Minimum penalty perturbation heuristics for curriculum-based timetables subject to multiple disruptions

https://doi.org/10.1016/j.cor.2021.105306Get rights and content

Highlights

  • Finding minimum cost perturbations of timetables with constraint on perturbation size.

  • Assumed multiple disruptions affecting period and room availability.

  • Simulated annealing and integer programming-based algorithms proposed.

Abstract

Course timetables are often rendered infeasible due to unexpected changes in requirements and must be repaired. Given an initial timetable, planners prefer a repaired timetable whose quality is worsened as little as possible while its structural difference is also as little as possible. This paper formulates this problem as one of minimizing the quality degradation subject to an upper limit on the structural difference and presents one simulated annealing algorithm and a set of integer programming-based algorithms to solve it. These algorithms are tested on instances comprised of the ITC-2007 curriculum-based course timetabling test-bed and a set of randomly generated disruption scenarios, where each disruption scenario is based on a set of four disruption types that affect teacher and room availability. The results are analyzed with respect to two main goals, one being repairing a timetable after a single disruption scenario, and the other being the calculation of an estimated robustness of a solution based on a sample of disruption scenarios.

Introduction

In timetabling of university courses, lectures are assigned to periods and rooms to arrive at a weekly schedule. In the curriculum-based version of the problem, these assignments are made before students enroll in courses. To make sure students enrolled in a degree program can take the courses of the curriculum associated with that degree, constraints are introduced that require courses of the same curriculum not to be scheduled in the same period. In addition to these curriculum constraints, the problem has a set of other hard-constraints (those that cannot be violated) along with some soft-constraints (those that can be violated). The curriculum-based university course timetabling problem (CB-CTP) is formulated to minimize the total penalty associated with violating the soft constraints, subject to meeting all the hard-constraints. The specific timetabling problem we address is the CB-CTP defined for the International Timetabling Competition, ITC-2007 (see Di Gaspero et al. (2007)). These instances are based on the real-world timetabling data from University of Udine, with some modifications to make them more general while also ensuring that they contain different types of constraints common in real world timetabling. Hence, they are as relevant today as they were when designed. A recent review of the main works on the CB-CTP of ITC-2007, with focus on mathematical models, lower bounds, and exact and heuristic algorithms is given in Bettinelli et al. (2015).

In their comments on Bettinelli et al. (2015), Burke et al. (2015) point to the importance of having dynamic, interactive, robust scheduling algorithms for the CB-CTP. As in most practical scheduling contexts, some types of uncertainties can easily render an initial schedule infeasible or more generally change the constraints on available resources (teachers, rooms, etc.), forcing the planners to re-optimize. We refer to such uncertainties as disruptions. At the universities that use curriculum-based timetabling, an initial timetable is prepared based on constraints that are provided by the professors and administrators. This initial timetable is announced to the university staff, giving them a few days to submit requests for change, due to new constraints or data corrections. For example, a period to which a course has been scheduled could now be infeasible for the professor teaching it. The schedule should then be re-optimized taking these new constraints into account, while ensuring that the changes to the initial timetable are kept to a minimum. This second timetable is announced to the entire university, including the students who enroll in courses based on this timetable.

Inherent in the dynamic nature of such re-optimization problems is the need to keep the changes to a minimum while still obtaining high quality solutions. These two concerns are equally relevant in many rescheduling contexts. For example, in project scheduling under uncertainty the former one is known as schedule stability. An initial schedule that can be repaired with maximum schedule stability is said to have high solution robustness. On the other hand, one that can be repaired with minimal decline in the solution quality is said to have high quality robustness (see Herroelen and Leus (2004)). In line with this research stream, the work presented here addresses both solution stability and solution quality.

Given an initial timetable, S0, with total penalty P0, and a set of updated constraints due to disruptions that affect δ events (we use the terms lecture and event interchangeably), we define the minimum cost constrained perturbation problem as one where the objective is to find a new timetable S1 with minimum total penalty P1, subject to D(S0,S1)k×δ, where D(S0,S1) is the distance between the two solutions, defined as the number of lectures whose assigned period is different in S0 and S1. A more detailed discussion of this problem definition is provided in Section 2.

