terveecd Ya

ral anic inoive eved secat evg theinterdr senssensinum

ment;

through improved resilience e.g., recovery at an acceptable time extent to which a model accurately represents the real system

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d.and cost in both physical systems Haimes et al. 2008 and soci-etal systems Boin and McConnell 2007. There is a need to quan-tify the efficacy of such preparedness and other risk managementstrategies Haimes and Chittister 2005. Further, measuring theefficacy of critical infrastructure preparedness strategies must ac-count for interdependencies among these infrastructures Zimmer-man 2005; Heal et al. 2006. An approach to quantifying theefficacy of preparedness strategies for interconnected sectors ofthe economy is the dynamic inoperability input-output modelDIIM Haimes et al. 2005a,b; Lian and Haimes 2006.

Successfully modeling the effects of natural and malevolentman-made hazards on a set of interdependent infrastructure sec-tors can be hampered by uncertainty. Haimes 2004 provides twomajor sources of uncertainty in modeling a system: uncertainty in

Haimes and Hall 1977. The results of the DIIM, described indetail in a subsequent section, are particularly susceptible to er-rors in model parameters that describe interdependencies amonginfrastructure sectors.

Addressing uncertainty in modeling system interdependenciesleads to several benefits. Accounting for uncertainty in modelparameters can result in a more accurate representation of a com-plex, large-scale system, as behavioral uncertainty is a commoncharacteristic of such systems Haimes 1982. Modeling the ef-fects of a disruptive event among several interdependent infra-structure sectors, or the interdependent components in aneconomic system, with a model containing only deterministic pa-rameters may be unrealistic. We advocate the use of the DIIM notnecessarily for predictive purposes but as an approximation withwhich to compare risk management strategies aimed at reducingthe adverse effects of disruptive events. Indeed, accounting foruncertainty in the DIIM allows for more informed risk manage-ment policymaking, as strategies are more accurately assessed.Pannell 1997 discusses a number of benefits of sensitivity analy-sis in decision making, including the ability to test the robustnessof an optimal solution, investigating suboptimal policies, and de-veloping recommendations that are flexible in different scenariose.g., disruptive events.

This paper analyzes the uncertainty associated with modelingthe recovery of interdependent sectors of the economy followinga disruption. We evaluate the sensitivity to uncertain interdepen-

1Lecturer, School of Industrial Engineering, Univ. of Oklahoma, 202West Boyd St., Rm. 124 Norman, OK 73019 corresponding author.E-mail: kashbarker@ou.edu

2L.R. Quarles Professor of Systems and Information Engineering Di-rector, Center for Risk Management of Engineering Systems, Univ. ofVirginia, 112 Olsson Hall, Charlottesville, VA 22903.

Note. This manuscript was submitted on February 28, 2008; approvedon April 7, 2009; published online on November 13, 2009. Discussionperiod open until May 1, 2010; separate discussions must be submittedfor individual papers. This paper is part of the Journal of InfrastructureSystems, Vol. 15, No. 4, December 1, 2009. ASCE, ISSN 1076-0342/2009/4-394405/$25.00.

394 / JOURNAL OF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2009Uncertainty Analysis of InInfrastructure Reco

Risk-Based DKash Barker1 an

Abstract: The development of preparedness strategies for natucompare such investments is fraught with uncertainty. The dynamthat propagates through interdependent sectors following a disruptto quantify the efficacy of preparedness strategies for interconnectDIIM with a multiobjective approachthe uncertainty DIIMthdency and its impact on projected economic loss calculated usinprojected economic loss and on their sensitivity to changes in theanalyses are discussed, where sectors are ranked according to theition of key sectors allows decision makers to focus on certainstrategies. The models developed in this paper are illustrated with

DOI: 10.1061/ASCE1076-0342200915:4394

CE Database subject headings: Infrastructure; Risk manage

Introduction

Recent homeland security planning documents e.g., Departmentof Homeland Security 2004, 2006 motivate the development ofpreparedness strategies for critical infrastructure against naturaland malevolent man-made hazards. The focus on preparednessactivities promotes rapid recovery of infrastructure systemsJ. Infrastruct. Syst. 200dependencies in Dynamicry: Applications inision Makingcov Y. Haimes2

d malevolent man-made hazards and approaches with which toperability input-output model DIIM quantifies the inoperabilityent and then diminishes with time. This approach has been showntors of the economy. Work presented in this paper strengthens thealuates the inherent uncertainty in the parameters of interdepen-DIIM. Preparedness strategies can then be compared based on

ependent nature of infrastructure sectors. Additionally, key sectoritivity to changes in interdependent relationships. Such enumera-tive infrastructure sectors for the development of preparednesserical examples.

Decision making; Economic models; Disasters.

the system itself and uncertainty in the ability of the modeler tocapture the behavior of the system. This second source of uncer-tainty, also referred to as epistemic uncertainty Pate-Cornell1996, is of interest in this discussion and is characterized byseveral types of errors, including those in model topology, modelscope, data, and, most important for this work, model parametersHaimes 2004. The choice of such parameters determines the9.15:394-405.

Table 1. General Representation of Interindustry Flows of Goods in Dollars

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d.dency parameters in the DIIM, which is discussed in the Inoper-ability Input-Output Model: Foundation and Extensions section,with a multiobjective approach that strengthens the model andallows for structural changes in its interdependencies. The multi-objective approach, referred to as the uncertainty DIIM U-DIIMand discussed in the Sensitivity of DIIM Metrics to UnspecifiedChanges to Matrix section, provides a general means for robustrisk-based decision making by minimizing total economic lossand sensitivity of this loss with respect to changes in sector inter-dependencies by integrating the DIIM with the uncertainty sensi-tivity index method USIM Haimes and Hall 1977; Li andHaimes 1988. The USIM approach is described in the Uncer-tainty Sensitivity Index Method section. Another useful result ofthe USIM is the calculation of sensitivity indices for each sector,allowing for the ranking of sectors according to their sensitivitywith respect to changes in sector interdependencies. Such insightinto sector behavior provides a key sector analysis approach, pro-vided in the Key Sector Analysis section. The approaches areillustrated with examples exploring transportation, utilities, andother infrastructure sectors.

