Review on Energy Resilience

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Inoperabilityinput-outputmodelforinterdependentinfrastructuresectors.pdf

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Inoperability Input-Output Model for Interdependent Infrastructure Sectors. I: Theory and Methodology

Yacov Y. Haimes, F.ASCE1; Barry M. Horowitz2; James H. Lambert, M.ASCE3; Joost R. Santos4; Chenyang Lian5; and Kenneth G. Crowther6

Abstract: The paper discusses the theory and methodology supporting the development of the inoperability input-output mo~IIM !. The IIM is based on Leontief’s input-output model, which characterizes interdependencies among sectors in the economy an initial disruptions to a set of sectors and the resulting ripple effects. An advantage of building on Leontief’s model is that it is s by publications of the Bureau of Economic Analysis. Independent computer runs of the IIM can represent the entire nation within particular U.S. regions. A dynamic extension of the IIM analyzes different temporal frames of recovery and characte required sector adjustments for achieving new production levels. The IIM can systemically prioritize and manage the sectors be economically critical and also identify those sectors whose continued operability is critical during recovery. A compani demonstrates applying the IIM to attacks on electric power and telecommunications.

DOI: 10.1061/~ASCE!1076-0342~2005!11:2~67!

CE Database subject headings: Interactive systems; Perturbation; Dynamic analysis; Economic factors; Mathematical m Communication systems; Risk analysis.

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Introduction

The industry sectors of the economy are physically and logi interdependent systems. Today, critical infrastructures~e.g., tele communications, power, transportation, banking, etc.! are marked by immense complexity, characterized predominantly by st intra- and interdependencies as well as hierarchies. These connections take many forms, including flows of informat shared security, physical flows of commodities, and others. T is a need for a modeling framework capable of describing risks to our nation’s critical infrastructures and industry sec focusing on risks arising from interdependencies.

In assessing a system’s vulnerability, it is important to ana

1Quarles Professor, Depts. of Systems and Information Engine and Civil Engineering, and Founding Director, Center for R Management of Engineering Systems, Box 400736, Univ. of Virg Charlottesville, VA 22904~corresponding author!. E-mail: haimes@ virginia.edu

2Professor, Department of Systems and Information Enginee Univ. of Virginia. E-mail: [email protected]

3Research Associate Professor, Dept. of Systems and Inform Engineering, and Associate Director, Center for Risk Manageme Engineering Systems, Univ. of Virginia. E-mail: [email protected]

4Research Scientist, Dept. of Systems and Information Engine Univ. of Virginia. E-mail: [email protected]

5Graduate Research Assistant, Dept. of Systems and Inform Engineering, Univ. of Virginia. E-mail: [email protected]

6Graduate Research Assistant, Dept. of Systems and Inform Engineering, Univ. of Virginia. E-mail: [email protected]

Note. Discussion open until November 1, 2005. Separate discus must be submitted for individual papers. To extend the closing da one month, a written request must be filed with the ASCE Mana Editor. The manuscript for this paper was submitted for review and sible publication on May 11, 2004; approved on November 19, 2 This paper is part of theJournal of Infrastructure Systems, Vol. 11, No.

2, June 1, 2005. ©ASCE, ISSN 1076-0342/2005/2-67–79/$25.00.

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J. Infrastruct. Syst., 200

both the intraconnectedness of its subsystems and its inte nectedness with other external systems. The importance of connectedness can be addressed by modeling the way “in ability” propagates throughout our critical infrastructure syst or industry sectors. The inoperability caused by willful atta accidental events, or natural causes can set off a complex ch cascading impacts on other interconnected systems.

The paper is organized as follows:~1! introduction of the clas sic Leontief input-output~I-O! model to lay the foundation for a inoperability input-output model~IIM !; ~2! a general framewor for addressing different time frames of recovery;~3! available data sources;~4! derivations of the IIM formulation as well model extensions;~5! capabilities of the IIM and the scope of applicability; and~6! summary and conclusions of the paper.

