An Optimal Modeling Approach for the Interdiction Median Problem with Fortification

Systematic approaches to security investment decisions, from intelligence and surveillance to fortification, are crucial for improved homeland security. We present an optimization modeling approach for allocating protection resources among a system of facilities so that the disruptive effects of possible intentional attacks to the system are minimized. This paper is based upon the p-median service protocol for an operating set of p-facilities. The primary objective is to identify the subset of q facilities which, when fortified, provides the best protection against the worst-case loss of r non-fortified facilities. This problem, known as the r-interdiction median problem with fortification (IMF), was first formulated as a mixed integer program by Church and Scaparra [6]. In this paper, we reformulate the IMF as a maximal covering problem with precedence constraints, which is amenable to a new solution approach. This new approach produces good approximations to the best fortification strategies. Furthermore, it provides upper and lower bounds that can be used to reduce the size of the original model. The reduced model can readily be solved to optimality by CPLEX. Computational results on two geographical data sets with different structural characteristics show the effectiveness of the proposed methodology for solving IMF instances of considerable size.

[1]  B. Golden A problem in network interdiction , 1978 .

[2]  J. Salmeron,et al.  Analysis of electric grid security under terrorist threat , 2004, IEEE Transactions on Power Systems.

[3]  Richard L. Church,et al.  COBRA: A New Formulation of the Classic p-Median Location Problem , 2003, Ann. Oper. Res..

[4]  Zvi Drezner,et al.  An Efficient Genetic Algorithm for the p-Median Problem , 2003, Ann. Oper. Res..

[5]  Richard D. Wollmer,et al.  Removing Arcs from a Network , 1964 .

[6]  Richard L. Church,et al.  Identifying Critical Infrastructure: The Median and Covering Facility Interdiction Problems , 2004 .

[7]  R. Kevin Wood,et al.  Shortest‐path network interdiction , 2002, Networks.

[8]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[9]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[10]  Delbert Ray Fulkerson,et al.  Maximizing the minimum source-sink path subject to a budget constraint , 1977, Math. Program..

[11]  Alan W. McMasters,et al.  Optimal interdiction of a supply network , 1970 .

[12]  Matteo Fischetti,et al.  Algorithms for the Set Covering Problem , 2000, Ann. Oper. Res..

[13]  R. Kevin Wood,et al.  Deterministic network interdiction , 1993 .

[14]  W. C. Turner,et al.  Optimal interdiction policy for a flow network , 1971 .

[15]  Pierre Hansen,et al.  New Branch-and-Bound Rules for Linear Bilevel Programming , 1989, SIAM J. Sci. Comput..

[16]  Jonathan F. Bard,et al.  The Mixed Integer Linear Bilevel Programming Problem , 1990, Oper. Res..

[17]  Richard L. Church,et al.  Protecting Critical Assets: The r-interdiction median problem with fortification , 2007 .