IIM: Foundation and Extensions

The inoperability input-output model IIM Santos and Haimes2004; Santos 2006 serves as the fundamental model for interde-pendency analysis in this work. This section discusses the IIM, itsfoundation, the input-output model, and an extension, the DIIMHaimes et al. 2005a,b; Lian and Haimes 2006.

Input-Output ModelWassily W. Leontiefs first discussion of input-output economicswas published in 1936, providing a numerical description of theAmerican economic structure Dorfman 1973. The view of Le-ontief, who was awarded the Nobel Prize in Economics in 1973Leontief 1936, 1951a,b, 1966, of this economic structure hasindividual industry sectors being interconnected with commoditytransactions. More than sixty countries maintain current input-output accounts of their economies Lahr and Dietzenbacher2001. Input-output models have long been advocated in the studyof large-scale systems e.g., Sage 1977.

The observed monetary value of commodity flow in dollarsfrom Sector i selling sector to Sector j purchasing sector isdenoted by zij. Table 1 provides the general representation offlows, demand, value added, and total output in an economy of nsectors. Note that the n x n area of Table 1 is referred to as Z, thematrix of commodity flows, and will become important in subse-quent discussions.

Selling sector

Purchasing sector

1 2 . . . j1 z11 z12 z1j2 z21 z22 z2j] ] ] ]i zi1 zi2 zij] ] ] ]n zn1 zn2 znjValue added w1 w2 . . . wjTotal input x1 x2 . . . xjJOURNAL O

J. Infrastruct. Syst. 200The technical coefficient, or the ratio of inputs, zij, to outputs,xj, is defined in Eq. 1

aij =zijxj

1

This relationship for all n sectors is provided in matrix form inEq. 2, where x is the vector of total outputs, A is the matrix oftechnical coefficients, and c is the vector of final demand. This isthe traditional input-output model

x = Ax + c x = I A1c 2

The discussion of the foundational input-output model is lim-ited here. The reader is encouraged to consult Miller and Blair1985 for more details.

Inoperability Input-Output ModelThe seminal work in modeling risk, or inoperability, using aninput-output framework is attributed to Haimes and Jiang 2001and Jiang and Haimes 2004. The demand-reduction IIM Santosand Haimes 2004; Santos 2006 developed the IIM methodologyfurther for the specific purpose of using Bureau of EconomicAnalysis BEA data to quantitatively assess the reduction of out-put from each sector resulting from a perturbation in demand. Thedemand-reduction IIM is presented in Eq. 3. Note that all gen-eral input-output equations provided in this paper assume aneconomy of n sectors, resulting in matrices of size nn andvectors of length n, and that all vectors, unless otherwise noted,are column vectors

q = Aq + c q = I A1c 3

Mathematical relationships exist between the components ofthe IIM and the components of the traditional Leontief input-output model, which allow for the use of the extensive commod-ity flow data collected by the BEA. The vector q represents theinoperability vector, the elements of which measure the propor-tion of unrealized production per as-planned production result-ing from a reduction in demand. Inoperability, which alsoconnotes the level of dysfunctionality, is analogous to the idea ofunreliability from the field of reliability engineering see, e.g.,McCormick 1981. The demand perturbation is expressed byvector c, whose elements represent the difference in as-planneddemand and perturbed demand divided by as-planned production,quantifying the reduced final demand as a proportion of totalas-planned output. The economy could experience a reduction indemand following a disruption for a number of reasons, includingthe result of forced diminished supply e.g., Miller and Blair1985 or lingering consumer fear or doubt e.g., Taylor and

Final demand Total output. . . n

z1n c1 x1z2n c2 x2] ] ]zin ci xi] ] ]

znn cn xn. . . wn

. . . xnF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2009 / 395

9.15:394-405.

McNabb 2007. The matrix A is the normalized interdepen- A nonrecursive formulation of the DIIM, derived in Barker

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d.dency matrix describing the extent of economic interdependencebetween sectors of the economy. The row elements of A indicatethe proportions of additional inoperability that are contributed bya column sector to the row sector. More detail on IIM compo-nents, including numerical examples, is provided by Santos andHaimes 2004 and Santos 2006.

A number of extensions and case studies have arisen from IIMwork, including terrorist attacks Haimes et al. 2005a,b; Santos2006, cyber security Haimes and Chittister 2005; Andrijcic andHorowitz 2006, natural disasters and accidents Anderson et al.2007; Crowther et al. 2007, supply perturbations Leung et al.2007, and multiregional analysis Crowther and Haimes 2009among others.

Dynamic Inoperability Input-Output Model

Despite the perceived need to model postdisaster recovery and tomeasure the efficacy of risk management strategies that promoterecovery, few attempts have been made to use dynamic input-output models to describe recovery from disruptive eventsOkuyama 2007. An exception is the DIIM. The DIIM, whosediscussion is found in Haimes et al. 2005a,b and Lian andHaimes 2006, extends the demand-reduction IIM form to thefollowing widely accepted dynamic input-output model found inMiller and Blair 1985. The DIIM, shown in Eq. 4, describesthe recovery of industry sectors following a disruption:

qt = KAqt + ct qt 4

Definitions of qt, A, and ct are the same as those for theircounterparts in the demand-reduction IIM, except that qt andct describe those values at a specific time t. K is a matrix withresilience coefficients k1 , . . . ,kn on the diagonals and zeroes else-where. The resilience coefficient ki represents the ability of Sectori to recover following a disruption, where the greater ki valuescorrespond to a faster response by the sector to a perturbation.Assuming that qt=qt+1qt and that t is discrete, then arecursive form of Eq. 4 can be rewritten as follows:

qt + 1 = qt + KAqt + ct qt 5

If it is assumed that Sector i recovers from some initial inop-erability, qi00, to some inoperability qiTi0 at known timeTi, then ki is defined in Eq. 6, where aii are the diagonal ele-ments of A. The calculation of ki is derived from the continuous-time DIIM in Eq. 4 Lian and Haimes 2006. Note that therelationship between resilience and recovery time, among theother parameters in Eq. 6, has not been developed by empiricalmeans. Because such an empirical relationship is outside thescope of this work dealing with uncertainty in interdependencyparameters, we use Eq. 6, the published results of Lian andHaimes 2006 in modeling sector resilience

ki =lnai0/qiTi

Ti 11 qii 6396 / JOURNAL OF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2

