MILP for Synchronized-mTSPTW: Application to Home HealthCare Scheduling

The Multiple Traveling Salesmen Problem is to deliver goods to a set of customers thanks to several travelling salesmen with known demands within minimizing the total salesman transportation cost while satisfying the delivery demands. Here, each salesman´s subset of customers and service or visit´ processing time are determined in advance. Many variants of this problem with different formulations and solving methods are provided in literature. In particular, the Synchronized m-TSP is defined where some customers are common for more than one salesman and must be entirely visited at the same time, simultaneously and not in sequence. The Synchronized m-TSPTW considering synchronization and time windows constraints is defined in Dohn (2011). This paper deals with Synchronized-mTSPTW within an application to Home Health Care (HHC) scheduling problem. Vehicles and customers are modified respectively by caregivers and patients. Time windows and synchronization are given as patients´ availability preferences and caregivers´ synchronization. The Synchronized-mTSPTW is NP-hard since TSPTW is NP-Hard. In this paper, we provide a new MILP considering the HHC routing problem as a special Resource-Constrained Project Scheduling Problem (RCPSP) with fixed resources (patients) and mobile resources (caregivers). A derived two-stage MILP heuristic is also provided to improve the computation times, especially for big instances.