A General Formulation for the Teaching Assignment Problem: Computational Analysis Over a Real-World Dataset

The Teacher Assignment Problem is a combinatorial optimization problem that involves assigning teachers to courses while guaranteeing that all courses are covered, teachers do not teach too few or too many hours, teachers do not switch assigned courses too often and possibly teach the courses they favor. Typically the problem is solved manually, a task that requires several hours every year. In this work we present a mathematical formulation for the problem and an experimental evaluation of the model implemented using state-of-the-art SMT, CP, and MILP solvers. The implementations are tested over a real-world dataset provided by the Division of Systems and Control at Chalmers University of Technology, and produce teacher assignments with smaller workload deviation, a more even workload distribution among the teachers, and a lower number of switched courses.

💡 Research Summary

The paper addresses the Teacher Assignment Problem (TAP) faced by the Division of Systems and Control at Chalmers University of Technology, where teaching assistants (TAs) must be allocated to a set of courses and associated tasks over an academic year. The authors first formulate a comprehensive mathematical model that captures both hard constraints (e.g., availability, workload limits, minimum staffing per task, and continuity of courses) and soft constraints (e.g., deviation from target workload, number of new courses, preferences, and soft bounds on staffing). The model introduces binary decision variables for TA‑course assignment (w), TA‑task assignment (y), and continuous variables for hours allocated (x). Hard constraints ensure feasibility, while soft constraints are weighted to reflect their relative importance.

To evaluate the practical usefulness of the model, the authors implement it using three distinct optimization paradigms:

1. SMT (Satisfiability Modulo Theories) – Z3 is used, with soft constraints encoded as weighted optimization objectives.
2. CP‑SAT (Constraint Programming – SAT) – OR‑Tools’ CP‑SAT solver, which combines constraint propagation with SAT‑based search.
3. MILP (Mixed‑Integer Linear Programming) – Both the commercial solver Gurobi and the open‑source solver SCIP are employed; soft constraints are linearized and incorporated into a single objective function.

The experimental data consist of five yearly instances (2022‑2026), each containing roughly 50 TAs, 45 courses, and up to ten tasks per course. No explicit TA preferences were available, so all preference values were set to neutral. Parameter values for hard and soft bounds were tuned each year based on expert input and comparison with the historical manual schedule.

Key Findings

• Computation Time: Gurobi solved every instance in under 34 seconds, demonstrating the superior speed of a modern commercial MILP engine. CP‑SAT required between 1,349 and 3,600 seconds, with a timeout on the 2025 instance. Z3 hit the 3,600‑second limit on all five instances, and SCIP timed out on three instances.
• Solution Quality (RMSE of workload deviation): CP‑SAT achieved the lowest root‑mean‑square error, ranging from 0.22 to 47.76, dramatically outperforming both MILP solvers (RMSE 22.5–52.8) and the manual schedule (RMSE 92.97–144.94). Gurobi consistently outperformed SCIP, but both were far behind CP‑SAT. Z3’s RMSE (68.7–119.9) was sometimes worse than the manual solution.
• Interpretation of Results: The TAP combines discrete assignment decisions with continuous workload allocation, a structure that CP‑SAT handles efficiently through strong propagation and heuristic search. SMT, while expressive, struggled with the mixed integer‑linear nature of the problem, leading to poor scalability. MILP offers excellent raw speed, especially with a commercial solver, but the linearized objective does not capture the nuanced trade‑offs that CP‑SAT’s weighted soft constraints do, resulting in higher workload deviation.
• Practical Implications: The authors argue for a “human‑in‑the‑loop” workflow where planners can quickly adjust constraint parameters and re‑solve. CP‑SAT’s relatively modest sensitivity to parameter changes makes it well‑suited for iterative planning sessions, whereas Gurobi is advantageous when large instances must be solved within tight time budgets.

Conclusions

The study demonstrates that automated optimization can substantially improve the fairness and efficiency of TA assignments compared with traditional manual methods. While all three paradigms produce feasible schedules, CP‑SAT delivers the highest quality in terms of workload balance, and Gurobi provides the fastest turnaround. The choice of solver should therefore be guided by the specific operational priorities of the department—solution quality versus time constraints. Future work is suggested to integrate multi‑year continuity more tightly and to explore richer preference modeling within the same framework.