J. Infrastruct. Syst. 2002008, is introduced in Eq. 7. For the purposes of this paper, thenonrecursive form is more advantageous than the recursive formin Eq. 5 as the vector of inoperabilities calculated at time t is afunction of initial inoperability, q0

qt = I = KA Ktq0 + i=0

t1

I + KA KiKct 1 i

7

Of interest in a DIIM analysis is the total economic loss for theentire economy over some span of time, caused by the sectorinoperabilities at each time period. Lian and Haimes 2006 in-troduce the calculation of total economic loss, Q, found in Eq.8, which essentially sums over time periods the economic lossfor each time period, where economic loss represents the amountof total output lost due to the inoperability experienced in eachsector. Eq. 8 assumes that total output, x, is time-invariant. Themetric Q is of interest to decision makers at a national level,likely not at a sector or company level, as it measures economicloss experienced for all n sectors combined

Q = xTj=1

qj 8

It is assumed that qi0, Ti, and cit are controllable throughthe risk management decision-making process e.g., preparednessstrategies addressing system hardening could alter the value ofqi0, prepositioned recovery supplies could reduce Ti, inventory,and storage activities could affect cit. Thus, as these values areinput into the DIIM, the value of Q can be calculated for each ofa number of risk management strategies and is a metric withwhich to measure the efficacy of such strategies that alter theeffects of a disruptive event, manifested quantitatively in the val-ues of several model parameters in Eq. 7. Note that althoughqi0, Ti, and cit are referred to here as decision variables, theyare actually pseudodecision variables, or the quantitative out-comes manifested as a result of sets of decisions that make up aparticular risk management strategy. An exact causal relationshipbetween risk management options and the values of these deci-sion variables is assumed known, and, because the objective ofthis work is to evaluate uncertainty in interdependency param-eters, an explicit causal relationship is not sought here. Minimiz-ing Q can then be one of a number of objectives in amultiobjective optimization framework e.g., Chankong andHaimes 1983, 2008 for performing a trade-off analysis for pre-paredness strategies.

Assumptions of the IIM and DIIM

Crowther and Haimes 2005 and Santos 2006 discuss a numberof assumptions made when calculating inoperability with the IIMand DIIM. One assumption, of particular importance in this paper,involves the stationarity of A.

Assumption 1. The values in the interdependency matrix, A,are deterministic and time-invariant. The BEA data collection ef-fort occurs on a small scale annually and a large-scale every fiveyears. Such infrequency of data collection is due to the resourcesrequired and because the interdependency matrix does not deviatedramatically from year to year.009

9.15:394-405.

Uncertainty in Interdependency Parameters simulation-based approach to study structural change. Our workintegrates the DIIM with the USIM Haimes and Hall 1977; Li

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d.The uncertainty dimension addressed in this paper is associatedwith the ability of the modeler to select the systems interdepen-dency model parameters accurately. A potential error in modelparameters can cause the violation of Assumption 1 of the IIMand DIIM, namely that the A matrix does not vary with timefollowing a disruptive event, thus limiting the accuracy of theresults of those models.

The violation of Assumption 1 is typically due, in the econom-ics literature, to structural change, or temporal variability in theinterdependency matrix due to, for example, technology develop-ing at a faster pace than interindustry data collection. A number ofapproaches have been used to loosen the deterministic assumptionof the input-output model, including works by Quandt 1958,1959, Sebald and Bullard 1976, Bullard and Sebald 1977,Goicoechea and Hansen 1978, West 1986, and Percoco et al.2006, among others. Computable general equilibrium modelingRose 1995 is a nonlinear approach to input-output modeling thataccounts for substitution.

In work involving the IIM and its extensions, such structuralchange would likely be the result of forced substitution followinga disruptive event, whether by strategy a priori or by necessity aposteriori. That is, the inoperability of Infrastructure i may requireother infrastructures that depend on i to look elsewhere to satisfyinput necessities. Kujawski 2006 challenged Assumption 1, sug-gesting that it is not realistic for disruptive events. Percoco 2006suggests that extreme events can lead to changes in both finaldemand and the technical coefficients in A resulting from infra-structure service provision.

This paper develops an approach for evaluating uncertainty ininfrastructure interdependencies when modeling recovery withthe DIIM. Titled the Uncertainty DIIM, the approach minimizesthe sensitivity of DIIM output metrics with respect to unspecifiedsubstitution changes. The U-DIIM enhances robust risk-based de-cision making by incorporating sensitivity in infrastructure inter-dependency parameters, a potential source of error in themodeling activity, when comparing multiple preparedness strate-gies.

Uncertainty in the A matrix was first discussed in Lian2006, who introduced specific changes in infrastructure interde-pendencies resulting from a predetermined risk management strat-egy that altered the interdependency behavior of a set ofinfrastructure sectors, with the ultimate goal of reducing eco-nomic loss relative to a baseline strategy with no explicit substi-tution. Lian 2006 provided a multiobjective approach thatmaximizes the reduction in economic loss from a baseline donothing strategy, calculated from an IIM analysis, while mini-mizing the cost of the substitution strategy.

Sensitivity of DIIM Metrics to Unspecified Changesto A Matrix

The U-DIIM accounts for potential substitution effects but, unlikeLian 2006, does not attempt to directly evaluate the uncertaintyin the elements of the static A matrix, as defined in Eq. 13, byexamining a particular substitution policy, as a particular substi-tution policy is likely an unknown a priori. The U-DIIM mini-mizes not only total economic loss, Q, calculated from the DIIMbut also minimizes the sensitivity of Q with respect to changes inA. Sensitivity analysis has been performed previously, includingwork by Percoco et al. 2006, which provides a Monte CarloJOURNAL O

J. Infrastruct. Syst. 200and Haimes 1988; Haimes 2004 to account for potential changesin A for the purpose of comparing risk management strategies.The U-DIIM aids in determining what risk management strategiesbetter absorb the changes that could occur in the interdependentrelationships between sectors, that is, add more resilience to thesystem.

