Chapter 3: New Exact Algorithms for the Capacitated Vehicle Routing Problem

Abstract

Since the seminal work by Desrosiers, Soumis, and Desrochers [15], column generation has been the dominant approach for building exact algorithms for the Vehicle Routing Problem with Time Windows (VRPTW). This technique performed very well on tightly constrained instances (those with narrow time windows). As the Capacitated Vehicle Routing Problem (CVRP) can be regarded as the particular case of VRPTW where time windows are arbitrarily large, column generation was viewed as a non-promising approach for the problem. In fact, in the early 2000's, the best performing algorithms for the CVRP were Branch-and-Cut algorithms that separated quite complex families of cuts identified by polyhedral investigation (see Naddef and Rinaldi [31] and Chapter 2). In spite of their sophistication, some instances from the literature with only 50 customers could not be solved to optimality. At that moment, the Branch-and-Cut-and-Price algorithm (BCP) by Fukasawa et al. [19] showed that the combination of cut and column generation could be much more effective than each of those techniques taken alone. Since then, the most performing exact algorithms proposed for the CVRP are based on that combination.According to the classification proposed in Poggi de Aragão and Uchoa [36], the BCP algorithm in Fukasawa et al. [19] only uses robust cuts. A cut is said to be robust when the value of the dual variable associated with it can be translated into costs in the pricing subproblem. Therefore, the structure and the size of that subproblem remain unaltered, regardless of the number of robust cuts added. On the other hand, non-robust cuts are those that change the structure and/or the size of the pricing subproblem; each additional cut makes it harder.

Optimisation of maintenance routing and scheduling for offshore wind farms

“…In fact the overview of new exact solution methods for the capacitated VRP given by Poggi and Uchoa (2014) shows that all recent advances in solution methods are built on a similar separation of the problem into a master problem and sub-problem(s). It is therefore natural to use such decomposition also when solving the routing and scheduling problem of maintenance vessels at offshore wind farms.…”

A critical analysis of the “improved Clarke and Wright savings algorithm”

“…The CVRP is among the most studied problems in the field of Operations Research, and a very large number of algorithms have been developed to solve it (see, e.g., Toth and Vigo, , for an overview). Exact algorithms—that guarantee to find the optimal solution—can reliably solve instances with up to 200 customers in a reasonable amount of computing time (Poggi and Uchoa, ). For larger instances, a large number of heuristics are available.…”

A unified framework for rich routing problems with stochastic demands

“…In terms of solution methodology, an important area of future research is the development of theoretical lower bounds. Our optimization models rely on arc-based formulations which are known to provide weak lower bounds (Semet et al, 2014;Poggi and Uchoa, 2014). Such models quickly become intractable even for moderate instance sizes.…”