Risk Analysis DOI: 10.1111/risa.12223

Inherent Costs and Interdependent Impacts of InfrastructureNetwork Resilience

Hiba Baroud,1 Kash Barker,1,∗ Jose E. Ramirez-Marquez,2 and Claudio M. Rocco3

Recent studies in system resilience have proposed metrics to understand the ability of sys-tems to recover from a disruptive event, often offering a qualitative treatment of resilience.This work provides a quantitative treatment of resilience and focuses specifically on mea-suring resilience in infrastructure networks. Inherent cost metrics are introduced: loss of ser-vice cost and total network restoration cost. Further, “costs” of network resilience are oftenshared across multiple infrastructures and industries that rely upon those networks, particu-larly when such networks become inoperable in the face of disruptive events. As such, thiswork integrates the quantitative resilience approach with a model describing the regional,multi-industry impacts of a disruptive event to measure the interdependent impacts of net-work resilience. The approaches discussed in this article are deployed in a case study of aninland waterway transportation network, the Mississippi River Navigation System.

KEY WORDS: Infrastructure; interdependent impacts; networks; resilience

1. INTRODUCTION AND MOTIVATION

Infrastructure networks in today’s global land-scape can be characterized as exhibiting the charac-teristics of many complex and large-scale systems: (i)a large number of interacting components and sub-systems, (ii) a large number of decision and statevariables, (iii) complicated, complex, and often non-linear functional relationships, (iv) uncertainty andvariability, (v) hierarchical and/or networked inter-dependencies, (vi) multiple and often conflicting per-formance objectives, (vii) multiple decisionmakers,and (viii) dynamic changes, among others.(1) As such,disruptive events, whether malevolent attacks, natu-

1School of Industrial and Systems Engineering, University ofOklahoma, Norman, OK, USA.

2System Development and Maturity Lab, School of Systemsand Enterprises, Stevens Institute of Technology, Hoboken, NJ,USA.

3Facultad de Ingenieria, Universidad Central de Venezuela,Caracas, Venezuela.

∗Address correspondence to Kash Barker, School of Industrialand Systems Engineering, University of Oklahoma, Norman,OK, USA; tel: 405-325-3721; kashbarker@ou.edu.

ral disasters, manmade accidents, or common causefailures, can have devastating, widespread, and oftenunpredictable, results.

Consider, for example, the August 2003 U.S.blackout, which “contributed to at least 11 deathsand cost an estimated $6 billion,”(2) or the largestblackout in history experienced by India duringAugust 2012, affecting over 600 million people.Against the effects of these events, most researchefforts have been devoted to developing traditionalmeasures of protection (hardening)(3–5) and poli-cies that can be expensive, degrade typical perfor-mance, and are nonreactive. Recent attention hasbeen placed on preparedness, response, and recoveryfrom these events (e.g., in the large-scale homelandsecurity preparedness domain(6)). A perspective re-cently has been collectively referred to in the systemsengineering community as designing for resilience,considered an essential component in the design ofsystems and enterprises.(7)

The importance of having robust and resilient in-frastructure systems has gained the attention of de-cisionmakers and government officials in the past

1 0272-4332/14/0100-0001$22.00/1 C© 2014 Society for Risk Analysis

2 Baroud et al.

decade. Critical infrastructure systems, such as powergrids and transportation systems, have been vulner-able to numerous disruptive events, including natu-ral disasters, willful attacks, and accidents. The De-partment of Homeland Security (DHS) announced aset of grant programs targeting different areas proneto willful attacks or natural disasters,(8) aiming toprovide resources helpful in supporting the NationalPreparedness Goal (NPG) in succeeding in its mis-sion of ensuring “a secure and resilient Nation withthe capabilities required across the whole commu-nity to prevent, protect against, mitigate, respond to,and recover from the threats and hazards that posethe greatest risk.”(9) Further motivation comes fromBoin et al:(10)

If we accept that dominant trends such as globalization,increasing interdependence and complexity, the spreadof potentially dangerous technologies, new forms of ter-rorism, and climate change create new and unimaginablethreats to modern societies, it is only a small step to rec-ognizing and accepting the inherent shortcomings of con-temporary approaches to prevention and preparation. Ifwe cannot predict or foresee the urgent threats we face,prevention and preparation become difficult. The con-cept of resilience holds the promise of an answer.

Resilience is often thought of as the ability ex-hibited by a system to “bounce back” following a dis-turbance. In material science, a modulus of resilienceis also defined to represent the energy absorbed perunit volume of material when stressed to the propor-tional limit.(11) In socioecological systems, resilienceis defined as the magnitude of disturbance that canbe absorbed before the system changes its structureby changing the processes that control behavior.(12,13)

Regarding enterprise systems, Jackson(14,15) definesresilience as the ability of organizational, hardware,and software systems to mitigate the severity andlikelihood of failures or losses, to adapt to changingconditions, and to respond appropriately. In businessterms, resilience has been defined as the ability of anorganization to sustain the impact of a business inter-ruption and recover and resume operations to con-tinue to provide minimum services.(16) With respectto critical infrastructure, the Infrastructure SecurityPartnership(17) noted that a resilient infrastructuresector would “prepare for, prevent, protect against,respond, or mitigate any anticipated or unexpectedsignificant threat or event” and “rapidly recover andreconstitute critical assets, operations, and serviceswith minimum damage and disruption.” In an en-gineering context, Hollnagel et al.(18) promote re-silience engineering as a new paradigm for safety en-

gineering. And many others have defined resiliencein various ways.(19–22)

Several qualitative schema have recently beenoffered for describing resilience,(23–26) with a quanti-tative treatment by Henry and Ramirez-Marquez(27)

describing how system performance is affected bythe change of the state of the system in the pres-ence of a disruption and throughout the recoveryprocess. It is this resilience paradigm, developedfor deterministic(27) and stochastic(28) analyses, thatdrives the resilience cost metrics in this article.