Background: Leontief Input-Output Model

Wassily Leontief won a Nobel prize in economics in 1973 what became known as the input-output~I-O! model for econom ~Leontief 1951a,b, 1966!. Miller and Blair ~1985! provide a com prehensive introduction of the model and its applications. L tief’s I-O model describes the equilibrium behavior of both gional and national economies~Lahr and Stevens 2002; Lie 2000; Isard 1960!. The I-O model is a useful tool in econom decision-making processes used in many countries~Miller et al. 1989!. Leontief’s I-O model presents a framework capable describing the degree of interconnectedness among variou nomic sectors. Recent literature in the area of cascading fa through interconnected sectors can be found in U.S. Depar of Commerce~2003! and Embrechts et al.~1997!. Many notable extensions were later created based on the original Leo model, including the nonlinear Leontief model~Krause 1992!, energy I-O analysis~Griffin 1976; Proops 1984!, and environ

mental I-O analysis~Converse 1971; Lee 1982!. Haimes and

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Nainis ~1974! and Haimes~1977! developed an I-O model supply and demand in a regional water resources system. Ol al. ~1997! developed an I-O model for risk analysis of distribu flood protection. Extensions of I-O analysis are described by and Dietzenbacher~2001!.

The formulation of the original Leontief I-O model is sho in Eq. ~1!. The notationxi refers to the total production output industryi. On the other hand, the Leontief technical coefficienaij indicates the ratio of the input of industryi to industry j , with respect to the total production requirements of industryj . Thus, given n industries,aij can tell the distribution of inputs contri uted by various industriesi = 1 , 2 , . . . ,n to the total inputs require by industryj . Finally, the notationci refers to the final demand f the i th industry—the portion of industryi’s total output for fina consumption by end-users~i.e., the excess of all intermediate co sumptions by various industriesj = 1 , 2 , . . . ,n!.

x = Ax + c ⇔ Hxi = o j

aij xj + cjJ ∀ i s1d Grounded on Leontief’s work, a first-generation inoperab

I-O model of interconnected systems was developed by Ha and Jiang~2001!. This model, which we call thephysical-base IIM , considers multiple intra- and interconnected systems. output is the inoperability that can be triggered by one or mul failures due to their inherent complexity or to external pertu tions ~e.g., natural hazards, accidents, or acts of terrorism!. Inop- erability is defined asthe inability of the system to perform intended natural or engineered functions. In the model, the term inoperability can denote the level of the system’s dysfunct expressed as a percentage of the system’s “as-planned” le operation. Alternatively, inoperability can be interpreted as a radation of a system’s capacity to deliver its intended outpu~or supply! due to internal failures or external perturbations.

Although inoperability in its current scope applies to phys and economic losses, it can be extended to assess impacts information failure. In addition, other factors for assessing ures ~e.g., loss of lives, environmental quality, etc.! can supple ment the economic factors used in the context of inoperab The primary purpose of the model is to improve understandin the impact of complexity on the continued and sustained ope ity of these systems under adverse conditions. Other related on infrastructure interdependencies and risks of terrorism is sented in Haimes~2002, 2004!, Haimes and Horowitz~2004!, Santos and Haimes~2004!, and Jiang and Haimes~2004!.

For the IIM we consider a system consisting ofn critical interconnected sectors. The output is their inoperability, w can be triggered by the input of one or multiple failur accidents, or acts of terrorism. Inoperability is assumed to continuous variable evaluated between 0 and 1, with 0 c sponding to a flawlessly operable system state and 1 corres ing to the system being completely inoperable. Inoperability take different forms, depending upon the nature of the pro and the type of system. In circumstances when the produ level is of major concern, inoperability may well be defined unrealized production~i.e., the actual production level subtrac from the desired production level!. For instance, for a pow plant, inoperability may be defined as the ratio of the ac amount of power produced to the desired production leve~in appropriate units!. The concept of inoperability also attempts capture the quality of a system’s function. Assuming that qu can be measured numerically, a defective system whose p mance is of degenerate quality as opposed to a perfect sys

considered partially operable and thus has nonzero inoperability.

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J. Infrastruct. Syst., 200

Regimes of Recovery

Several time frames—or regimes—exhibit different feature interdependencies following an attack or other extreme eve fecting infrastructure. The nature and extent of sector interac will vary from one time frame to the next. Moreover, the met of outcomes will be allowed to vary from time frame to ti frame ~Tsang et al. 2002; Lambert and Patterson 2002!. Within each time frame, the inoperability I-O risk model can descri conceptual situation of equilibrium. Before equilibrium reached, the system will have evolved to a distinct and new f of interactions. A sample of several time frames that will be dressed by IIM is presented in Fig. 1. Further uses of the reg include comparing the physical versus psychological effects attack. While the physical-based inoperability I-O risk model lyzes the physical losses caused by either natural or hu caused disasters, it is important to consider psychological fa as well.

