Algorithms for fair repetitive scheduling based on total completion time criterion

We consider a single-machine scheduling problem consisting of n clients and q consecutive operational periods (e.g., days). Each client submits a single job for processing on each of the days and wishes for his jobs to be completed as early as possible. A solution is defined by a set of q schedules (one for each day) and it is classified as a K-fair solution if the total completion time for any of the clients, across the entire set of q days, does not exceed K. The scheduler's objective is to obtain a K-fair solution with the minimum possible K value. This problem is known to be strongly NP-hard (Hermelin et al., 2025), but no practical solution approaches have been developed and tested for solving the general problem. Our main goal is to close this gap in the literature by providing a set of practical tools to maximize the system's fairness. To do so, we design a mixed linear integer programming (MILP) formulation, two greedy algorithms, a simple day-insertion heuristic and a metaheuristic algorithm to solve the problem. We then experimentally evaluate the quality of each algorithm's solutions by comparing them against a lower bound. We show that the two greedy algorithms and the day-insertion algorithm are able to provide good solutions quickly, but the quality of the solution can be further increased by applying the more sophisticated and time-consuming metaheuristic algorithm. In addition to the experimental study, we provide some theoretical results, including approximability analysis of the greedy algorithms, and a simple polynomial time procedure to solve the unit processing-time case.¹