**4. Methods in quantifying resilience**

The range of methods for defining resilience include qualitative, quantitative and probabilistic. A quantitative method can be used to compare outcomes using data from different actual events. A number of researchers have explored quantifying resilience to move beyond qualitative representations. Henry and Ramirez-Marquez [7] proposed a quantitative approach for system resilience as a function of time. The formulation was a ratio of the recovery and losses using a figure-of-merit function. A disruptive event (e<sup>j</sup> ) at time, te, impacts the system until time, td.

$$\mathcal{H}\_{\boldsymbol{\varphi}}\left(\boldsymbol{t}|\boldsymbol{e}^{j}\right) = \frac{\boldsymbol{\varrho}\left(\boldsymbol{t}|\boldsymbol{e}^{j}\right) - \boldsymbol{\varrho}\left(\boldsymbol{t}\_{d}|\boldsymbol{e}^{j}\right)}{\boldsymbol{\varrho}\left(\boldsymbol{t}\_{0}\right) - \boldsymbol{\varrho}\left(\boldsymbol{t}\_{d}|\boldsymbol{e}^{j}\right)} \tag{1}$$

As shown, the numerator relates to the recovery until time t and the denominator represents the total loss due to disruption. Hosseini et al. [8] reviewed definitions and measures of system resilience. Their literature review was based on multiple domains including organizational, social, economic, and engineering using papers published between 2000 and April 2015. The major categories of assessment approaches are qualitative and quantitative with quantitative measures further defined as either probabilistic or deterministic.

The intent to analyze protracted subsystem disruptions leads to a focus on quantitative deterministic methods of calculating resiliency. The literature review by Hosseini et al. [8] included 11 deterministic methods of quantification. Bruneau et al. [9] utilized a method of integration based on the degradation in quality of infrastructure during recovery period of Eq. (5). Larger RL values indicate lower resilience while smaller RL imply higher resilience. Hosseini et al. [8] RL is calculated based on the formulation in Eq. (2).

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

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

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

The range of methods for defining resilience include qualitative, quantitative and probabilistic. A quantitative method can be used to compare outcomes using data from different actual events. A number of researchers have explored quantifying resilience to move beyond qualitative representations. Henry and Ramirez-Marquez [7] proposed a quantitative approach for system resilience as a function of time. The formulation was a ratio of the recovery and losses using a figure-of-merit

<sup>Я</sup>*<sup>φ</sup> <sup>t</sup>*j*e<sup>j</sup>* <sup>¼</sup> *<sup>φ</sup> <sup>t</sup>*j*e<sup>j</sup>* � *<sup>φ</sup> td*j*e<sup>j</sup>*

As shown, the numerator relates to the recovery until time t and the denominator represents the total loss due to disruption. Hosseini et al. [8] reviewed definitions and measures of system resilience. Their literature review was based on multiple domains including organizational, social, economic, and engineering using papers published between 2000 and April 2015. The major categories of assessment approaches are qualitative and quantitative with quantitative measures further

The intent to analyze protracted subsystem disruptions leads to a focus on quantitative deterministic methods of calculating resiliency. The literature review by Hosseini et al. [8] included 11 deterministic methods of quantification. Bruneau et al. [9] utilized a method of integration based on the degradation in quality of infrastructure during recovery period of Eq. (5). Larger RL values indicate lower resilience while smaller RL imply higher resilience. Hosseini et al. [8] RL is calcu-

) at time, te, impacts the system until time, td.

*<sup>φ</sup>*ð Þ� *<sup>t</sup>*<sup>0</sup> *<sup>φ</sup> td*j*ej* ð Þ (1)

have emerged from disaster scholarship and research [4].

*Operations Management - Emerging Trend in the Digital Era*

formalized by materials science using stress/strain testing.

**4. Methods in quantifying resilience**

defined as either probabilistic or deterministic.

lated based on the formulation in Eq. (2).