A comprehensive survey of the psychological effects of v ous types of disasters is documented in Norris et al.~2000!. Spe- cific empirical studies such as those by Susser et al.~2002! and Galea et al.~2002! show the significance of the “fear facto induced by the September 11, 2001, terrorist attacks. Fea cause the public to reduce their demand for the goods an vices produced by an attacked industry. For example, publi prehension after 9 / 11 about the safety of air transportation c a drastic reduction in the operations of the airlines and o airline-dependent industries. These retrenchments and chan demand can have large economic repercussions that com the physical losses~e.g., degraded production capacity!. Both physical and psychological considerations ought to be acco for in analyzing the long-term adverse economic impacts on as-planned operation levels of interconnected sectors.

Supporting Databases for IIM Analysis

An advantage of building on the Leontief I-O model is that supported by major ongoing data-collection efforts. These a able databases of interdependency statistics provide an es foundation for applying the IIM to model a terrorist attack. In section we review two main data resources:~1! the Bureau o Economic Analysis~BEA! database of national I-O accounts; ~2! the Regional Input-Output Multiplier System~RIMS II! ac- counts. The BEA database, which provides an overview o national economic I-O accounts, is a series of tables depictin production and consumption of commodities~i.e., goods and se vices! by various sectors in the U.S. economy. The BEA c

Fig. 1. Three temporal regimes of recovery considered in in output model analysis of attack impacts

sumption and production tables are combined to calculate the

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Leontief technical coefficient matrix for nearly 500 industry s tors of the U.S. economy and their corresponding interdepen cies with the workforce sector. RIMS II is a set of regional d maintained by the BEA’s, Regional Economic Analysis Divis Empirical tests suggest that regional multipliers can be use surrogates for time-consuming and expensive surveys wi compromising accuracy.

Utilizing the BEA database~U.S. Department of Commer 1998!, the demand-reduction inoperability I-O model~or demand reduction IIM! complements and supplements the physical-b model developed by Haimes and Jiang~2001!. While the physical-based model quantifies inoperability in terms degraded capacity to deliver the intended outputs, the dem based model quantifies the inoperability as the reduced pr tion resulting from perturbations to the demand~Santos an Haimes 2004; Santos 2003!. Logically, the demand reduction o perturbed sector produces further adverse impacts on the o tion of other dependent sectors. For example, the demand r tion of the airline industry—an industry primarily affected by 9 / 11 terrorism—caused the demand for other dependent i tries to decline as well~e.g., travel and hotel industries!. Specifi- cally, a 33.2% reduction in passenger enplanements~FAA 2002! and a 19.2% reduction in hotel occupancy~Ernst and Youn 2002! were realized in the aftermath of 9 / 11, relative to 2 figures. Integrating the concept of inoperability into Leonti economic I-O model makes it possible to analyze how dem reduction inoperability affects other interdependent infras tures.

Two motivations have driven the use of an economic mod study physical interactions. One deals with the general iss translating between economic and physical values, while the accounts for the effects of terrorist attacks on power sourc well as on equipment operated by the using sectors~e.g., comput ers, control systems!. An assumption made when applying IIM that the level of economic dependency is the same as the le physical dependency. That is, it is assumed that two comp with a large amount of economic interaction will have a simil large amount of physical interdependency. However crude assumption may be, it is founded on BEA data that reflect physical interactions between economic sectors. These are mensurated into dollar units by multiplying interactions of ph cal quantities by producers’ prices. In turn, these prices ind how a sector values the physical interdependencies.

However, based on the availability of economic data from BEA, the corresponding lack of data on physical interdepen cies, and the extraordinary cost required to collect such info tion on the scale of economic data collections, the degree o accuracy in IIM results becomes a question. Addressing question would require identifying sector pairs where finan couplings are roughly proportional to physical couplings, m greater than physical couplings, or much less. The IIM m would need to be adjusted for those cases where the ph couplings are much different from economic couplings.

The case study discussed in the companion paper~Haimes e al. 2005! determines the rank order of interdependent sec the loss in their production capacities, and the correspon economic impact. This can be used to determine the size o risk and where to invest to reduce it. One possible way to confidence in the results is to carry out a study of the top se resulting from an IIM analysis to determine how close the ph cal ties are relative to economic ties. Such a study m

be bounded enough to carry out at an acceptable cost when

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J. Infrastruct. Syst., 200

compared with costs of poor risk management, and could res a modification of the prioritizations.

