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Contextual Stochastic Vehicle Routing with Time Windows

Breno Serrano, Alexandre M. Florio, Stefan Minner, Maximilian Schiffer, Thibaut Vidal

TL;DR

The paper extends VRPTW to settings with stochastic travel times conditioned on contextual features, formulating a contextual stochastic VRPTW and proposing data-driven prescriptive models to approximate the conditional objective. It develops point-based, sample-average-based, and penalty-based approaches, including specialized branch-price-and-cut solvers with completion bounds, and evaluates them on Solomon-based instances up to 100 customers. Computational results show that feature-dependent sample-average approximations frequently yield the best out-of-sample performance, closely approaching full-information benchmarks. The work demonstrates tangible gains from incorporating contextual information into stochastic routing and outlines scalable exact solution methods and future research directions in contextual optimization for logistics.

Abstract

We study the vehicle routing problem with time windows (VRPTW) and stochastic travel times, in which the decision-maker observes related contextual information, represented as feature variables, before making routing decisions. Despite the extensive literature on stochastic VRPs, the integration of feature variables has received limited attention in this context. We introduce the contextual stochastic VRPTW, which minimizes the total transportation cost and expected late arrival penalties conditioned on the observed features. Since the joint distribution of travel times and features is unknown, we present novel data-driven prescriptive models that use historical data to provide an approximate solution to the problem. We distinguish the prescriptive models between point-based approximation, sample average approximation, and penalty-based approximation, each taking a different perspective on dealing with stochastic travel times and features. We develop specialized branch-price-and-cut algorithms to solve these data-driven prescriptive models. In our computational experiments, we compare the out-of-sample cost performance of different methods on instances with up to one hundred customers. Our results show that, surprisingly, a feature-dependent sample average approximation outperforms existing and novel methods in most settings.

Contextual Stochastic Vehicle Routing with Time Windows

TL;DR

The paper extends VRPTW to settings with stochastic travel times conditioned on contextual features, formulating a contextual stochastic VRPTW and proposing data-driven prescriptive models to approximate the conditional objective. It develops point-based, sample-average-based, and penalty-based approaches, including specialized branch-price-and-cut solvers with completion bounds, and evaluates them on Solomon-based instances up to 100 customers. Computational results show that feature-dependent sample-average approximations frequently yield the best out-of-sample performance, closely approaching full-information benchmarks. The work demonstrates tangible gains from incorporating contextual information into stochastic routing and outlines scalable exact solution methods and future research directions in contextual optimization for logistics.

Abstract

We study the vehicle routing problem with time windows (VRPTW) and stochastic travel times, in which the decision-maker observes related contextual information, represented as feature variables, before making routing decisions. Despite the extensive literature on stochastic VRPs, the integration of feature variables has received limited attention in this context. We introduce the contextual stochastic VRPTW, which minimizes the total transportation cost and expected late arrival penalties conditioned on the observed features. Since the joint distribution of travel times and features is unknown, we present novel data-driven prescriptive models that use historical data to provide an approximate solution to the problem. We distinguish the prescriptive models between point-based approximation, sample average approximation, and penalty-based approximation, each taking a different perspective on dealing with stochastic travel times and features. We develop specialized branch-price-and-cut algorithms to solve these data-driven prescriptive models. In our computational experiments, we compare the out-of-sample cost performance of different methods on instances with up to one hundred customers. Our results show that, surprisingly, a feature-dependent sample average approximation outperforms existing and novel methods in most settings.
Paper Structure (30 sections, 2 theorems, 53 equations, 6 figures, 13 tables, 1 algorithm)

This paper contains 30 sections, 2 theorems, 53 equations, 6 figures, 13 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. The Contextual Stochastic VRPTW
  5. Data-driven Prescriptive Models
  6. Point-based Approximation
  7. Sample Average Approximation
  8. Conditional SAA
  9. Residual-based SAA
  10. Late Arrival Penalty Approximation
  11. Solution Methods for the Data-driven Prescriptive Models
  12. Column Generation
  13. Pricing Algorithm
  14. Column Generation for Penalty-based Approximation
  15. Design of Experiments
  16. ...and 15 more sections

Key Result

Proposition 1

Let $\theta$ be a route starting at the depot and ending at customer $i$ with reduced cost $\overline{C}_{\theta}$ and cumulative demand $q_{\theta}$. Let $L_{\theta'}$ be a label extension associated with the route $\theta'=\theta \oplus\mathcal{E}$, where $\mathcal{E}=(u_1, u_2, \dots, u_L)$ is a

Figures (6)

  • Figure 1: Joint distribution and different realizations of travel times and feature variable.
  • Figure 2: Data set and solution structures of our illustrative example.
  • Figure 3: Distribution of test costs for instances with 25 customers from different generative models.
  • Figure 4: Average first-stage and second-stage test costs for instances from the exponential generative model.
  • Figure 5: Solution times of different methods for instances with 25 customers with a linear generative model.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2