**20**

function. A disruptive event (e<sup>j</sup>

$$RL = \int\_{t\_0}^{t\_1} [\mathbf{100} - \mathbf{Q}(t)] dt \tag{2}$$

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).

$$R(X,T) = \frac{T^\*-XT/2}{T^\*} \tag{3}$$

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

$$R = \frac{\text{\textdegree}\Delta Y^{\text{max}} - \text{\textdegree}\Delta Y}{\text{\textdegree}\Delta DY^{\text{max}}} \tag{4}$$

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

$$DR = \sum\_{i=1}^{N} SO\_{HR}(t\_i) - SO\_{WR}(t\_i) \tag{5}$$

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 operation (zi).

$$R = \max \sum\_{i=1}^{m} z\_i \frac{d\_i}{c\_i} \tag{6}$$

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 operations can be difficult to estimate.

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 after disruption (dw) for pair (w).

$$Resiliavec = E\left(\sum\_{w \in W} d\_w / \sum\_{w \in W} D\_w\right) \tag{7}$$

Orwin and Wardle [15] considered the instantaneous and maximum disturbance in the quantification of resilience. The maximum absorbable force without

upsetting system function (Emax) and effect of the disturbance on safety (Ej) at a given time (Tj) are used to define resilience.

$$Resiliavec = \left(\frac{2 \times |E\_{\max}|}{|E\_{\max}| + |E\_j|}\right) - 1\tag{8}$$

point calculations based on their ability to compensate for nonlinear restoration curves; however, complexity beyond the resilience triangle [9] would be necessary to capture differences in event magnitude and restoration response in disparate

normal operation was typically used to identify the end of the restoration time period; however, this approach does not set the time based on the aim of the subsystem. Evaluations of subsystems beyond a critical point with respect to use of the subsystem output could lead to poor decision-making. One of the main weaknesses of the current resilience metric is that they do not relate the effects of a disruptive event to any of the event characteristics, unlike materials science [19]. Materials science utilizes a change in length for evaluation of stress and strain; however, the difference in recovery response to a common cause and special cause event was not found in the literature review. These distinctions serve to highlight the differences between reliability for normally occurring events and resiliency to low frequency events. Additionally, the need for utilizing subjective variables

[10, 11, 12, 14, 15] does not lend well to empirical study.

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

recovery [17] are not representative of many empirical cases.

for improved response from the subsystem.

**23**

may fail to represent intermediate progress in recovery.

The concept of a yield point was not identified in existing literature. A return to

The ability to normalize responses to different events is beneficial for evaluating the resiliency of different subsystems or different events on the same subsystem. The literature reviewed began analysis of the event from the start of restoration [7] or by treating the entire curve from time of event to the completed restoration as a single integral [16]. This approach can confuse the quantities of force, stress and strain. An equal force can result in different stress and strain based on the subsystem being reviewed. As a result, the descending slope and associated area prior to the start of recovery may prove informative of stress. Strain is more associated with the total area under the curve. The review of literature did not identify a bifurcation of the curve to delineate stress (prior to start of recovery) and strain (total area). Therefore, the assumption of instantaneous loss and exponential

In reviewing the concepts of resilience, a force is applied to a subsystem, the subsystem absorbs a portion of the force, experiences stress, and adapts to recover to a pre-disruption state. These references highlight an importance of considering the stress on the subsystem in determining the resiliency of a subsystem. Three primary points of measure for use in quantifying resiliency were identified including: stress, total area of event and change in length. Stress is a foundational variable of resiliency, as the term resiliency implies a response to a significant disruption. Therefore, only events of significance from a subsystem level are commonly referred to in terms of resilience. Additionally, the ability to compare resiliency events needs some level of normalization based on the associated stress for each event. Force continues to be applied until the subsystem decay ceases, allowing for subsystem assessment and initiation of recovery. The rate of subsystem decay influences the stress applied to the subsystem and the subsystem ability to bounce back. This connection exists due to the role of adaptation in the resiliency process. A slow evolving scenario (i.e., slow subsystem decay) presents the subsystem opportunity to adapt, resist, and recover in ways an acute decay will not. Therefore, when considering the normalization process of resiliency both the decay (i.e., stress proxy) and recovery portion of the resiliency curve must be independently considered. The delayed decay provides an opportunity

Total area of recovery best quantifies recovery and resiliency by compensating for the nonlinearity in the response function. As the subsystem attempts to recover, disruptions in the recovery process may cause discontinuities not captured by linear slope calculations. Similarly, time to recovery (i.e., 3 days to recovery) calculations

events.

