A new lower bound for curriculum-based course timetabling

Abstract

In this paper, we propose a new method to compute lower bounds for curriculum-based course timetabling (CTT), which calls for the best weekly assignment of university course lectures to rooms and time slots. The lower bound is obtained by splitting the objective function into two parts, considering one separate problem for each part of the objective function, and summing up the corresponding optimal values (or, in some cases, lower bounds on these values), found by formulating the two parts as Integer Linear Programs (ILPs). The solution of one ILP is obtained by using a column generation procedure. Experimental results show that the proposed lower bound is often better than the ones found by the previous methods in the literature, and also much better than those found by other new ILP formulations illustrated in this paper. The proposed approach is able to obtain improved lower bounds on real-world benchmark instances from the literature, used in the international timetabling competitions ITC2002 and ITC2007, proving for the first time that some of the best-known heuristic solutions are indeed optimal (or close to the optimal ones).

Introduction

Curriculum-based course timetabling (CTT) is the problem of determining the best scheduling of university course lectures in a given time horizon of usually 1 week (i.e., five or six working days). A feasible solution must satisfy a set of “hard” constraints, and must also take into account a set of “soft” constraints, whose violation produces penalty terms in the objective function, which must be minimized. A commonly accepted definition of CTT has been stated in two international timetabling competitions ITC2002 and ITC2007 (see [16]).

Another important outcome of the competitions was that scientists were provided with a wide set of real-world instances to test their approaches. These are publicly available via a website, where the best-known heuristic solution and lower bound values are also reported and continuously updated. Looking at these results, one notices that there are a few instances for which the optimum is not known, and there is a fairly large gap between the reported upper and lower bounds. The wide amount of research conducted on finding heuristic solutions seems to suggest that the best-known heuristic solutions are close to the optimal ones. The aim of this paper is to reduce these gaps through the computation of improved lower bounds.

The CTT problem can be formally described as follows. We are given a set of courses to be delivered at a university in a given time period. Each course consists of a number of lectures to be taught by the teacher of the course, which can belong to some university curricula, and has a number of students that will attend it. A curriculum corresponds to a set of courses that can be taken by the same students. A set of rooms, a set of days (typically five) and a set of time slots in which each day is split are given. Each room is characterized by a capacity, equal to the number of seats in the room. Each pair day/time slot forms a time period. For each course, a set of unavailable time periods is given (i.e., time periods during which the teacher of the course is not available). In addition, for each course, a minimum number of days among which the lectures of the course should be spread is specified. The CTT consists of finding the best assignment of course lectures to rooms and time periods, so that a set of constraints is satisfied. These constraints consist of the so-called hard constraints, and are as follows:

• all the lectures of each course must be scheduled;

• courses belonging to the same curriculum cannot be scheduled in the same time period;

• each teacher can teach at most one lecture per time period;

• each room can host at most one lecture per time period;

• unavailable time periods for a course (or for the teacher of the course) cannot be used for scheduling a lecture of that course.

The soft constraints that determine the objective function are as follows:

• room capacity: A penalty is given for each student who cannot have a seat in the room assigned to the course lecture;

• minimum number of working days: A penalty is given for each day below the minimum number of working days for each course;

• curriculum compactness: It is preferable that the lectures of a curriculum are consecutive, without any empty time period in between; thus, a penalty is given for each isolated lecture, i.e., a lecture not adjacent to any other lecture of the same curriculum in the same day;

• room stability: It is preferable that a course is always taught in the same room; thus, a penalty is given for each additional room used for a course.

A set of formulations of the CTT problem (i.e., a specific set of soft constraints to be penalized in the objective function, if violated, along with the weights assigned to each of them) is proposed in Bonutti et al. [3], aiming to represent real-world problems. Two formulations are the most studied ones. The first one, called Basic Formulation or UD1 (UD for Udine since these problems are based on the system at the University of Udine), was introduced in Di Gaspero and Schaerf [10]. The second one, called Extended Formulation or UD2 is described in Di Gaspero et al. [9]. These formulations were used in two international timetabling competitions ITC2002 and ITC2007 (Track 3) (see [16]). The main difference between the two formulations is that in the Basic Formulation the soft constraint of room stability is not taken into account, whereas all the soft constraints appear in the Extended Formulation. Both formulations must satisfy the hard constraints defined above. In addition, different weights are given for the penalization of the violation of the soft constraints in the objective function. In Bonutti et al. [3], additional formulations of the CTT problem are proposed, taking into account the other soft constraints in order to capture other real-world features.

