Applied Soft Computing 68 (2018) 343–359 Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc A bi-objective partial interdiction problem considering different defensive systems with capacity expansion of facilities under imminent attacks Amir Mohammad Fathollahi Fard, Mostafa Hajiaghaei-Keshteli ∗ Department of Industrial Engineering, University of Science and Technology of Mazandaran, Behshahr, Iran a r t i c l e i n f o Article history: Received 1 December 2017 Received in revised form 2 March 2018 Accepted 11 April 2018 Available online 13 April 2018 Keywords: Partial interdiction problem Defensive systems Capacity expansion Demand outsourcing Bi-level metaheuristic a b s t r a c t Nowadays, the disordered caused by terrorist attacks is considered as a major treat by governments and international comities. Meanwhile, different types of defensive systems are to develop and to focus on protecting the critical system services in recent years. By this motivation and contrary to previous works, this study firstly proposes a bi-objective, bi-level formulation for the partial interdiction problem con- sidering different defensive systems. In the developed model, defender as a leader tries to optimize the two objectives i.e. different types of defensive systems to protect the facilities as well as the total cost for supply the demands of customers and outsourcing, simultaneously. Besides, the attacker seeks to maxi- mize destroy for the customers’ service system. Due to the nature of the problem, the solution approach is based on a nested approach, namely, bi-level metaheuristic. Accordingly, this study not only uses the old and the recent algorithms, but also utilizes two new hybrid algorithms according to the benefits of both groups. The performance of the algorithms is enhanced by Response Surface Method (RSM) to set the proper values for algorithms’ parameters. Finally, the approaches are evaluated by different criteria to identify the performance of algorithms as well as a set of sensitivity analyses are done to approve the efficiency of the proposed model. © 2018 Elsevier B.V. All rights reserved. 1. Motivation and literature review Todays, by advancing, expansion and diversification of defen- sive systems, the world has become an arena for militant groups to achieve a series of perverse ideals. In addition, the ease of com- mitting terrorist acts is resulted in increasing of the number of deliberate civilian and the military attacks in the current century. The attacks are cleverly designed to aim the most sensitive and critical facilities. With an increasing of the variety of different tech- niques and plans, this has made it more difficult to fight to the terrorism. The leading countries in fighting terrorism such as the United States, have utilized different defense and military systems to deal with terrorism in recent years [1]. Recent years have seen a numerous incidents of bombings in Palestine, Turkey, India, Afghanistan, etc. which shows the multi- plicity, complexity and sensitivity of planning to deal with these events [2]. The mentioned examples are numerous, where terror- ists (the attackers) try to stop services providing, create chaos and ∗ Corresponding author. E-mail address: mostafahaji@mazust.ac.ir (M. Hajiaghaei-Keshteli). to kill people. Accordingly, they choose the most sensitive facil- ities [3]. So, the role of defensive systems to protect the service systems is very strategic for defender’s level. A robust defense budget and policy strengthens the nations are so significant. A well-equipped defense enhances capabilities of governments and influences across virtually all other elements of national power: the economy, the diplomacy, the alliances, and the credibility and influence in the world. Conversely, an underestimated resourced national defense threatens to diminish government’s power across all of these other dimensions. A strong system of defense is thus indispensable for a peaceful, successful, and free world if a shot is never fired. 1 This is the main motivation of this study to get closer to real-world applications. So, this paper adds different defensive systems into the proposed model by considering a probabilistic protection of critical facilities and also a distinguished objective function for the defender’s level. It is estimated that after the attack on the World Trade Cen- ter in 2001, using operation research models to plan security 1 http://index.heritage.org/military/2016/essays/role-of-a-strong-national- defense/. https://doi.org/10.1016/j.asoc.2018.04.011 1568-4946/© 2018 Elsevier B.V. All rights reserved.