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Deep Learning for Data-Driven Districting-and-Routing

Arthur Ferraz, Cheikh Ahmed, Quentin Cappart, Thibaut Vidal

TL;DR

This work tackles data-driven districting-and-routing under stochastic demand by learning district-cost estimates with a Graph Neural Network and integrating them into an Iterated Local Search to rapidly generate high-quality, connected district partitions. The GNN uses BU-level features and district membership indicators to predict long-term district costs, enabling scalable optimization that outperforms continuous-approximation baselines. Across five UK metropolitan areas, the approach achieves an average economic gain of $10.12\%$ over baselines and reveals that district geometry beyond simple compactness critically influences routing efficiency. The results support practical decision-support by delivering fast, accurate cost estimates, strong solution quality, and insights into how learnable geometric features shape effective districting.

Abstract

Districting-and-routing is a strategic problem aiming to aggregate basic geographical units (e.g., zip codes) into delivery districts. Its goal is to minimize the expected long-term routing cost of performing deliveries in each district separately. Solving this stochastic problem poses critical challenges since repeatedly evaluating routing costs on a set of scenarios while searching for optimal districts takes considerable time. Consequently, solution approaches usually replace the true cost estimation with continuous cost approximation formulas extending Beardwood-Halton-Hammersley and Daganzo's work. These formulas commit errors that can be magnified during the optimization step. To reconcile speed and solution quality, we introduce a supervised learning and optimization methodology leveraging a graph neural network for delivery-cost estimation. This network is trained to imitate known costs generated on a limited subset of training districts. It is used within an iterated local search procedure to produce high-quality districting plans. Our computational experiments, conducted on five metropolitan areas in the United Kingdom, demonstrate that the graph neural network predicts long-term district cost operations more accurately, and that optimizing over this oracle permits large economic gains (10.12% on average) over baseline methods that use continuous approximation formulas or shallow neural networks. Finally, we observe that having compact districts alone does not guarantee high-quality solutions and that other learnable geometrical features of the districts play an essential role.

Deep Learning for Data-Driven Districting-and-Routing

TL;DR

This work tackles data-driven districting-and-routing under stochastic demand by learning district-cost estimates with a Graph Neural Network and integrating them into an Iterated Local Search to rapidly generate high-quality, connected district partitions. The GNN uses BU-level features and district membership indicators to predict long-term district costs, enabling scalable optimization that outperforms continuous-approximation baselines. Across five UK metropolitan areas, the approach achieves an average economic gain of over baselines and reveals that district geometry beyond simple compactness critically influences routing efficiency. The results support practical decision-support by delivering fast, accurate cost estimates, strong solution quality, and insights into how learnable geometric features shape effective districting.

Abstract

Districting-and-routing is a strategic problem aiming to aggregate basic geographical units (e.g., zip codes) into delivery districts. Its goal is to minimize the expected long-term routing cost of performing deliveries in each district separately. Solving this stochastic problem poses critical challenges since repeatedly evaluating routing costs on a set of scenarios while searching for optimal districts takes considerable time. Consequently, solution approaches usually replace the true cost estimation with continuous cost approximation formulas extending Beardwood-Halton-Hammersley and Daganzo's work. These formulas commit errors that can be magnified during the optimization step. To reconcile speed and solution quality, we introduce a supervised learning and optimization methodology leveraging a graph neural network for delivery-cost estimation. This network is trained to imitate known costs generated on a limited subset of training districts. It is used within an iterated local search procedure to produce high-quality districting plans. Our computational experiments, conducted on five metropolitan areas in the United Kingdom, demonstrate that the graph neural network predicts long-term district cost operations more accurately, and that optimizing over this oracle permits large economic gains (10.12% on average) over baseline methods that use continuous approximation formulas or shallow neural networks. Finally, we observe that having compact districts alone does not guarantee high-quality solutions and that other learnable geometrical features of the districts play an essential role.
Paper Structure (19 sections, 10 equations, 8 figures, 13 tables, 2 algorithms)

This paper contains 19 sections, 10 equations, 8 figures, 13 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Statement
  4. Graph Neural Network for District-Cost Estimations
  5. Features
  6. Architecture of the graph neural network
  7. Training
  8. Districting Solution Strategy
  9. Initial Solution
  10. Local Search
  11. Perturbation
  12. Experimental Analyses
  13. Data Collection and Experimental Setup
  14. Estimation Methods for Routing Costs
  15. Analysis -- Prediction Accuracy
  16. ...and 4 more sections

Figures (8)

  • Figure 1: Neural architecture dedicated to approximate delivery costs.
  • Figure 2: Local Search example
  • Figure 3: Visualization of the ground truth and estimated district costs on a subset of districts for Bristol and London (with a central depot).
  • Figure 4: Performance analysis of the different estimators for districts with varying compactness
  • Figure 5: Performance analysis of the different estimators for districts with varying elongation
  • ...and 3 more figures