal to one if room r is unavailable at somepoint during timeset t, zero otherwiseD1 An array where D1[r1, r2, t1, t2] is equal to one if there is not enough timebetween t1 and t2 to travel between r1 and r2, zero otherwiseD2 A matrix where D2[t1, t2] is equal to zero if the first meeting in t1 concludesbefore the start of the first meeting in t2, one otherwiseD3 A matrix where D3[t1, t2] is zero if the meetings in t1 and t2 do not occur onoverlapping weeks and days, if any meeting in t1 and t2 occurs on the sameday and week then D3[t1, t2] is equal to the number of timeslots betweenthe start of the earliest meeting and end of the latestD4 A matrix where D4[t1, t2] is equal to zero if t1 and t2 start at the same timeof day, one otherwiseD5 A matrix where D5[t1, t2] is equal to zero if t1 completely overlaps t2 in thetimes of day they meet or vice versa, one otherwiseD6 A matrix where D6[t1, t2] is equal to zero if t1 meets on a subset of daysthat t2 does or vice versa, one otherwiseD7 A matrix where D7[t1, t2] is equal to zero if t1 and t2 do not meet on anyof the same days, one otherwiseD8 A matrix where D8[t1, t2] is equal to one if t1 overlaps t2, zero otherwise.2.2 ConstraintsThere are various constraints that need to be included within any universitytimetabling model. Some are more specialised so that the timetable adheresto university policy or allows students and staff to travel comfortably betweenclasses. In this section, some constraints included within our model are described.4 M. Davison et al.Most of these constraints have been proposed in the existing timetabling litera-ture [4] and what is presented in this paper is our approach to modelling theseconstraints.Classes can only be assigned at most a single room and a single timeset∑t∈T∑r∈Rxc,r,t ≤ 1, ∀c ∈ C.Classes can only be assigned compatible rooms and times To ensureonly compatible rooms and times are used, for each c ∈ C add the followingconstraints: ∑r∈Rxc,r,t = 0, ∀t ∈ T \ Tc.∑t∈Txc,r,t = 0, ∀r ∈ R \Rc.Classes should not happen in a room when that room is not available∑t∈T∑r∈RD0[r, t]xc,r,t = 0, ∀c ∈ C.In-person classes should not use the same room at the same time∑c∈Rcr∑t∈Osxc,r,t ≤ 1, ∀r ∈ R \ r∗, s ∈ S.Group of classes should occur in the same room Let G be a set ofclasses that must occur in the same room. For each r ∈ RG define a binaryvariable sGr that takes the value one if every class in G uses room r and zerootherwise. Add the following constraints:∑c∈Gyc,r = |G|sGr , ∀r ∈ RG,∑r∈RGsGr ≤ 1,yc,r = 0, ∀r /∈ RG, c ∈ G.Attending a group of classes Staff and students have collections of classesthey must attend. Denote a collection of classes as G. For each (c1, c2) ∈ CG addthe following constraints:D1[r1, r2, t1, t2](xc1,r1,t1 + xc2,r2,t2) ≤ 1, ∀t1 ∈ Tc1 , t2 ∈ Tc2 , r1 ∈ Rc1 , r2 ∈ Rc2 .Group of classes should occur in a certain order Let G be a sequence ofclasses that should occur in order, meaning that the first meeting of a class shouldcompletely finish before the start of the next class. Suppose G = (c1, c2, . . . , ck),then for each pair (ci, ci+1) where i ∈ {1, . . . , k−1} add the following constraints:D2[t1, t2](yci,t1 + yci+1,t2) ≤ 1, ∀t1 ∈ Tci , t2 ∈ Tci+1 .Optimising Scheduling of Hybrid Learning 5Group of classes should be grouped within a period of time Let Gbe a set of classes that should all happen within H timeslots if they happen onthe same day and week. For each (c1, c2) ∈ CG add the following constraints:I(D3[t1, t2] > H)(yc1,t1 + yc2,t2) ≤ 1, ∀t1 ∈ Tc1 , t2 ∈ Tc2 ,where I is an indicator function that takes the value one if D3[t1, t2] > H holdsand zero otherwise.Timing constraints All timing specific constraints have the same form.Suppose G is the set of classes the constraint applies to. For each (c1, c2) ∈ CGadd the following constraints:D[t1, t2](yc1,t1 + yc2,t2) ≤ 1, ∀t1 ∈ Tc1 , t2 ∈ Tc2 ,where D is the appropriate parameter array for that constraint. Table 3 describesa constraint on a group of classes and the associated parameter array.Table 3. Constraints and associated parameter arrayConstraint Associated DClasses should start at the same time D4Classes should occur during the same time of day D5Classes should occur on the same days of the week D6Classes should occur on the different days of the week D7Classes should not overlap in time D8Course structure constraints Courses can be split into sections that teachidentical content. Sections are a collection of classes and it is possible for twosections of the same course to share classes. We define binary variables to indicateif a course section is offered:hk,l ={1 Section l ∈ Lk of course k ∈ K is offered,0 Otherwise.For a given course, k ∈ K, a section l ∈ Lk can be offered only if all of theclasses in that section are offered. This is modelled by the following constraints:hk,l ≤1|l|∑c∈l∑t∈T∑r∈Rxc,r,t, ∀l ∈ Lk, k ∈ K,where |l| is the number of classes