Annals of Operations Research

, Volume 283, Issue 1–2, pp 87–117 | Cite as

An integrated tri-level model for enhancing the resilience of facilities against intentional attacks

  • Sachuer Bao
  • Chi Zhang
  • Min OuyangEmail author
  • Lixin Miao
S.I.:Applications of OR in Disaster Relief Operations


It is paramount to enhance the resilience of many facilities that are known to be critical for modern societies and threatened by intentional attacks. For this purpose, it is important to not only protect them before disruptions, but also recover them after disruptions. To deal with this problem, this paper proposes a tri-level model explicitly integrating the decision making on recovery strategies of disrupted facilities with the decision making on protecting facilities from intentional attacks. The facilities studied in this research are assumed to be capacitated and a recovery strategy is considered to include repairing a subset of disrupted facilities and expanding the capacities of a subset of non-interdicted facilities. We are concerned with the defender’s objective of maximizing the resilience of the given set of facilities making profit within a prescribed time interval. To deal with the complexity of solving the proposed tri-level model, the ant colony system algorithm is employed with necessary adaptations.


Tri-level Facility location Intentional attack Recovery Resilience 

List of symbols


Set of customers


Set of facilities


Index of customers


Index of facilities


The number of customers


The number of original facilities


The number of protected facilities


The number of interdicted facilities


The length of concerned time interval


Time required to repair a disrupted facility


Amount of budget


Transportation cost per distance


Cost required to repair each disrupted facility


Capacity expansion cost per demand unit

\(\mu \)

Maximal incremental value by which a facility’s capacity can be expanded


Revenue earned by satisfying each unit of customer demand


Penalty cost paid for each unit of unmet demand per unit of time


Total demand of customer i


Distance between facility j and customer i


Length of a supply cycle


Number of ants


A random number uniformly distributed in [0,1]

\(\eta _{j}\)

Heuristic value of choosing facility j

\(\alpha ,\beta \)

Indexes reflecting the importance of pheromone and heuristic value

\(\rho \)

Pheromone evaporating rate


Stage of decision making, with \(m = 0\) indicates the first (i.e., protection) stage, and \(m = 1\) indicates the second (i.e., interdiction) stage

\(\tau _{{\textit{mj}}}\)

Pheromone amount on facility j in stage m


Probability of choosing facility j in stage m



The authors gratefully acknowledge the support from the National Natural Science Foundation of China under Grants 71301085, 71332005, 71671074 and 71731008, and the Fundamental Research Funds for the Central Universities (Grant Number HUST: 2017KFYXJJ178).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Sachuer Bao
    • 1
  • Chi Zhang
    • 2
  • Min Ouyang
    • 3
    Email author
  • Lixin Miao
    • 1
  1. 1.Division of Logistics and TransportationTsinghua UniversityShenzhenPeople’s Republic of China
  2. 2.Department of Industrial EngineeringTsinghua UniversityBeijingPeople’s Republic of China
  3. 3.School of AutomationHuazhong University of Science and TechnologyWuhanPeople’s Republic of China

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