Frameworks for local and global resilience were introduced by Enjalbert et al. [16] for modeling system safety in public transportation systems. A safety indication function (S(t)) is used to calculate resilience either instantaneously or over time, representing local and global, respectively. Global resilience is calculated from the time of disturbance (tb) to the end of the disturbance (te). The calculations are as follows:

$$\text{Local\\_resile} = \frac{d\mathbf{S}(t)}{dt} \tag{9}$$

$$\text{Global\\_resilience} = \int\_{t\_b}^{t\_r} \frac{dS(t)}{dt} \tag{10}$$

Francis and Bekera [17] introduced a metric for dynamic resilience. The calculation uses the speed of recovery (Sp), original performance level (Fo), performance level at new stable level (*Fr*) and performance level immediately after disruption (Fd). The speed of recovery variable assumes exponential growth for a maximum acceptable recovery time (tδ), total recovery time (tr) to a new equilibrium state, time to complete initial recovery (*t* <sup>∗</sup> *<sup>r</sup>* ), and a decay in resilience (a). The resilience metric is calculated using Eq. (11).

$$\rho\_i = \mathcal{S}\_p \frac{F\_r}{F\_o} \frac{F\_d}{F\_o} \tag{11}$$

$$\mathcal{S}\_p = \left(\mathfrak{t}\_\delta / \mathfrak{t}\_r^\*\right) \exp\left[-a\left(\mathfrak{t}\_r - \mathfrak{t}\_r^\*\right)\right] for \, \mathfrak{t}\_r \ge \mathfrak{t}\_r^\* \tag{12}$$

Otherwise,

$$\mathbf{S}\_p = \begin{pmatrix} \mathfrak{t}\_\delta / \mathfrak{t}\_r^\* \end{pmatrix} \tag{13}$$

Cimellaro et al. [18] utilized quality of service to represent resilience. The method uses before disruption quality of service (Q1(t)), post disruption quality of service (Q2(t)), a control time (TLC) and a weighting factor (α) in developing a healthcare resilience metric.

$$R = a \int\_{T\_{LC}} \frac{Q\_1(t)}{T\_{LC}} dt + (1 - a) \int\_{T\_{LC}} \frac{Q\_2(t)}{T\_{LC}} dt \tag{14}$$

Aside from the works investigated by Hosseini et al. [8], Dessavre et al., [19] introduced a new model and visual tools adding a stress dimension representing the force and stress of disruptive events. Defining the stress of the events is not a trivial task and completely domain dependent [19].

A review of the concepts found in literature was completed for elements consistent with the modulus of resilience. Methods were limited to quantitative approaches which could be utilized with empirical data sets. Although the use of scaling factors was identified in literature [13, 18], such methods are not desired in the development of subsystem-based methods due to the subjectivity associated with them. A ratio-based approach has merit in its ability to normalize event effects and resulting recovery. Area-based calculations using integration are preferred to

upsetting system function (Emax) and effect of the disturbance on safety (Ej) at a

j jþ *Emax E <sup>j</sup>* � � � �

Frameworks for local and global resilience were introduced by Enjalbert et al. [16] for modeling system safety in public transportation systems. A safety indication func-

representing local and global, respectively. Global resilience is calculated from the time of disturbance (tb) to the end of the disturbance (te). The calculations are as follows:

*Local resilience* <sup>¼</sup> *dS t*ð Þ

Francis and Bekera [17] introduced a metric for dynamic resilience. The calculation uses the speed of recovery (Sp), original performance level (Fo), performance level at new stable level (*Fr*) and performance level immediately after disruption (Fd). The speed of recovery variable assumes exponential growth for a maximum acceptable recovery time (tδ), total recovery time (tr) to a new equilibrium state,

ð*te tb*

*dS t*ð Þ

*<sup>r</sup>* ), and a decay in resilience (a). The resilience

� � (13)

!

