Theory and practice of the minimum shift design problem

The min-SHIFT DESIGN problem 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 the workforce requirements. Our research considers both theoretical and practical aspects of the min-SHIFT DESIGN problem. First, we establish a complexity result by means of a reduction to a network flow problem. The result shows that even a logarithmic approximation of the problem is NP-hard. However, the problem becomes polynomial if the issue of minimizing the number of shifts is neglected. On the basis of these results, we propose a hybrid heuristic for the problem which relies on a greedy construction phase, 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 an existing commercial implementation and highlights the respective merits of the composing heuristics for different performance parameters.