Annals of Operations Research

, Volume 196, Issue 1, pp 411–442 | Cite as

Network interdiction to minimize the maximum probability of evasion with synergy between applied resources

Article

Abstract

In this paper, we model and solve the network interdiction problem of minimizing the maximum probability of evasion by an entity traversing a network from a given source to a designated terminus, while incorporating novel forms of superadditive synergy between resources applied to arcs in the network. Inspired primarily by operations to coordinate Iraqi and U.S. security forces seeking to interdict an evader attempting to avoid detection while transiting part of the nearly rectilinear street network in East Baghdad, this study motivates and examines either linear or concave (nonlinear) synergy relationships between the applied resources within our formulations. We also propose an alternative model for sequential overt and covert deployment of subsets of interdiction resources, and conduct theoretical as well as empirical comparative analyses between models for purely overt (with or without synergy) and composite overt-covert strategies to provide insights into absolute and relative threshold criteria for recommended resource utilization. Our empirical results confirm the value of tactical patience regarding decisions on the covert utilization of resources for network interdiction. Furthermore, considering non-integral and integral resource allocations, we identify (theoretically and empirically) parametric characteristics of instances that exhibit the relative worth of employing partially covert operations. Under the relatively more practical scenario involving integral resource allocations, we demonstrate that the composite overt-covert strategy of deploying resources has a greater potential to improve over a purely overt resource deployment strategy, both with and without synergy, particularly when costs are positively correlated, resources are plentiful, and a sufficiently high ratio of covert to overt resources exists. Moreover, should an interdictor be able to ascertain an optimal evader path, the potential and magnitude of this relative improvement for the overt-covert resource allocation strategy is significantly greater.

Keywords

Resource allocation Minimax flow problems Synergy Network evasion Network interdiction Overt and covert strategies 

