Neural Embedded Mixed-Integer Optimization for Location-Routing Problems

TL;DR

The paper tackles the CLRP by introducing NEO-LRP, a framework that embeds a neural surrogate hat{R}_d for the CVRP routing cost into a simple, off-the-shelf MIP to guide location-allocation decisions. The surrogate is trained offline on a large, independent CVRP dataset to avoid information leakage, and the final routes are computed post-hoc with a CVRP solver on induced subgraphs. Empirical results across four benchmark sets show competitive solution quality with substantial speedups (2x–120x) relative to state-of-the-art heuristics, especially on larger instances, with a one-time training cost around 6 hours. The approach emphasizes modularity, scalability, and practical applicability, and provides open-source code and data for reproducibility.

Abstract

We present a novel framework that combines machine learning with mixed-integer optimization to solve the Capacitated Location-Routing Problem (CLRP). The CLRP is a classical NP-hard problem that integrates strategic facility location with operational vehicle routing decisions, aiming to minimize the sum of fixed and variable costs. The proposed method trains a neural network to approximate the optimal cost of a Capacitated Vehicle Routing Problem (CVRP) for serving any subset of customers from a candidate facility. Crucially, the neural network is trained on an independently generated dataset of CVRP instances from the literature, entirely separate from any CLRP test instances, thereby avoiding the overfitting and information leakage that can affect learning-based methods. The trained network is then embedded as a surrogate within a mixed-integer optimization model for location-allocation decisions, which is solved using off-the-shelf solvers, thus leveraging decades of advances in vehicle routing and the availability of mature solvers. Computational experiments across four benchmark sets demonstrate competitive solution quality compared to best-known solutions while providing computational speedups of 2x to 120x over state-of-the-art heuristics. After a one-time training cost of only 6 hours, per-instance solve times range from under a second to under five minutes, even for the largest instances with 600 customers and 30 depots, where the method achieves a 1% median gap compared to over four hours for leading heuristics. Our results demonstrate the value of routing cost approximations from the neural surrogate in informing high-quality location-allocation decisions. Our code and data are publicly available.

Figures (10)

• Figure 1: An illustrative CLRP solution with 5 candidate depots and 9 customers. The solution opens depots 2, 3, and 4, while depots 1 and 5 remain closed. The customers allocated to each depot are as follows: $S_1(\hat{w}) = \emptyset$, $S_2(\hat{w}) = \{\text{a}, \text{b}\}$, $S_3(\hat{w}) = \{\text{c}, \text{d}, \text{i}\}$, $S_4(\hat{w}) = \{\text{e}, \text{f}, \text{g}, \text{h}\}$, and $S_5(\hat{w}) = \emptyset$. The vehicle routes are depicted as directed arrows between nodes.
• Figure 2: Overview of the NEO-LRP framework. Top: A pre-trained neural network processes node-level input features through a feature extractor ($\phi$) to produce embeddings, which are combined with the allocation variables $w$ and aggregated to form a depot-level representation $Z$; a regressor ($\rho$) then predicts the routing cost $\hat{R}_d$. The neural network architecture and notation are detailed in Section \ref{['sec:learning']}. Bottom: The regressor is embedded within a MIP that determines location-allocation decisions, which are then completed by an off-the-shelf VRP solver to obtain the final routes.
• Figure 3: Empirical cumulative distribution of the gap to the best-known solution ($E^{\text{gap}}_{\text{BKS}}$): (a) $\mathbb{P}$ benchmark of prins2004nouveaux, (b) $\mathbb{T}$ benchmark of tuzun1999two, (c) $\mathbb{B}$ benchmark of barreto_2004, and (d) $\mathbb{S}$ benchmark of schneider2019large. Higher and to the left is better.
• Figure 4: Empirical cumulative distribution of computational time ($T_{\text{total}}$): (a) $\mathbb{P}$ benchmark of prins2004nouveaux, (b) $\mathbb{T}$ benchmark of tuzun1999two, (c) $\mathbb{B}$ benchmark of barreto_2004, and (d) $\mathbb{S}$ benchmark of schneider2019large. Higher and to the left is better.
• Figure 5: Effect of problem size on the $\mathbb{S}$ benchmark set of schneider2019large. (a) displays the median $E^{\text{gap}}_{\text{BKS}}$, while (b) shows the median $T_{\text{total}}$, both as a function of problem size (number of customers). Shaded regions represent the interquartile range.