Detour Dual Optimal Inequalities for Column Generation with Application to Routing and Location

We consider the problem of accelerating column generation (CG) for logistics optimization problems using vehicle routing as an example. Without loss of generality, we focus on the Capacitated Vehicle Routing Problem (CVRP) via the addition of a new class of dual optimal inequalities (DOI) that incorporate information about detours from the vehicle routes. These inequalities extend the Smooth-DOI recently introduced in the literature for the solution of certain classes of set-covering problems by CG. The Detour-DOI introduced in this article permit low cost swap operations between items on a given active route with items near to other items on that route to estimate (and bound) the values of the dual variables. Smooth-DOI in contrast only permit low cost swap operations between nearby items. The use of Detour-DOI permits a faster convergence of CG without weakening the linear programming relaxation. We then argue that these DOI can also be conveniently applied to single source capacitated facility location problems. These problems have been shown to be equivalent to a broad class of logistics optimization problems that include, for example telecommunication network design and production planning. The importance of developing vastly more efficient column generation solvers cannot be overstated. Detour-DOI, which permit large numbers of columns to be expressed with a finite set of variables, contributes to this important endeavor.