Uncertainty Sensitivity Index MethodThe USIM was developed by Haimes and Hall 1977 and ex-tended by Li and Haimes 1988 to address the sensitivity ofoptimal model response to errors in model parameters. The USIMis described generally with a single-objective problem of interestto the decision maker in Eq. 9

minu

fu,

subject to gu, 9In Eq. 9, fu , is an objective function to be minimized

subject to a vector function gu , of constraints. Vector u rep-resents a vector of decision variables, and represents a vector ofm model parameters. Sensitivity analysis of an optimization prob-lem such as that in Eq. 9 would typically involve the study ofchanges to the optimal solution of fu , with respect to changesin the bounds of gu , see, e.g., Luenberger 2003. However,with the USIM, sensitivity of fu , is measured with respect tochanges in model parameters, . As such, the USIM addressesuncertainty in with a multiobjective optimization problem thatseeks to minimize fu , while minimizing u ,, an indexrepresenting the sensitivity of fu , to uncertainty in .

For a single parameter , the choice of u , made inHaimes and Hall 1977 squares the partial derivative of the ob-jective function with respect to . Li and Haimes 1988 showsthat a change in an objective function of multiple uncertain pa-rameters, vector = 1 ,2 , . . . ,m, can be minimized when thesensitivity index in Eq. 10 is minimized. That is, for m uncertainparameters, the sensitivity index is defined as the sum of m partialderivatives of fu , with respect to each parameter. Note thateach partial derivative is equally weighted in Eq. 10

u, = i=1

m i

fu,2 10The resulting multiobjective optimization formulation is pro-

vided in Eq. 11. The objective functions and sensitivity indicesare evaluated at the nominal value , or initial parameter esti-mates. It is assumed that , though its true values may be uncer-tain, varies in the neighborhood of

minu

fu;

u;

subject to gu; 11The above optimization problem simultaneously optimizes the

objective function of interest to the decision maker while mini-mizing the sensitivity of the system output to the unknown pa-rameters 1 ,2 , . . . ,m as they vary around their nominal values1 , 2 , . . . , m. Using a multiobjective programming algorithm,F INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2009 / 397

9.15:394-405.

trade-offs between the objective function and the sensitivity indexn n

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d.can be calculated and depicted graphically with curves of nonin-ferior solutions in functional space. See Li and Haimes 1988 fora more detailed development of the USIM and other extensions tothe general formulation provided here.

Derivation of the Uncertainty DIIM

The analysis of risk management strategies with the DIIM pro-vides an appropriate use of the USIM, as violation of Assumption1 may lead to uncertain interdependency parameters. AnalyzingDIIM with the USIM approach provides a means for comparingpreparedness alternatives to appropriately balance a reduction ineconomic loss while minimizing the effects of errors in interde-pendency parameters. Uncertainty in the parameters is assumed,but it is not directly quantified in this approach as substitutionpolicies are not identified a priori.

Suppose that minimizing the value of economic loss, Q, fol-lowing a disruptive event is the primary objective of a macro-level decision maker, e.g., a Department of Homeland Securityplanner. To minimize sensitivity of this economic loss, the partialderivatives of Q with respect to each of the elements of the Zmatrix are taken, resulting in the sum of n2 partial derivatives.This number of partial derivatives would most likely be unrealis-tic, especially when modeling an economic system with highgranularity, or a large number of sectors. Many of the n2 elementswill likely differ very little following a disruption, and will there-fore not be meaningful in an analysis of parameter uncertainty.Following the general discussion, an approach for reducing thenumber of partial derivatives is provided.

Although Percoco 2006 suggests that infinitesimal changesin the elements of the A matrix will not result in model instabil-ity, changes to the matrix following a disruption may not neces-sarily be infinitesimal. Therefore, recalling Table 1, the DIIMsubstitution analysis will be performed with respect to changes inthe commodity flow matrix, Z, the origin of A. That is, assumethat changes in interdependencies among sectors are reflected inthe elements of the Z matrix. Eq. 1 defined the scalar relation-ship between the technical coefficient, commodity flow, and de-mand. Eq. 12 provides the matrix form of this definition. It isassumed that xj stays constant, therefore zij serves as an appropri-ate surrogate for aij

aij =zijxj

A = Zdiagx1 12

The normalized technical coefficient matrix, A, a function ofthe traditional A matrix, and its relationship with the matrix ofcommodity flows, Z, is provided in Eq. 13

A = diagx1Adiagx = diagx1Z 13

The calculation of sensitivity index, Z, should then reflect thesensitivity of total economic loss to change in commodity flow, aspresented in Eq. 14

Z = i=1

n

j=1

n Qzij

2 14Combining Eqs. 7, 8, and 13. results in the more specific

calculation of Z found in Eq. 15, providing the sensitivity in Qwith respect to changes in sector interdependencies398 / JOURNAL OF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2

J. Infrastruct. Syst. 200Z = i=1

j=1 zij xTt=1 I + Kdiagx1Z Ktq0 + l=0 I

+ Kdiagx1Z KlKct 1 l2 15Note that, given the relationship between aij and zij shown in

matrix form in Eq. 13, the calculation of resilience coefficientsincludes commodity flow, as shown in Eq. 16

ki =lnqi0/qiTi

Ti 11 zii/xi 16

Another metric for comparing risk management strategies isthe cost of implementation for each strategy. The implementationcost function, adapted from Lian 2006 and also referred to ash , is provided in Eq. 17 and is measured in monetary units. Tis an n1 vector of times to recovery, Ti

cost = hq0,T,c0,c1, . . . ,c 1 17

Crowther and Haimes 2005 introduce a framework address-ing the likely competing objectives of minimizing economic im-pact from IIM and minimizing risk management costs. Bothobjectives would appear to be commensurate as viewed by na-tional decision makers, as both are measured in monetary units.That is, it would seem that both economic impact and risk man-agement costs could be added together in a single-objective opti-mization problem. However, Lian and Haimes 2006 alsomotivate a multiobjective approach, citing that risk managementcosts likely originate from the government whereas economic im-pact is experienced over a large number of sectors and that eco-nomic loss is a measurable surrogate for other noncommensuratepolitical, social, and environmental impacts. One would expect acostly investment in sound risk management strategies to reducethe economic impact experienced as a result of a disruptive event.Thus, the larger the investment, the larger is the reduction ineconomic impact. Such are competing objectives that requiremultiobjective analysis. Fundamental to multiobjective analysis isthe notion of the Pareto optimum, or noninferior solution. A non-inferior solution is defined as a solution to a multiobjective prob-lem where any improvement in one objective comes only at theexpense of another objective Chankong and Haimes 1983,2008.