CTT has received large attention in the literature since it is an interesting problem from both the applied and the theoretical points of view, and also because of the organization of the two above-mentioned international competitions ITC2002 and ITC2007 (Track 3). The organizers proposed a set of real-world benchmark instances from the University of Udine [3], so the problem is also known as the Udine Course Timetabling.

Many pieces of work present heuristic algorithms for solving CTT (see [8], [15], [17], [19]). Since in this paper we propose new lower bounds and ILP models for the CTT, we focus on the literature dealing with mathematical formulations.

The three-index ILP model proposed by Burke et al. [5], [6], [7] presents binary variables x_prc, assuming value 1 if a lecture of course c is given in room r in time period p. Additional variables are then used in order to describe the soft constraints in the objective function: integer variables to count the number of days below the minimum number of working days for each course, integer variables to count the number of isolated lectures for each curriculum and day, and binary variables to express if a course is given in a certain room at any time period, in order to determine the number of rooms used by each course.

Burke et al. [4] describe a way, also used in Burke et al. [5], [7], of taking into account the curriculum compactness (pattern penalization): in daily timetables, isolated lectures in the first and in the last period of the day are checked and then triples of consecutive periods with only the middle one occupied by a lecture are checked as well. Cuts are derived in Burke et al. [7] and a branch-and-cut algorithm is developed. In particular, the authors generate a set of necessary cuts that is used to express the curriculum compactness constraints (combined with the pattern penalization). Then, additional cuts from implied bounds are introduced: for example, each course must be assigned to at least one room and to no more rooms than the number of lectures of the course. Finally, general cuts from graph coloring are introduced, and in particular clique based constraints: given a time period and a subset of courses that correspond to a complete subgraph in a conflict graph, at most one lecture of such courses can be scheduled in that time period. In Burke et al. [5], an ILP model, called Surface is derived, which takes into account only a subset of the soft constraints (minimum number of working days and curriculum compactness), neglecting the other ones (room capacity and room stability). It presents binary variables w_pc assuming value 1 if a lecture of course c is assigned to time period p, neglecting the room assignment. In this model, the authors introduce an additional constraint for each time period in order to prevent scheduling more lectures than the number of available rooms. In the same paper, an alternative ILP model, called Surface2, is also considered: while in the Surface model all the rooms are joined into a single room of multiplicity equal to the number of available rooms and capacity equal to the size of the largest room, in the Surface2 model rooms are divided into two groups, those larger and those smaller than an intermediate size.

Lach and Lübbecke [13] propose an ILP model based on the decomposition of the problem into two stages. The decomposition is exact with respect to the hard constraints, and guarantees an optimal solution to the entire problem if room stability is not considered, as in the Basic Formulation. The first stage aims at assigning courses to time periods. The second stage aims at assigning pairs (course, time period) to rooms. The ILP model of the first stage is characterized by binary variables x_cp, assuming value 1 if a lecture of course c is given in time period p. In this stage, all the conflict-avoidance constraints are imposed, together with constraints taking into account the room capacity, derived through a modification of Hall's condition (see [12]), and with constraints on the minimum number of working days and on the curriculum compactness. Additional variables are used in the first stage in order to express the soft constraints. When room stability is neglected, the second stage consists of solving a set of minimum weight bipartite perfect matching problems. If room stability is considered, then an ILP model is also used in the second stage.

Hao and Benlic [11] develop a partition-based approach in order to derive lower bounds for the CTT. The original problem is partitioned into k subproblems, and the constraints linking the subproblems are relaxed. In particular, the relaxation concerns the soft constraints for curriculum compactness and the hard constraints of room occupancy and time period conflict (i.e., all the lectures belonging to the same curriculum or taught by the same teacher must be scheduled in distinct periods). Each subproblem is then formulated as an ILP model following the approach proposed by Lach and Lübbecke [13] and solved by COIN-OR solver. The sum of the lower bounds of the subproblems gives a lower bound for the original problem. The main ingredient is how to partition the original problem: the authors propose an Iterated Tabu Search in order to derive an effective partition.

Asín Achá and Nieuwenhuis [2] apply novel techniques based on propositional satisfiability (SAT) solvers and optimizers. In particular, the authors present different encodings into variants of SAT (MaxSAT, Partial MaxSAT or Weighted MaxSAT), in which soft constraints are alternatively made hard or soft. New best lower bounds are derived and new best solutions are obtained for some ITC2007 benchmark instances.

