Stochastic Home Health Care Planning and Scheduling: Models, Relaxations, and Bi-level Exact and Multi-Level Approximate Methods

Abstract

We investigate several practical features of the multi-period home health care planning and scheduling (HHCPS) problem, including full-assignment space, synchronization among caregivers' visits, and uncertainty. First, we develop a mixed-integer program (MIP) that extends an existing pattern-based MIP model, which considers a partial-assignment space for tractability. Utilizing a dataset from the literature, we examine the trade-off between tractability and optimality of these two MIPs. Our MIP remains as tractable as the pattern-based MIP, but it achieves solutions with 11.3% lower costs. We then develop logic-based Benders decomposition (LBBD) approaches combining both mathematical and constraint programming models, leveraging strong relaxations to accelerate convergence. MIP and the optimized LBBD achieve average optimality gaps of 26% and 4.4%, respectively. Lastly, we extend our deterministic MIP to a scenario-based stochastic MIP (SMIP) model that captures uncertainty in caregivers' transit and service times. We solve the SMIP with novel bi-level exact and multi-level approximate LBBD methods to incorporate uncertainty scenarios into different levels of decomposition and to show algorithms with varying performances and convergence profiles. The SMIP could not find even a single integer feasible solution for our instances, whereas our best LBBD method finds optima in 33% of instances with an average optimality gap of 5.1%. Additionally, we determine the average value of stochasticity to be 2.04%. While we recommend stochastic LBBD for smaller instances and deterministic LBBD for larger ones to improve computational efficiency and solution accuracy, due to low caregiver utilization in small instances, our final recommendation is to use the deterministic model for our problem.

Keywords: Home Health Care, Multi-Period Planning and Scheduling, Visit Patterns, Synchronization, Stochasticity, Mixed-Integer Program, Exact and Approximate Techniques, Relaxations