� 1 (8)

*dt* (9)

*dt* (10)

(11)

*<sup>r</sup>* (12)

*dt* (14)

*Resilience* <sup>¼</sup> <sup>2</sup> � j j *Emax*

tion (S(t)) is used to calculate resilience either instantaneously or over time,

*Global resilience* ¼

*ρ<sup>i</sup>* ¼ *Sp*

� � *exp* �*a tr* � *<sup>t</sup>* <sup>∗</sup>

*Sp* <sup>¼</sup> *<sup>t</sup>δ=<sup>t</sup>* <sup>∗</sup>

Cimellaro et al. [18] utilized quality of service to represent resilience. The method uses before disruption quality of service (Q1(t)), post disruption quality of service (Q2(t)), a control time (TLC) and a weighting factor (α) in developing a

*r*

*dt* þ ð Þ 1 � *α*

Aside from the works investigated by Hosseini et al. [8], Dessavre et al., [19] introduced a new model and visual tools adding a stress dimension representing the force and stress of disruptive events. Defining the stress of the events is not a trivial

A review of the concepts found in literature was completed for elements consis-

ð*: TLC*

*Q*2ð Þ*t TLC*

*Fr Fo Fd Fo*

*r* � � � � *for tr* ≥*t* <sup>∗</sup>

given time (Tj) are used to define resilience.

*Operations Management - Emerging Trend in the Digital Era*

time to complete initial recovery (*t* <sup>∗</sup>

*Sp* <sup>¼</sup> *<sup>t</sup>δ=<sup>t</sup>* <sup>∗</sup>

*R* ¼ *α*

task and completely domain dependent [19].

ð*: TLC*

*Q*1ð Þ*t TLC*

tent with the modulus of resilience. Methods were limited to quantitative approaches which could be utilized with empirical data sets. Although the use of scaling factors was identified in literature [13, 18], such methods are not desired in the development of subsystem-based methods due to the subjectivity associated with them. A ratio-based approach has merit in its ability to normalize event effects and resulting recovery. Area-based calculations using integration are preferred to

*r*

metric is calculated using Eq. (11).

Otherwise,

**22**

healthcare resilience metric.

point calculations based on their ability to compensate for nonlinear restoration curves; however, complexity beyond the resilience triangle [9] would be necessary to capture differences in event magnitude and restoration response in disparate events.

The concept of a yield point was not identified in existing literature. A return to normal operation was typically used to identify the end of the restoration time period; however, this approach does not set the time based on the aim of the subsystem. Evaluations of subsystems beyond a critical point with respect to use of the subsystem output could lead to poor decision-making. One of the main weaknesses of the current resilience metric is that they do not relate the effects of a disruptive event to any of the event characteristics, unlike materials science [19]. Materials science utilizes a change in length for evaluation of stress and strain; however, the difference in recovery response to a common cause and special cause event was not found in the literature review. These distinctions serve to highlight the differences between reliability for normally occurring events and resiliency to low frequency events. Additionally, the need for utilizing subjective variables [10, 11, 12, 14, 15] does not lend well to empirical study.

The ability to normalize responses to different events is beneficial for evaluating the resiliency of different subsystems or different events on the same subsystem. The literature reviewed began analysis of the event from the start of restoration [7] or by treating the entire curve from time of event to the completed restoration as a single integral [16]. This approach can confuse the quantities of force, stress and strain. An equal force can result in different stress and strain based on the subsystem being reviewed. As a result, the descending slope and associated area prior to the start of recovery may prove informative of stress. Strain is more associated with the total area under the curve. The review of literature did not identify a bifurcation of the curve to delineate stress (prior to start of recovery) and strain (total area). Therefore, the assumption of instantaneous loss and exponential recovery [17] are not representative of many empirical cases.

