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A Stochastic Electric Vehicle Routing Problem under Uncertain Energy Consumption

Andrea Spinelli, Dario Bezzi, Ola Jabali, Francesca Maggioni

TL;DR

This paper addresses EV routing under energy uncertainty by formulating SEVRP-T, a two-stage stochastic programming problem with a threshold-based recourse policy that triggers detours to charging stations during arc traversal. It contributes a robust matheuristic (ILS-SP) that combines an Iterated Local Search route generator with a Set Partitioning assembly, augmented by a stochastic charging subproblem solver (SFRVCP-T) and lower bounds to prune moves. To tackle large scenario sets, it introduces an optimal-transport–based scenario-reduction technique (FFS) and demonstrates substantial computational advantages over exact solvers while maintaining solution quality. Comprehensive experiments on EVRP-benchmarked instances reveal the value of incorporating energy-uncertainty, quantify the impact of threshold parameters, and provide managerial guidance on planning EV-enabled logistics under uncertainty.

Abstract

The increasing adoption of Electric Vehicles (EVs) for service and goods distribution operations has led to the emergence of Electric Vehicle Routing Problems (EVRPs), a class of vehicle routing problems addressing the unique challenges posed by the limited driving range and recharging needs of EVs. While the majority of EVRP variants have considered deterministic energy consumption, this paper focuses on the Stochastic Electric Vehicle Routing Problem with a Threshold recourse policy (SEVRP-T), where the uncertainty in energy consumption is considered, and a recourse policy is employed to ensure that EVs recharge at Charging Stations (CSs) whenever their State of Charge (SoC) falls below a specified threshold. We formulate the SEVRP-T as a two-stage stochastic mixed-integer second-order cone model, where the first stage determines the sequences of customers to be visited, and the second stage incorporates charging activities. The objective is to minimize the expected total duration of the routes, composed by travel times and recharging operations. To cope with the computational complexity of the model, we propose a heuristic based on an Iterated Local Search (ILS) procedure coupled with a Set Partitioning problem. To further speed up the heuristic, we develop two lower bounds on the corresponding first-stage customer sequences. Furthermore, to handle a large number of energy consumption scenarios, we employ a scenario reduction technique. Extensive computational experiments are conducted to validate the effectiveness of the proposed solution strategy and to assess the importance of considering the stochastic nature of the energy consumption. The research presented in this paper contributes to the growing body of literature on EVRP and provides insights into managing the operational deployment of EVs in logistics activities under uncertainty.

A Stochastic Electric Vehicle Routing Problem under Uncertain Energy Consumption

TL;DR

This paper addresses EV routing under energy uncertainty by formulating SEVRP-T, a two-stage stochastic programming problem with a threshold-based recourse policy that triggers detours to charging stations during arc traversal. It contributes a robust matheuristic (ILS-SP) that combines an Iterated Local Search route generator with a Set Partitioning assembly, augmented by a stochastic charging subproblem solver (SFRVCP-T) and lower bounds to prune moves. To tackle large scenario sets, it introduces an optimal-transport–based scenario-reduction technique (FFS) and demonstrates substantial computational advantages over exact solvers while maintaining solution quality. Comprehensive experiments on EVRP-benchmarked instances reveal the value of incorporating energy-uncertainty, quantify the impact of threshold parameters, and provide managerial guidance on planning EV-enabled logistics under uncertainty.

Abstract

The increasing adoption of Electric Vehicles (EVs) for service and goods distribution operations has led to the emergence of Electric Vehicle Routing Problems (EVRPs), a class of vehicle routing problems addressing the unique challenges posed by the limited driving range and recharging needs of EVs. While the majority of EVRP variants have considered deterministic energy consumption, this paper focuses on the Stochastic Electric Vehicle Routing Problem with a Threshold recourse policy (SEVRP-T), where the uncertainty in energy consumption is considered, and a recourse policy is employed to ensure that EVs recharge at Charging Stations (CSs) whenever their State of Charge (SoC) falls below a specified threshold. We formulate the SEVRP-T as a two-stage stochastic mixed-integer second-order cone model, where the first stage determines the sequences of customers to be visited, and the second stage incorporates charging activities. The objective is to minimize the expected total duration of the routes, composed by travel times and recharging operations. To cope with the computational complexity of the model, we propose a heuristic based on an Iterated Local Search (ILS) procedure coupled with a Set Partitioning problem. To further speed up the heuristic, we develop two lower bounds on the corresponding first-stage customer sequences. Furthermore, to handle a large number of energy consumption scenarios, we employ a scenario reduction technique. Extensive computational experiments are conducted to validate the effectiveness of the proposed solution strategy and to assess the importance of considering the stochastic nature of the energy consumption. The research presented in this paper contributes to the growing body of literature on EVRP and provides insights into managing the operational deployment of EVs in logistics activities under uncertainty.

Paper Structure

This paper contains 31 sections, 3 theorems, 22 equations, 6 figures, 31 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Problem description and formulation
  4. Observation
  5. Solution strategy
  6. Route generator and route assembler
  7. Perturbation phase
  8. Variable Neighborhood Descent search phase
  9. The stochastic fixed route vehicle charging problem with a threshold policy
  10. Set Partitioning
  11. A scenario reduction technique for the SEVRP-T
  12. Computational results
  13. Instances and parameters
  14. Scenario generation
  15. Performance of the ILS-SP heuristic in the deterministic setting
  16. ...and 16 more sections

Key Result

Proposition 1

Let $i\in\mathcal{I}^+$, $j\in\mathcal{I}$ and $k_1,k_2 \in \mathcal{K}$. Suppose that CSs $k_1,k_2$ have the same technology, i.e. $\mathcal{B}_{k_1}=\mathcal{B}_{k_2}=\mathcal{B}=\{0,1,2\}$ and $c_{k_1b}=c_{k_2b}=c_b$, $a_{k_1b}=a_{k_2b}=a_b$ for all $b\in\mathcal{B}$. The corresponding piecewise with $\beta^{1}=a_1/c_1\geq 1$ and $\beta^{2}=(Q^{max}-a_1)/(c_2-c_1)<\beta^{1}$. Assume that when

Figures (6)

  • Figure 1: An example with three customers $(1,2,3)$ and two CSs $(k_1,k_2)$. Left panel: the first-stage route $\{0,1,2,3,0\}$ is determined. Middle panel: after the realization of the uncertainty under a certain scenario, the threshold recourse policy is implemented, prescribing a stop at CS $k_2$ between customer nodes 2 and 3 to recharge the battery. Right panel: SoC when implementing the threshold recourse policy in the second stage.
  • Figure 2: Example in which the chosen CS is not the closest one to the detour point $f$. Left panel: graph with relevant nodes and distances. Customers are represented as circles, while CSs are squares. Right panel: charging function of the CSs $k_1,k_2$.
  • Figure 3: Relevant distances between nodes and CSs in the computation of the lower bound \ref{['eq:lowerboundlengthdetour']}. Left panel: the case with $\underline{k}^i \equiv \underline{k}^j$. Right panel: the case with $\underline{k}^i \not\equiv \underline{k}^j$.
  • Figure 4: Example of the FFS algorithm application. Left panel: the original scenario tree with 50 scenarios and their corresponding probabilities. The 10 selected scenarios are highlighted in black. Right panel: the reduced scenario tree, consisting of the 10 scenarios selected by the FFS algorithm, along with their reassigned probabilities.
  • Figure 5: Approximate in-sample stability analysis for different energy consumption probability distributions.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1: HeiRom2003 and DupGro-KusRom2003