Learning to Deliver: a Foundation Model for the Montreal Capacitated Vehicle Routing Problem

In this paper, we present the Foundation Model for the Montreal Capacitated Vehicle Routing Problem (FM-MCVRP), a novel Deep Learning (DL) model that approximates high-quality solutions to a variant of the Capacitated Vehicle Routing Problem (CVRP) that characterizes many real-world applications. The so-called Montreal Capacitated Vehicle Routing Problem (MCVRP), first formally described by Bengio et al. (2021), is defined on a fixed and finite graph, which is analogous to a city. Each MCVRP instance is essentially the sub-graph connecting a randomly sampled subset of the nodes in the fixed graph, which represent a set of potential addresses in a real-world delivery problem on a given day. Our work exploits this problem structure to frame the MCVRP as an analogous Natural Language Processing (NLP) task. Specifically, we leverage a Transformer architecture embedded in a Large Language Model (LLM) framework to train our model in a supervised manner on computationally inexpensive, sub-optimal MCVRP solutions obtained algorithmically. Through comprehensive computational experiments, we show that FM-MCVRP produces better MCVRP solutions than the training data and generalizes to larger sized problem instances not seen during training. Even when compared to near-optimal solutions from state-of-the-art heuristics, FM-MCVRP yields competitive results despite being trained on inferior data. For instance, for 400-customer problems, FM-MCVRP solutions on average fall within 2% of the benchmark. Our results further demonstrate that unlike prior works in the literature, FM-MCVRP is a unified model, which performs consistently and reliably on a range of problem instance sizes and parameter values such as the vehicle capacity.

Predicting Drivers' Route Trajectories in Last-Mile Delivery Using A Pair-wise Attention-based Pointer Neural Network

In last-mile delivery, drivers frequently deviate from planned delivery routes because of their tacit knowledge of the road and curbside infrastructure, customer availability, and other characteristics of the respective service areas. Hence, the actual stop sequences chosen by an experienced human driver may be potentially preferable to the theoretical shortest-distance routing under real-life operational conditions. Thus, being able to predict the actual stop sequence that a human driver would follow can help to improve route planning in last-mile delivery. This paper proposes a pair-wise attention-based pointer neural network for this prediction task using drivers' historical delivery trajectory data. In addition to the commonly used encoder-decoder architecture for sequence-to-sequence prediction, we propose a new attention mechanism based on an alternative specific neural network to capture the local pair-wise information for each pair of stops. To further capture the global efficiency of the route, we propose a new iterative sequence generation algorithm that is used after model training to identify the first stop of a route that yields the lowest operational cost. Results from an extensive case study on real operational data from Amazon's last-mile delivery operations in the US show that our proposed method can significantly outperform traditional optimization-based approaches and other machine learning methods (such as the Long Short-Term Memory encoder-decoder and the original pointer network) in finding stop sequences that are closer to high-quality routes executed by experienced drivers in the field. Compared to benchmark models, the proposed model can increase the average prediction accuracy of the first four stops from around 0.2 to 0.312, and reduce the disparity between the predicted route and the actual route by around 15%.