A branch-and-price-and-cut algorithm for the home health care routing and scheduling problem with multiple prioritized time windows

Abstract

This paper addresses a variant of home health care routing and scheduling problem with endogenous time window options for patients. The objective is to minimize the total operational cost while ensuring a specified level of service satisfaction regarding patients’ prioritized time windows. We formulate the problem as a set-covering model and propose a branch-and-price-and-cut (BPC) algorithm to tackle it. Specifically, we devise a column generation procedure to obtain the lower bounds in the branch-and-bound framework, in which we employ a labeling algorithm with a multiple time windows processing mechanism to deal with the pricing sub-problems. Additionally, the lower bound is strengthened by incorporating 2-path inequalities and subset-row inequalities. The effectiveness of the BPC algorithm is demonstrated through numerical experiments conducted on the modified Solomon benchmark instances. Furthermore, we analyze the problem and derive the following managerial insights: i) Incorporating the assignment decisions of time windows for patients into the problem can significantly reduce nurse costs and travel costs, even when accounting for patients’ preferences for time windows. ii) There is a negative correlation between cost savings and the overall satisfaction levels of patients’ preferred time windows, with marginal cost savings decreasing as the overall satisfaction level declines. iii) Besides the total satisfaction level, the operational cost is also influenced by factors such as the distribution of patient locations, the width of time windows, and the proportion of patients with multiple time windows.