in a section. A course is only offered if there isat least one section being offered. This is modelled by the following constraints:gk ≤∑l∈Lkhk,l, ∀k ∈ K.6 M. Davison et al.2.3 ObjectivesThere are many objectives in the timetabling literature. In this section some ofthe objectives that could be used within this model are presented.Maximise number of courses offeredmax z1 =∑k∈Kgk,Maximise number of classes heldmax z2 =∑c∈C∑t∈Tc∑r∈Rcxc,r,t.Whilst courses are important, offering as many classes as possible providesflexibility within these courses.Minimise number of classes held onlinemin z3 =∑c∈C∑t∈Tcxc,r∗,t.Ideally, there would be no need for online classes but it cannot be ignoredthat they can help increase z1 because they are not subject to physical spacelimitations, a common limit to teaching capacity.Minimise cost of assignmentmin z4 =∑c∈C(∑r∈RcPc,ryc,r +∑t∈TcPc,tyc,t),where Pc,r and Pc,t are non-negative penalties for assigning class c ∈ C to roomr ∈ Rc and timeset t ∈ Tc respectively. The exact value of these penalties aresubjective in practice. They could represent:– Actual monetary cost of using the resource.– Approximation of preference (low penalty indicating higher preference).– Arbitrarily large penalties to deter solution from using a resource.Weighted objective approachmax z = w1z1 + w2z2 + w3z3 + w4z4,where w1 and w2 are non-negative weights, and w3 and w4 are non-positiveweights. By properly tuning these weights the model is able to determine anoptimal mix of in-person and online teaching.Optimising Scheduling of Hybrid Learning 73 ResultsTo verify that this model produces feasible timetables, three instances fromthe 2019 International Timetabling Competition (ITC-2019) were used [4]. De-scribed in Table 4 is the number of classes, timesets and rooms for each instance.The problem defined in the ITC-2019 involves creating a complete timetableand allocating students to classes based on their course requests. Our modeldoes not consider student allocation nor requires a complete timetable to beconstructed. Therefore, in this experiment, we maximise z2 only. Using this so-lution, it is possible to evaluate z3 and z4.For our experiments we used an internal computing node running CentOSLinux with an Intel Xeon E5-2699 v3 CPU running at 2.30GHz and 528GB ofRAM. The model was implemented in Python 3.5 and solutions were found usingthe commercial solver Gurobi 9.0, which successfully produced valid timetables.Table 4 provides information about each solution.Table 4. The problem instances and quantities of key features. In the “Type” col-umn, “T” indicates “test” instances and “C” indicates “competition” instances. The z3recorded is the number of classes that does not require a room, which we treat as“online-only” classes. The z4 recorded here is the same value reported by the ITC-2019validation tool [4]Instance Type |K| |C| |T | |R| z2 z3 z4 Time (s)lums-sum17 T 19 20 93 62 20 0 73 0.003bet-sum18 T 48 127 50 46 127 6 3502 0.011tg-fal17 C 36 711 1645 23 711 15 9610 58757.559As can be seen from Table 4 in all three instances |C| = z2 meaning that oursolutions are optimal for this objective. Since we are offering all possible classes,it is clear that it is possible to offer every course.Acknowledgements We gratefully acknowledge the support of the EPSRCfunded EP/S022252/1 STOR-i Centre for Doctoral Training.References1. Al-Yakoob, S.M., Sherali, H.D.: A mixed-integer programming approach to a classtimetabling problem: A case study with gender policies and traffic considerations.European Journal of Operational Research 180(3), 1028–1044 (2007). https://doi.org/https://doi.org/10.1016/j.ejor.2006.04.0352. Barnhart, C., Bertsimas, D., Delarue, A., Yan, J.: Course Scheduling Under SuddenScarcity: Applications to Pandemic Planning. Manufacturing & Service OperationsManagement 24(2), 727–745 (2022). https://doi.org/10.1287/msom.2021.0996,https://doi.org/10.1287/msom.2021.09963. Lederman, D.: What’s the future of the physical college campus? In-side Higher Ed (2021), https://www.insidehighered.com/news/2021/07/16/whats-future-physical-college-campus8 M. Davison et al.4. Müller, T., Rudová, H., Müllerová, Z.: University course timetabling and Interna-tional Timetabling Competition 2019. Proceedings of the 12th International Con-ference on the Practice and Theory of Automated Timetabling (PATAT 2018) pp.5–31 (2018)5. Vermuyten, H., Lemmens, S., Marques, I., Beliën, J.: Developing compact coursetimetables with optimized student flows. European Journal of Operational Re-search 251(2), 651–661 (2016). https://doi.org/https://doi.org/10.1016/j.ejor.2015.11.028

Optimising Scheduling of Hybrid Learning using Mixed Integer Programming

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Optimising Scheduling of Hybrid Learning using Mixed Integer Programming

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