When applying the IIM to a potential terrorist attack, BEA’s data can be used to determine the expenditures of al nomic sectors on items that use electricity~i.e., how much a se tor spends on computers and other electrical equipment!. Using the percentage of each sector’s total resources spent on ele equipment to estimate the production-focused level of de dence on electric power, we can estimate the percentage l production level that each sector would suffer due to its electrical devices failing. This permits us to create an input ve for inoperability that includes not only the unavailability of pow sources, but also the production losses of power-dependen tors even with power restored~e.g., due to dysfunctional equ ment!. This use of the IIM provides a direct approach for un standing interdependencies, however limited it may be substituting economic data for physical data, as discussed ously.

National and Regional Databases for IIM Analysis

Bureau of Economic Analysis Database The Bureau of Economic Analysis~BEA! publishes the nation economic input-output accounts, which are a series of table picting the production and consumption of commodities~i.e., goods and services! of various sectors of the U.S. economy. T detailed national tables are composed of hundreds of indus organized according to the Standard Industry Classification~SIC!, or more recently, the North American Industry Classification tem ~NAICS! codes.

In the original Leontief model formulation, each industry assumed to produce a distinct commodity. The termcommodityin this paper refers to the output of an industry, which can be i form of goods or services. Realistically, however, it is poss that a given industry can produce more than one commodity the other hand, a given commodity may not be a unique outp a given industry. The BEA recognizes that the one-to-one c spondence assumption between an industry and a commo generally not true. The BEA makes a distinction between a dustry and a commodity in its published I-O data via theindustry- by-commodity and commodity-by-industrymatrices. Fig. 2 adapted from Miller and Blair~1985!, summarizes the types national input-output accounts maintained by the BEA.

The make matrix in Fig. 2, denoted byV, would show the monetary values of the different column commoditiesproduced

Fig. 2. Summary of economic input-output accounts

by the different row industries. A sample ofmakematrix data is

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shown in Table 1. Theusematrix, on the other hand, denoted U, would show the monetary values of the different row c moditiesconsumedby the different column industries. A samp of usematrix data is shown in Table 2. Note that Fig. 2 does directly specify the I-O matrix representing the industry- industry transactions.

This matrix, denoted byA in the Leontief formulation, i called the industry-by-industry technical coefficient matrix in ontief parlance. It would show the input of industryi to j , ex- pressed as a proportion of the total production inputs to indusj . The BEA does not publish the elements of theA matrix becaus this task is left to the analyst. Typically, theA matrix is estab lished from the make and use matrices using various assum @e.g., commodity-technology assumption~CTA! and industry technology assumption~ITA !; see Guo et al.~2002!#. One ap proach is carried out by first normalizing the values of themake and use matrices. The following sections discuss the operat

for deriving thenormalized makematrix sV& d from the makema- trix sVd, and thenormalized usematrix sU& d from the usematrix sUd.

Coefficients of Production in National and Regional Economies The make matrixsVd in BEA I-O reports shows the itemize production of commodities by various industries. Each eleme the makematrix svij d, shows industryi’s production of commod ity j ~typically measured in millions of dollars!. Suppose there a m commodities andn industries; then the total industry output the i th industry sxid must follow the balance equation below~Fig. 2!.

xi = vi1 + vi2 + ¯ + vim = o j øm

vij ; ∀ i = 1,2, . . . ,n s2d

Denotingx as the vector of total industry outputs,S as a unity vector ~i.e., a vector whose elements are all 1s, also known summation vector!, and V as the make matrix, it can be sho that Eq.~2! can be written in the following matrix form:

x = VS s3d

Table 1. Sample Make Matrix for 1992 U.S. Economy

Industry ~SIC code!

Commodity output ~SIC code!

Value ~in million $!

1.0100......... ............... 20,285

1.0100 19,646

4.0001 86

14.0600 365

76.0206 188

Note: Excerpt from U.S. Department of Commerce, p. 47~1998!!

Table 2. Sample Use Matrix for 1992 U.S. Economy

Commodity Using

industry Value

~in million $!

2.0502......... ................ 2,162

2.0502 55

14.1900 2,099

93.0000 4

94.0000 4

Note: Excerpt from U.S. Department of Commerce, p. 83~1998!.

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Sample data from themake matrix are depicted in Table Due to the volume of data, the BEA does not present themake matrix in the format ofvij ~i.e., with the industries arranged alo the rows and the commodities along the columns!. Rather, on industry is given at a time~see first column of Table 1!; the second column enumerates the commodities produced by th dustry; and the third column gives the value of those comm ties. For example, the dairy farm productsindustry in Table 1 ~1.0100! produces $19,646 million worth of the dairy farm pr ucts commodity~1.0100!; $86 million worth of the agricultura forestry, and fishery servicescommodity~4.0001!; $365 million worth of the fluid milk commodity~14.0600!; and $188 million worth of the other amusement and recreation servicescommodity ~76.0206!.