Figs. 1 and 2 illustrate a general approach for vi-sualizing system performance over time when facedwith a disruptive event e j . The states of the sys-tem are depicted across the bottom of Fig. 1: theoriginal system state S0 transitions to disrupted stateSd following event e j , and then to recovered stateSf following a recovery effort. The performance ofthe system is quantified with system service func-tion ϕ (t) (e.g., ϕ (t) could represent commodity flowsalong a waterway network). In Fig. 1, larger valuesof ϕ (t) are preferred (therefore, e j leads to reducedϕ (t)), with the opposite (smaller values of ϕ (t) pre-ferred) depicted in Fig. 2. The reliability, vulnerabil-ity, survivability, and recoverability system descrip-tors, among other details of the state transition, canbe found in Baroud et al.(29) Note that recovered sys-tem performance need not necessarily be the same asthe initial performance prior to the disruptive event.For example, the state of infrastructure following the2010 earthquake in Haiti may be improved relativeto predisruption levels, as part of the recovery activ-ities aiming at helping the infrastructure system re-gain its functionality might improve the deterioratedstate of the system. Further, system performancecould fluctuate over time even when no disruptionoccurs.

We consider resilience to be a time-dependentproportional measure of how the system is perform-ing relative to an as-planned performance level ϕ (t0),namely, in how different the disrupted performancelevel ϕ (td) is from ϕ (t0). Given that R is historicallyreserved for quantifying reliability, resilience is giventhe notation of Я and computed at time t as the ra-tio of the network performance recovered by timet over the loss of the performance after the disrup-tion occurred (Equation (1)). As such, resilience is afunction of the extent of loss experienced at time td(or vulnerability) and the speed at which the systemrecovers (or recoverability).

Я (t) = Recovery (t) /Loss (td) (1)

Interdependent Impacts of Network Resilience 3

Fig. 1. Graphical depiction of state transitions over time with respect to an increasing system service function, φ(t).

A thorough analysis of the recovery dynamics ina network is useful to accurately assess the time(29)

and cost needed for a system to regain its function-ality and improve the risk-informed decision makingin regards to allocating resources after the disruption.Such costs resulting from a disrupted system are dueto inherent costs such as (i) the loss of service, (ii) thecost of restoring the system back to a functional state,and other losses such as (iii) the interdependent ef-fects on industries relying on the disrupted system.(30)

For example, if the power grid was disrupted in a cer-tain region, (i) the loss of service could be quantifiedin terms of energy not supplied (megawatts-hours)to a number of customers, (ii) the cost of restora-tion involves the work crews and equipment neededfor repair, and (iii) one of the interdependent im-pacts could be the loss of production in industriesrelying on the power generation in that area (andbroader to the interdependent relationships outsideof the directly impacted area). Other examples in-clude a disruption in certain river links of a waterwaytransportation network impacting the commodityflow throughout the entire system and in particular,disrupting the functionality of neighboring ports andindustries relying on the import and export of com-modities through the port.

This work deals with cost and impact metricsaimed at assisting risk managers in accurately iden-tifying and quantifying the multiple costs of a dis-ruptive event in the context of building resilience,with particular emphasis given to decision makingunder uncertainty. The primary contribution of thisarticle is the integration of (i) the resilience model-ing paradigm in Figs. 1 and 2 for a disrupted net-work with (ii) an economic interdependency modelto more accurately quantify the resilience trajectoryof indirectly impacted sectors and more effectivelymake network recovery decisions. Section 2 pro-vides the methodological background with respectto stochastic resilience metrics and interdependencymodels. Inherent cost metrics are introduced inSection 3, with metrics of interdependent impacts ofresilience discussed in Section 4. Section 5 providesan application using a case study of an inland wa-terway transportation network, the Mississippi RiverNavigation System. Concluding remarks are given inSection 5.

2. METHODOLOGICAL BACKGROUND

This section details the resilience metrics derivedfrom the depiction of resilience in Figs. 1 and 2, as

4 Baroud et al.

Fig. 2. Alternative depiction of state transitions describing a decreasing system service function, φ(t).

well as a risk-informed interdependency model. Thebackground on these methodologies will motivatethe development of the inherent cost and interdepen-dent impact metrics proposed thereafter. The discus-sion of resilience that follows will be centered aroundnetworks as opposed to general systems.

2.1. Metrics of Network Resilience

Assume that disruptive event e j affects the orig-inal network state S0 at time te. The effect on thenetwork is assessed by quantifying the damage tothe network service function ϕ (·). For example, ifthe network under study is an inland waterway net-work, ϕ (·) could measure commodity flows, notingthat lower service function values are considered un-desirable. After a period of degradation of length(td − te), the network service function is damagedfrom its original state, S0 (with corresponding ϕ (t0)),to a disrupted state, Sd (with corresponding ϕ (td)).That is, the disruptive effect of such an event is quan-tified via the analysis of a function ϕ (t) describing thebehavior of the network as a function of time. Af-ter a disrupted state of length (ts − td), the networkrestoration commences until it reaches a stable sys-tem state, Sf, with corresponding ϕ (tf). Equation (2)provides a more specific quantification of the value of

resilience Яϕ( tr | e j ) evaluated at time tr ∈ (td, tf).(27)

Set D is the set of possible disruptive events.

Яϕ( tr | e j ) =

[ϕ( tr | e j ) − ϕ( td| e j )

][ϕ (t0) − ϕ ( td| e j )]

∀e j ∈ D (2)

This model enables quantifying and tracking thechanges in the network state as a function of time andaccurately observing the network response to therecovery strategies employed. Using this metric,decisionmakers can dynamically assess theirresilience-building decisions during the after-math of a disruption. It could also be used as apreparedness decision tool, whereby risk managersdecide on investments in vulnerability reductionand/or increased recoverability.

We operationalize Яϕ

(tr | e j

)from Equation (2)

with a set of three metrics describing the time re-quired to achieve different resilience and restorationgoals.(28,29,31)

First, the metric time to total network restora-tion, TT(e j ), records the total time spent fromthe point when recovery activities commence, attime ts, up to the time when all recovery activi-ties finalize, TT(e j ). Since these are stochastic met-rics, one can calculate the probability that total

Interdependent Impacts of Network Resilience 5

system restoration is finished before mission time tas PR (t) = P(TT(e j ) ≤ t).

