Robust Optimization of Vehicle Routing Problems under Uncertainty

Abstract

Vehicle routing problems are a broad class of combinatorial optimization problems that seek to determine the optimal set of routes to be performed by a fleet of vehicles to satisfygiven transportation requests. They lie at the heart of transportation operations in supply chains, and constitute one of the most important and intensely studied problems incomputational mathematics and operations research. Traditional methods to solve these problems have focused on finding solutions in a deterministic context, in which all input parameters are assumed to be known precisely and in advance, an assumption that is difficult to justify in practical applications. Ignoring uncertainty can lead to solutions thatare infeasible or highly suboptimal, and may result in significant economic repercussions when implemented in practice. The primary goal of this thesis is to develop mathematical tools for the systematic treatment of uncertainty in vehicle routing problems arising at the operational, tactical and strategic levels of planning. A major distinguishing focus of our work is the use of robust optimization, a paradigm for optimization under uncertainty that has received only minor attention in this context. We argue and demonstrate that robust optimizationoffers a flexible and computationally tractable way to deal with uncertainty in vehicle routing problems. At the operational level, we develop a unified and scalable algorithmto generate vehicle routes that can be feasibly executed when visiting customers with unknown demands. At the tactical level, we study multi-period problems where thegoal is to serve customers whose service requests are not entirely known in advance. We introduce a dynamic model which adaptively chooses which requests to serve ineach period, as a function of past realizations of the unknown service requests, and develop an algorithm that significantly outperforms traditional methods. At the tacticallevel, we also contribute the first exact algorithms to design vehicle routes that remain consistent when satisfying variable demands across multiple time periods, and expositthat modest increases in costs can be translated to high levels of service consistency. Finally, at the strategic level, we present a method that enables distributors to postulategeneric scenarios of operational uncertainty when allocating long-term delivery time windows to their customers.The secondary goal of this thesis is to contribute to the broader field of optimization under uncertainty, through the development of theory and algorithms, that are currentlylacking but are adequately motivated in vehicle routing applications. In particular, we study dynamic two-stage robust optimization problems with mixed discrete-continuous recourse decisions and present a finite adaptability approximation for their solution. Wecharacterize the geometry of these problems, and present an algorithmic scheme that enjoys strong convergence properties both in theory as well as experiments.