The multiobjective formulation, provided generally in Eq. 18and in formula form in Eq. 19, includes the minimization oftotal economic loss, Q, sensitivity of economic loss, Z, and costof implementation, h . Q and Z are evaluated at the nominalvalue of the Z matrix, Z . Note that ensuring stability in the input-output system requires structural change in the Z matrix, not A,and that indices i and j represent the rows and columns of the Zmatrix. Assume that x varies negligibly with any changes in theelements of Z. The minimization is performed with respect todecision variables q0, T, and ct

minq0,T,ct

QZ=Z

i=1

n j=1

n Qzij

2Z=Z

h 18009

9.15:394-405.

+ l=0

t1

tq0

0,c

xT I + Kdiagx1Z Ktq0 + t1 I + Kdiagx1Z KlKct 1 lZ=Z

jective function is then rewritten in Eq. 21

+ MsMMls1 K ct 1 l 22

demand perturbation, c t, that are associated with each riskmanagement option. Instead, a discrete set of risk management

jective function of total economic loss and represents the sen-Dow

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d.l=0 s=0 zij zij

Since both K /zij and M /zij are both straightforward cal-culations due to the nature of the K and M matrices, Eq. 22 iscomputable, and therefore, the multiobjective optimization prob-lem in Eq. 20 is computable.

The original presentation of the USIM follows traditional mul-tiobjective optimization literature see, e.g., Chankong andHaimes 1983, 2008 where trade-offs between objectives arecalculated as functions of decision variables. Note that an optimal

sitivity index defining the sensitivity of total economic loss to achange in zij parameters

Q,A,DN = Q

=

QOption A QOption DNOption A Option DN

23

The ultimate purpose of the U-DIIM is to: 1 measure theefficacy of risk management options using the total economic lossmetric calculated from the DIIM; 2 account for uncertainty ininterdependency parameters using principles from the USIM; 3i,jS

n zij xTt=1 Mtq0 + l=0t1 MlKct 1 l2 21Using the product rule in matrix calculus see, e.g., Turkington

2002, it follows that the partial derivative calculation in Eq.21 would be performed as in Eq. 22

i,jS

n xTt=1

r=0

t1

MrMzij

Mtr1q0

t1 l1 2JOURNAL O

J. Infrastruct. Syst. 200options is developed and comparison among the options in thepredetermined set is made. For each option, the decision variablevalues are entered into the three objectives in Eq. 20, and adiscrete set of Pareto-optimal solutions is generated. Therefore,trade-offs among risk management options must be calculateddifferently.

One approach is to calculate trade-offs between two discreteoptions, as illustrated in Eq. 23. The trade-off, for example,between Option A and Option DN, the do nothing option, isfound with the following calculation, where Q represents the ob-minq0,T,c t=1 l=0 i,jSn zij xTt=1 I + Kdiagx1Z Ktq0 + l=0t1 I + Kdiagx1Z KlKct 1 l2Z=Z

hq0,T,c0,c1, . . . ,c 1 20

Showing that the second objective function, the sensitivityindex, is computable is a straightforward task. First, for ease,make the substitution M=I+Kdiagx1ZK. The second ob-

decision is only optimal if the model significantly describes thesystem being modeled. For this reason, we do not advocate solv-ing for values of initial inoperability, q0, recovery time, T, and

minq0,T,c x

Tt=1I + Kdiagx1Z Ktq0

i=1

n

j=1

n zij xTt=1 I + Kdiagx1Z K

hq0,T,c

Alluded to previously, the analysis will surely become compu-tationally intensive when a large number of sectors are studied.For example, the benchmark commodity flow data published bythe BEA every 5 years consist of 483 sectors, resulting in 4832=233,289 elements of the Z matrix. It is likely that only a hand-ful of these elements will be meaningful in an uncertainty analy-sis. That number of sectors can be reduced using a sectoraggregation scheme, though that may reduce the quality of analy-sis of specific sector interdependencies. An alternative that re-duces the number of partial derivatives is to focus the sensitivityanalysis on a subset of sector interdependencies. This subset, re-

I + Kdiagx1Z KlKct 1 lZ=Z+

l=0

t1

I + Kdiagx1Z KlKct 1 l2Z=Z

1, . . . ,c 1 19

ferred to as S, consists of particular i , j combinations that pointto specific elements of the Z matrix for which derivatives are tobe taken. This is represented in Eq. 20. The choice of set S couldbe based on the set of seventeen critical infrastructure and keyresources identified by the Department of Homeland Security2006 or on the interests of the decision maker. A particularlyinteresting approach to choose the elements of set S could comefrom the fields of influence approach discussed by Sonis andHewings 1992, Percoco et al. 2006, and Percoco 2006,among others, wherein the elements of the interdependency ma-trix with greatest impact on the rest of the economy are found.F INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2009 / 399

9.15:394-405.

tions of individual options, including the preplacement of recov-Table 2. Daily As-Planned Commodity Flows for the Two-Sector Ex-ample in 103 Dollars

Strateg

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d.compare a number of options in a multiobjective framework thatincludes competing objectives of economic loss, sensitivity, andimplementation cost; and 4 calculate trade-offs among the ob-jectives for each of the options. The following two-industry ex-ample illustrates these four desired characteristics of the U-DIIMon a small scale for the Transportation and Utilities sectors.

Illustrative Example of the U-DIIMThe U-DIIM, which constitutes a multiobjective comparison ofrisk management strategies that include minimal economic lossand sensitivity effects, is illustrated here with a two-industry ex-ample. The commodity flows, adapted from Miller and Blair1985, are found in Table 2, which follows the form of Table 1for two infrastructure sectors, say, Transportation and Utilities.Table 2 constitutes the nominal commodity flows matrix, Z . Com-modity flows are assumed here to occur daily. The units of timefor the DIIM are days, therefore daily as-planned commodityflows are provided. The total output column and row in Table 2describe as-planned production for the two sectors. For example,daily production in Utilities requires $500,000 in inputs fromTransportation, and the Utilities sector produces $2 million worthof total output daily.

