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The stochastic interdiction median problem with disruption intensity levels

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Abstract

In this paper we introduce a stochastic interdiction problem for median systems in which the operational state of the system’s disrupted elements in the aftermath of the disruption is uncertain as it is based on the intensity of the disruption. We assume that a disruption disables a facility with a given probability and this probability depends on the intensity of the disruption. The objective of this problem is to identify which disruption scenario entails a maximum overall traveling distance in serving all customers. We show that the initial two stage stochastic formulation can be reformulated into a deterministic counterpart whose size is polynomial in the number of facilities and intensity levels. Then, our ensuing efforts to solve the problem efficiently focus on studying alternative deterministic formulations that allow the solution of realistic size instances of the model. We observe that the most efficient of the deterministic formulations provide great scalability with respect to variations in the input parameters and size of the instances solved. Finally, we analyze the robustness of the optimal solutions due to misestimations in the probability functions that relate disruption intensity levels with the probabilities of facility survivability.

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References

  • Aksen, D., & Aras, N. (2012). A bilevel fixed charge location model for facilities under imminent attack. Computers & Operations Research, 39, 1364–1381.

    Article  Google Scholar 

  • Aksen, D., Piyade, N., & Aras, N. (2010). The budget constrained r-interdiction median problem with capacity expansion. Central European Journal of Operations Research, 18, 269–291.

    Article  Google Scholar 

  • Aksen, D., Aras, N., & Piyade, N. (2011). A bilevel p-median model for the planning and protection of critical facilities. Journal of Heuristics. doi:10.1007/s10732-011-9163-5.

    Google Scholar 

  • Arroyo, J. M., & Galiana, F. D. (2005). On the solution of the bilevel programming formulation of the terrorist threat problem. IEEE Transactions on Power Systems, 20, 789–797.

    Article  Google Scholar 

  • Bayrak, H., & Bailey, M. D. (2008). Shortest path network interdiction with asymmetric information. Networks, 52, 133–140.

    Article  Google Scholar 

  • Berman, O., Drezner, T., Drezner, Z., & Wesolowsky, G. O. (2009). A defensive maximal covering problem on a network. International Transactions in Operational Research, 16, 69–86.

    Article  Google Scholar 

  • Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.

    Google Scholar 

  • Boros, E., Borys, K., Gurvich, V., & Rudolf, G. (2006). Inapproximability bounds for shortest-path network interdiction problems (DIMACS Technical Report 13).

  • Brooks, A., Bennett, B., & Bankes, B. (1999). An application of exploratory analysis: the weapon mix problem. Military Operations Research, 4, 67–80.

    Article  Google Scholar 

  • Brown, G., Carlyle, M., Salmeron, J., & Wood, K. (2006). Defending critical infrastructure. Interfaces, 36(6), 530–544.

    Article  Google Scholar 

  • Brown, G. G., Carlyle, W. M., Harney, R. C., Skroch, E. M., & Wood, R. K. (2009). Interdicting a nuclear-weapons project. Operations Research, 57(4), 866–877.

    Article  Google Scholar 

  • Cappanera, P., & Scaparra, P. M. (2011). Optimal allocation of protective resources in shortest-path networks. Transportation Science, 45, 64–80.

    Article  Google Scholar 

  • Carr, R. D., Greenberg, H. J., Hart, W. E., Konjevod, G., Lauer, E., Lin, H., Morrison, T., & Phillips, C. A. (2005). Robust optimization of contaminant sensor placement for community water systems. Mathematical Programming, 107, 337–356.

    Article  Google Scholar 

  • Church, R. L., & Scaparra, M. P. (2007). Analysis of facility systems’ reliability when subject to attack or a natural disaster. In: Reliability and vulnerability in critical infrastructure: a quantitative geographic perspective. Berlin: Springer.

    Google Scholar 

  • Church, R. L., Scaparra, M. P., & Middleton, R. S. (2004). Identifying critical infrastructure: the median and covering facility interdiction problems. Annals of the Association of the American Geographers, 94(3), 491–502.

    Article  Google Scholar 

  • Cormican, K. J., Morton, D. P., & Wood, R. K. (1998). Stochastic network interdiction. Operations Research, 46(2), 184–197.

    Article  Google Scholar 

  • Cui, T., Ouyang, Y., & Shen, Z. M. (2010). Reliable facility location design under the risk of disruptions. Operations Research, 58(4), 998–1011.

    Article  Google Scholar 

  • Dempe, S. (2002). Foundations of bilevel programming. The Netherlands: Kluwer Academic.

    Google Scholar 

  • Dimitrov, N. B., Michalopoulos, D. P., Morton, D. P., Nehme, M. V., Pan, F., Popova, E., Schneider, E. A., & Thoreson, G. G. (2011). Network deployment of radiation detectors with physics-based detection probability calculations. Annals of Operations Research, 187(1), 207–228.

