Production, Manufacturing, Transportation and LogisticsTime-dependent stochastic vehicle routing problem with random requests: Application to online police patrol management in Brussels
Graphical abstract
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 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 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
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2022, Applied GeographyCitation 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.