Assume that four-risk management strategies A, B, C, D aredevised to improve, relative to the do nothing strategy DN,the ability of a region to recover from a disruptive event e.g., aCategory 3 hurricane. Such strategies could include combina-

Sector Transportation Utilities Final demand Total output

Transportation 150 500 350 1,000Utilities 200 100 1,700 2,000Value added 650 1,400Total input 1,000 2,000

Table 3. Decision Variables and Parameters for Each Risk Management

Strategy Sector qi0 Ti qiTi

A 1 0.65 24 0.012 0.72 25 0.01

B 1 0.77 12 0.012 0.75 20 0.01

C 1 0.56 18 0.012 0.68 23 0.01

D 1 0.82 9 0.012 0.71 18 0.01

DN 1 0.87 27 0.012 0.90 28 0.01

Table 4. Economic Loss, Sensitivity Index, and Cost Associated with Ea

StrategyQ

103 dollars with respect to z11 with respect to z12A 39,941 436 236B 37,602 413 295C 40,627 517 264D 36,187 477 258DN 50,016 588 379400 / JOURNAL OF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2

J. Infrastruct. Syst. 200ery supplies and inventory and storage capabilities, among others.Associated with each strategy are vector values for the decisionvariables of initial inoperability, q0, time to recovery, T, andtime-varying demand perturbation, ct. Table 3 provides the de-cision variable values and other parameters for each strategy. Re-covery over =30 days is examined. Assume that demand isperturbed in three different time epochs from t=0 to t=9, fromt=10 to t=19, and from t=20 to t=29 and is constant duringeach of the intervals.

For the calculation of the sensitivity index in the U-DIIM, thenominal zij values are those from Table 2. Table 4 summarizes theresults of U-DIIM metrics. Sensitivity was calculated from Eq.15, where the partial derivatives with respect to all zij only 22=4 values in this two-sector case are taken, then added. Sensi-tivity with respect to each zij is provided in columns three throughsix of Table 4, representing the square of the slope of economicloss in relation to a change in zij. Essentially, these individualsensitivity values represent a measure of steepness, where asteeper slope results in great sensitivity to changes in zij. Theindividual sensitivities could be normalized i.e., forced to rangeon 0, 1 over the set of strategies for ease in comparison. Totalsensitivity is shown in column seven. The cost of implementationof each strategy is also provided in column eight. For example,after entering the various parameters and decision variables fromTable 2 and 3 into Eq. 19, the total economic loss experiencedafter a Category 3 hurricane given the implementation of StrategyA would be $39,941,000, cost of implementation would be$2,284,000, and the unitless sensitivity index with respect to allzij is 1,241.

The competing objectives of economic loss, the sensitivityindex, and implementation cost are depicted graphically in thethree plots in Fig. 1. Note that a multiobjective problem withmore than two objectives, much like Eq. 19, becomes moredifficult to show graphically as the number of objectives in-creases. Fig. 1a provides three Pareto-optimal risk management

y

ci0 , . . . ,ci

9 ci10 , . . . ,ci

19 ci20 , . . . ,ci

29

0.40 0.21 0.180.34 0.19 0.100.28 0.19 0.150.39 0.24 0.060.46 0.25 0.090.37 0.20 0.150.37 0.28 0.060.35 0.21 0.080.40 0.32 0.110.42 0.28 0.10

k Management Strategy

nsitivityCost=h

103 dollarsith respect to z21 with respect to z22 All four

458 111 1,241 2,284418 146 1,272 1,234498 140 1,419 3,041487 133 1,355 2,835666 175 1,808 0009

9.15:394-405.

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d.options A, B, D given the total economic loss and sensitivityobjectives. For example, deciding between Option A and OptionB requires that implementation cost be increased for some amountof change in total economic loss. Fig. 1b depicts Pareto-optimalsolutions DN, B, D for total economic loss and implementationcost objectives. Options DN, B, and A are Pareto-optimal in acomparison of economic loss and sensitivity, as depicted in Fig.1c. Strategy C is dominated in all three graphs and is not aPareto-optimal solution when all three objectives are accountedsimultaneously.

Table 5 provides trade-offs among the three objectives for onlythe Pareto-optimal solutions for each two-objective comparisonfrom Fig. 1. Columns one and two numerical describe the slopesof the Pareto-optimal frontier depicted in Fig. 1a, the same asfor columns three and four and Fig. 1b, and columns five andsix and Fig. 1c. For example, from Table 5, Q,A,B=75.5 wouldbe interpreted to mean that, by changing from Option A to OptionB, the decision maker is willing to increase economic loss sensi-

Table 5. Trade-Offs among the Three Objectives for the Risk ManagemQ versus Q

Options Q, Options

A, B 75.5 DN, BA, D 32.9 DN, D

Fig. 1. Graphical comparison of the five-risk management strategihurricane with the strategy in place, the sensitivity of the economic lostrategy. Economic loss and implementation cost are measured in 10JOURNAL O

J. Infrastruct. Syst. 200tivity by one unit to reduce the amount of economic loss by$75,500. Thus, the slope between points A and B in Fig. 1a is75.5.

Key Sector Analysis

The term key sector, in an economic sense, describes an indus-try with strong influence on the expansion of others in aneconomy Lahr and Dietzenbacher 2001, and several metricshave been proposed to identify these key sectors, e.g., Cheneryand Watanabe 1958, Rasmussen 1956, Cella 1984, Dietzen-bacher 1992. Though a substantial amount of research has beendone in the area of key sector analysis, there has been no consen-sus on a best practice Cai and Leung 2004. Several key sectoranalysis applications are found in the risk analysis context, wherekey sectors refer to those most impacted by a particular disrup-

ategies Units Vary Depending on the Objectives Being Comparedh versus h

Q,h Options ,h10.1 DN, B 0.434.9 DN, A 0.25

icting trade-offs among total economic loss experienced during aric to changes in interdependency, and the implementation cost of thers.ent Str

versus

es depss met3 dollaF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2009 / 401

9.15:394-405.

tive event. Examples include IIM extensions and case studiesfrom the Inoperability Input-Output Model. Further, using the

should a disruption change the magnitude of interdependencies ofthe interconnected sectors. That is, sectors are ranked by the po-

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d.U-DIIM approach, key sectors can by identified based on sensi-tivity metrics.

The USIM has been demonstrated as an approach for compar-ing risk management strategies in a multiobjective framework,where each strategy has a different value of Z, thus affecting Qeach to a differing extent. However, Z may have more value asa single-objective relative measure for key sector analysis. Thatis, Z can be used to measure the sensitivity of each sector toprovide a ranking of sectors for two cases: those sectors that aremost sensitive to the changes in the interdependent relationshipswith other sectors Sensitivity due to Changes in OriginatingSector section and those that cause the most sensitivity in othersectors due to their interdependent relationships Sensitivity dueto Changes in Other Sectors section.

