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.