  When vehicle routing decisions are intertwined with higher-level decisions,
the resulting optimization problems pose significant challenges for
computation. Examples are the multi-depot vehicle routing problem (MDVRP),
where customers are assigned to depots before delivery, and the capacitated
location routing problem (CLRP), where the locations of depots should be
determined first. A simple and straightforward approach for such hierarchical
problems would be to separate the higher-level decisions from the complicated
vehicle routing decisions. For each higher-level decision candidate, we may
evaluate the underlying vehicle routing problems to assess the candidate. As
this approach requires solving vehicle routing problems multiple times, it has
been regarded as impractical in most cases. We propose a novel
deep-learning-based approach called Genetic Algorithm with Neural Cost
Predictor (GANCP) to tackle the challenge and simplify algorithm developments.
For each higher-level decision candidate, we predict the objective function
values of the underlying vehicle routing problems using a pre-trained graph
neural network without actually solving the routing problems. In particular,
our proposed neural network learns the objective values of the HGS-CVRP
open-source package that solves capacitated vehicle routing problems. Our
numerical experiments show that this simplified approach is effective and
efficient in generating high-quality solutions for both MDVRP and CLRP and has
the potential to expedite algorithm developments for complicated hierarchical
problems. We provide computational results evaluated in the standard benchmark
instances used in the literature.
