University Timetabling Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. University Timetabling Lecture 19 Formalization and Modelling University Timetabling An Example Timetabling in Practice Marco Chiarandini 2 Modelling Modelling University Timetabling An Example University Timetabling An Example Outline Course Timetabling Practice Practice The weekly scheduling of the lectures/events/courses of courses avoiding students, teachers and room conflicts. Input: A set of courses C = { C 1 , . . . , C n } each consisting of a set of lectures C i = { L i 1 , . . . , L il i } . Alternatively, A set of lectures L = { L 1 , . . . , L l } . 1. University Timetabling A set of curricula S = { S 1 , . . . , S r } that are groups of courses with Formalization and Modelling common students (curriculum based model). Alternatively, An Example A set of enrollments S = { S 1 , . . . , S s } that are groups of courses Timetabling in Practice that a student wants to attend (Post enrollment model). a set of time slots T = { T 1 , . . . , T p } (the available periods in the scheduling horizon, one week). All lectures have the same duration (say one period) Output: An assignment of each lecture L i to some period in such a way that no student is required to take more than one lecture at a time. 3 5
Modelling Modelling University Timetabling An Example University Timetabling An Example Graph model IP model Practice Practice Including the assignment of indistinguishable rooms m t rooms ⇒ maximum number of lectures in time slot t Graph G = ( V, E ) : V correspond to lectures L i Variables E correspond to conflicts between lectures due to curricula or x it ∈ { 0 , 1 } i = 1 , . . . , n ; t = 1 , . . . , p enrollments Number of lectures per course Time slots are colors ➜ Graph-Vertex Coloring problem ➜ NP-complete (exact solvers max 100 vertices) p � x it = l i ∀ i = 1 , . . . , n t =1 Typical further constraints: Number of lectures per time slot Unavailabilities n � Preassignments x it ≤ m t ∀ t = 1 , . . . , p The overall problem can still be modeled as Graph-Vertex Coloring. How? i =1 6 7 Modelling Modelling University Timetabling An Example University Timetabling An Example Practice Practice Further complications: Teachers that teach more than one course (not really a complication: treated similarly to students’ enrollment) Number of lectures per time slot (students’ perspective) A set of rooms R = { R 1 , . . . , R n } n � with eligibility constraints x it ≤ 1 ∀ i = 1 , . . . , n ; t = 1 , . . . , p (this can be modeled as Hypergraph Coloring [de Werra, 1985] : C i ∈ S j introduce an (hyper)edge for events that can be scheduled in the If some preferences are added: same room � p � n the edge cannot have more colors than the rooms available of that max i =1 d it x it i =1 type) Moreover, Corresponds to a bounded coloring. [de Werra, 1985] Students’ fairness Logistic constraints: not two adjacent lectures if at different campus Max number of lectures in a single day and changes of campuses. Precedence constraints Periods of variable length 8 9
Modelling Modelling University Timetabling An Example University Timetabling An Example IP approach Practice Practice 2D IP model including room eligibility [Lach and Lübbecke, 2008] 3D IP model including room eligibility [Lach and Lübbecke, 2008] R ( c ) ⊆ R : rooms eligible for course c Decomposition of the problem in two stages: G conf = ( V conf , E conf ) : conflict graph (vertices are pairs ( c, t ) ) Stage 1 assign courses to timeslots Stage 2 match courses with rooms within each timeslot X min d ( c, t ) x ctr ∀ c ∈ C solved by bipartite matching ctr X x ctr = l ( c ) ∀ c ∈ C t ∈ T Model in stage 1 r ∈ R ( c ) X x ctr ≤ 1 ∀ t ∈ T, r ∈ R Variables: course c assigned to time slot t c ∈ R − 1 ( r ) X X x ct ∈ { 0 , 1 } c ∈ C , t ∈ T x c 1 t 1 r + x c 2 t 2 r ≤ 1 ∀ (( c 1 , t 1 )( c 2 , t 2 )) ∈ E conf r ∈ R ( c 1 ) r ∈ R ( c 2 ) x ctr ∈ { 1 , 0 } ∀ ( c, t ) ∈ V conf , r ∈ R Edge constraints (forbids that c 1 is assigned to t 1 and c 2 to t 2 simultaneously) This 3D model is too large in size and computationally hard to solve x c 1 ,t 1 + x c 2 ,t 2 ≤ 1 ∀ (( c 1 , t 1 ) , ( c 2 , t 2 )) ∈ E conf 10 11 Modelling Modelling University Timetabling An Example University Timetabling An Example Practice Practice Hall’s constraints Hall’s constraints are exponentially many (guarantee that in stage 1 we find only solutions that are feasibile for stage 2) [Lach and Lübbecke] study the polytope of the bipartite matching and G t = ( C t ∪ R t , E t ) bipartite graph for each t find strengthening conditions G = ∪ t G t (polytope: convex hull of all incidence vectros defining subsets of C n � perfectly matched) x ct ≤ | N ( U ) | ∀ U ∈ C , t ∈ T Algorithm for generating all facets not given but claimed efficient c ∈ U If some preferences are added: Could solve the overall problem by branch and cut (separation problem is easy). p n � � However the the number of facet inducing Hall inequalities is in max d it x it practice rather small hence they can be generated all at once i =1 i =1 12 13