To the best of our knowledge, the earliest research done on minimal perturbation of invalid timetables used constraint programming (El Sakkout and Wallace, 2000). Then, Elkhyari et al. (2003) showed that the problem can be solved as a minimum perturbation problem on Resource-Constrained Project Scheduling Problems (RCPSP). Again using constraint programming, Zivan et al. (2011) developed a hybrid search algorithm, which was later outperformed by Fukunaga (2013). Verfaillie and Jussien (2005) give a more general overview of the approaches used in the domain of constraint satisfaction to deal with uncertain and dynamic environments.

Although a large number of articles have been published on the university course timetabling problems (see Pillay (2014)), there have been relatively few articles on perturbation algorithms for them. To the best of our knowledge, the earliest such work is Barták et al. (2004), in which they formally define the minimal perturbation problem as finding a solution that is as close as possible to the solution of the original problem. In other words, their objective is to minimize D(S0,S1). They describe a branch-and-bound-like algorithm for solving the problem. They later develop a local search-based metaheuristic that outperforms the previous algorithm (Müller et al., 2005). They report an application of this methodology at the course timetabling process of Purdue University in Rudová et al. (2011). Phillips et al. (2017) present an integer programming (IP) based algorithm for the minimal perturbation problem in university course timetabling. Among these algorithms, the last one bears the most similarity with our IP-based approach in terms of algorithm design, however they all address the minimal perturbation problem that minimizes the distance between the solutions. We, on the other hand, minimize the increase in the penalty function value subject to an upper limit on the distance. One reason for taking this different perspective is because we plan to use this problem formulation in evaluating the robustness of timetables, taking into account both solution and quality robustness.

Recently, Banbara et al. (2019) report a very competitive answer set programming (ASP) approach, called the teaspoon system, to solve the CB-CTP of ITC-2007, which they also extend to find Pareto optimal solutions for a minimal perturbation problem associated with CB-CTP. They use an off-the-shelf ASP system, clingo, which, authors explain, has a very efficient lexicographic optimization capability. The two criteria used for Pareto optimality are solution quality and stability, as measured by the Manhattan distance between the initial and repaired solutions. Each assignment difference between the initial and repaired solutions counts as 1 unit distance, even if the difference is only in the room assignments. We take a different approach, and count only the differences in time assignments (day/period) in the distance measurement. Their disruptions are comprised of removal or addition of constraints or both, including an entire category of constraints such as room stability. Actually, computational experiments are done by adding only room stability constraints, using initial solutions for 42 instances that did not have such constraints. Although we have not experimented with adding room stability constraints, our experiments involved adding a new category of constraints (room unavailability), removal of individual constraints (period unavailability for a specific course), adding of individual constraints (new period unavailability for a course) and change in data that affect soft constraint violations (increased number of students expected for a course). They find the Pareto frontier by lexicographic optimization, in which restrictions are placed on the solution quality and stability, iteratively. We did not attempt to identify the Pareto frontier, rather a single point on the frontier by constraining the distance between initial and repaired solutions.

Lindahl et al. (2019) also address the problem of repairing course timetables formulated as a bi-criteria optimization problem with solution quality and perturbation as the two objectives. They use the CB-CTP of ITC-2007 as their initial (static) timetabling problem and measure perturbation as the Hamming distance between two solutions. They search for Pareto-optimal solutions so that planners can trade-off perturbation with solution quality. The search procedure uses the three-indexed mixed-integer programming formulation of Burke et al. (2008) to find the best quality timetable with a constraint on the allowed perturbation. The problem they solve with this IP formulation is the problem we address here as well. However, there are some important differences between our work and theirs. The first difference is in how they measure the Hamming distance between solutions. They include room differences in addition to period differences, whereas, we only measure period differences in calculating the distance. The second difference is in our definitions of disruptions. All of their disruptions reduce the feasible solution space (so the quality of the repaired solution can never be better than the quality of the initial solution), whereas, one of our disruptions simultaneously takes away a set of periods and adds a different set to the available periods for a course. The third difference comes from the fact that we define disruption scenarios in which disruptions of multiple types occur simultaneously. Although their formulation of the problem allows repairing multiple disruptions, they limit their experimental work to repairing one disruption at a time. The fourth difference is in our algorithmic approaches. We implement a Simulated Annealing algorithm as well as an IP formulation. Since the goal of their work is to quantify the trade off between perturbation and solution quality, they formulate and solve an IP model to find the optimal penalty given the perturbation constraint. One of our main motivations is to obtain a fast algorithm that can repair a solution with multiple disruptions so that we can use this repair algorithm to estimate the robustness of a timetable by repairing it for a large sample of disruptions. Thus, our IP-based algorithms are designed as heuristics. Finally, due to the difference in the underlying motivations of these two studies, the computational experiments are designed differently. In Lindahl et al. (2019) only one disruption is randomly generated for each ITC-2007 instance and for each disruption type and they assume the initial solution is the optimal solution to the ITC-2007 instance. We, on the other hand, use three initial solutions for each ITC-2007 instance, and for each ITC-2007 instance and initial solution combination, generate 12 disruption scenarios, each with multiple disruptions.