Eq. ~8!, which shows the formulation for thenormalized mak

matrix V& = fv&ij g, is an industry-by-commodity matrix because shows the industries along the rows and the commodities the columns. To better understand how Eq.~8! is derived, we dissect the elements of the underlying make matrixsVd and the total commodity output vectorsyTd as follows~Fig. 2!:

V = 3 v11 ¯ v1j ¯ v1m ] ] ]

vi1 ¯ vij ¯ vim ] ] ]

vn1 ¯ vnj ¯ vnm

4 s4d yT = Fy1 = o

vi1 ¯ yj = o i

vij ¯ ym = o i

vimG s5d The normalized makematrix, whose elements are denoted byv&ij , can be obtained by dividing each element of the make matrixsvij d by the respective column sumsyjd as follows:

V& = 3 v11/y1 ¯ v1j/yj ¯ v1m/ym ] ] ]

vi1/y1 ¯ vij /yj ¯ vim/ym ] ] ]

vn1/y1 ¯ vnj/yj ¯ vnm/ym 4 s6d

As Eq. ~7! shows, Eq.~6! can be written in a compact mat notation by first denoting the operator diagsud as the resultin diagonal matrix constructed from a given vectoru ~note that this notation will also be used later!.

diagsud = diag3 u1

]

um 4 = 3

u1 0 ¯ 0

0 u2 � ]

] � � 0

0 ¯ 0 um 4 s7d

Thus, from Eqs.~6! and ~7!,

V& = Vfdiagsydg−1 ⇔ Hv&ij = vij yj J ∀ i, j s8d

Coefficients of Consumption in National and Regional Economies The use matrixsUd in the BEA I-O reports shows the itemiz consumption of commodities by various industries. Each ele of the use matrixsuij d, shows industryj ’s consumption of thei

commodity ~typically measured in millions of dollars!. Suppos

there arem commodities andn industries. Denotingei as exog-

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enous consumptions for commodityi ~or final commodity de mands!, the total commodity output for thei th commodity syid must follow balance Eq.~9! ~the notationc or ci throughout this paper refers to final industry demand and should be distingu from e or ei, which refers to the exogenous or final commo demand! ~Fig. 2!:

yi = ui1 + ui2 + ¯ + uin + ei = o j øn

uij + ei ; ∀ i = 1,2, . . . ,m

s9d

Denoting the total commodity output vector byy, a summation vector byS, and the use matrix byU, Eq.~9! can be written in th following matrix notation~the notationx or xi throughout this paper refers to total industry output and should be distingui from y or yi, which refers to the total commodity output!:

y = US + e s10d

Sample data from theusematrix are depicted in Table 2. D to the volume of data, the BEA does not present theusematrix in the format ofuij ~i.e., the commodities arranged along the ro and industries along the columns!. Rather, one commodity listed at a time~see the first column of Table 2!, the secon column enumerates the industries that use that commodity the third column gives the amount of that commodity used by industries. For example, the usage of the sugar cropscommodity in Table 2~2.0502! is as follows: $55 million by the sugar cro industry (2.0502!; $2,099 million by the sugar industry~14.1900!; $4 million as change in business inventories~93.0000!; and $4 million as exports of goods and services~94.0000!. Note that the last two codes, 93.0000 and 94.0000, are not industriesper se. Rather, they are the final commodity consumptionsseid in balance Eq. ~9!.

Eq. ~14!, which shows the formulation for the normalizeduse

matrix U& = fu&ij g, is a commodity-by-industry matrix because shows the commodities along the rows and the industries the columns. To better understand how Eq.~14! is derived, we dissect the elements of the underlyingusematrix sUd and the tota industry output vectorsxd as follows~Fig. 2!:

U = 3 u11 ¯ u1j ¯ u1n ] ] ]

ui1 ¯ uij ¯ uin ] ] ]

um1 ¯ umj ¯ umn

4 s11d

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v1j

]

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vij

]

xn = o j

vnj 4 ⇔ xT = fx1 ¯ xj ¯ xng s12d

The normalized usematrix, whose elements are denoted byu&ij , can be obtained by dividing each element of theusematrix suij d by its respective column sum, which happens to bexj ~Fig. 2!. The normalized usematrix is represented by the following mat

notations:

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J. Infrastruct. Syst., 200

U& = 3 u11/x1 ¯ u1j/xj ¯ u1n/xn ] ] ]

ui1/x1 ¯ uij /xj ¯ uin/xn ] ] ]

um1/x1 ¯ umj/xj ¯ umn/xn

4 s13d Thus,

U& = Ufdiagsxdg−1 ⇔ Hu&ij = uij xj J ∀ i, j s14d

Technical Coefficient Matrix The technical coefficientmatrix, denoted byA, has industrie along the rows as well as the columns. It can be shown thatA is the product of thenormalized makeand thenormalized usema- trices.

