We study the periodic assignment problem, in which a set of periodically repeating tasks must be assigned to workers within a repeating schedule. The classical efficiency objective is to minimize the number of workers required to operate the schedule. We propose a (n log n) algorithm to solve this problem. Next, we formalize a notion of fairness among workers, and impose that each worker performs the same work over time. We analyze the resulting trade-off between efficiency and fairness, showing that the price of fairness is at most one extra worker, and that such a fair solution can always be found using the Nearest Neighbor heuristic. We characterize all instances that admit a solution that is both fair and efficient, and use this result to develop a (n log n) exact algorithm for the fair periodic assignment problem. Finally, we show that allowing aperiodic schedules never reduces the price of fairness.

There is growing interest in including fairness in optimization models. In particular, the concept of fairness over time, or, long-term fairness, is gaining attention. In this paper, we focus on fairness over time in online optimization problems involving the assignment of work to multiple homogeneous workers. This encompasses many real-life problems, including variants of the vehicle routing problem and the crew scheduling problem. The online assignment problem with fairness over time is formally defined. We propose a simple and interpretable assignment policy with some desirable properties. In addition, we perform a case study on the capacitated vehicle routing problem. Empirically, we show that the most cost-efficient solution usually results in unfair assignments while much more fair solutions can be attained with minor efficiency loss using our policy.