The minimum shift design problem

The min-SHIFT DESIGN problem (MSD) is an important scheduling problem that needs to be solved in many industrial contexts. The issue is to find a minimum number of shifts and the number of employees to be assigned to these shifts in order to minimize the deviation from workforce requirements. Our research considers both theoretical and practical aspects of the min-SHIFT DESIGN problem. 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 ofMECF and, exploiting this reduction, we prove a logarithmic hardness of approximation lower bound forMSD . On the basis of these results, we propose a hybrid heuristic for the problem which relies on a greedy heuristic with a min-cost max-flow subroutine based on the network flow analogy followed by a local search algorithm that makes use of multiple neighborhood relations. An experimental analysis on structured random instances shows that the hybrid heuristic clearly outperforms our previous commercial implementation and highlights the respective merits of the composing heuristics for different performance parameters.