Mathematical Programming and Heuristic Approaches for Three Novel Vehicle Routing Problems

Abstract

Vehicle routing problems (VRPs) arise in key real-world domains including mail delivery, police patrol, and snow plowing. Due to the inherent complexity of the VRP, no algorithm currently exists that can solve the VRP exactly in polynomial time. In the meantime, high quality solutions are needed to tackle challenging issues such as rising mail and package delivery costs and surging crime rates. In this dissertation, we design efficient routing solutions for three applications of the VRP. First, we consider the application of the hot spot police patrol, where hot spots refer to locations with high crime rates. We examine the routing problem for a patrol car that visits locations in a region and remains close to the center of the region modeling a high-crime neighborhood (HCN). We formulate this problem as the traveling salesman problem with a center (TSPC), in which we minimize an energy function including L (the length of a tour) and C (the distance from a tour to the center, defined by some metric). To address the TSPC, we propose a metric to measure C rather accurately and also introduce the idea of a triangular path, in which the patrol car no longer travels in a straight line between two nodes. We show that under identical circumstances, the tour with triangular paths remains closer to the center than a tour using all direct edges in both a Euclidean graph and a grid network. Second, we extend the hot spot police patrol to a fleet of patrol cars. Given the disorder and chaos in the HCN, we require at least one patrol car in the HCN at any given time during the patrol. We call this routing problem the hot spot coverage patrol problem (HSCPP). In the HSCPP, the importance of a patrol location is quantified by a prize and our objective is to maximize the sum of prizes collected by the patrol cars while obeying all operations requirements. We propose mathematical formulations and develop several solution approaches for the HSCPP. These solution approaches are based on integer programming and column generation. We then conduct a detailed computational study to compare these approaches using real crime data from Montgomery County, Maryland. We point out the best solution approaches in terms of optimality gap, runtime, and workload balancing. Third, we consider mail deliveries. To reduce the fleet size and carbon emissions of a postal service, we allow two mail carriers per truck. We call this problem the paired mail carrier problem (PMCP). Our objective is to minimize the route completion time while ensuring that each service stop is fully serviced and obeys all feasibility constraints. We develop a mixed integer programming formulation and two fast heuristics for the PMCP. We evaluate the impact of the paired mail carriers (PMC) setting on both a one-truck situation and a fleet (multiple trucks) situation, relative to the single mail carrier (SMC) setting. Overall, the PMC setting can not only accomplish over 50% more work within the same shift hours but can also lead to a 22% cost savings.