In reviewing the concepts of resilience, a force is applied to a subsystem, the subsystem absorbs a portion of the force, experiences stress, and adapts to recover to a pre-disruption state. These references highlight an importance of considering the stress on the subsystem in determining the resiliency of a subsystem. Three primary points of measure for use in quantifying resiliency were identified including: stress, total area of event and change in length. Stress is a foundational variable of resiliency, as the term resiliency implies a response to a significant disruption. Therefore, only events of significance from a subsystem level are commonly referred to in terms of resilience. Additionally, the ability to compare resiliency events needs some level of normalization based on the associated stress for each event. Force continues to be applied until the subsystem decay ceases, allowing for subsystem assessment and initiation of recovery. The rate of subsystem decay influences the stress applied to the subsystem and the subsystem ability to bounce back. This connection exists due to the role of adaptation in the resiliency process. A slow evolving scenario (i.e., slow subsystem decay) presents the subsystem opportunity to adapt, resist, and recover in ways an acute decay will not. Therefore, when considering the normalization process of resiliency both the decay (i.e., stress proxy) and recovery portion of the resiliency curve must be independently considered. The delayed decay provides an opportunity for improved response from the subsystem.

Total area of recovery best quantifies recovery and resiliency by compensating for the nonlinearity in the response function. As the subsystem attempts to recover, disruptions in the recovery process may cause discontinuities not captured by linear slope calculations. Similarly, time to recovery (i.e., 3 days to recovery) calculations may fail to represent intermediate progress in recovery.

Consideration of a failure point based on the aim of the subsystem aids in representing real-world scenarios. Recovery which occurs after a critical point of the subsystem would indicate a lack of resiliency. As an example, if a water subsystem requires 10 days to restore operation post contingency but the consumers of the water can only survive 4 days without water; the subsystem lacks resiliency. Attempts to quantify the subsystem's resilience should stop at 4 days. Calculations beyond the 4-day time period no longer support the aim of the subsystem or the practical operation of the subsystem.

The fracture point should be set based on the aim of the subsystem. For example, if a drinking water subsystem failure requires a 7-day restoration period but 4 days is the survival period without water; the calculation of subsystem resiliency should be limited to a 4-day period. In some cases, the acknowledgement of a fracture point will result in the calculation of resiliency stopping prior to the subsystem returning to pre-disruption output levels. **Figure 6** represents a case where the

Calculations to quantify resiliency which consider values beyond the failure point are theoretical as opposed to practical in nature. The failure point should be

An operational definition is derived from the combination of literature review and isomorphic adaptation of the modulus of resilience. Hence, resiliency is defined as the ability to limit proportional stain from abnormal stress to less than the subsystem yield point, through the achievement of recovery in less than the subsystem critical timeframes. This definition allows the use of quantitative

subsystem recovery takes longer than the subsystem failure point.

given priority in quantifying resiliency.

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

**Figure 5.**

**Figure 4.**

**Figure 6.**

**25**

*Representation of failure point.*

*Recovery curves with dissimilar characteristics.*

*Recovery curves with similar characteristics.*

Lastly, change in length was included in the materials science calculation of the modulus of resilience. The change in length from the original length to the length under stress could be translated to a subsystem resilience construct to allow consideration of how subsystem recovery under lower stress common cause events and high stress special cause events are related. The consideration of a change in length may aid in joining concepts associated with reliability in the quantification of resilience.

Comparing these constructs with the reviewed literature results in the identification of conceptual gaps. The resulting resiliency values should reflect the subsystem performance for practical cases. Units are required based on subsystem parameters. The x-axis utilizes units of time, while the y-axis measures the units associated with the aim of the subsystem.

The methods of quantification reviewed begin the process of quantification at the point of recovery or assume no time delta between the initiating event and start of recovery. To support the incorporation of stress in the quantification of resilience, a bifurcation of the event curve is used as shown in **Figure 3**.

The use of ratio methods may provide consistency in scenarios of similar characteristics. When disparate characteristics are present, computed values may prove inconsistent with event outcomes. Depending on the event characteristics, either ratio methods or area-based methods may identify a less resilient subsystem response as more resilient. **Figure 4** depicts the concept of less recovery time for less disruption. The scenario of **Figure 4** is representative of a minor difference in subsystem response and would provide consistent rankings for resilience outcomes in many cases, where less area is representative of increased resilience.

Conversely, cases may exist where a longer recovery results from a less impactful initial event. The delayed recovery to a less impactful event could result from many factors including a lack of preparedness, inability to adapt, etc. In such cases, observation would assume that the subsystem which took longer to recover from a less impactful event is less resilient. However, present formulations may suggest the opposite. **Figure 5** illustrates this scenario, where the smaller area is not representative of the more resilient outcome.

