Improved Branch-Cut-and-Price for Capacitated Vehicle Routing

Abstract

Abstract. The best performing exact algorithms for the Capacitated Vehicle Routing Problem are based on the combination of cut and column generation. Some authors could obtain reduced duality gaps by also using a restricted number of cuts over the Master LP variables, stopping separation before the pricing becomes prohibitively hard. This work introduces a technique for greatly decreasing the impact on the pricing of the Subset Row Cuts, thus allowing much more such cuts to be added. The newly proposed Branch-Cut-and-Price algorithm also incorporates and combines for the first time (often in an improved way) several elements found in previous works, like route enumeration and strong branching. All the instances used for benchmarking exact algorithms, with up to 199 customers, were solved to optimality. Moreover, some larger instances with up to 360 customers, only considered before by heuristic methods, were solved too.

Strong Lower Bounds for the CVRP via Column and Cut

“…And finally, also in 2014, Pecin et al [19] introducted the Limited Memory Subset Row Cuts. They are a weakening of the traditional Subset Row Cuts, introduced in Jepsen et al [20], which are Chvátal-Gomory rank-1 cuts.…”

“…The algorithm proposed by Pecin et al [21] was the first one to solve all the classical instances from the literature with up to 200 vertices. However, this algorithm was a combination of several methods and it is not clear what was the improvement made by each one.…”

“…Therefore, the objective of this work is to devise a pure implementation of the algorithm proposed in Pecin et al [21], using only the lm-SRC combined with Rounded Capacity Cuts on the SP formulation considering ng-routes. This has not…”

“…To evaluate this approach, we run the algorithm for the same instances that Pecin et al [21] and we also present the lower bounds for different sizes of ng-sets.…”

“…19) and π 0 is the dual variable associated to constraint (II.20). Thus, the reduced cost of a route is its original cost c r minus the dual variable π 0 and the dual variable π i for each customer i visited by this route.…”