    Article  Google Scholar 

  • Frauendorfer, K. (1992). Lecture notes in economics and mathematical systems: Vol. 392. Stochastic two-stage programming. Berlin: Springer.

    Book  Google Scholar 

  • Fulkerson, D. R., & Harding, G. C. (1977). Maximizing the minimum source-sink path subject to a budget constraint. Mathematical Programming, 13, 116–118.

    Article  Google Scholar 

  • Goel, V., & Grossmann, I. E. (2006). A class of stochastic programs with decision dependent uncertainty. Mathematical Programming, 108(2–3), 355–394.

    Article  Google Scholar 

  • Golden, B. (1978). A problem of network interdiction. Naval Research Logistics Quarterly, 25, 711–713.

    Article  Google Scholar 

  • Hakimi, S. L. (1964). Optimum location of switching centers and the absolute centers and medians of a graph. Operations Research, 113, 544–559.

    Google Scholar 

  • Held, H., & Woodruff, D. L. (2005). Heuristics for multi-stage interdiction of stochastic networks. Journal of Heuristics, 11, 483–500.

    Article  Google Scholar 

  • Held, H., Hemmecke, R., & Woodruff, D. L. (2005). A decomposition algorithm applied to planning the interdiction of stochastic networks. Naval Research Logistics, 52, 321–328.

    Article  Google Scholar 

  • Hemmecke, R., Schultz, R., & Woodruff, D. L. (2003). Interdiction stochastic networks with binary interdiction effort. In D. L. Woodruff (Ed.), Network interdiction and stochastic integer programming (pp. 69–84). Boston: Kluwer.

    Chapter  Google Scholar 

  • Higle, J. L., & Sen, S. (1996). Stochastic decomposition: a statistical method for large scale stochastic linear programming. Norwell: Kluwer Academic.

    Google Scholar 

  • Holmgren, A. J., Jenelius, E., & Westin, J. (2007). Evaluating strategies for defending electric power networks against antagonistic attacks. IEEE Transactions on Power Systems, 22(1), 76–84.

    Article  Google Scholar 

  • Israeli, E., & Wood, R. K. (2002). Shortest-path network interdiction. Networks, 40(2), 97–111.

    Article  Google Scholar 

  • Janjarassuk, U., & Linderoth, J. (2008). Reformulation and sampling to solve a stochastic network interdiction problem. Networks, 52, 120–132.

    Article  Google Scholar 

  • Kall, P., Ruszczyński, A., & Frauendorfer, K. (1988). Approximation techniques in stochastic programming. In Y. Ermoliev & R. J.-B. Wets (Eds.), Numerical techniques for stochastic optimization, Berlin: Springer.

    Google Scholar 

  • Khachiyan, I., Boros, E., Borys, K., Elbassioni, K., Gurvich, V., Rudolf, G., & Zhao, J. (2008). On shortest paths interdiction problems: total and node-wise limited interdiction. Theory of Computing Systems, 43, 204–233.

    Article  Google Scholar 

  • Liberatore, F., & Scaparra, M. P. (2011). Optimizing protection strategies for supply chains: comparing classic decision making criteria in an uncertain environment. Annals of the Association of American Geographers, 101, 1–17.

    Article  Google Scholar 

  • Liberatore, F., Scaparra, M. P., & Daskin, M. S. (2011). Analysis of facility protection strategies against an uncertain number of attacks: the stochastic r-interdiction median problem with fortification. Computers & Operations Research, 38(1), 357–366.

    Article  Google Scholar 

  • Liberatore, F., Scaparra, M. P., & Daskin, M. (2012). Optimization methods for hedging against disruptions with ripple effects in location analysis. Omega, 40(1), 21–30.

    Article  Google Scholar 

  • Lim, C., & Smith, L. C. (2007). Algorithms for discrete and continuous multicommodity flow network interdiction problems. IIE Transactions, 39, 15–26.

    Article  Google Scholar 

  • Losada, C., Scaparra, M. P., & Church, R. L. (2009). Interdiction of p-median systems with facility recovery time and disruptions frequency: analysis of resiliency (Working Paper No. 220). University of Kent, Kent Business School, Canterbury, Kent, UK.

  • Losada, C., Scaparra, M. P., & O’Hanley, J. R. (2012). Optimizing system resilience: a facility protection model with recovery time. European Journal of Operational Research, 217, 519–530.

    Article  Google Scholar 

  • Lulli, G., & Sen, S. (2004). A branch-and-price algorithm for multistage stochastic integer programming with application to stochastic batch-sizing problems. Management Science, 50(6), 786–796.

    Article  Google Scholar 

  • Lulli, G., Messina, E., Archetti, F., & Lanzeni, S. (2010). A mathematical model for optimal functional disruption of biochemical networks. Journal of Mathematical Modelling and Algorithms, 9(1), 19–37.

    Article  Google Scholar 

  • Matisziw, T. C., Murray, A. T., & Grubesic, T. H. (2010). Strategic network restoration. Networks and Spatial Economics, 10, 345–361.