The second metric, time to full network ser-vice resilience, Tϕ(t0)(e j ), records the total time spentfrom the point when recovery activities commence,at time ts, up to the exact time, tf, when networkservice is completely restored. From Tϕ(t0), one candefine the probability that network service restora-tion is finished before mission time tf as PF (tf) =P(Tϕ(t0)(e j ) ≤ tf). Note that TT(e j ) ≥ Tϕ(t0)(e j ), or thetime at which the network is fully restored, is at leastas lengthy as the time until a desired network re-silience, say Я

ϕ(tr |e j ) = 1 (though a different target,

either better or worse than ϕ(t0), may be desired),is achieved. For example, flows along a network canoccur with full capacity despite not all arcs being re-stored: full capacity would suggest full network ser-vice resilience without all network components beingcompletely restored.

Finally, the metric time to α×100% resilience,Tα(e j ), records the total time spent from the pointwhen recovery activities commence, at time ts,up to the exact time, tα , when the system ser-vice is restored to αϕ(t0). From Tα , one can de-fine the probability that network service is re-stored by α×100%, or αϕ(t0), before mission timetα as Pα(tα) = P(Tα(e j ) ≤ tα). This metric providesa means to compare different recovery strategies,determining which strategy achieves α×100% re-silience the quickest holding resilience constant. Sim-ilarly, Я

ϕ(tr |e j ) can be found for different strategies,

holding tr constant.A comparison of the different distributions for

the different resilience time metrics can help deci-sionmakers choose the best recovery strategy thatwould lead to the optimal time to full recovery.

2.2. Model of Interdependent Behavior

Adapted from Hernandez-Fajardo and Duenas-Osorio,(32) the term cascading effects used through-out this article refers to the inoperability that oc-curs internally to a network, often induced by flowredistribution resulting from capacity exceedances,and the term interdependent effects refers to the in-operability that occurs external to the network asa result of a disruption to the network, often ex-perienced in other infrastructure and industry sec-tors. This work integrates the resilience paradigm ofEquation (2) with a risk-informed interdependencymodel to quantify the economic impact of a disrup-tive event on the interdependent industries that rely

on the directly impacted and disrupted network. Thissection reviews the interdependency model used inthis approach.

A widely accepted model for describing the in-terconnected relationships among infrastructure sys-tems and industry sectors is the Nobel-Prize-winningeconomic input-output model,(33) as shown in Equa-tion (3). For a set of n infrastructure and industrysectors, n × 1 vector x quantifies production outputsin each sector, n × n matrix A represents the pro-portional interdependence among sectors (that is, Axrepresents intermediate demand resulting from theproduction of x), and n × 1 vector c provides finalexogenous consumer demand. As such, Equation (3)describes how changes in consumer demand lead towidespread changes in sector production. The exten-sive usage of input-output models is due, in part, tothe availability of interdependency data describingthe interconnected nature of infrastructures and in-dustries in a number of countries,(34) including an ex-tensive data collection effort by the U.S. Bureau ofEconomic Analysis (BEA), which maintains input-output tables at different levels of aggregations.(35)

x = Ax + c ⇒ x = [I − A]−1 c (3)

The input-output model was extended to de-scribe the propagation of inoperability, or the pro-portional extent to which sectors are not performingin an as-planned manner (e.g., reduced productioncapability), through several interdependent infras-tructure and industry sectors.(36) This model, the in-operability input-output model (IIM), is expressed inEquation (4).

q = A∗q + c∗ ⇒ q = [I − A∗]−1 c∗ (4)

Vector q is a vector of infrastructure and indus-try inoperabilities, proportional reductions in pro-duction, describing the extent to which ideal func-tionality is not realized following a disruptive event.An inoperability of 0 suggests that an industry is op-erating at normal production levels, while an inoper-ability of 1 means that the industry is not producing atall. Normalized interdependency matrix A∗ is a mod-ified version of the original BEA-driven A matrixdescribing the extent of economic interdependenceamong a set of infrastructure and industry sectors.A∗ is an interdependency matrix in which every en-try represents how much inoperability is contributedby the column industry to the corresponding row in-dustry due to the interdependent nature of indus-try interactions. The effects of inoperability across

6 Baroud et al.

multiple sectors can be expressed with total eco-nomic losses, Q = xTq, or the amount of productionmade inoperable due to a disruption.

A discrete-time dynamic version of this modelcalculates the inoperability at any point in time us-ing the recursive formula in Equation (5), quantify-ing the temporal nature of how inoperability propa-gates across sectors, then dissipates with recovery.(37)

The inoperability vector, q(t), as well as the vectorof demand perturbation, c∗(t), change in time. An n× n resilience matrix, K, represents the capability ofa certain sector to recover from the disruptive eventand reach a desired performance state.

q (t + 1) = (I − K) q (t) + K [A∗q (t) + c∗ (t)]

(5)

One means to estimate the entries in matrix K isin Equation (6), a result of the dynamic version rep-resented in Equation (5) with no temporal demandperturbations.(37) In this case, K is a diagonal matrixwith zero nondiagonal entries and kii as its diagonalentries. Value qi (0) is the initial inoperability experi-enced in sector i following a disruptive event, qi (Ti )is the desired inoperability state after recovery (as-sumed to be small but nonzero), which requires Ti

time periods to achieve, and a∗i i is the diagonal en-

try in the interdependency matrix. An alternative ap-proach to estimate K that incorporates the resilienceparadigm of Equation (2) is proposed in Section 4.

kii =ln

(qi (0)qi (Ti )

)Ti (1 − a∗

i i )(6)

The IIM and its extensions, included amongthe top 10 contributions to risk analysis from1980 to 2010,(38) have been found in a num-ber of risk-related applications, including inventorydecision making,(39,40) transportation closures,(41,42)

workforce disruptions,(43,44) and electric power out-ages,(45,46) among others.