The general premise for the use of sensitivity as a key sectormetric is to hold all scenario-specific parameters demand pertur-bation, initial inoperability, time to recovery constant whiledeeming particular interdependency parameters uncertain. Thevalue of the time epoch of interest, whose length is time peri-ods, also remains constant for each calculation. The sensitivityindex can then be calculated for each sector. The two cases areeach discussed in subsequent subsections.

Sensitivity due to Changes in Originating SectorThe value zij refers to the commodity flow from Sector i to Sectorj. That is, zij dollars worth of output from Sector i is used in theproduction of Sector j. A row of the Z matrix is shown as a vectorin Eq. 24. The commodity flows in Eq. 24 describe the flowfrom Sector i to all other sectors, including itself. Measuring thesensitivity of Q with respect to the row vector in Eq. 24 willdescribe how sensitive total economic loss is to changes in de-pendencies of other sectors on the commodity flow originating inSector i. The notation for the row vector in Eq. 24 has an arrowabove zi pointing to the right representing that the commodityflow is originating from Sector i and flowing out to other sec-tors

zi = zi1 zij zin 24Isolating the particular row of Z describing the dependencies

on Sector i in the U-DIIM results in the sensitivity index in Eq.25. Again, the notation for this sensitivity index is Zi, signify-ing the measure of sensitivity projected outward from changes inSector i

Zi = j=1n Q

zij2 25

Inserting the DIIM calculation for Q results in Eq. 26. Notethat Eq. 26 differs from Eq. 15 in the overarching summation.The evaluation of Eq. 26 occurs at Z

Zi = j=1

n zij xTt=1 I + Kdiagx1Z Ktq0

+ l=0

t1

I + Kdiagx1Z KlKct 1 l 26Ranking the individual Zi values provides a rough summary

of the sectors that may lead to most sensitivity in other sectors402 / JOURNAL OF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2

J. Infrastruct. Syst. 200tential sensitivity that they may cause.

Sensitivity due to Changes in Other SectorsConversely, sectors can be ranked by the potential impacts theymay receive. A column vector associated with Sector j, repre-sented in Eq. 27, describes how commodity flows originatingfrom other sectors contribute to the production of Sector j. Thenotation for this vector has an arrow above z j pointing to the left,representing that the commodity flow is originating from othersectors and flowing into Sector j

z j = z1j]zij]znj

27Focusing on the elements of the particular column of Z, one

can calculate the sensitivity of Q with respect to only those ele-ments. This resulting sensitivity index is found in Eq. 28. Thenotation for this sensitivity index is Zj. Eq. 28 is evaluated atnominal commodity flow matrix, Z

Zj = i=1

n Qzij

2 28Inserting the DIIM calculation for Q results in Eq. 29

Zj = i=1

n zij xTt=1 I + Kdiagx1Z Ktq0

+ l=0

t1

I + Kdiagx1Z KlKct 1 l 29If the calculation in Eq. 29 is repeated for each sector, a

ranking of sectors can be performed based on how those sectorsare affected by their dependencies on all other sectors. Such aranking would characterize the individual sensitivity of each sec-tor to changes in its incoming commodity flow from other sectors.That is, Zj measures the sensitivity of a sector given changes inall other originating sectors.

Illustrative ExamplesTo better illustrate the ranking approaches, consider an illustrativeexample of five interdependent infrastructure sectors. The table ofcommodity flows for these sectors is provided in Table 6. Thesevalues constitute nominal commodity flows, or Z . Assume thatthe Total Output column describes daily total output.

A disruptive event is expected to result in the initial inoper-abilities and times to recovery found in Table 7. Assume for sim-plicity that no additional demand perturbation is anticipated tooccur, that is, c1=c2= =0. Time to recovery, Ti, is ex-pressed as days anticipated to reach desired inoperability qiTi=0.01 for all i. Assume that the time span of interest to the deci-sion maker is =12 days. That is, the interpretation of here isthat the decision maker is interested in the sectors that are sensi-tive in the first twelve days following a disruptive event.009

9.15:394-405.

Table 6. Daily As-Planned Commodity Flows for the Five-Sector Example in 103 Dollars

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d.Given the interdependency parameters in Table 6 and thescenario-specific parameters in Table 7, Eqs. 26 and 29 can beused to rank sectors by their sensitivity to interdependencychanges. Table 8 provides the metrics Zi and

Zj and an ordering

for both metrics. The total economic loss, Q, for the five-sectorexample is calculated to be $11,655,000.

The ordering of the Zi values for the five-sector example inTable 8 results in the Mining sector having the largest outwardsensitivity. This is interpreted to mean that Mining could have thelargest impact on total economic loss across the five sectors,given how the commodity flow from the Mining sector to othersectors may vary in the neighborhood of the Mining row in Table6. In other words, the Mining sector has the potential to impactthe economy the most if the sectors depending on Mining mustsubstitute for its goods and services after a disruptive event, giventhe data in the example. As a result, the Mining sector may be atarget for risk-based decision making to ensure that its servicesare maintained following a disruption as its loss may significantlyimpact the operations of other sectors.

The Utilities sector is found overwhelmingly to be the mostsensitive to interdependency changes in the sectors upon whichthe utilities rely, given the example data. Whereas the Miningsector was found to be the most significant source of sensitivity,as determined by Zi, the Utilities sector is the most sensitive topotential substitution brought about by the sectors that supplyinputs to it, as determined by Zj in Table 8. That is, utilities canleast absorb changes in interdependencies.

Sector Transportation Utilities Construction

Transportation 150 500 250Utilities 200 100 150Construction 350 400 200Manufacturing 100 250 300Mining 400 150 500Value added 1,800 600 1,100Total input 3,000 2,000 2,500

Table 7. Initial Inoperabilities and Times to Recover from Those Inop-erabilities for Each of Five Sectors

Sector qi0 TiTransportation 0.05 5Utilities 0.30 25Construction 0.20 15Manufacturing 0.05 5Mining 0.05 5

Table 8. Initial Inoperabilities and Times to Recover from Those Inop-erabilities for Each of Five Sectors

Sector

Outward sensitivity Inward sensitivity

Zi Ranking

Z j Ranking

Transportation 11.9464 3 3.7877 3Utilities 3.6461 5 31.8895 1Construction 8.9386 4 14.9682 2Manufacturing 12.5919 2 3.6120 4Mining 18.9499 1 2.6833 5JOURNAL O

J. Infrastruct. Syst. 200Note that the rankings are dependent upon the time span ofinterest to the decision maker and the extent of the disruptiveevent as manifested in the scenario-specific DIIM parametersq0, T, and ct. The results in Table 8, and the interpretation ofthose results, are based on a illustrative example and not on actualBEA data.

