A Novel and Efficient Exact Technique for Integrated Staffing, Assignment, Routing, and Scheduling of Home Care Services Under Uncertainty

Abstract

We model and solve an integrated multi-period staffing, assignment, routing, and scheduling for home care services under uncertainty. The goal is to construct a weekly schedule that adheres to related operational considerations and determines optimal staffing of caregivers by minimizing caregivers’ fixed- and overtime costs. For tractability, we incorporate a priori generated visit patterns—an existing practical approach that deals effectively with hard assignment decisions in. First, we propose a novel mixed-integer program (MIP) for the nominal problem. We then incorporate uncertainty in service and travel times and develop a robust counterpart by hybridizing interval and polyhedral uncertainty sets. Second, we show that there is a special mathematical structure within the model that allows us to develop a novel logic-based Benders branching-decomposition algorithm that systematically delays the resolution of difficult routing/ scheduling problems and efficiently solves both the nominal and robust MIP models. Using a dataset from the literature, we show that CPLEX can solve our nominal and robust models with an average optimality gaps of 44.56% and 45.53%, respectively. Using the same dataset, we demonstrate that our new exact technique can solve our nominal and robust mixed-integer models to an average optimality gap of 2.8% and 4.5%, respectively. Third, we provide practical insights into (i) the price of robustness and (ii) the impacts of nurse flexibility and overtime. The average total cost does not increase beyond 12.7% than the nominal solution and the cost-savings of nurse flexibility is approximately five times higher than that of overtime.

Keywords: Combinatorial Optimization, Home Health Care Planning and Scheduling, Multi period, Logic-Based Benders, New Exact Branching-Decomposition.