Production, Manufacturing, Transportation and Logistics
Time-dependent stochastic vehicle routing problem with random requests: Application to online police patrol management in Brussels

https://doi.org/10.1016/j.ejor.2020.11.007Get rights and content

Highlights

  • •

    Stochastic Vehicle Routing Problem with random Requests and Time-Dependent travel durations.

  • •

    A priori optimization: recourse strategy and two-stage stochastic program.

  • •

    Closed-form exact computation of the second-stage expected cost.

  • •

    Real world police patrol case study in urban area.

Abstract

The Static and Stochastic Vehicle Routing Problem with Random Requests (SS-VRP-R) describes realistic operational contexts in which a fleet of vehicles has to serve customer requests appearing dynamically. Based on a probabilistic knowledge about the appearance of requests, the SS-VRP-R seeks a priori sequences of vehicle relocations, optimizing the expected responsiveness to the requests. In this paper, an existing computational framework, based on recourse strategies, is adapted to meet the objectives of the SS-VRP-R. The resulting models are applied to a real case study of the management of police units in Brussels. In this context, the expected average response time is minimized. To cope with the reality of the urban context, a time-dependent variant is also studied (TD-SS-VRP-R) in which the travel time between two locations is a function that depends on the departure time at the first location. Experiments confirm the contribution and the adaptability of the recourse strategies to a real-life, complex operational context. Provided an adequate solution method, simulation-based results show the high quality of the a priori solutions designed, even when compared to those designed by field experts. Finally, the experiments provide evidence that there is no potential gain in considering time-dependency in such an operational context.

Section snippets

Introduction and background

The Vehicle Routing Problem (VRP) and its variants have become increasingly popular in the academic literature. The VRP consists of determining an optimal set of routes to be carried out by a fleet of vehicles to perform a set of customer requests (Toth & Vigo, 2014) at a minimum distance or some measure more or less functionally related to distance (e.g., travel time or cost). Whereas deterministic VRPs assume that input data are known with certainty, in real-world applications, some input

Problem definition and methodology

The SS-VRP-R tackled in this article is to serve a set of online requests with a given set of available vehicles while minimizing the expected average response time (or response time). For this purpose, we develop an algorithm to generate a first-stage solution based on some probabilistic knowledge about the missing data. Moreover, a recourse strategy is provided to adapt the first-stage solution to the requests revealed dynamically. The recourse strategy aims at describing which vehicle

Recourse strategy and expected cost computation

Given a first-stage solution x, a recourse strategy states how the requests, which appear dynamically, are handled by the vehicles. In other words, it defines how the second-stage solution is gradually constructed, based on x and depending on these online requests. A more formal description of recourse strategies is provided in Saint-Guillain et al. (2017).

Ideally, whenever a request appears, the right vehicle should be selected to optimize the operational performances. Furthermore, if several

Case study: Brussels police department

Our methodology developed to tackle the SS-VRP-R is applied to the case study faced by a specific team of mobile police units in Brussels in charge of urgent interventions only. We propose the best relocations strategies for every unit to minimize the average expected response time.

In this section, we describe how we derived the problem input data (graph nodes, travel durations, and request probabilities) for the historical data provided by the police department. The historical data are

Experiments and results

Our experiments aim at answering the following questions. Under a two-stage assumption, that is, when excluding online reoptimization, is it possible to identify first-stage solutions that beat a simple, intuitive operational policy? Also, what is the impact of the time-dependent travel durations on the quality of the first-stage decisions?

Based on the stochastic knowledge that we were able to extract from our historical records, the question is to determine whether the SS-VRP-R model is useful

Conclusions

In this paper, we described the SS-VRP-R as a practical modeling framework for the real-world problem of the management of police units in Brussels. Relying on the intervention requests observed from 2013 to 2016, we showed how to minimize the expected average response time. Experiments are conducted while considering 2017’s observations as a validation benchmark.

Coupled with a simple recourse strategy, the first-stage solutions obtained under the SS-VRP-R framework reduce the average response

Acknowledgments

Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11, and by the Walloon Region (Belgium). We thank Christine Solnon for the useful remarks on the first stage of this study, as well as the anonymous reviewers for their valuable comments and suggestions. Finally, such a realistic case study would not have been possible without the enthusiastic

References (39)

  • J. Naoum-Sawaya et al.

    A stochastic optimization model for real-time ambulance redeployment

    Computers & Operations Research

    (2013)
  • V. Pillac et al.

    A review of dynamic vehicle routing problems

    European Journal of Operational Research

    (2013)
  • J.-Y. Potvin et al.

    Vehicle routing and scheduling with dynamic travel times

    Computers & Operations Research

    (2006)
  • V. Schmid

    Solving the dynamic ambulance relocation and dispatching problem using approximate dynamic programming

    European Journal of Operational Research

    (2012)
  • B.-H. Ahn et al.

    Vehicle-routeing with time windows and time-varying congestion

    Journal of the Operational Research Society

    (1991)
  • R.W. Bent et al.

    Waiting and relocation strategies in online stochastic vehicle routing.

    Proceedings of the IJCAI

    (2007)
  • D.J. Bertsimas

    A vehicle routing problem with stochastic demand

    Operations Research

    (1992)
  • N.L. Boland et al.

    Perspectives on integer programming for time-dependent models

    Top

    (2019)
  • A.A. Braga et al.

    Hot spots policing and crime reduction: An update of an ongoing systematic review and meta-analysis

    Journal of Experimental Criminology

    (2019)

    • Cited by (17)

    • Heterogeneous instant delivery orders scheduling and routing problem

      2023, Computers and Operations Research

    • A GV-drone arc routing approach for urban traffic patrol by coordinating a ground vehicle and multiple drones

      2023, Swarm and Evolutionary Computation

    • Efficient feasibility checks and an adaptive large neighborhood search algorithm for the time-dependent green vehicle routing problem with time windows

      2023, European Journal of Operational Research

    • Hot spots and burning times: A spatiotemporal analysis of calls for service to establish police demand

      2022, Applied Geography
      Citation Excerpt :

      In other words, they do not take into account the priority of the CFS themselves and the impact on the available resources. Quite often only the most urgent CFS are considered in a patrol strategy (e.g., Saint-Guillain et al., 2021), which does not allow for a correct estimation of the daily demand for police (i.e., underestimation of the actual demand for police). As we demonstrate in this paper, the urgency of a CFS has repercussions on the spatial and temporal distribution as well as on the operation of the police.

    • A Decision Support System for Predictive Crime Analytics and a Patrol System

      2024, SSRN

    • A Joint Optimization Algorithm for Trajectory Planning and Resource Allocation of Vehicle Mobile Base Stations for On-Demand Coverage Networks

      2024, Processes
    View all citing articles on Scopus
    View full text
    © 2020 Elsevier B.V. All rights reserved.