An optimization model for the vehicle routing problem with practical three‐dimensional loading constraints

Abstract

In this paper, we present an integer linear programming model for the vehicle routing problem that considers real-world three-dimensional (3D) loading constraints. In this problem, a set of customers make requests of goods that are wrapped up in boxes, and the objective is to find minimum cost delivery routes for a set of identical vehicles that, departing from a depot, visit all customers only once and return to the depot. Apart from the usual 3D container loading constraints that ensure the boxes are packed completely inside the vehicles and the boxes do not overlap each other in each vehicle, the problem also takes into account constraints related to the vertical stability of the cargo, multidrop situations, and load-bearing strength of the boxes (including fragility). Computational tests with the proposed model were performed using an optimization solver embedded into a modeling language. The results validate the model and show that it is only able to handle problems of a moderate size. However, this model will be useful to motivate other researchers to explore approximate solution approaches to solve this problem, such as decomposition methods, relaxation methods, heuristics, among others, as well as to treat other variants of the problem, such as when time windows or a heterogeneous fleet are present, among others.

Vehicle routing problems with loading constraints: state-of-the-art and future directions

“…Corner points follow the concept of envelope and are introduced by Martello et al (2000) for 3D bin packing. Junqueira et al (2013) are to the authors' knowledge the first to propose an exact method to solve the 3L-CVRP. They assume a homogeneous vehicle fleet, sequencebased loading, stacking constraints, orientation constraints and stability constraints.…”

“…Vertical stability constraints specify the minimum supporting area of each item (e.g., as a percentage of the base area of the item). Horizontal (or dynamic) stability of the cargo refers to the support of the lateral faces of items in the container to prevent items moving around in the container (Junqueira et al 2013). The literature concerning three-dimensional VRPs often takes vertical stability constraints into account (e.g., Gendreau et al 2006;Fuellerer et al 2010;Bortfeldt 2012;Zhu et al 2012;Ruan et al 2013).…”

“…Stacking constraints (also denoted as loadbearing strength constraints or fragility constraints) prevent this from happening. The load-bearing strength of an item is the maximum pressure that can be applied on this item before damage takes place (Junqueira et al 2013). The load-bearing strength may vary across different vertical orientations of this item (Ratcliff and Bischoff 1998).…”

“…Stacking constraints have been considered in several papers concerning threedimensional loading VRPs (e.g. Gendreau et al 2006;Tarantilis et al 2009;Fuellerer et al 2010;Ruan et al 2013;Junqueira et al 2013). …”

A matheuristic algorithm for the three-dimensional loading capacitated vehicle routing problem (3L-CVRP)

“…All the set of packing constraints are modeled and linked with the transportation model in [12], but due to the high complexity of the model, the proposed methodology is inadequate for solving medium and large size problems. In this work, a GRASP algorithm within the mathematical model guarantees these constraints, allowing the solution of real problems of the companies.…”