A = V& U& ⇔ Haij = o k

v&iku&kjJ ∀ i, j s15d On the other hand, the vector of industry final demandsscd can be shown to be the product of the normalized make matrix and exogenous commodity demandvector.

c = V& e ⇔ Hci = o k

v&ikekJ ∀ i s16d Deriving Eqs.~15! and ~16! involves the following steps. Subs tuting Eq.~8! for Eq. ~3!, we have

x = VS = fV& diagsydgS s17d

Eq. ~17! can be simplified further by using the fact t diagsydS = y.

x = V& y s18d

Similarly, we substitute Eq.~14! for Eq. ~10! to form the follow- ing equation:

y = fU& diagsxdgS + e s19d

Eq. ~19! can be simplified further by using the fact t diagsxdS = x:

y = U& x + e s20d

Premultiply V& to Eq. ~20!:

V& y = V& U& x + V& e s21d

Substitute Eq.~18! for Eq. ~21!:

x = V& U& x + V& e s22d

For Eq. ~22! to become equivalent to the usual Leontief ance Eq.~1!, then Eqs.~15! and~16! must be true. Thus we ha shown that the Leontief industry-by-industry coefficient ma sAd can be calculated on the bases of thenormalized makeand normalized usematrices as described in Eq.~15!. In addition, the industry final demand can be constructed from the exoge commodity demand by premultiplying it by thenormalized mak matrix, as described in Eq.~16!.

Relevant Data for Workforce Sector Vulnerability Analysis We have added a new row and column to the original nat technical coefficient matrixsAd. By extracting the household po

tion of the exogenous demand~measured in terms of personal

RNAL OF INFRASTRUCTURE SYSTEMS © ASCE / JUNE 2005 / 71

5, 11(2): 67-79

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consumption expenditures! and the household portion of the va added~measured in terms of personnel compensations!, we were able to generate an updatedA matrix. This integrates informatio on additional interdependency impacts contributed by the ho hold sector. The extraction of household portions from exoge demand and value-added vectors is described in Fig. 3. household sector, a standard BEA sector classification, i source of labor inputs in various sectors of the economy. T from here on, this paper will refer to the husehold sector a workforce.

Regional Input-Output Multiplier System „RIMS II…

Regional decomposition enables a more focused and thus accurate analysis of interdependencies for regions of intere the United States. Miller et al.~1989! and Lahr and Dietzenbach ~2001! discuss the validity of closing the I-O analysis to a p ticular region~i.e., a single regional I-O framework as oppose multiregional! since interregional feedbacks empirically are fo to be small. The RIMS II division of the U.S. Department Commerce is responsible for releasing multipliers for variou gions of the United States. Empirical tests suggest that reg multipliers can be used as surrogates for time-consuming expensive surveys without compromising accuracy~Brucker et al 1990!. With the availability of national I-O tables and locati quotients~U.S. Department of Commerce 1997, 1998!, analysts can convert and customize the national data according to th gion of interest.

RIMS II utilizes location quotients derived from personal come data and wage-and-salary data to regionalize the na Leontief technical coefficient matrix~i.e., theA matrix!. A loca- tion quotient indicates how well an industry’s production capa satisfies the regional local demand. In addition, as the value industry’s location quotient tends to 1, its relative concentratio the region approaches that of the national level

l i = x̂i

R/x̂s R

x̂i/x̂s s23d

wherex̂i R is the regional output for thei th industry; x̂s

R is the tota regional output for all regional-level industries;x̂i is the nationa output for thei th industry; andx̂s is the total national output for a national-level industries.