To conclude, we wish to mention that the only column generation based approach which we are aware of is that of Qualizza and Serafini [18], even if a different version of the CTT is studied. The main difference lies in the objective function: preferences are given to the time periods due to teaching reasons and to teachers themselves, and the goal is to maximize these preferences. Thus, they do not consider the soft constraints used in the Basic and in the Extended Formulations. They propose a model with exponentially many binary variables, each one associated with a weekly timetable of a single course. Constraints in the master problem do not allow the use of more than the number of rooms available for each time period, to schedule no more than one lecture of any course belonging to a given curriculum in a given time period, and to assign a weekly timetable to each course.

In this paper, we compute a lower bound to the CTT by splitting the objective function into two parts, considering one separate problem for each part of the objective function, and summing up the corresponding optimal values (or, in some cases, lower bounds on these values), found by formulating the two parts as Integer Linear Programs (ILPs). In this respect our approach is partly similar to that proposed by Hao and Benlic [11], though in that case the problem was decomposed into smaller CTTs, each with the same original objective function.

In the first part of the objective function, we take into account the penalties for room capacity and room stability derived from the assignment of lectures to rooms. For the benchmark instances, the associated problem turns out to be easily solvable in practice, and a descriptive ILP formulation can be solved fairly quickly by a general-purpose solver. In most cases the associated optimal value is equal to zero. Nevertheless, in all the cases in which its optimal value is not zero, the proposed lower bound proves the optimality of the corresponding best-known heuristic solution.

In the second part of the objective function, we take into account the penalties for violating curriculum compactness and minimum number of working days, which are derived from the assignment of lectures to time slots. The associated problem turns out to be fairly hard to be solved in practice, but its solution gives the most significant contribution to the final lower bound. In fact, it naturally decomposes into independent subproblems (depending on which courses are common to more curricula) that can be separately solved. If the size of each subproblem is small, then the subproblem is solved to optimality through a descriptive ILP formulation; otherwise, by using a column generation procedure, the Linear Programming (LP) relaxation of an ILP formulation with exponentially many variables is solved.

Experimental results on the benchmark instances used in the international timetabling competitions ITC2002 and ITC2007 show that the proposed approach finds lower bounds which are often better than the ones found by the previous methods in the literature. In particular, for the Basic Formulation the proposed method is able to improve the best-known lower bounds of 9 out of 13 instances of ITC2007. In addition, for the first time we prove optimality for two instances, and in a few other cases we prove that the best-known solution value is, at most, some units away from the optimum value. For the Extended Formulation the proposed method is able to obtain the best-known or better lower bounds in 14 out of 26 instances, improving over the known lower bounds in four cases. Furthermore, we prove optimality of one instance for the first time. We also consider a few other new ILP formulations for the CTT, all with exponentially many variables, discussing the associated column generation problems. The computational results show that all these formulations appear to be of little practical use for the benchmark instances.

We use a notation which is as close as possible to the wide CTT literature. Let $\mathcal{C}$ be the set of courses, $\mathcal{R}$ the set of rooms, $\mathcal{H}$ the set of time periods, $\mathcal{D}$ the set of days of the planning period, $\mathcal{Q}$ the set of curricula, and $\mathcal{T}$ the set of teachers. For each course $c\in \mathcal{C}$, let $\mathit{lect}(c)$ be the number of lectures that must be scheduled, $\mathit{mwd}(c)$ the minimum number of working days, and $\mathit{stud}(c)$ the number of students that attend the course. For each room $r\in \mathcal{R}$, let $\mathit{cap}(r)$ be the number of seats of the room. Finally, let $\mathcal{C}(q)$ be the set of courses belonging to curriculum $q\in \mathcal{Q}$, $\mathcal{C}(t)$ the set of courses taught by teacher $t\in \mathcal{T}$, and $\mathcal{H}(d)$ the set of time periods of day $d\in \mathcal{D}$.

In addition, let $W^{\mathit{RCap}}$, $W^{\mathit{Mwd}}$, $W^{\mathit{CComp}}$, and $W^{\mathit{RStb}}$ be the weights corresponding to the penalization of room capacity, minimum number of working days, curriculum compactness, and room stability, respectively. In the Basic Formulation and in the Extended Formulation these weights are, respectively, $(1,5,1,0)$ and $(1,5,2,1)$.

For the sake of simplicity, in every reported ILP model, we do not explicitly insert the set of constraints for the unavailable time periods for each course: these constraints are taken into account by setting to zero the upper bound of the variables expressing the assignment of a lecture of a course to an unavailable time period.

The paper is organized as follows. In Section 2, we present the lower bound we propose. Section 3 is dedicated to the description of other new ILP models, whose LP relaxations are solved through column generation procedures. In Section 4, we present computational results on benchmark instances of the literature and compare the derived lower bounds with the best-known ones. Finally, in Section 5, we draw some conclusions and discuss guidelines for future research.