Lemos et al. (2020b) provide a maximum satisfiability (MaxSAT) encoding to solve the university timetabling problem subject to disruptions in which the objective is to find the minimal perturbation solution (they also solve the original problem without disruptions). Perturbation is measured by the Hamming distance between the assignments, thus both time and room differences contribute to the distance. They assume two commonly studied disruption types, namely, period and room unavailability for a specific course (they use the terms invalid time and invalid room). The performance of the approach is tested on ITC-2019 instances (Müller et al., 2018), where only one of the disruption types is assumed to occur. The disruptions are randomly generated following a uniform distribution with the probability of an assignment being invalid equal to 21% and 25% for invalid time and invalid room, respectively. Since this results in significant disruptions, they report that the time to find the optimal solution often took as much as finding the best solution without disruptions.

Our main contribution is positioning the minimal cost constrained timetable perturbation (MCCTP) problem associated with the CB-CTP within the context of evaluating the robustness of a timetable subject to multiple disruptions of different types, and developing algorithms which we then test through a comprehensive computational experiment. Since we argue that measuring the robustness can be done by sample average approximation, SAA, (see Kleywegt et al. (2002)), which requires repairing a given timetable subject to a large random sample of disruption scenarios, finding a fast repair algorithm is critical. To the best of our knowledge, (Gülcü and Akkan, 2020) is the first work done to address the problem of finding robust timetables CB-CTP of ITC-2007 subject to multiple disruptions. They use a SAA approach and assume a disruption scenario is comprised of a set of single-period unavailability constraints for individual courses. The size of the set is assumed to come from the Binomial distribution. As discussed in Section 2, we assume four different types of disruptions, not only period unavailability, but also room unavailability and an increase in the number of students expected to be enrolled in a course. To evaluate the quality of the Pareto frontiers found by their Multi-Objective Simulated Annealing (MOSA) algorithm, with an integrated SAA approach (MOSA-SAA) that utilizes a network of solutions comprised on the approximation frontier and solutions to a small sample of disruption scenarios. To evaluate the quality of the final frontier, they run their Algorithm 3 (without the UpdateFront() step), which is a part of MOSA-SAA, to identify repairing solutions for a random sample of approximately 300 disruption scenarios and for all solutions on the frontier, simultaneously. Essentially, a local search algorithm is called iteratively and all the solutions found feasible for the disruption scenarios are used to populate a network that links solutions that satisfy the constraints of each disruption scenario with the solutions on the frontier that meet the Hamming distance constraint. Then, for any solution on the frontier, the lowest penalty solution feasible for a disruption scenario and connected to the frontier solution with an edge (i.e. satisfying the distance constraint) is used in the robustness calculation. As opposed to this local search-based algorithm, the algorithms presented here are designed to find the lowest penalty repairing solution that satisfies the distance constraint for a single initial (frontier) solution for a single disruption scenario. We experimented with two very different methodologies, namely simulated annealing and integer programming and a combined use of the two.

The rest of the paper is organized as follows. In the next section, we provide a more formal definition of the addressed problem. In Section 3, we define the types of disruptions that we assume trigger the need for repairing a timetable. In Section 4, we describe a simulated annealing (SA) algorithm for the problem. Then, in the following section we present our Integer Programming-based repair heuristics. In Section 6 we describe how we generated the random disruption scenarios for the computational study and provide the results of this study, comparing the performances of the heuristics. Finally, in Section 7, we provide some concluding remarks and suggestions for future research. Information on how the algorithms were tuned is provided in Appendix.