3. INHERENT COSTS AND IMPACTSOF NETWORK RESILIENCE

Alluded to previously, two primary inherentcosts of network resilience are (i) the cost of lost ser-vice and (ii) the cost of restoring the system back to adesired state. Stochastic measures of these two costsare described here.

3.1. Loss of Service Cost

When a disruptive event occurs, a loss in the ser-vice of the disrupted network is expected, and thechange in the service function quantifies the extent ofloss. For example, in the case of a road transportationnetwork, the performance measure could be the traf-fic flow. Different problems might use different per-formance measures for the same network, with rel-evant performance measures determined by the riskmanager or the decisionmaker.

A disruptive event is assumed to impact a cer-tain number of components in the network. For roadnetworks, components would be bridges or roads;for inland waterway networks, river links, ports, ordams/locks. Impacted components are assumed tocease functioning for a certain period of time, dur-ing which the network is in a disrupted state (Fig. 1).It is assumed that the length of the disrupted state de-pends on the severity of the event. In Baroud et al.,(47)

the intensity of a disruptive event is assumed to fol-low a power-law distribution, suggesting that moresevere events resulting in longer disrupted state peri-ods of time have a lower probability of occurring.

The power-law distribution has been used inseveral studies to model extreme events. Studiesaimed at determining the probability distribution ofthe severity of a terrorist attack agree that accord-ing to empirical data on worldwide terrorist eventsfrom 1968 to 2008, the probability of a terroristevent claiming a number deaths follows a power-lawdistribution.(48–50) Power-law relationships are alsoused to model the severity of natural disasters suchas tornadoes,(51) other severe weather conditions,(52)

and large earthquakes,(53) among others. The distri-bution used to model the severity of a disruptiveevent is presented in Equation (7), where d is thenumber of days the impacted components are notfunctional (i.e., the period of time during which thesystem is in a disrupted state). C is a normalizationconstant ensuring that the area under the curve ofthe probability distribution function is equal to 1 andλ is a scaling parameter.

f (d) = Cd−λ (7)

The loss of service cost is then computed as afunction of the severity of the disruptive event. If theperformance measure is the commodity flow, thenthe loss of service is the aggregate commodity thatwas supposed to flow across the disrupted compo-nents for the duration of the disruption. Equation (8)

Interdependent Impacts of Network Resilience 7

is the performance measure of the network system atthe disrupted state.

ϕ (td) = ϕ (t0) −m∑

i=1

ϕi (t0) × d365

(8)

ϕi (t0) is the original performance measure ofcomponent i prior to disruption e j . It is assumedthat annual flow ϕ(t0) across the network is the sumof individual ϕi (t0) such that flows are not countedmore than once in the sum. If daily commodity flowis available and if a disruption renders m componentscompletely inoperable for d days, then the disruptedflow across the network, ϕ(td), can be measured withEquation (8). Since d follows a probability distribu-tion, simulation techniques (e.g., Monte Carlo) canbe used to construct the probability distribution ofthe loss of service.

3.2. Total Network Restoration Costs

Restoring a disrupted network is not only timeconsuming but costly as a consequence. Careful pre-paredness strategies and accurate resource allocationshould be made to determine the right amount of re-sources invested at the right time.

To develop a probability distribution for the costof network restoration, it is assumed that the cost ofrepairing one component is stochastic. More specifi-cally, we introduce Ci (e j ) = C j

i to be the cost of re-pairing link i disrupted by event e j . C(e j ) defines thevector of costs for all the links disrupted by the sameevent. The individual component’s restoration costprobability distribution is described in Equation (9).Note that this distribution could also be a functionof the severity of the event; the relationship betweenthe component restoration cost distribution and theseverity of the event would be case dependent. Oneparticular example will be explored in the case studyof this article.

C ji =

⎧⎨⎩C j

i |P(

cs < C ji ≤ cr

)=

cr∫cs

f(

c ji

)dc j

i

⎫⎬⎭ (9)

The total network restoration cost would thenbe the sum of the individual components’ restorationcosts. However, note that taking the sum assumesthat component repair is performed in series. To ac-count for potential parallel recovery activities, a con-stant factor θi is introduced that would depend on theorder in which components are repaired. This factorwould have higher values in cases where more com-

ponents are repaired in parallel and would be multi-plied by the individual component’s cost of restora-tion, as shown in Equation (10). This is similar to theidea of a weighted average, where more weight (inthis case higher cost) is given to components repairedin parallel.

Ctotal(e j) =

∑i

θi Cj

i (10)

Similar to the loss of service cost, the total net-work restoration cost’s probability distribution isconstructed by means of simulation, such as MonteCarlo simulation.

Ultimately, decisionmakers would be interestedin a metric describing the overall aggregate cost ofthe entire disruptive event that covers both the lossof service and the cost of restoration. Such a metriccan also be computed using simulation of each of theother metrics, provided that they are expressed in thesame unit, either dollars, tons of commodity flow, orothers.

3.3. Interdependent Impacts of Network Resilience

A disruptive event impacting an infrastructurenetwork does not only have impacts on the networkitself but also on the surrounding regional infrastruc-ture systems and industries related to and relyingupon it. The costs discussed previously are consid-ered to be inherent costs related directly to the dis-rupted network, and indirect impacts would be lossesand costs incurred by infrastructures and industriesrelated to the disrupted system that were not neces-sarily directly impacted by the disruptive event.

In order to quantify for those indirect losses, anintegration of the resilience paradigm in Equation (2)with the discrete-time dynamic interdependencymodel in Equation (5) is developed. The original in-terdependency model considered a resilience matrixK that is held constant throughout the recovery time.That matrix represented the capability of the econ-omy to restore its functionality without taking intoconsideration the resilience of the underlying dis-rupted physical infrastructure. Hence, the K matrixin Equation (5) assumed Я(t) = 1 for the disruptedsector, and is not effectively accounting for the re-covery of the underlying physical infrastructure thatcaused the economic perturbation in the first place.