The regional industry-by-industry technical coefficient ma A R, whose elements are denoted byaij

R, is then established

Fig. 3. Economic input-output accounts reconfigured for workfo analysis

follows:

72 / JOURNAL OF INFRASTRUCTURE SYSTEMS © ASCE / JUNE 2005

J. Infrastruct. Syst., 200

aij R = Haij sl id l i , 1

aij l i ù 1 J s24d

When l is used to denote a vector of location quotients andS a unity vector, Eq.~24! can be written in the following matrix n tation:

A R = diagfMinsI ,SdgA ⇔ haij R = Minsl i,1daij j ∀ i, j s25d

RIMS II issues a series of multipliers for various sectors specified region generated via the region’s location quotients@see Eq. ~23!#. Some examples are as follows: • Output multiplier: gives the change in the production outpu

a sector resulting from a $1 change in the demand for an sector’s output;

• Earnings multiplier: gives the change in the workforce e ings of a sector resulting from a $1 change in the deman another sector’s output; and

• Employment multiplier: gives the change in the numbe workers of a sector resulting from a $1 million change in demand for another sector’s output. RIMS II multipliers are presented in the form of 383 490

matrices. The columns in Fig. 4 represent detailed sectors~e.g., Column 420 ~C420!, electric services/utilities; Column 4 ~C421!, natural gas transportation; Column 422~C422!, natura gas distribution; and so on!. On the other hand, the rows in t matrix of RIMS II multipliers represent aggregated [email protected]., Row 26 ~R26!, electric, gas, and sanitary services#. Furthermore a specific row corresponds to an aggregation of several co [email protected]., R26~electric, gas, and sanitary services! is the ag gregated version of C420–C424#.

An extreme event such as a terrorist attack degrades the bility of a sector to supply its as-planned level of output. A s tor’s supply reduction necessarily leads to demand reduction~e.g., consumption adjusts when available supply is below the planned demand level!. The RIMS II multipliers can be utilize for predicting the impact of reduced demand or supply on va interconnected sectors of a region, due to extreme events.

Development of IIM and Its Extensions

Physical-Based IIM

A first-generation, physical-based inoperability I-O model~or physical IIM, for simplicity! was developed by Haimes and Jia

Fig. 4. Sample interpretation of RIMS II multipliers

~2001!, Jiang ~2003!, and Jiang and Haimes~2004! to describe

5, 11(2): 67-79

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how the impact of willful attacks can cascade through a syste interconnected infrastructures. Inoperability connotes degrad in the system’s functionality~expressed as a percentage relativ the intended state of the system!. The formulation of the physica based model is as follows:

xi P = o

aij P xj

P + ci P ⇔ xP = A PxP + cP s26d

The superscriptP in Eq. ~26! is added to the original formu lation by Haimes and Jiang~2001! to distinguish it from Leon tief’s model. Although the mathematical construct of the models is similar, the interpretation of the model paramete fundamentally different. The supply and demand concepts i Leontief economy model now assume different interpreta and have been inverted in the physical IIM. In Leontief’s mo c and x represent commodities typically measured in produc or monetary units. In the physical-based model, the vectocP

represents theinput to the interconnected infrastructures perturbations in the form of natural events, accidents, or w attacks. Theoutput is defined as the resulting vector of inop ability of the different infrastructures, denoted byxP, due to thei connections to the perturbed infrastructure and to one an The long-run inoperabilities of the interconnected infrastruct following an attack can be calculated using Eq.~26!.

The inoperability vectorxP describes the degree of functio ality of interconnected infrastructures. Thus it takes on va between 0 and 1, where flawless operation corresponds toxP= 0 or x1

P= x2 P= ¯ = xn

P= 0 for n interconnected infrastructures. Wh this condition is in effect, the infrastructures are said to be at as-plannedor ground state. A perturbation inputcP will cause a departure from this as-planned state. In addition, a perturb can intuitively set off a chain of effects leading to higher-or inoperabilities.

For example, a power infrastructure~the kth infrastructure! would initially lose 10% of its functionality due to an attack t delivers a perturbation ofck

P= 0. This means that the perturbat can be interpreted as the resulting inoperability of the powe frastructure right after an attack. In addition, the inoperab propagated by the power infrastructure to other power-depe infrastructures will in turn cause more inoperabilities and mately may cause additional inoperability in the power infrast ture itself. In general, we expect the long-run inoperability o attacked infrastructure to increase from its postattack value~i.e., the perturbation!.

Demand-Reduction IIM

The demand-reduction IIM is derived by combining the ins and intuition gained from the physical IIM with the rigor a proven BEA databases that accompany the original Leo model. The BEA data are a record of the physical exchang commodities between various interconnected industrial secto the economy that have been scaled by producers’ prices int common unit of dollars and therefore will be the foundation our measure of interdependency.

Using the definition of normalized production loss we de the demand-based model on the basis of the Leontief mode first define an as-planned production scenario based on the

tief balance:

JOU

J. Infrastruct. Syst., 200

x̂ = Ax̂ + ĉ s27d

The variables in Eq.~27! are defined as follows:x̂ = as-planne total production vector;A = Leontief coefficient matrix; andĉ = as-planned final demand vector.