Section snippets

Problem definition

As different universities have different requirements and rules, over the years researchers worked on many variants of the course timetabling problem which made it difficult to compare the performance of these algorithms (Schaerf, 1999, Kingston, 2013, Pillay, 2014). To address this challenge, several international timetabling competitions have been organized. We have chosen to use the curriculum-based course timetabling problem definition and instances developed for ITC-2007 (Di Gaspero et

Disruption scenarios

As discussed before, we limit our attention to changes before students enroll in the courses, which could be due to unforeseen changes as well as data errors. It is assumed that these changes/errors are observed after the first draft timetable is announced to the faculty members and administrators. This has direct implications on the types of disruptions we have assumed. For instance, we do not assume a disruption could be in the form of a new curriculum (e.g. as in Lindahl et al. (2019)) since

Simulated annealing-based repair heuristic

Meta-heuristics have been widely used to solve problems in dynamic, uncertain environments. Jin and Branke (2005) and Bianchi et al. (2009) provide two surveys on this topic, the latter focusing on combinatorial optimization, with a section devoted to dynamic environments. Simulated Annealing (SA) is a well-established (see Kirkpatrick et al. (1983)) meta-heuristic, which has been shown to perform very well for the university course timetabling problem (see Bellio et al. (2016) and Lewis (2007)

Mixed integer programming-based repair heuristics

The Curriculum-based Course Timetabling problem is NP-hard (see Burke et al. (2010)), and since the disruptions simply change the data associated with the constraints without the changing the structure of the problem, optimally repairing a timetable is still NP-hard. The recent review of the problem by Bettinelli et al. (2015) also highlights the difficulty of solving these problems in practice. Since our motivation is to ultimately develop repair algorithms that can be used to measure

Generating the problem instances

A problem instance is comprised of three related sets of data, namely the initial timetabling problem instance (the ITC-2007 instances), an initial solution that provides a feasible solution to the ITC-2007 instance and a randomly generated disruption scenario. As discussed in Section 3, a disruption scenario is comprised of a set of disruptions, where each disruption belongs to one of the four types of disruptions defined. For each timetabling problem instance and an associated solution, these

Concluding remarks

The need to repair a timetable due to changes in constraints is a common need in timetabling. Prior research focused mostly on minimal perturbation problems where the goal is to have the difference between the disrupted and repaired solutions as small as possible, whereas here we aimed at obtaining a new timetable with the best solution quality (i.e. with minimal penalty) and put a constraint on the difference between the initial and repaired timetables. We called this the minimal cost

CRediT authorship contribution statement

Can Akkan: Conceptualization, Methodology, Software, Formal analysis, Resources, Writing - original draft, Visualization, Writing - review & editing, Supervision, Project administration, Funding acquisition. Ayla Gülcü: Methodology, Software, Validation, Resources. Zeki Kuş: Software, Validation.

References (41)

  • BonuttiA. et al.

    Benchmarking curriculum-based course timetabling: formulations, data formats, instances, validation, visualization, and results

    Ann. Oper. Res.

    (2012)
  • BurkeE.K. et al.

    Comments on: An overview of curriculum-based course timetabling

    Top

    (2015)
  • BurkeE. et al.

    Hybrid variable neighbourhood approaches to university exam timetabling

    (2006)
  • Burke, E.K., Mareček, J., Parkes, A.J., Rudová, H., 2008. Penalising patterns in timetables: novel integer programming...
  • BurkeE.K. et al.

    A supernodal formulation of vertex colouring with applications in course timetabling

    Ann. Oper. Res.

    (2010)
  • ConoverW.

    Practical Nonparametric Statistics

    (1999)
  • Di GasperoL. et al.

    The second international timetabling competition (ITC-2007: Curriculum-based course timetabling (track 3)

    (2007)
  • El SakkoutH. et al.

    Probe backtrack search for minimal perturbation in dynamic scheduling

    Constraints

    (2000)
  • ElkhyariA. et al.

    Scheduling problems using new constraint

  • FukunagaA.

    An improved search algorithm for min-perturbation

  • This research was supported by TÜBİTAK, Turkey grant 217M475.

    View full text