The minimum shift design problem: theory and practice

We study the minimum shift design problem (MSD) that arose in a commercial shift scheduling software project: Given a collection of shifts and workforce requirements for a certain time interval, we look for a minimum cardinality subset of the shifts together with an optimal assignment of workers to this subset of shifts such that the deviation from the requirements is minimum. This problem is closely related to the minimum edge-cost flow problem (MECF), a network flow variant that has many applications beyond shift scheduling. We show that MSD reduces to a special case of MECF. We give a logarithmic hardness of approximation lower bound. In the second part of the paper, we present practical heuristics for MSD. First, we describe a local search procedure based on interleaving different neighborhood definitions. Second, we describe a new greedy heuristic that uses a min-cost max-flow (MCMF) subroutine, inspired by the relation between the MSD and MECF problems. The third heuristic consists of a serial combination of the other two. An experimental analysis shows that our new heuristics clearly outperform an existing commercial implementation.