A novel and efficient exact technique for integrated staffing, assignment, routing, and scheduling of home care services under uncertainty

We model and solve integrated multi-period staffing, assignment, routing, and scheduling of caregivers for home care services and obtain insights for the case 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 pattern—an existing practical approach that deals effectively with hard assignment decisions. First, we propose a novel mixed-integer program (MIP) for the nominal (deterministic) 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, i.e., our solution approach for the deterministic case allowed us to solve the robust model. Using a benchmark and newly generated instances from the literature, we show that CPLEX can solve our nominal model with average optimality gap of 56%. On the other hand, our new exact technique can solve our nominal model to an average optimality gap of 5%. 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 18% than the nominal solution and the cost-savings of nurse flexibility is about 19% on average and higher than overtime’s. We also demonstrate that nurse flexibility impact is high and comparable in both cases, deterministic and stochastic.