Modelling Modelling University Timetabling An Example University Timetabling An Example Examination Timetabling Practice Practice So far feasibility. By substituting lecture with exam we have the same problem! However: Preferences (soft constraints) may be introduced [Lach and Lübbecke, 2008b] Course Timetabling Exam Timetabling Compactness or distribution limited number of time slots unlimited number of time slots, Minimum working days seek to minimize Room stability conflicts in single slots, seek to conflicts may involve entire days Student min max load per day compact and consecutive days,seek to Travel distance spread Room eligibility one single course per room possibility to set more than one Double lectures exam in a room with capacity constraints Professors’ preferences for time slots Different ways to model them exist. lectures have fixed duration exams have different duration Often the auxiliary variables have to be introduced 14 15 Modelling University Timetabling An Example 2007 Competition Practice Constraint Programming is shown by [Cambazard et al. (PATAT 2008)] to be not yet competitive Integer programming is promising [Lach and Lübbecke] and under active development (see J.Marecek http://www.cs.nott.ac.uk/~jxm/timetabling/ ) however it was not possible to submit solvers that make use of IP commericial programs Two teams submitted to all three tracks: [Ibaraki, 2008] models everything in terms of CSP in its optimization counterpart. The CSP solver is relatively very simple, binary variables + tabu search [Tomas Mueller, 2008] developed an open source Constraint Solver Library based on local search to tackle University course timetabling problems ( http://www.unitime.org ) All methods ranked in the first positions are heuristic methods based on local search 17
Modelling University Timetabling An Example Heuristic Methods Practice Assemblage Basic Metaheuristics components Hybrid Heuristic Methods Guided Local Search Some metaheuristic solve the general problem while others or exact Integer Variable Neighborhood Search Programming algorithms solve the special problem Beam Search Constraint Tabu Search Programming algorithm Replace a component of a metaheuristic with one of another or of Simulated Annealing configurations an exact method (ILS+ SA, VLSN) Iterated Local Search Construction Heuristics Iterated Greedy Treat algorithmic procedures (heuristics and exact) as black boxes Ant Colony Optimization and serialize Neighborhood Search Evolutionary Algorithm ... Let metaheuristics cooperate (evolutionary + tabu search) Testable units Testable Solving sub−problems units Use different metaheuristics to solve the same solution space or a Graph Coloring, Bipartite Matching, Testable Solving the Hard constraints, Soft Constraints units global problem partitioned solution space 18 Modelling Modelling University Timetabling An Example University Timetabling An Example Post Enrollment Timetabling Practice Practice Definition Find an assignment of lectures to time slots and rooms which is Configuration Problem Feasible Algorithms must be configured and tuned and the best selected. 9 rooms are only used by one lecture at a time, This has to be done anew every time because constraints and their > > each lecture is assigned to a suitable room, > > Hard = density (problem instance) are specific of the institution. no student has to attend more than one lecture at once, Constraints lectures are assigned only time slots where they are available; > > > > Appropriate techniques exist to aid in the experimental assessment of precedences are satisfied; ; algorithms. Example: F-race [Birattari et al. 2002] and Good (see: http://www.imada.sdu.dk/~marco/exp/ for a full list of 9 no more than two lectures in a row for a student , references) Soft = unpopular time slots avoided (last in a day) , Constraints students do not have one single lecture in a day. ; 20 22
Recommend
More recommend
