Abstract
University timetabling is a real-world problem frequently encountered in higher education institutions. It has been studied by many researchers who have proposed a wide variety of solutions. Measuring the variation of the performance of solution approaches across instance spaces is a critical factor for algorithm selection and algorithm configuration, but because of the diverse conditions that define the problem within different educational contexts, measurement has not been formally addressed within the university timetabling context. In this paper, we propose a set of metrics to predict the performance of combinatorial optimization algorithms that generate initial solutions for university timetabling instances. These metrics, derived from the fields of enumerative combinatorics and graph coloring, include size-related instance properties, counting functions, feature ratios and constraint indexes evaluated through a feature selection methodology that, based on regression algorithms, characterizes the empirical hardness of a subspace of synthetically generated instances. The results obtained with this methodology show the current need not only to develop solution strategies for particular cases of the problem, but also to produce a formal description of the conditions that make instance spaces hard to solve, in order to improve and integrate the available solution approaches.
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Notes
From the available 16 types of XHSTT constraints described in Table 1, Distribute Split Events Constraint and Order Events Constraint are not used by the instance generator.
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We gratefully acknowledge the support of CONACYT-Mexico (Reg. 618204/461410) and the helpful suggestions of the reviewers.
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Conditions that can be modeled by the CB-CTT instance generator
Conditions that can be modeled by the CB-CTT instance generator
Condition | Description | |
|---|---|---|
1 | Assign teachers | Classes must have an assigned teacher |
2 | Assign rooms | Classes must have an assigned room |
3 | Assign times | Classes must be scheduled for the number of time slots required by their durations |
4 | Avoid clashes | Resources (i.e., curricula, teachers, and rooms) must not be assigned to different lectures at the same time |
5 | Split theory event | Specifies the valid set of lecture configurations in which classes of theory courses (which do not require a laboratory) can be scheduled |
6 | Split practice event | Specifies the valid set of configurations in which classes of practice courses (which require a laboratory) can be scheduled |
7 | Prefer teachers | Defines the subset of classes to which teachers can be allocated |
8 | Prefer rooms | Defines the subset of classes to which rooms and laboratories can be allocated |
9 | Prefer times | Limits a randomly selected percentage of classes to be scheduled only on a user-defined set of days |
10 | Teachers stability | Requires that all lectures derived from a class must be allocated to the same teacher |
11 | Rooms stability | Requires that all lectures derived from a class must be allocated to the same room |
12 | Courses stability | Requires that all lectures derived from a class which requires both a classroom (for theory instruction) and a laboratory (for practice activities) are allocated the same teacher |
13 | Single lecture | Requires that for each class only one lecture be scheduled per day |
14 | Daily lecture | Requires that a randomly selected percentage of classes be scheduled in a daily basis, within a user-defined set of days. |
15 | Link events | Requires that all lectures derived from a set of classes be scheduled simultaneously. This set of classes is defined by randomly selecting a class from each one of the active terms in the curricular plan |
16 | Working shifts | Requires that teachers be allocated only in the set of time slots defined by their work shifts |
17 | Study shifts | Requires that curricula be allocated only in the set of time slots defined by their study shifts |
18 | Idle times of part-time teachers | Specifies the range of daily idle time slots that part-time teachers must have |
19 | Idle times of curricula | Specifies the range of daily idle time slots that curricula must have |
20 | Busy days of full-time teachers | Specifies the number of working days that full-time teachers must give classes |
21 | Busy days of part-time teachers | Specifies the number of working days that part-time teachers must give classes |
22 | Busy days of curricula | Specifies the number of days that curricula must be assigned classes |
23 | Daily workload of full-time teachers | Specifies the number of daily time slots that full-time teachers must give classes |
24 | Daily workload of part-time teachers | Specifies the number of daily time slots that part-time teachers must give classes |
25 | Daily workload of curricula | Specifies the number of daily time slots that curricula must attend classes |
26 | Weekly workload of full-time teachers | Limits the number of weekly time slots that full-time teachers can be allocated |
27 | Weekly workload of part-time teachers | Limits the number of weekly time slots that part-time teachers can be allocated |
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de la Rosa-Rivera, F., Nunez-Varela, J.I., Puente-Montejano, C.A. et al. Measuring the complexity of university timetabling instances. J Sched 24, 103–121 (2021). https://doi.org/10.1007/s10951-020-00641-y
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DOI: https://doi.org/10.1007/s10951-020-00641-y