Editorial Board
Publishing Ethics
Author Guide
Submission Prepare Your ManuscriptPeer review
Archive
English version:
Journal of Applied and Industrial Mathematics, 2018, 12:3, 417-425
Volume 25, No 3, 2018, P. 5-22
UDC 519.8
V. L. Beresnev, I. A. Davydov, P. A. Kononova, and A. A. Melnikov
Bilevel “defender–attacker” model with multiple attack scenarios
Abstract:
We consider a bilevel “defender-attacker” model built on the basis of the Stackelberg game. In this model, given is a set of the objects providing social services for a known set of customers and presenting potential targets for a possible attack. At the first step, the Leader (defender) makes a decision on the protection of some of the objects on the basis of his/her limited resources. Some Follower (attacker), who is also limited in resources, decides then to attack unprotected objects, knowing the decision of the Leader. It is assumed that the Follower can evaluate the importance of each object and makes a rational decision trying to maximize the total importance of the objects attacked. The Leader does not know the attack scenario (the Follower’s priorities for selecting targets for the attack). But, the Leader can consider several possible scenarios that cover the Follower’s plans. The Leader’s problem is then to select the set of objects for protection so that, given the set of possible attack scenarios and assuming the rational behavior of the Follower, to minimize the total costs of protecting the objects and eliminating the consequences of the attack associated with the reassignment of the facilities for customer service. The proposed model may be presented as a bilevel mixed-integer programming problem that includes an upper-level problem (the Leader problem) and a lower-level problem (the Follower problem). The main efforts in this article are aimed at reformulation of the problem as some one-level mathematical programming problems. These formulations are constructed using the properties of the optimal solution of the Follower’s problem, which makes it possible to formulate necessary and sufficient optimality conditions in the form of linear relations.
Bibliogr. 16.
Keywords: bilevel programming, complementarity slackness, optimality criteria.
DOI: 10.17377/daio.2018.25.612
Vladimir L. Beresnev 1,2
Ivan A. Davydov 1,2
Polina A. Kononova 1,2
Andrey A. Melnikov 1,2
1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
2. Novosibirsk State University,
2 Pirogov St., 630090 Novosibirsk, Russia
e-mail: beresnev@math.nsc.ru, vann.davydov@gmail.com, anilopko@gmail.com, melnikov@math.nsc.ru
Received 19 March 2018
References
[1] I. A. Davydov, A. A. Melnikov, and P. A. Kononova, Local search for load balancing problems for servers with large dimension, Avtom. Telemekh., No. 3, 34–50, 2017 [Russian]. Translated in Autom. Remote Control, 78, No. 3, 412–424, 2017.[2] D. Aksen and N. Aras, A bilevel fixed charge location model for facilities under imminent attack, Comput. Oper. Res., 39, No. 1, 1364–1381, 2012.
[3] D. Aksen, N. Piyade, and N. Aras, The budget constrained $r$-interdiction median problem with capacity expansion, Cent. Eur. J. Oper. Res., 18, No. 3, 269–291, 2010.
[4] B. An, F. Ordóñez, M. Tambe, E. Shieh, R. Yang, C. Baldwin, J. DiRenzo, K. Moretti, B. Maule, and G. Meyer, A deployed quantal response-based patrol planning system for the U. S. Coast Guard, Interfaces, 43, No. 5, 400–420, 2013.
[5] J. S. Angelo and H. J. C. Barbosa, A study on the use of heuristics to solve a bilevel programming problem, Int. Trans. Oper. Res., 22, 861–882, 2015.
[6] V. Beresnev and A. Melnikov, Facility location in unfair competition, in Discrete Optimization and Operations Research (Proc. 9th Int. Conf. DOOR, Vladivostok, Russia, Sept. 19–23, 2016), pp. 325–335, Springer, Cham, 2016 (Lect. Notes Comput. Sci., Vol. 9869).
[7] L. Brotcorne, S. Hanafi, and R. Mansi, One-level reformulation of the bilevel Knapsack problem using dynamic programming, Discrete Optim., 10, No. 1, 1–10, 2013.
[8] F. M. Delle Fave, F. M. Jiang, Z. Yin, C. Zhang, M. Tambe, S. Kraus, and J. P. Sullivan, Game-theoretic security patrolling with dynamic execution uncertainty and a case study on a real transit system, J. Artif. Intell. Res., 50, 321–367, 2014.
[9] S. Dempe, Foundations of Bilevel Programming, Kluwer Acad. Publ., Dordrecht, 2002.
[10] M. Jain, J. Tsai, J. Pita, C. Kiekintveld, S. Rathi, M. Tambe, and F. Ordóñez, Software assistants for randomized patrol planning for the LAX airport police and the federal air marshal service, Interfaces, 40, No. 4, 267–290, 2010.
[11] A. X. Jiang, Z. Yin, C. Zhang, M. Tambe, and S. Kraus, Game-theoretic randomization for security patrolling with dynamic execution uncertainty, in Proc. 12th Int. Conf. Auton. Agents Multiagent Syst., Saint Paul, MN, USA, May 6–10, 2013, pp. 207–214, Int. Found. Auton. Agents Multiagent Syst., Richland, SC, 2013.
[12] S. Martello and P. Toth, Knapsack Problems: Algorithms and Computer Implementations, John Wiley & Sons, New York, USA, 1990.
[13] M. T. Melo, S. Nickel, and F. Saldanha-Da-Gama, A tabu search heuristic for redesigning a multi-echelon supply chain network over a planning horizon, Int. J. Prod. Econ., 136, No. 1, 218–230, 2012.
[14] M. P. Scaparra and R. L. Church, A bilevel mixed-integer program for critical infrastructure protection planning, Comput. Oper. Res., 35, 1905–1923, 2008.
[15] H. von Stackelberg, The Theory of the Market Economy, Oxf. Univ. Press, Oxford, 1952.
[16] Y. Zhu, Z. Zheng, X. Zhang, and K. Y. Cai, The $r$-interdiction median problem with probabilistic protection and its solution algorithm, Comput. Oper. Res., 40, 451–462, 2013.
