CIP and MIQP Models for the Load Balancing Nurse-to-Patient Assignment Problem

The load balancing nurse-to-patient assignment problem requires the assignment of nurses to patients to minimize the variance of the nurses’ workload. This challenging benchmark is currently best solved exactly with constraint programming (CP) using the spread constraint and a problem-specific heuristic. We show that while the problem is naturally modelled as a mixed integer quadratic programming (MIQP) problem, the MIQP does not match the performance of CP. We then develop several constraint integer programming (CIP) models that include bounds propagation, linear relaxations, and cutting planes associated with the quadratic, gcc, and spread global constraints. While the quadratic and gcc techniques are known, our additions to the spread constraint are novel. Our empirical results demonstrate that the CIP approach substantially out-performs the MIQP model, but still lags behind CP. Finally, we propose a simple problem-specific variable ordering heuristic which greatly improves the CIP models, achieving performance about an order of magnitude faster than CP and establishing a new state of the art.