Solving capacitated vehicle routing problem with three-dimensional loading and relocation constraints

Abstract

In capacitated vehicle routing problem with three-dimensional loading constraints (3L-CVRP) that combines the container loading and capacitated vehicle routing problems, the relationship between vehicle visiting sequence and item loading sequence is reflected through relocation-ban constraint. This widely applied constraint prohibits the temporarily unloading and repositioning of loaded items during the entire transportation process, simplifying loading operations but also limiting the volume utilization of each vehicle and increasing transportation costs. To address this issue and obtain a trade-off between transportation cost and operational complexity (reflected in relocation cost), two improved relocation constraints that seek to allow necessary and restricted relocations are developed in this study. Under pickup scenario, a mixed integer-linear programming model is developed to describe the 3L-CVRP with the relocation constraints. An improved branch-and-price algorithm is employed to solve the model. Two loading algorithms, incorporated with a backward dynamic programming method, are proposed to simultaneously generate loading and relocation plans. Enhancement strategies, including an improved label-correcting-based algorithm and memory components that collect loading feasibility and relocation cost information, are developed. Numerical experiments were designed to test the performance of the proposed algorithms and validate the significance of necessary relocations. In pure loading instances, necessary relocations bring an increase in volume utilization by 3.75 % on average and 36.48 % at maximum. In large-scale benchmark instances, allowing necessary relocations decrease the overall costs by 4.86 % on average and 13.08 % at maximum.

Abbreviations: 3L-CVRP, capacitated vehicle routing problem with three-dimensional loading constraints; CLP, container loading problem; CVRP, capacitated vehicle routing problem; RC, relocation constraints; MILP, mixed integer linear programming; B&P, branch and price; BDP, backward dynamic programming; D–W, Dantizig–Wolfe; ESPPRLC, elementary shortest path problem with resource and loading constraints; LCA, label-correcting-based algorithm; 3L-PDP, pickup and delivery vehicle routing problem with three-dimensional loading constraints; RMP, restricted-master-problem; SP, sub-problem; IGHA, improved greedy heuristic algorithm; ITRSA, improved tree search algorithm; TN, tree node; BR, Bischoff & Ratcliff.