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Location-coverage models for preventing attacks on interurban transportation networks

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Abstract

Interurban roads are constantly used by transient vehicles. In some places, however, network users are subject to attacks, resulting in assaults to drivers and cargo theft. In an attempt to solve this problem, a binary integer programming model is developed, whose objective is to maximize the expected vehicle coverage across the network. The model dynamically locates patrol units through a fixed time horizon, subject to movement and location constraints, considering a probability of not being able to attend to an attack, due to a distance factor. A chronological decomposition heuristic is developed, and achieves an optimality gap of less than 1 %, in less than 5 min. The problem is also solved by developing a geographical decomposition heuristic. To introduce a measure of equity, two sets of constraints are proposed. Three measures are considered: total vehicle coverage, inequity and network coverage. A trade-off between these three measures is observed and discussed. Finally, scalability of the model is explored, using decomposition in terms of patrolling units, until we obtain subproblems of equal size as the original instance. All of these features are applied to a case study in Northern Israel. In the last section, some adaptations and additions to the model that can be made in further research are discussed.

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Notes

  1. Several connections (usually highways) between cities/towns in a given region of a country.

  2. As mentioned in Sect. 1, this work address the problem of prevention/deterrence rather than response to attacks. Also, attacks may occur once at a time, and not simultaneously.

  3. The reason for this assumption is that allowing more than one patrolling unit per location node/traveling segment would require to include combinatorial nonlinear terms in the objective function, which would make the model formulation (and hence its tractability) too complex (see Sect. 8 for a more elaborated explanation)

  4. Along with distance, traffic volume between the attack site and unit k may increase. As a way to consider this, we choose f such that \(\frac{df}{dx}<0\) and \(\frac{d^2f}{dx^2}>0\), for \(0\le x < H\), such as the function selected in the formulation.

  5. As the example consists in single road, the vehicle flow for each subsegment is the same (this implies every \(\lambda _{jlt}\) has the same value), and because the situation represents a given time period t, the flow is also considered homogeneous across a given road (and hence the \(\lambda _{jlt}\) term may be taken out of the integral).

  6. This fraction is calculated over \(T-2\), as we do not consider the first and last time period, because each unit is forced to be at their police station (starting point) at that time.

  7. Not considering the time periods \(t=1\) and \(t=T\), because unit k does not exerts any protection during those time periods.

  8. Time period \(t=1\) and \(t=T\) are not considered, because at those time periods there is no protection exerted from the units.

  9. This is because the location of every unit at the beginning of any subproblem must be physically consistent with the location at the end of the previous subproblem.

  10. This is guaranteed because of the use of dummy nodes.

  11. This results are based on this particular case study. On other instances, the result may differ from these.

  12. This results are based on this particular case study. On other instances, the result may be different from these.

  13. This would still imply an increase in solving times, but a significantly more acceptable time than not using this procedure, being the latter case even unsolvable in some cases

  14. This results are based on this particular case study. Results of applying this procedure in other instances may differ.

  15. Modifications in region size should be subject of further research, in order to evaluate more precisely the behavior of the scaled model in such situations.

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Acknowledgments

The authors would like to thank the Associate Editor and anonymous referees for their comments on earlier versions of this paper. This proved to be of great help, specially for improving many explanations in this work.

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Authors and Affiliations

  1. Departamento de Ingenieria Industrial, Universidad Catolica del Norte, Av. Angamos 0610, Antofagasta, Chile

    Ramon Auad

  2. Department of Industrial and Systems Engineering, University at Buffalo (SUNY), 410 Bell Hall, Buffalo, NY, 14260, USA

    Rajan Batta

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  1. Ramon Auad
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Correspondence to Rajan Batta.

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Auad, R., Batta, R. Location-coverage models for preventing attacks on interurban transportation networks. Ann Oper Res 258, 679–717 (2017). https://doi.org/10.1007/s10479-015-2087-y

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