    Article  Google Scholar 

  • McMasters, A., & Mustin, T. M. (1970). Optimal interdiction of a supply network. Naval Research Logistics Quarterly, 17(3), 261–268.

    Article  Google Scholar 

  • Moore, J. T., & Bard, J. F. (1990). The mixed integer linear bilevel programming problem. Operations Research, 38, 911–921.

    Article  Google Scholar 

  • Morton, D. P., Pan, F., & Saeger, K. J. (2007). Models for nuclear smuggling interdiction. IIE Transactions, 39, 3–14.

    Article  Google Scholar 

  • Murray, A. T., & Grubesic, T. H. (2007). Critical infrastructure: reliability and vulnerability. Berlin: Springer.

    Google Scholar 

  • O’Hanley, J., & Church, R. L. (2011). Designing robust coverage to hedge against worst-case facility losses. European Journal of Operational Research, 209, 23–36.

    Article  Google Scholar 

  • Pan, F., Charlton, W., & Morton, D. (2003). A stochastic program for interdicting smuggled nuclear material (pp. 1–20). Dordrecht: Kluwer Academic.

    Google Scholar 

  • Royset, J. O., & Wood, R. K. (2007). Solving the bi-objective maximum-flow network-interdiction problem. INFORMS Journal on Computing, 19(2), 175–184.

    Article  Google Scholar 

  • Salmeron, J., Wood, R. K., & Baldick, R. (2004). Analysis of electric grid security under terrorist threat. IEEE Transactions on Power Systems, 19(2), 905–912.

    Article  Google Scholar 

  • Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochastic programming approach for supply network design under uncertainty. European Journal of Operational Research, 167, 96–115.

    Article  Google Scholar 

  • Scaparra, M. P. (2009). Hardening facilities to minimize expected costs given random failures (Working paper, KBS Working Paper No. 185). University of Kent.

  • Scaparra, M. P., & Church, R. L. (2008a). A bilevel mixed integer program for critical infrastructure protection planning. Computers & Operations Research, 35, 1905–1923.

    Article  Google Scholar 

  • Scaparra, M. P., & Church, R. L. (2008b). An exact solution approach for the interdiction median problem with fortification. European Journal of Operational Research, 189, 76–92.

    Article  Google Scholar 

  • Silva, E. F., & Wood, R. K. (2006). Solving a class of stochastic mixed-integer programs with branch and price. Mathematical Programming, 108(2), 395–418.

    Article  Google Scholar 

  • Singh, K., Philpott, A., & Wood, K. (2005). Column-generation for design of survivable networks (Working Paper). Department of Engineering Science, University of Auckland, Auckland, New Zealand.

  • Stackelberg, H. (1952). The theory of market economy. London: Oxford University Press.

    Google Scholar 

  • Van Slyke, R. M., & Wets, R. J. B. (1969). L-shaped programs with applications to optimal control and stochastic linear programming. SIAM Journal on Applied Mathematics, 17, 638–663.

    Article  Google Scholar 

  • Waters, N. M. (1977). Methodology for servicing the geography of urban fire: an exploration with special reference to London, Ontario. Unpublished Ph.D. thesis. Department of Geography, University of Western Ontario, London, Ontario, Canada.

  • Wets, R. J.-B. (1974). Stochastic programs with fixed recourse: the equivalent deterministic program. SIAM Review, 16, 309–339.

    Article  Google Scholar 

  • Whiteman, P. S. (1999). Improving single strike effectiveness for network interdiction. Military Operations Research, 4, 15–30.

    Article  Google Scholar 

  • Wollmer, R. (1964). Removing arcs from a network. Operations Research, 12(6), 934–940.

    Article  Google Scholar 

  • Wood, R. K. (1993). Deterministic network interdiction. Mathematical and Computer Modelling, 17(2), 1–18.

    Article  Google Scholar 

  • Yates, J. T., Batta, R., & Karwan, M. (2011). Optimal placement of sensors and interception resource assessment for the protection of regional infrastructure from covert attack. Journal of Transportation Security, 4, 145–169.

    Article  Google Scholar 

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Acknowledgements

This research was supported by the Engineering and Physical Sciences Research Council (EPSRC) Grant EP/E048552/1. This support is gratefully acknowledged.

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Authors and Affiliations

  1. Kent Business School, University of Kent, Canterbury, UK

    Chaya Losada & M. Paola Scaparra

  2. Geography Department, University of California, Santa Barbara, USA

    Richard L. Church

  3. Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, USA

    Mark S. Daskin

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  1. Chaya Losada
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  2. M. Paola Scaparra
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  4. Mark S. Daskin
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Correspondence to M. Paola Scaparra.

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Losada, C., Scaparra, M.P., Church, R.L. et al. The stochastic interdiction median problem with disruption intensity levels. Ann Oper Res 201, 345–365 (2012). https://doi.org/10.1007/s10479-012-1170-x

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