A robust-MILP for synchronized-mTSPTW: Application to home health care under uncertainties

In this paper, we focus on the Synchronized Multiple Traveling Salesmen Problem with Time Windows (Synchronized-m-TSPTW) within an application to Home Health Care (HHC) scheduling problem under uncertain processing and setup times. The general Synchronized-m-TSPTW consists on delivering goods to a set of customers thanks to several traveling salesmen with known demands. In the considered application, each salesmans subset of customers and service are known in advance, but visit processing times and setup times are uncertain. 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 with known parameters is NP-hard. An efficient Mixed Integer Linear Program is proposed in (Masmoudi and Mellouli 2014) to solve the HHC scheduling problem in its certain version. In this paper, we extend the considered problem by considering the uncertainty of processing times and setup times. Thus, we formulate and solve the problem as a robust Resource-Constrained Project Scheduling Problem (Robust-RCPSP) with fixed resources (patients), and mobile resources (caregivers). We use a scenario-based model to represent the uncertain processing and setup times and min-max objective to generate robust solutions that withstand the uncertainty in the worst-case. The experimental results show that the CPU time of the Robust-MILP formulation for the uncertain problem is closed to the CPU time of the MILP for nominal problem. The feasibility of the solution is insured for all scenarios for a set of instances. The cost of robustness increases as the level of uncertainties increases.