**3. Quantifying high impact, low probability events**

HILP events require a subsystem to bounce back to normalcy following major disruption. The goal is to regain pre-disruption levels of output as quickly as possible; however, recovery time is not the only metric of importance. The shape of the recovery curve is also of significance. Resiliency aids in defining a disaster response paradigm which differs from previous approaches such as resistance and sustainability by emphasizing return to normal. Nonetheless, the literature frequently uses the concept of resilience to imply the ability to recover or bounce back to normalcy after a disaster occurs [3]. Review of scholarly work related to the resiliency concept identified three main ideals: no assumption that disaster prevention is always possible,

recognition of the need to include social variables, and the necessity to include disciplines outside the physical sciences and engineering. The term resiliency has increased in usage over the past decades. A multitude of definitions have been proposed whose interpretations can align with either resistance or sustainability. Although the resilience construct offered advantages in many areas relative to competing paradigms, the ambiguity associated with its meaning and scope hindered consensus. The multiplicity of definitions is a reflection of the philosophical and methodological diversities that have emerged from disaster scholarship and research [4].

*RL* ¼

*The Modulus of Resilience for Critical Subsystems DOI: http://dx.doi.org/10.5772/intechopen.93783*

precisely develop; however, the formulation is shown in Eq. (4).

ð*t*1 *t*0

Zobel [10] proposed a method based on the total possible loss over some suitably long-time interval (T\*), percentage of functionality lost after disruption (X), and time required for full recovery (T). An effort was made to analyze different combinations of X and T which result in the same level of resilience as shown in Eq. (3).

*R X*ð Þ¼ , *<sup>T</sup> <sup>T</sup>*<sup>∗</sup> � *XT=*<sup>2</sup>

*<sup>R</sup>* <sup>¼</sup> %*ΔYmax* � %*Δ<sup>Y</sup>*

Alternatively, Rose [12] considered time effects using a concept of dynamic resilience. The quantification of dynamic resilience is the difference in system recovery with hastened system recovery (SOHR) and without hastened system recovery (SOWR). This calculation is utilized over the total number of time steps

Wang et al. [13] explored resilience in information systems based on the number of operations in the enterprise information system (m). The ratio of the demand time (di) and completion time of operation (ci) are weighted by the importance of

> *i*¼1 *zi di ci*

The larger the value of the metric the more resilient the system is determined to be. The calculation requires the assignment of a weight and assumes the number of operations is known. When attempting to quantify unknown events the number of

Chen and Miller-Hooks [14] quantifies the "post-disruption expected fraction of demand that, for a given network, can be satisfied within pre-determined recovery budgets" (Hosseini et al.). The measure was based on transportation networks and compares the maximum demand that can be satisfied before disruption (Dw) and

*w* ∈*W*

Orwin and Wardle [15] considered the instantaneous and maximum disturbance

*dw=* X *w* ∈*W*

!

*Dw*

*<sup>R</sup>* <sup>¼</sup> max <sup>X</sup>*<sup>m</sup>*

(N) considered. The dynamic resilience calculation is shown in Eq. (5).

*i*¼1

*Resilience* <sup>¼</sup> *<sup>E</sup>* <sup>X</sup>

in the quantification of resilience. The maximum absorbable force without

*DR* <sup>¼</sup> <sup>X</sup> *N*

operation (zi).

**21**

operations can be difficult to estimate.

after disruption (dw) for pair (w).

This metric is based on a linear recovery making it unrealistic for some scenarios. Alternative methods were proposed by Cox et al. [11] based on economic resilience using the difference in disruption (%ΔDYmax) between the expected disruption (%ΔY) and maximum potential disruption (%ΔYmax). Therefore, an estimate of performance degradation is required. Such an estimation may be a challenge to

½ � 100 � *Q t*ð Þ *dt* (2)

*<sup>T</sup>*<sup>∗</sup> (3)

%*ΔDYmax* (4)

*SOHR*ð Þ� *ti SOWR*ð Þ*ti* (5)

(6)

(7)

Resilience first came to prominence in the English language in the early 19th century when Tredgold used the term to describe a property of timber [5]. In his essay "On the transverse strength and resilience of timber," Tredgold tested the properties of timber to be used in ship making. Tredgold cites resilience as the power of resisting a body in motion [5]. The statement is foundational in establishing the concept of resilience as more than recovery but instead as an ability to first withstand an applied force. Furthermore, Tredgold varied the weight and height of objects dropped on the test samples and recorded the effects to different forces on various wood pieces. These effects ranged from no effect, broke to curved. A second reference to the consideration of force can be found in the 1858 work, "On the Physical Conditions Involved in the Construction of Artillery, and on Some Hitherto Unexplained Causes of the Destruction of Cannon in Service," by Robert Mallet. He states the modulus of resilience of other writers, referred to hereafter, depends, is much greater for gunmetal, and hence a given force produces a greater proportional distortion of form [6]. The modulus of resilience was further formalized by materials science using stress/strain testing.