**Figure 3.** *Bifurcation of event curve.*

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

Consideration of a failure point based on the aim of the subsystem aids in representing real-world scenarios. Recovery which occurs after a critical point of the subsystem would indicate a lack of resiliency. As an example, if a water subsystem requires 10 days to restore operation post contingency but the consumers of the water can only survive 4 days without water; the subsystem lacks resiliency. Attempts to quantify the subsystem's resilience should stop at 4 days. Calculations beyond the 4-day time period no longer support the aim of the

Lastly, change in length was included in the materials science calculation of the modulus of resilience. The change in length from the original length to the length under stress could be translated to a subsystem resilience construct to allow consideration of how subsystem recovery under lower stress common cause events and high stress special cause events are related. The consideration of a change in length may aid in joining concepts associated with reliability in the quantification of

Comparing these constructs with the reviewed literature results in the identifi-

The methods of quantification reviewed begin the process of quantification at the point of recovery or assume no time delta between the initiating event and start of recovery. To support the incorporation of stress in the quantification of resil-

The use of ratio methods may provide consistency in scenarios of similar characteristics. When disparate characteristics are present, computed values may prove inconsistent with event outcomes. Depending on the event characteristics, either ratio methods or area-based methods may identify a less resilient subsystem response as more resilient. **Figure 4** depicts the concept of less recovery time for less disruption. The scenario of **Figure 4** is representative of a minor difference in subsystem response and would provide consistent rankings for resilience outcomes

Conversely, cases may exist where a longer recovery results from a less impactful initial event. The delayed recovery to a less impactful event could result from many factors including a lack of preparedness, inability to adapt, etc. In such cases, observation would assume that the subsystem which took longer to recover from a less impactful event is less resilient. However, present formulations may suggest the opposite. **Figure 5** illustrates this scenario, where the smaller area is not represen-

cation of conceptual gaps. The resulting resiliency values should reflect the subsystem performance for practical cases. Units are required based on subsystem parameters. The x-axis utilizes units of time, while the y-axis measures the units

ience, a bifurcation of the event curve is used as shown in **Figure 3**.

in many cases, where less area is representative of increased resilience.

subsystem or the practical operation of the subsystem.

*Operations Management - Emerging Trend in the Digital Era*

associated with the aim of the subsystem.

tative of the more resilient outcome.

**Figure 3.**

**24**

*Bifurcation of event curve.*

resilience.

The fracture point should be set based on the aim of the subsystem. For example, if a drinking water subsystem failure requires a 7-day restoration period but 4 days is the survival period without water; the calculation of subsystem resiliency should be limited to a 4-day period. In some cases, the acknowledgement of a fracture point will result in the calculation of resiliency stopping prior to the subsystem returning to pre-disruption output levels. **Figure 6** represents a case where the subsystem recovery takes longer than the subsystem failure point.

Calculations to quantify resiliency which consider values beyond the failure point are theoretical as opposed to practical in nature. The failure point should be given priority in quantifying resiliency.

An operational definition is derived from the combination of literature review and isomorphic adaptation of the modulus of resilience. Hence, resiliency is defined as the ability to limit proportional stain from abnormal stress to less than the subsystem yield point, through the achievement of recovery in less than the subsystem critical timeframes. This definition allows the use of quantitative

#### **Figure 4.** *Recovery curves with similar characteristics.*

#### **Figure 5.**

*Recovery curves with dissimilar characteristics.*

**Figure 6.** *Representation of failure point.*

measures in the calculation of resilience in a deterministic and normalized approach based on concepts from materials science.

