Formulations and Valid Inequalities for the Heterogeneous Vehicle Routing Problem

Abstract

We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift some of the constraints to improve the lower bounds. We generalize and strengthen subtour elimination and generalized large multistar inequalities.

Cumulative Vehicle Routing Problems

“…, which produce constraints (10).□ Similar constraints for classical CVRP may be seen in (Gouveia (1995), Baldacci et al(2004), Letchford & Salazar-Gonzalez (2006), Yaman (2006)). Due to Observation 1, the delivery problem for the symmetric case need not be discussed.…”

“…The CVRP has been studied extensively in the literature (for recent publications, see e.g. Achutan et al (1996), Toth & Vigo (2002b), Ralphs et al (2003), Baldacci et al (2004), Kara et al (2004), Letchford & Salazar-Gonzalez (2006), Yaman (2006)). CVRP was first defined by Dantzig and Ramser in 1959. In that study, the authors used distance as a surrogate for the cost function.…”

Industrial aspects and literature survey: Fleet composition and routing

“…A heuristic procedure based on space-filling curves [6] is used to create a giant tour which is split into single routes and assigned to appropriate vehicles with an algorithm for finding several shortest paths in the network. An exact approach to the FSMVRP is proposed by Yaman [159] when deriving formulations and valid inequalities to compute lower bounds to the problem. Six different formulations are developed; the first four are based on Miller-Tucker-Zemlin constraints [101] and the last two are based on flows.…”

Robust Branch-Cut-and-Price Algorithms for Vehicle Routing Problems

“…We used a set of instances proposed in Golden et al [20], the same set was used in the experiments in Yaman [39] and Choi and Tcha [9] which are compared with our approach. Bound comparisons are presented in table 8 where the columns with header t and C contain the number of different types of vehicles and the largest capacity, respectively.…”

“…Bound comparisons are presented in table 8 where the columns with header t and C contain the number of different types of vehicles and the largest capacity, respectively. It worth noting that the bounds from Yaman [39], given in the columns Yam., come from a cutting plane algorithm on a flow formulation. Her times were obtained in a Sun Ultra 400 MHz machine.…”