The emergency department physician rostering problem: obtaining equitable solutions via network optimization

In this study, we address workload balancing in Emergency Department Physician Rostering Problems. We propose a two-phase approach to deal with two common workload balancing issues: (1) the even distribution of worked weekends and weekend night shifts across physicians in the long term, and (2) the even distribution of morning and afternoon shifts in the medium term. To implement such an approach, we have developed two Integer Linear Programming (ILP) models, one for each phase. In the first phase, we determine the weekends that each physician will be on duty over the long term planning horizon (6-months) while evenly distributing the workload (worked weekends and weekend night shifts) across physicians. In the second phase, month by month, we iteratively determine the workday shifts of each physician while pursuing the even distribution of workload (morning and afternoon shifts) across physicians. The second phase relies on the solution of the first phase, i.e., the weekend shifts assigned to each physician in the first phase are considered preassigned shifts in the second phase. In both phases, we consider the constraints deriving from collective as well as individual contractual agreements (e.g. constraints limiting the maximum number of night shifts each physician can work every month, their maximum weekly and monthly workload, etc.) as well as individual physician’s preferences and desiderata. The problems addressed in the two phases differ in terms of the planning horizon, objective function, and constraints, yet they are both modeled as multicommodity ow problems and share the same network structure. Also, we define some families of simple yet effective, valid inequalities that are crucial to address the computational complexity of the first-phase problem. The proposed optimization models have been tested on real data from a leading European Hospital and on benchmark instances from the literature. The models’ effectiveness has been assessed through six key performance indicators purposely defined. Results demonstrate that the presented models allow considering the complex nature of physicians rostering problems and obtaining well-balanced and thus equitable work schedules.