An evaluation between two groups can result in an isomorphic application of findings from one structure to another. This mapping between groups can yield opportunities to apply known methodologies in an inter-disciplinary manner. The process of verifying an isomorphism requires the identification of elements in each structure and evaluating their equivalence. If equivalence is identified an opportunity for applying the computational framework may exist. The quantification of subsystem resilience was compared to resiliency as used in materials science. Materials science's definition of resiliency includes the concepts of per unit volume, maximum energy, and integration from zero to the elastic limit. The modulus of resilience (Ur) is found from the stress-strain curve measured during the tensile test. Stress (σ) in the stress-strain curve is "the applied force per unit original undeformed cross-sectional area of the specimen" [20] as delineated in Eq. (15).

$$
\sigma = \frac{F}{A\_0} \tag{15}
$$

The area under the curve will then be applied to the maximum percentage of

Protracted subsystem disruptions create stress and strain due to an inability to complete the subsystem aim. The similarities between tensile strength test used in materials science and the need to measure stress and strain subsystems create an isomorphic relationship. **Table 1** shows the parallels between materials science and

The application of the modulus of resilience to a specific subsystem requires the identification of an aim the subsystem exist to accomplish. "Without an aim, there is no system" [21]. The aim should be quantifiable with metrics available for analysis. The data must be accessible in order to serve as the basis for the resilience calculations and will vary based on the subsystem under study. Examples include percentage of successful operations or percentage of end users receiving service. The next section provides an empirical example in applying the modulus of

The power industry was selected to provide an example for applying the modulus of resiliency using empirical data. The aim of the electric subsystem is to deliver electricity to all end use customers; therefore, data regarding the number of customers out of service can be used to quantify subsystem performance. The use of customers out of service in quantifying subsystem performance was supported by a review of regulatory reliability metrics used by Public Utility Commissions. For major electric utility disruptions, DOE situation reports provide customer outage information for and are publicly available from the DOE website. One of the most prominent events to challenge utilities is hurricane, and as a result, multiple

2

0 @

subsystem

protracted event

disruptions events

ð Þ *SD* <sup>2</sup> ð Þ *SD*�*At =Anl* ð Þ *De*�*Da =Da*

1

Percent out of service is equivalent to stress

Area from zero to peak subsystem out of service is point where curve loses linearity

Represents total strain experienced by

Accounts for average non-protracted

Use of change in duration accounts for the change in length between average and

A (20)

Resiliency index RI ð Þ¼ <sup>1</sup>

where SD = % of subsystem disrupted; Anl = Area under the curve to nonlinearity; At = Total area under the curve; Da = Duration of average disruption;

**Protracted subsystem disruption Comparisons**

Stress applied Peak percent of subsystem out of service

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

disruption duration

Original length System's average duration of disruptions

event

Area under curve from the origin to peak subsystem out of service

Area under curve for entire disruptive

Delta between subsystem's average duration of disruptions and event

*Parallels between materials science test and protracted subsystem disruptions.*

subsystem disrupted.

**Materials science.**

Cross-sectional

Actual crosssectional area

Change in length

area

**Table 1.**

De = Duration of event disruption.

protracted subsystem disruptions.

**5. Application of the modulus of resiliency**

resilience.

**27**

where F = force; A0 = cross sectional area.

Young's modulus (E) serves as a measure of stiffness for a solid material. "Because of the difficulty in determining the elastic limit, it is commonly replaced by the proportional limit, which is the stress at which the stress-strain curve is out of linearity" [20].

$$E = \frac{F/A}{\Delta L/L\_0} \tag{16}$$

And,

$$F = \sigma \times A \tag{17}$$

where F = force; A = actual cross-sectional area; ΔL = amount of change in length; L0 = original length of the object.

"The modulus of resilience is the strain energy per unit volume absorbed up to the elastic limit for a tensile test and equals the area under the elastic part of the stress-strain curve" [20].

$$\mathbf{U}\_r = \mathbf{1}\_{\{2 \: \vert \: \sigma^2 \vert\_E\}} \tag{18}$$

"This quantity indicates how much energy a material can absorb without deforming plastically" [20]. Plastic deformation occurs when a material undergoes non-reversible changes in response to applied forces. The use of the stress-strain curve from materials testing is similar to conditions faced by disrupted subsystems regardless of type. Stress is the impact to the material under test, while strain is the resulting effects of the stress.

Based on the desire of applying a consistent methodology to quantify resilience regardless of disruption magnitude or subsystem size, the percentage of subsystem disrupted is proposed to achieve a per unit value. The area under the curve will then be integrated from the beginning to end of the disruptive event. Calculus to determine area under the curve is shown in Eq. (19).

$$\text{Total Area under the Curve} = \int\_{E\_i}^{E\_\epsilon} \mathbf{f}(\mathbf{x})d\mathbf{x} \tag{19}$$

where Ei = Event initial; Er = Event restored.