References

  1. Bailey, M. D., Shechter, S. M., & Schaefer, A. J. (2006). SPAR: stochastic programming with adversarial recourse. Operations Research Letters, 34(3), 307–315. CrossRefGoogle Scholar
  2. Bayrak, H., & Bailey, M. (2008). Shortest path network interdiction with asymmetric information. Networks, 52(3), 133–140. CrossRefGoogle Scholar
  3. Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2006). Nonlinear programming: theory and algorithms (3rd ed.). Hoboken: Wiley. CrossRefGoogle Scholar
  4. Brown, S. S. (1980). Optimal search for a moving target in discrete time and space. Operations Research, 28(6), 1275–1289. CrossRefGoogle Scholar
  5. Brown, G., Carlyle, M., Salmerón, J., & Wood, R. K. (2006). Defending critical infrastructure. Interfaces, 36(6), 530–544. CrossRefGoogle Scholar
  6. Brown, G. G., Harney, R. C., Skroch, E. M., & Wood, R. K. (2009). Interdicting a nuclear-weapons project. Operations Research, 57(4), 866–877. CrossRefGoogle Scholar
  7. Cormican, K. J. (1995). Computational methods for deterministic and stochastic network interdiction problems. Master’s Thesis, US Naval Postgraduate School, Monterey, CA. Google Scholar
  8. Cormican, K. J., Morton, D. P., & Wood, R. K. (1998). Stochastic network interdiction. Operations Research, 46(2), 184–197. CrossRefGoogle Scholar
  9. Dempe, S. (2002). Foundations of bilevel programming. Dordrecht: Kluwer Academic. Google Scholar
  10. Ford, L. R., & Fulkerson, D. R. (1956). Maximal flow through a network. Canadian Journal of Mathematics, 8, 399–404. CrossRefGoogle Scholar
  11. Ford, L. R., & Fulkerson, D. R. (1955). A simple algorithm for finding maximal network flows and an application to the Hitchcock problem. Project rand research memorandum, RM-1604, Santa Monica, CA. Google Scholar
  12. Fulkerson, D. R., & Harding, G. C. (1977). Maximizing the minimum source-sink path subject to a budget constraint. Mathematical Programming, 13(1), 116–118. CrossRefGoogle Scholar
  13. Golden, B. (1978). A problem in network interdiction. Naval Research Logistics Quarterly, 25(4), 711–713. CrossRefGoogle Scholar
  14. Hausken, K. (2011). Strategic defense and attack of series systems when agents move sequentially. IIE Transactions, 43(7), 483–504. CrossRefGoogle Scholar
  15. Held, H., Hemmecke, R., & Woodruff, D. L. (2005). A decomposition algorithm applied to planning the interdiction of stochastic networks. Naval Research Logistics, 52(4), 321–328. CrossRefGoogle Scholar
  16. Hemmecke, R., Schultz, R., & Woodruff, D. L. (2003). Interdicting stochastic networks with binary effort. In D. L. Woodruff (Ed.), Network interdiction and stochastic integer programming (pp. 69–84). Norwell: Kluwer Academic. CrossRefGoogle Scholar
  17. Israeli, E., & Wood, R. K. (2002). Shortest-path network interdiction. Networks, 40(2), 97–111. CrossRefGoogle Scholar
  18. Koopman, B. O. (1979). Search and its optimization. The American Mathematical Monthly, 86, 527–540. CrossRefGoogle Scholar
  19. Lim, C., & Smith, J. C. (2008). Algorithms for network interdiction and fortification games. In A. Chinchuluun, P. M. Pardalos, A. Migdalas, & L. Pitsoulis (Eds.), Pareto optimality, game theory and equilibria (pp. 609–644). New York: Springer. Google Scholar
  20. Lim, C., & Smith, J. C. (2007). Algorithms for discrete and continuous multicommodity flow network interdiction problems. IIE Transactions, 39(1), 15–26. CrossRefGoogle Scholar
  21. Lunday, B. J. (2010). Resource allocation on networks: nested event tree optimization, network interdiction, and game theoretic methods. Doctoral Dissertation, Virginia Tech, Blacksburg, VA. Google Scholar
  22. Lunday, B. J., & Sherali, H. D. (2011a). A dynamic network interdiction problem. Informatica, 21(4), 553–574. Google Scholar
  23. Lunday, B. J., & Sherali, H. D. (2011b). Network flow interdiction models and algorithms with resource synergy considerations. Manuscript, Grado Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA. Google Scholar
  24. Meijers, E. (2005). Polycentric urban regions and the quest for synergy: is a network of cities more than the sum of the parts? Urban Studies, 42(4), 765–781. CrossRefGoogle Scholar
  25. Morton, D. P., Pan, F., & Saeger, K. J. (2007). Models for nuclear smuggling interdiction. IIE Transactions, 39(1), 3–14. CrossRefGoogle Scholar
  26. Napier, R. W., & Gershenfeld, M. K. (1993). Groups: theory and experiences. Boston: Houghton Mifflin Company. Google Scholar
  27. Nagurney, A., & Woolley, T. (2010). Environmental and cost synergy in supply chain network integration in mergers and acquisitions. In M. Ehrgott, B. Naujoks, T. Stewart, & J. Wallenius (Eds.) Lecture notes in economics and mathematical systems: Vol. 634. Multiple criteria decision making for sustainable energy and transportation systems. Proceedings of the 19th international conference on multiple criteria decision making (pp. 51–78). Berlin: Springer. Google Scholar
  28. Nehme, M. V. (2009). Two-person games for stochastic network interdiction: models, methods, and complexities. Doctoral Dissertation, University of Texas, Austin, TX. Google Scholar
  29. Pan, F., Charlton, W. S., & Morton, D. P. (2003). A stochastic program for interdicting smuggled nuclear material. In D. L. Woodruff (Ed.), Network interdiction and stochastic integer programming (pp. 1–19). Norwell: Kluwer Academic. CrossRefGoogle Scholar
  30. Royset, J. O., & Wood, R. K. (2007). Solving the bi-objective maximum-flow network-interdiction problem. INFORMS Journal on Computing, 19(2), 175–184. CrossRefGoogle Scholar
  31. Sherali, H. D., & Lunday, B. J. (2010). Equitable apportionment of railcars within a pooling agreement for shipping automobiles. Transportation Research. Part E, 47, 263–283. CrossRefGoogle Scholar
  32. Unsal, O. (2010). Two-person zero-sum network-interdiction game with multiple inspector types. Master’s Thesis, Naval Postgraduate School, Monterey, CA. Google Scholar
  33. von Eye, A., Schuster, C., & Rogers, W. M. (1998). Modelling synergy using manifest categorical variables. International Journal of Behavioral Development, 22(3), 537–557. CrossRefGoogle Scholar
  34. Washburn, A., & Wood, R. K. (1995). Two-person zero-sum games for network interdiction. Operations Research, 43(2), 243–251. CrossRefGoogle Scholar
  35. Wood, R. K. (1993). Deterministic network interdiction. Mathematical and Computer Modelling, 17(2), 1–18. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Grado Department of Industrial and Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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