The Quadratic Capacitated Vehicle Routing Problem

In this article we introduce the Quadratic Capacitated Vehicle Routing Problem (QCVRP), a combinatorial optimization problem that arises in practical applications in logistics and transportation. The QCVRP extends two other known problems, the Capacitated Vehicle Routing Problem (CVRP) and the Quadratic Symmetric Traveling Salesman Problem (QSTSP), the former by considering a modified cost matrix in which traveling costs are now associated to pairs of two consecutive edges, and the latter by introducing customer demands and vehicle capacities. We present a three-index vehicle-flow formulation for the problem in which variables represent q-edges (pairs of consecutive edges), and strengthen it with some classes of valid inequalities. We present efficient separation routines for the inequalities used in this paper and derive an exact solver based on the branch-and-cut paradigm. We conduct computational experiments to show the efficiency of the modeling and solution approaches by providing tight lower bounds and optimal solutions for smalland medium-size instances.