The approach considers a dynamic version of theresilience matrix that governs the trajectory of in-terdependent recovery, introducing a matrix whosevalues are functions of time, K(t). Further, the

8 Baroud et al.

resilience matrix is updated with information regard-ing the trajectory of resilience as a function of time,as shown in Equation (11). Matrix K is considered tobe a baseline matrix of recovery trajectory whose en-tries are computed according to Equation (6) that up-dates the resilience matrix at each point in time withthe cumulative resilience of the physically disruptedsystem. It is also assumed that the perturbation is ex-pressed through a production inoperability; hence,c�(t) = 0,∀t . The new resilience-based dynamic in-terdependency model is expressed in Equation (12),where 0 < Я

ϕ(t |e j ) ≤ 1.

K (t) = KeЯ

ϕ( t |e j ) (11)

q (t + 1) = (I − K (t)) q (t) + K (t) [A�q (t)] (12)

Also, the disrupted system might recover be-fore the rest of the economy does, for which caseЯ(t) = 1 for all t ≥ tr , where tr is the time at whichthe physically disrupted infrastructure network is re-covered. In some other instances, the economy’s re-covery does not start until the physically disruptedsystem is fully recovered, which is the case of a portclosure, for example. In others cases in which com-ponents of the system (nodes or links) are disrupted,the recovery of the economy would overlap with therecovery of the physically disrupted system; the rela-tion in Equation (11) takes into account the quanti-tative analysis of such overlap and how the recoverystrategy of the physically disrupted system impactsthe recovery trajectory of the entire economy.

4. CASE STUDY: INLAND WATERWAYNETWORK

The methodology discussed in this article is ap-plied to an inland waterway transportation network,specifically the Mississippi River Navigation System.The system is a series of river links connecting thedifferent inland ports in the United States. This typeof transportation system has a special feature thatdifferentiates it from other regular network systems.Generally, there is usually single point to point ac-cess: there is no redundancy due to the nature of thelinks being a part of the river.

The nation’s economy depends strongly on thiswaterway network, with nearly 25 billion tons of an-nual commodity flow circulating through the net-work. The National Waterway Network (NWN) iscomposed of a large number of links and nodes. Alink represents either a shipping lane or simply a path

in open water, and a node could be a facility suchas a port, lock, dam, or perhaps another intermodalterminal. The case study analyzes the resilience ofa known number of links that might become com-pletely inoperable due to a disruptive event. A U.S.Army Corps of Engineers(54) database was used toconstruct the 3,046 links of the Mississippi River Nav-igation System network, as shown in Fig. 3.

The Mississippi River is prone to different typesof disruptive events, including periods of drought(55)

and flooding.(56) Closing sections of the river impactsthe nation’s economy by incurring losses to a largenumber of industries relying on the shipments thatare being delayed, with macrolevel, interdependentlosses becoming quite significant.(41,42) As such, re-silience planning for inland waterway networks is ofhigh importance.

4.1. Parameters and Assumptions

A few assumptions were made in this case study.There is one disruptive event impacting four specificlinks of the river, highlighted in red in Fig. 3 (col-ors visible in online version). Note that the four linksrepresent in fact 10 segments of the river accordingto the data of the Army Corps of Engineers. Thosesegments were condensed into four links due to sim-ilar commodity flow capacities and proximity of theirlocation; the event in this case is disrupting opera-tions along a stretch of 141.6 miles of the river lo-cated in the area surrounding the port of Catoosa inOklahoma, resulting in delays in the flow of importsand exports to and from Oklahoma. The individualcomponent restoration cost follows a uniform dis-tribution, with a multiplicative relationship with theseverity of the event (Equation (13)). The severity isexpressed in terms of days of disrupted state, d, andfollows a power-law distribution (Equation (14)).

Ci ∼ UNI (0, 1) × d (13)

P (d) = (1.5 × d1.5

min

)d−2.5 (14)

Generally, extreme events that are modeled witha power-law distribution result in an estimated scaleparameter λ ranging between 2 and 3 (as discussedin Section 3.1). It is assumed that λ = 2.5. It is alsoassumed that the severity of the disruptive event isbounded with a minimum and a maximum numberof days during which the links are disrupted, betweendmin = 5 and dmax = 30 days. The values chosen forthe parameters can be thought of as starting values,and sensitivity analysis should be done to observe the

Interdependent Impacts of Network Resilience 9

Fig. 3. Inland waterway network of the Mississippi River Navigation System.

model’s outcome over a specific range for the pa-rameters. For example, λ controls the spread of theprobability distribution, with larger values resultingin a larger spread of the likelihood of more severeevents. Risk-averse decisionmakers might considerlarger values to design preparedness options that ac-count for extreme events expressed by the upper tailof the distribution. Also, since the application per-tains to a transportation network measured with thenumber of days transportation flow capability is re-duced, there is a need to specify a threshold for theseverity of the event to avoid unreasonable impactsthat alter the preparedness decision and resource al-location without a significant tradeoff of protection.

In the interdependency model, industry inoper-ability is assumed to be the result of imports failingto reach the port and causing a shortage in the ma-terial needed to produce in the industries relying onthat commodity. As such, inoperability in industry iis then the ratio of its imports to its total produc-tion output, as an industry can only be as productive

as its most disrupted supplier.(41,42) Equation (15) il-lustrates this thought process as the maximum initialinoperability experienced in industry i. The yearly es-timate of imports from industry j is mj , the total pro-duction output for industry j is xj , and the numberof industries that typically circulate on the disruptedlinks is h. This maximum perturbation correspondsto a disruption lasting for a year; therefore, for a dis-ruption of duration d days, the initial inoperability iscomputed in Equation (16). This approach has beenused in previous interdependent impact analyses fordisruption of inland waterway ports.(42,57)

qmaxi = max

(m1

x1, . . . ,

mj

xj, . . . ,

mh

xh

)(15)

qi (0) =(

d365

)qmax

i (16)

Commodity flow data for each link in the Mis-sissippi River Navigation System are provided bythe U.S. Army Corps of Engineers, composed of the

10 Baroud et al.

Table I. Annual Commodity Flow (in Tons) Across the FiveLinks Chosen from the Mississippi Navigational System Network

Link ID Total Annual Commodity Flow

500 5,007,904600 240,925709 4,766,979810 6,236,462

Table II. Four Recovery Sets Considered for the Restoration ofDisrupted Waterway Links

Recovery Set Description

W1 Repair links in series in the order: 500–600–709–810W2 Repair link 500 first, then 600 and 709 in parallel in

the second order, and 810 in the third orderW3 Repair link 500 first, then links 600, 709, and 810 in

parallel in the second order

yearly tonnage of commodity flow, with the commod-ity flows of five chosen links provided in Table I. Weconsider the daily commodity flow to be the annualflow divided by 365 days.

