Neural Network Estimators for Optimal Tour Lengths of Traveling Salesperson Problem Instances with Arbitrary Node Distributions

It is essential to solve complex routing problems to achieve operational efficiency in logistics. However, because of their complexity, these problems are often tackled sequentially using cluster-first, route-second frameworks. Unfortunately, such two-phase frameworks can suffer from suboptimality due to the initial phase. To address this issue, we propose leveraging information about the optimal tour lengths of potential clusters as a preliminary step, transforming the two-phase approach into a less myopic solution framework. We introduce quick and highly accurate Traveling Salesperson Problem (TSP) tour length estimators based on neural networks (NNs) to facilitate this. Our approach combines the power of NNs and theoretical knowledge in the routing domain, utilizing a novel feature set that includes node-level, instance-level, and solution-level features. This hybridization of data and domain knowledge allows us to achieve predictions with an average deviation of less than 0.7% from optimality. Unlike previous studies, we design and employ new instances replicating real-life logistics networks and morphologies. These instances possess characteristics that introduce significant computational costs, making them more challenging. To address these challenges, we develop a novel and efficient method for obtaining lower bounds and partial solutions to the TSP, which are subsequently utilized as solution-level predictors. Our computational study demonstrates a prediction error up to six times lower than the best machine learning (ML) methods on their training instances and up to 100 times lower prediction error on out-of-distribution test instances. Furthermore, we integrate our proposed ML models with metaheuristics to create an enumeration-like solution framework, enabling the improved solution of massive-scale routing problems. In terms of solution time and quality, our approach significantly outperforms the state-of-the-art solver, demonstrating the potential of our features, models, and the proposed method. History: This paper has been accepted for the Transportation Science Special Issue on Machine Learning Methods and Applications in Large-Scale Route Planning Problems. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0015 .