Cyclic preference scheduling of nurses using a Lagrangian-based heuristic

This paper addresses the problem of developing cyclic schedules for nurses while taking into account the quality of individual rosters. In this context, quality is gauged by the absence of certain undesirable shift patterns. The problem is formulated as an integer program (IP) and then decomposed using Lagrangian relaxation. Two approaches were explored, the first based on the relaxation of the preference constraints and the second based on the relaxation of the demand constraints. A theoretical examination of the first approach indicated that it was not likely to yield good bounds. The second approach showed more promise and was subsequently used to develop a solution methodology that combined subgradient optimization, the bundle method, heuristics, and variable fixing. After the Lagrangian dual problem was solved, though, there was no obvious way to perform branch and bound when a duality gap existed between the lower bound and the best objective function value provided by an IP-based feasibility heuristic. This led to the introduction of a variable fixing scheme to speed convergence. The full algorithm was tested on data provided by a medium-size U.S. hospital. Computational results showed that in most cases, problem instances with up to 100 nurses and 20 rotational profiles could be solved to near-optimality in less than 20 min.