We examine three possible recovery activitiessets, described in Table II, for the four disrupted linkswith IDs 500, 600, 709, and 810.

4.2. Resilience and Restoration Time Results

We first observe one possible realization for theresilience trajectory over time. One observation isdrawn from a triangular distribution with parametersrandomly selected for each link, and we compute theresilience at each point in time using Equation (2)based on the three strategies in Table II. Note the dif-ference in the time required to achieve full networkresilience, portrayed in Fig. 4. W1 requires approxi-mately 15 additional time units of recovery activitieswhen compared with W2, and almost 35 additionaltime units when compared with W3. Also, W2 andW3 differ by 20 time units. Clearly, as more links arerepaired in parallel, the trajectory toward a fully re-silient network tends to be faster, leading to a shorterrecovery time. Note that the step function is used inthis article to describe the resilience over time, whileother types of linear and nonlinear functions can beinvestigated.(58)

To more effectively represent the variability inthe underlying model parameters, we simulate 2,000scenarios of possible disruptive events and look atthe cumulative distribution function (cdf) of time to

full restoration under the three strategies. The con-clusions align with the observations from Fig. 4 asmore links are repaired in parallel, the overall timeto full network restoration decreases. Fig. 5 suggeststhat W3 generally dominates the other two strategiesfor much of the length of the cdfs. For example, theprobability that full network restoration occurs be-fore 20 days is approximately 0.2 for W1, 0.60 forW2, and 0.80 for W3. Only the average point esti-mate of the simulated times to full recovery suggestsa similar outcome. There is, on average, a decreaseof 9.5 time units between W1 and W2, 15.4 betweenW1 and W3, and 5.9 between W2 and W3. It is clearthat going from strategy W1 to strategy W2 has agreater impact, almost double, than going from strat-egy W2 to strategy W3. To make a better decision re-garding which strategy to choose, risk managers lookat other objectives, such as costs and interdepen-dent impacts, to determine the tradeoff between thestrategies.

4.3. Inherent Cost Results

Mentioned previously, the cost of a disruptiveevent has several dimensions: the loss of service cost,the cost of network restoration, and the cost incurredby interdependent impacts. Using the 2,000 simula-tions of possible scenarios for disruptive events, weconstruct the probability distribution function (pdf)and cumulative probability distribution (cdf) for thetwo cost metrics: the loss of service cost in Figs. 6(A)and (C), and the total restoration cost in Figs. 6(B)and (d) for recovery strategy W1. Given the natureof the power-law distribution, smaller values of costare more likely to occur with extreme values becom-ing increasingly less likely. Also, note the differencein the units of the cost between the loss of service costand the network restoration cost: the loss of servicecost is measured in terms of tons of commodity flowand the restoration cost is measured in thousands ofdollars. To commensurate costs, the commodity flowin tons would be converted into dollar amounts (ifavailable) before adding the two random variablesand generating a distribution for the total cost of thedisruption.

The loss of service cost depends solely on theseverity of the event and is not impacted by the dif-ference in the recovery strategies. However, fromEquation (10), different strategies have differentcosts that increase as more links are repaired in par-allel, which impacts the cost incurred by the recov-ery activities. The individual cost for repairing each

Interdependent Impacts of Network Resilience 11

Fig. 4. Inland waterway network of the Mississippi River Navigation System.

Fig. 5. Cumulative distribution function for the time to full network restoration.

link is multiplied by the number of links that arebeing repaired in parallel with this particular link.Given the manner in which this is modeled in Equa-tion (10), Fig. 7 suggests that W3 incurs the highestcost with three out of the four disrupted links be-ing repaired in parallel, while W1 has the lowest costwith all links being repaired in series. Also, note thatgoing from W2 to W3 involves a larger investmentthan going from W1 to W2. In fact, on average, W2costs more than W1 by 9.4 cost units and by 19.2 forW3, this will help decisionmakers in choosing a re-covery strategy by assessing the tradeoff between thestrategies.

4.4. Interdependent Impact Results

We now examine the interdependent impacts ofa disruptive event in the waterway network on theindustries relying on the commodities flowing on thenetwork, in light of the resilience quantification andthe three strategies considered above.

Table AI in the Appendix lists the combined es-timates for the annual exports and imports in tonsof the eight primary industries using the Port ofCatoosa among the states that do the most com-merce using the port.(41) Sixty-three BEA sectors,listed in Table AII of the Appendix, were consid-ered in the multi-industry impact analysis.(35) Fig. 8

12 Baroud et al.

Fig. 6. Approximate pdf results for 2,000 simulations of (A) the loss of service cost and (B) the network restoration cost, along with theirrespect cdfs in (C) and (D).

Fig. 7. Cumulative distribution function for the network restoration cost.

depicts the relationship among (i) network resilience,Я

ϕ( t | e j ), represented as a dark line, and (ii) inop-

erability of a selected number of industries, qi (t),represented with lighter gray lines. The selected in-

dustries are those with imports flowing primarilyon the disrupted links: food and beverage and to-bacco products (FBT), petroleum and coal products(PC), chemical products (CH), nonmetallic mineral

Interdependent Impacts of Network Resilience 13

Fig. 8. Resilience (black line, right vertical axis) and sector inoperabilities (gray lines, left vertical axis) over 50 time periods for the threerecovery strategies.