Key Sector DiscussionAlthough the sensitivity of infrastructure sectors to changes ininterdependencies is a potentially important concept with whichto rank these sectors for preparedness risk management planning,it is likely not the only objective or attribute of importance. Thatis, sectors ranked only according to sensitivity to interdependencyuncertainty perhaps do not provide all appropriate information.Further attributes with which to rank sectors for priority in riskmanagement decision making could include sector economic lossexperienced following a disruption, time required to recover tonormal or near-normal conditions, and inoperability experiencedat for choice time periods e.g., early stages where t=1,2 ,3 , . . .,or certain milestones in the recovery process where t=Ti /4,Ti /2 , . . ., among others. The notion of sector sensitivity tochanges in interdependencies can fit into a risk filtering and rank-ing method e.g., Haimes et al. 2002, Haimes 2004 or multi-attribute ranking procedure e.g., Larichev and Moshkovich1995, Cardoso and Sousa 2003.

Recall that q0, T, and ct remain constant for the rankingof sectors. This is done to reflect the ranking of sectors to aparticular scenario e.g., natural disaster, terrorist attack withwhich is associated particular values of q0, T, and ct. That is,an analyst determines, from historical data or expert elicitationmeans, that a hurricane, for example, would lead to particularvalues of q0, T, and ct, and for that hurricane, a ranking ofsectors could be performed. Perhaps a more holistic ranking couldbe performed for a set of scenarios, where sector sensitivity isweighted according to some level of concern given to each sce-nario. An overall quantification of sensitivity for a set of scenarioscould be established for each sector. Combining this concept withEqs. 25 and 28 results in the formulations in Eqs. 30 and31, respectively. Individual sensitivity indices are found foreach of a set of m scenarios and are combined by concern factors,wk, resulting in Zi,0 and

Zj,0, whose 0 subscript refers to the

weighted sum of scenarios with k subscripts. The potential listof scenarios could come from the set of fifteen planning scenariosdefined by Department of Homeland Security 2005 or scenariosof geographical interest to preparedness decision makers of agiven region

Zi,0 = k=1

m

wkZi,k 30

Manufacturing Mining Final demand Total output

300 450 1,350 3,00050 550 950 2,000

150 600 800 2,500100 350 900 2,000250 100 1,100 2,500

1,150 4502,000 2,500F INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2009 / 403

9.15:394-405.

m

be used as an approximation with which to measure the efficacyof and to make comparisons among risk management options.

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d.Zj,0 = k=1

wkZj,k 31

Concluding Remarks

Models are built to answer specific questions. For example, wemake use of the DIIM to answer the following question: what arethe impacts e.g., proportion of inoperability/dysfunctionality,economic loss on all infrastructure sectors, given a natural ormalevolent man-made hazard affecting multiple sectors? How-ever, to answer such specific questions, models must be built withsufficient complexity to capture the essence of the system. Thequality and effectiveness of the IIM and its derivatives, as a mod-eling enterprise, are indeed subject to model assumptions, topol-ogy, and selection of parameters, among others, whose associateduncertainties hinder the complexity required to realistically dealwith the specific questions that we seek to answer. Therefore,short of modifying the basic, widely accepted Leontief-based in-operability model for which considerable data are collected, thispaper attempts to compensate for the inherent uncertainties in themodel and its assumptions through the proposed U-DIIM. Theparticular Leontief model assumption addressed in this paper isthat interdependent relationships among the interconnected sec-tors of the economy, as represented by the matrix A, are time-invariant.

This paper develops a quantitative approach, the U-DIIM, thatbuilds on the dynamic recovery modeling capabilities of the DIIMand also accounts for the inherent uncertainty in interdependentrelationships by measuring the sensitivity of the effects of a dis-ruptive event to changes in these interdependencies due to theabove inherent uncertainties. In particular, parameter uncertaintiescan arise due to the inability to accurately represent in the DIIMthe impact of the substitution of commodities i.e., given a supplyshortage on sectors of the economy following a disruptive event.Parameter uncertainties markedly affect the results of models andultimately the usefulness of the models.

The U-DIIM is capable to evaluate and quantify parameteruncertainties in interdependent models to improve the risk man-agement policymaking process by more accurately measuring theefficacy of risk management strategies. It also promotes the use ofa multiobjective framework for comparing strategies and the cal-culation of trade-offs among competing objectives for each strat-egy. The U-DIIM minimizes the sensitivity of DIIM metricsabout the nominal values of the elements of A with respect tounforeseeable substitution strategies. It is advantageous because itdoes not limit the modeler to a predefined substitution policy.

The U-DIIM enables the identification of those infrastructuresectors that are most sensitive to economic perturbations due totheir inherent interdependencies with other critical sectors. De-pending on the uncertainty in modeling their interdependencies,the key sector sensitivity metric offers insight to those sectors thathave the most impact on and that can be most impacted by othersectors of the economy following a disruption To achieve sustain-able resiliency, such knowledge is important for decision makersin making sector-specific preparedness strategies.

Finally, Taleb 2007 advises against taking the results of pre-dictive models too seriously, and we take a similar stance. Due tothe uncertainties in representing interdependencies in the model,we do not advocate the use of the DIIM as a forecasting approachfor predicting the results of a disruptive event; rather, it ought to404 / JOURNAL OF INFRASTRUCTURE SYSTEMS ASCE / DECEMBER 2

J. Infrastruct. Syst. 200Acknowledgments

The writers wish to thank professor Joost R. Santos and ProfessorJames H. Lambert of the Center for Risk Management of Engi-neering Systems and Dr. Ed Hall of the Research ComputingSupport Group, all from the University of Virginia, for their valu-able comments and assistance during the development of theideas described in this paper. They also thank the reviewerswhose comments contributed greatly to their work.

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