#### **Table 1.**

measures in the calculation of resilience in a deterministic and normalized approach

An evaluation between two groups can result in an isomorphic application of findings from one structure to another. This mapping between groups can yield opportunities to apply known methodologies in an inter-disciplinary manner. The process of verifying an isomorphism requires the identification of elements in each structure and evaluating their equivalence. If equivalence is identified an opportunity for applying the computational framework may exist. The quantification of subsystem resilience was compared to resiliency as used in materials science. Materials science's definition of resiliency includes the concepts of per unit volume, maximum energy, and integration from zero to the elastic limit. The modulus of resilience (Ur) is found from the stress-strain curve measured during the tensile test. Stress (σ) in the stress-strain curve is "the applied force per unit original undeformed cross-sectional area of the specimen" [20] as delineated in Eq. (15).

> *<sup>σ</sup>* <sup>¼</sup> *<sup>F</sup> A*<sup>0</sup>

Young's modulus (E) serves as a measure of stiffness for a solid material. "Because of the difficulty in determining the elastic limit, it is commonly replaced by the proportional limit, which is the stress at which the stress-strain curve is out

> *<sup>E</sup>* <sup>¼</sup> *<sup>F</sup>=<sup>A</sup> ΔL=L*<sup>0</sup>

where F = force; A = actual cross-sectional area; ΔL = amount of change in

"This quantity indicates how much energy a material can absorb without deforming plastically" [20]. Plastic deformation occurs when a material undergoes non-reversible changes in response to applied forces. The use of the stress-strain curve from materials testing is similar to conditions faced by disrupted subsystems regardless of type. Stress is the impact to the material under test, while strain is the

Based on the desire of applying a consistent methodology to quantify resilience regardless of disruption magnitude or subsystem size, the percentage of subsystem disrupted is proposed to achieve a per unit value. The area under the curve will then be integrated from the beginning to end of the disruptive event. Calculus to deter-

Total Area under the Curve ¼

"The modulus of resilience is the strain energy per unit volume absorbed up to the elastic limit for a tensile test and equals the area under the elastic part of the

*F* ¼ *σ* � *A* (17)

*Ur* ¼ <sup>1</sup> <sup>2</sup> *<sup>σ</sup>*<sup>2</sup> *=* ð *=E*Þ (18)

ðEr Ei

f xð Þdx (19)

(15)

(16)

based on concepts from materials science.

*Operations Management - Emerging Trend in the Digital Era*

where F = force; A0 = cross sectional area.

length; L0 = original length of the object.

stress-strain curve" [20].

resulting effects of the stress.

**26**

mine area under the curve is shown in Eq. (19).

where Ei = Event initial; Er = Event restored.

of linearity" [20].

And,

*Parallels between materials science test and protracted subsystem disruptions.*

The area under the curve will then be applied to the maximum percentage of subsystem disrupted.

$$\text{Resiliaency index } \left( \text{RI} \right) = \frac{1}{2} \left( \frac{\left( \text{SD} \right)^2}{\frac{\left( \text{SD} \times A\_t \right) / A\_{\text{nl}}}{\left( D\_t - D\_a \right) / D\_a}} \right) \tag{20}$$

where SD = % of subsystem disrupted; Anl = Area under the curve to nonlinearity; At = Total area under the curve; Da = Duration of average disruption; De = Duration of event disruption.

Protracted subsystem disruptions create stress and strain due to an inability to complete the subsystem aim. The similarities between tensile strength test used in materials science and the need to measure stress and strain subsystems create an isomorphic relationship. **Table 1** shows the parallels between materials science and protracted subsystem disruptions.

The application of the modulus of resilience to a specific subsystem requires the identification of an aim the subsystem exist to accomplish. "Without an aim, there is no system" [21]. The aim should be quantifiable with metrics available for analysis. The data must be accessible in order to serve as the basis for the resilience calculations and will vary based on the subsystem under study. Examples include percentage of successful operations or percentage of end users receiving service. The next section provides an empirical example in applying the modulus of resilience.