14 Baroud et al.

Fig. 9. Average economic losses experienced in each of the six primary waterway industries for the three network recovery strategies.

products (NMM), primary metal products (PM), andfabricated metal products (FM). Originally consid-ered over 120 days in Fig. 5, we reduce the numberof days along the horizontal axis in Fig. 8 to betterillustrate (i) when Я

ϕ( t | e j ) approaches 1, or full net-

work resilience, and (ii) when qi (t) approaches 0 forthe selected industries. Inoperability is depicted onthe vertical axis on the left, while resilience, rang-ing from 0 to 1, appears on the vertical axis on theright.

Naturally, resilience as measured by ϕ (t) is in-creasing with time, while inoperability is decreas-ing. For these particular recovery examples, the indi-vidual sectors recover faster with strategies W2 andW3 with steeper decreasing trajectories for the in-dividual industries inoperability. This is due to theparallel recovery activities in strategies W2 and W3speeding the recovery of the disrupted network andhence resulting in a faster recovery for the rest ofthe economy. Also, note that for recovery strate-gies W1 and W2, the full recovery of the economicsectors almost aligns with the full recovery of thedisrupted system, while for strategy W3, the dis-rupted network recovered very fast and the econ-omy fully recovers shortly after that. Finally, for

one of the sectors, the inoperability continues to in-crease at the beginning, primarily due to the rel-atively high initial inoperability resulting in an in-finitesimal impact of the resilience matrix on therecovery process. Such observation might trigger aneed for system hardening or extra resources if de-cisionmakers want this sector to start to recoversooner.

When economic losses are aggregated acrosstime, that is, the cumulative effect of inoperabilitymultiplied by the as-planned output of each indus-try, the effect of recovery strategy can impact indus-tries in different ways. The average economic lossesfrom 2,000 simulations are depicted in Fig. 9 for thethree strategies. The same pattern is seen across allthree strategies: the petroleum and coal products in-dustry experiences the most economic losses by far,with fabricated metal products next, and the chemi-cal products industry affected the least in economicterms. However, the extent to which these industriesare impacted does depend on the strategy, with W3resulting in the fewest losses across industries. Sucha breakdown can point a decisionmaker in the ap-propriate direction when patterns point to key indus-tries.

Interdependent Impacts of Network Resilience 15

Fig. 10. Economic losses across the six primary waterway industries (darker curvilinear function, left vertical axis) and network resilience(lighter step function, right vertical axis) over time for the three recovery strategies and four disruptive scenarios. Note that the economicloss axis is held constant for the same disruptive scenario.

Figs. 8 and 9 provide average behavior when d,the number of days the waterway links are disrupted,is treated as a random variable. Fig. 10 focuses onfour particular disruption lengths: d = 5, d = 10, d =15, and d = 20. Behavior of economic losses acrossall industries is depicted with the blue curve, corre-sponding to the left axis. The trajectory of resilienceover time is depicted with the green curve and theright axis ranging from 0 to 1. Note that resiliencereaches 1 in a shorter time as we go from W1 to W3for one particular disruptive scenario, and it takeslonger for scenarios with larger impacts. Also, eco-nomic losses increase as the disruptive event’s sever-ity increases, but they decrease as recovery strategiesbecome faster. Total economic losses range roughlyfrom $45 to $1,200 million for disruptions lastingfrom 5 to 20 days under strategy W1, for example.The decrease in total economic losses for faster re-

covery strategies can be observed by examining plotsof the same disrupted scenario as the y-axis of the to-tal economic loss is the same across different recov-ery strategies. Hence, observations in the plots be-low align with the conclusions from Figs. 8 and 9. Afaster recovery strategy means that at a certain pointin time, the resilience of the disrupted system is largerthan one of a slower recovery strategy, and this re-sults in a more resilient industries being able to re-cover faster and this is portrayed by larger entries inthe resilience matrix, according to Equation (11).

In order to compare the three recovery strate-gies W1, W2, and W3, interdependent impacts fora particular disruption scenario (d = 10) are exam-ined further. Consider first the total economic loss,an aggregation of the losses incurred across all sec-tors at each point in time from the disruption throughrecovery. This function is cumulative; therefore,

16 Baroud et al.

Fig. 11. Total economic loss computed under three recovery activities strategies.

Fig. 12. Impact on individual sector inoperability for adopting strategy W2 as opposed to W1.

naturally exhibits an increasing pattern, as illustratedin Fig. 11. Note that the total economic loss is as-sessed to be larger under strategy W1, and W3 re-sults in the lowest estimate for the total economicloss. Note that the different recovery strategies areimpacting the interdependency model through (i) theresilience trajectory and (ii) the time to full recov-ery. A faster recovery of the disrupted system leadsto less total economic losses incurred as the econ-omy can recover faster and more effectively. How-ever, the faster the strategy is, the more costly it is.

Rather than a multi-industry economic loss per-spective, focus could be given to a particular industry.In particular, we look at the industries whose com-modities flow along the disrupted waterway links:

food and beverage and tobacco products (FBT),petroleum and coal products (PC), chemical prod-ucts (CH), nonmetallic mineral products (NMM),primary metal products (PM), and fabricated metalproducts (FM). Figs. 12 and 13 are plots of the dif-ference in the inoperability of the individual sectorswhen considering strategies W1 and W2, and strate-gies W2 and W3, respectively.

In both cases, the sector of primary metal prod-ucts is the most impacted by the change of strate-gies, while the rest of the sectors have comparableand a smaller change in the estimated inoperability.Note that all observe the same pattern in the differ-ence in inoperability over the time to full network re-covery. This pattern suggests that opting for a faster

Interdependent Impacts of Network Resilience 17

Fig. 13. Impact on individual sector inoperability for adopting strategy W3 as opposed to W2.

Fig. 14. Sector inoperability over time for the primary metal products industry for the three recovery strategies.

recovery strategy is not always beneficial. After timet = 15, the magnitude of the tradeoff of switch-ing to W2 from W1 starts to decrease, suggestingthat this might be a good time for decisionmak-ers to switch back to a cheaper strategy. A similarconclusion can be drawn from the comparison be-tween W3 and W2, with an overall much smallerdifference in the inoperability between the twostrategies.

