A Multi-Objective Approach to Curriculum-Based Course Timetabling with Continuity Across Semesters

Abstract: We study a curriculum-based university course timetabling problem in which the preferences of two key stakeholder groups - lecturers and students - must be balanced while maintaining continuity across semesters in a weekly repeating timetable. While existing approaches typically rely on single-objective formulations or aggregate multiple objectives into a weighted sum, this can obscure the underlying trade-offs between conflicting stakeholder preferences. We therefore propose a multi-objective mixed-integer programming approach that explicitly separates lecturer and student objectives and incorporates timetable continuity by limiting the number of changes, called perturbations, in the time period assignments of selected courses relative to the corresponding semester of the previous academic year. To explore the resulting trade-offs, we develop a multi-objective solution approach based on the lexicographic $\varepsilon$-constraint method, enabling the computation of a representative set of solutions whose images, i.e., their vectors of objective values, cover different regions of the objective space. The approach is evaluated on real-world instances from the Straubing Campus of the Technical University of Munich. The computational results reveal a clear and consistent trade-off between lecturers' and students' objectives across all instances. Moreover, the number of allowed perturbations is identified as a key decision parameter: relaxing this constraint significantly improves timetable quality for both stakeholder groups, although diminishing returns are observed beyond certain thresholds. Overall, the proposed approach provides decision support by generating a diverse set of optimized timetables and enabling a transparent analysis of stakeholder trade-offs and continuity for the practical timetable planning process.

• The paper introduces a multi-objective MIP model that separately optimizes lecturer and student timetabling preferences while managing continuity across semesters.
• It employs a lexicographic ε-constraint method to generate efficient trade-off solutions over varying perturbation bounds, revealing pronounced conflicts between stakeholder objectives.
• Empirical results using TUM Campus data demonstrate that relaxing continuity constraints improves student schedules significantly, with diminishing gains for lecturers.

Multi-Objective Curriculum-Based Timetabling with Semester Continuity

Problem Formulation and Motivation

This paper addresses a curriculum-based university course timetabling (CB-CTT) problem, explicitly considering the conflicting preferences of lecturers and students while maintaining continuity across academic semesters. Traditional practices in university timetabling often aggregate multiple objectives via weighted sums, masking the underlying trade-offs. This work instead formulates the CB-CTT as a multi-objective optimization problem, leveraging mixed-integer programming (MIP) to separately model the lecturer and student objectives. The notion of continuity is operationalized by constraining the number of perturbations—changes in course time assignments relative to previous semesters.

The study is grounded in real-world timetabling data and requirements from the TUM Campus Straubing, incorporating constraints associated with room capacities, lecturer availabilities, course types (lectures, tutorials, exercises), curriculum structure, and inter-semester continuity. The model reflects practical scheduling bottlenecks, including overlap avoidance among mandatory courses within curricula, preferential scheduling, and the preservation of routines for stakeholders.

MIP Model for Multi-Stakeholder Timetabling

The proposed MIP model integrates a complex set of constraints (room capacities, period availabilities, course-module relations, exclusion and preference periods, etc.) and sector-specific objectives. Lecturer preferences are captured in terms of favored periods, consecutive/non-consecutive teaching slots, teaching load distribution, and avoidance of undesirable periods. Student objectives focus on overlap minimization, avoidance of unpopular periods (survey-derived penalties), break preservation, and sequencing/tutorial accessibility.

The continuity constraint is enforced by limiting the number of course assignments to periods different from those in the previous corresponding semester, allowing the explicit quantification and management of institutional stability versus schedule quality. The model utilizes auxiliary variables for soft constraint violations across both objectives.

Multi-Objective Solution Approach

The paper advances a bi-objective optimization strategy utilizing the lexicographic $\varepsilon$-constraint method. Solution generation proceeds as follows:

• Anchor Point Calculation: For a given perturbation bound, solutions are generated by optimizing the lecturers' and students' objectives alternately, setting new bounds based on the optimal values from previous runs.
• $\varepsilon$-Constraint Generation: Additional representative trade-off solutions are produced by tightening the bounds incrementally in both directions, yielding a set of efficient (nondominated) timetables whose images cover the relevant objective space.
• Perturbation Bounds: Feasibility and trade-off exploration are carried out across a range of bounded values for perturbations, reflecting practical acceptability thresholds derived from campus planners.

Hypervolume metrics are used for quality evaluation, computed relative to the Nadir point in objective space.

Empirical Evaluation and Numerical Results

The computational study employs three semester datasets from TUM Campus Straubing, with 162-181 courses, substantial curriculum and module diversity, extensive lecturer information, and room heterogeneity. Constraint and weight calibration is based on dedicated surveys involving 91 students and 60 lecturers. Up to ten solutions per instance and perturbation bound are generated: anchor points and eight $\varepsilon$-constraint solutions.

The results consistently reveal a pronounced trade-off structure: optimizing one stakeholder’s objective typically degrades the other. e.g., improvements in lecturer period preferences and consecutive teaching increase overlaps and consecutive blocks for students, and vice versa. Quantitative hypervolume analysis demonstrates monotonic improvements as perturbation constraints are relaxed, with substantial early gains followed by diminishing returns—particularly for lecturer-centric objectives. In some instances, increasing allowed perturbations beyond moderate values yields almost exclusive improvements for the student objective, indicating saturation in achievable lecturer-side gains.

Significant numerical results include:

• Efficient solution sets that span the relevant objective trade-off frontier.
• Clear and consistent hypervolume increases as continuity constraints are loosened (e.g., hypervolume for Instance 1 increases from 125.64 to 177.05 × 10⁵ across allowed perturbation bounds).
• Saturation effects where further continuity relaxation provides marginal improvements for lecturers but significant gains for students.
• Practical solver performance characterization with Gurobi, leveraging up to eight-hour runs and employing warm-starts.

Practical and Theoretical Implications

The explicit modeling and optimization of stakeholder trade-offs fundamentally advances practical decision support for university timetabling. The separation of objectives enables transparent institutional negotiation, departing from opaque aggregated weighted-sum models. Continuity across semesters, a critical operational requirement for institutional stability, is treated as a tunable planning dimension rather than a hard constraint.

Future developments are likely to include:

• Scalability improvements for larger instances and more complex institutional settings.
• Enhanced interactive tools for planners, facilitating exploration and communication of trade-offs.
• Extension to richer stakeholder models (e.g., including administrative constraints, hybrid teaching, further granular student preference modeling).
• Application and benchmarking with international datasets and broader educational environments.

On the theoretical side, the proposed framework demonstrates the efficacy of multi-objective MIP and lexicographic constraint methods for operational research in education, providing a blueprint for further multi-stakeholder optimization problems where continuity and change management must be balanced.

Conclusion

This paper presents a rigorous multi-objective mixed-integer programming approach for curriculum-based university course timetabling, explicitly separating lecturer and student objectives and embedding continuity management across semesters. The empirical results confirm persistent conflicts between objectives and highlight the necessity of transparent decision support in timetable planning. Relaxation of continuity constraints enables significant improvements for both groups, with diminishing returns for lecturers. The methodology can be directly integrated into institutional planning workflows, supporting informed negotiation amongst stakeholders and improving operational outcomes for universities.