Fig. 14 shows the inoperability trajectory for thesector of primary metal products under the three dif-ferent recovery strategies. The results shown complywith Fig. 11, total economic loss under the three re-covery strategies. Adopting strategy W1 results in anestimation of a largest inoperability, while W3 results

in a much steeper decreasing trajectory for the inop-erability of the primary metal products sector.

With such an extensive analysis, decisionmakershave a range of metrics to consider in order to choosethe best recovery strategy that balances cost and riskmanagement. While the example above consideredthree recovery strategies, a more exhaustive set ofstrategies could be considered and compared witha stochastic ordering technique (e.g., the Copelandscore method(29)). Using the metrics above, decision-makers can examine the possible strategies that min-imize the cost (W1), or minimize the total economiclosses (W3), or a combination of the two based onhow a strategy is significantly better than the otherover time.

18 Baroud et al.

5. CONCLUDING REMARKS

Risk managers preparing for all such sourcesof disruptive events must plan for the intercon-nected relationship of infrastructure networks withthe industries that rely upon them. Most work ininfrastructure networks addresses esoteric graphtheoretic measures of topology (e.g., centrality,betweenness) that may provide little insight fordecision making.(59) However, the ultimate useful-ness of understanding interdependent effects for asustainment decisionmaker is not just a descriptorof physical damage, but of economic interruption(the result of a lack of functionality).(60,61) That is,the benefit of physical models of interdependence islost unless they ultimately translate into (i) dollarsof losses incurred, and (ii) extent of and duration ofsystem inoperability.

This article, which progresses from previouswork by the authors, presents a stochastic approachto compute three metrics of the resilience of an in-frastructure network following a disruption: (i) theloss of service cost, (ii) the total network restora-tion cost, and (iii) the cost of interdependent im-pacts. These three metrics extend from prior workin stochastic network resilience.(29,31) The first twometrics are modeled using simulation of probabil-ity distributions. The economic impacts of a dis-ruptive event is a well-studied topic,(22,62-65) but thethird metric developed here represents a first step inmeasuring the broader multi-industry impacts of re-

silience in infrastructure networks, integrating a net-work resilience model and an economic interdepen-dency model.

A case study involving the disruption of linkson a waterway infrastructure network illustratesthese concepts and results demonstrate the impor-tance of considering such measures in risk-informeddecision-making problems. Incorporating resiliencein the interdependency model is helpful to accu-rately assess the patterns in the cost metrics overtime as the system is recovering. Strategies dif-fer in their cost of implementation and interdepen-dent impact, allowing a decisionmaker to under-stand tradeoffs among different objectives. In thisparticular example, the petroleum and coal prod-ucts industry was most impacted, on average, by thedisruption, measured by a stochastic duration ofcommodity flow stoppage, and the interdependentinoperability in the primary metal products indus-try was most affected by the change in recoverystrategy.

Future work includes the resilience-based anal-ysis of a more extensive set of disruptive scenariosand recovery strategies, as well as the exploration ofnetwork examples where disruptive events are not aslocalized as inland waterways. Further, particularlyfor the inland waterway case study, further valida-tion of the model and its usefulness in guiding net-work recovery strategies through conversations withstakeholders (e.g., U.S. Army Corps of Engineers) issought.

Table AI. Estimates of Annual Tonnage of Exports and Import Through the Port of Catoosa for 2007

Industry

Food/ Petro/ Chem Primary Fab Misc.State beverage coal products Nonmetallic metals metal Machinery manfg Total

Exports from AL 36,247 36,247OK IL 38,267 38,267

KY 60,000 60,000LA 547,496 127,057 3,200 677,753MS 237,900 237,900TX 20,967 8,236 1,600 30,803

Imports to AL 153,068 12,595 165,663OK AR 7,620 7,620

IA 6,426 6,426IL 4,269 7,210 11,479LA 8,488 45,928 438,338 86,090 6,917 585,761MS 5,377 5,377OH 50,399 3,973 54,312

Total 602,926 193,952 748,825 45,887 289,557 23,485 11,436 1,600 1,917,668

APPENDIX

Interdependent Impacts of Network Resilience 19

Table AII. The 63 BEA Industry and Infrastructure Sectors Comprising the Illustrative Examples

Industry Description Industry Description

1 Farms 33 Transit and ground passenger transportation2 Forestry, fishing, and related activities 34 Pipeline transportation3 Oil and gas extraction 35 Other transportation and support activities4 Mining, except oil and gas 36 Warehousing and storage5 Support activities for mining 37 Publishing industries (includes software)6 Utilities 38 Motion picture and sound recording industries7 Construction 39 Broadcasting and telecommunications8 Food and beverage and tobacco products 40 Information and data processing services9 Textile mills and textile product mills 41 Federal Reserve banks, credit intermediation, and related activities10 Apparel and leather and allied products 42 Securities, commodity contracts, and investments11 Wood products 43 Insurance carriers and related activities12 Paper products 44 Funds, trusts, and other financial vehicles13 Printing and related support activities 45 Real estate14 Petroleum and coal products 46 Rental and leasing services and lessors of intangible assets15 Chemical products 47 Legal services16 Plastics and rubber products 48 Miscellaneous professional, scientific, and technical services17 Nonmetallic mineral products 49 Computer systems design and related services18 Primary metals 50 Management of companies and enterprises19 Fabricated metal products 51 Administrative and support services20 Machinery 52 Waste management and remediation services21 Computer and electronic products 53 Educational services22 Electrical equipment, appliances, and components 54 Ambulatory health-care services23 Motor vehicles, bodies and trailers, and parts 55 Hospitals and nursing and residential care facilities24 Other transportation equipment 56 Social assistance25 Furniture and related products 57 Performing arts, spectator sports, museums, and related activities26 Miscellaneous manufacturing 58 Amusements, gambling, and recreation industries27 Wholesale trade 59 Accommodation28 Retail trade 60 Food services and drinking places29 Air transportation 61 Other services, except government30 Rail transportation 62 Federal government31 Water transportation 63 State and local government32 Truck transportation

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