We also define a degraded production scenario based o Leontief balance equation:

x̃ = Ax̃ + c̃ s28d

The variables in Eq.~28! are defined as follows:x̃ = degraded tota production vector; A = Leontief coefficient matrix; and c̃ = degraded final demand vector.

A reduction in the final demand@denoted bydc in Eq. ~30!# is defined to be the difference between the as-planned and deg final demands. This reduction in final demand consequently gers reduction in production@denoted bydx in Eq. ~28!#, which is defined to be the difference between the as-planned and deg productions.

dx = x̂ − x̃ s29d

dc = ĉ − c̃ s30d

Subtracting Eq.~28! from Eq. ~27! will result in the following relationship betweendx and dc:

sx̂ − x̃d = Asx̂ − x̃d + sĉ − c̃d ⇔ dx = Adx + dc s31d

The transformations in Eqs.~32!–~34! are needed to derive t demand-based model in a form analogous to the balance eq of the Leontief model:

c* = fsdiagsx̂dd−1dcg s32d

A * = fsdiagsx̂dd−1Asdiagsx̂ddg s33d

q = fsdiagsx̂dd−1dxg s34d

Define the transformation matrix:

P = fdiagsx̂dg−1 s35d

Using the transformation matrix in Eq.~35!, Eq. ~31! become Eq. ~37! by the transformation defined in Eq.~36!:

fPdxg = fPAP−1gfPdxg + fPdcg s36d

q = A * q + c* s37d

Assuming that the demand-based interdependency matrixA * is nonsingular and stable, the demand-based inoperabilityq can be calculated as follows:

q = fI − A * g−1c* s38d

Regional IIM

At the national level, the derived form of the demand-reduc IIM is q = A * q + c* . The regional model takes a similar form:

qR = A * RqR + c* R s39d

The system of equations corresponding to Eq.~39! is as follows

q1 R = a11

* Rq1 R + a12

* Rq2 R + ¯ + a1n

* Rqn R + c1

* R

q2 R = a21

* Rq1 R + a22

* Rq2 R + ¯ + a2n

* Rqn R + c2

* R

]

RNAL OF INFRASTRUCTURE SYSTEMS © ASCE / JUNE 2005 / 73

5, 11(2): 67-79

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qn R = an1

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regional technicalaij R coefficient using the identity shown in E

~41!. This identity is analogous to the corresponding natio level formulaA * = fsdiagsx̂dd−1Asdiagsx̂ddg.

A * R = fdiagsx̂Rdg−1A Rfdiagsx̂Rdg ⇔ Haij* R = aijRS x̂jR x̂i

RDJ ∀ i, j s41d

Now we express the regional industry-by-industry techn coefficient matrixsA Rd in terms of the counterpart national mat sAd. The resulting AR matrix in Eq.~42! is obtained by substitu ing Eq. ~25! for Eq. ~41!. Thus, the regional interdependen matrix A* R can be established on the bases of the location tients, the national industry-by-industry technical coefficients, the as-planned production outputs of the regional industries

A * R = fdiagsx̂Rdg−1fdiagfMinsl,SdgAgfdiagsx̂Rdg

⇔ Haij* R = Minsl i,1daijS x̂jR x̂i

RDJ ∀ i, j s42d Dynamic IIM

To address more effectively the temporal dynamic behavio industry recoveries in the static IIM, a dynamic IIM is propo and formulated as an extension that supplements and co ments the static IIM. In this section, the concept of anindustry resilience coefficientis introduced as a key element in the namic extension of IIM. Fundamentals on how to define a r ience coefficient and its connection to parameters of recover also discussed. A comparison of dynamic and static models end of this section shows the consistency of the two models

Introduction to Dynamic IIM In the I-O literature, the classic Leontief dynamic I-O model ta the following form ~Miller and Blair 1985!:

xstd = Axstd + cstd + Bẋstd s43d

Matrix B in Eq. ~43!, which is a square matrix of capital c efficients, represents the willingness of the economy to inve capital resources. Blanc Diaz and Ramos Carvajal~2002! argue that the elements ofB must be either zero or negative for economic system to be stable. Such a condition will produc economic behavior consistent with the static model, indepen of initial conditions and final demand. Therefore, the capital efficient matrix,B, can be interpreted as an expression of sh term countercyclical policy instead of long-term growth. For tuition aboutB, consider the case investigated by Blanc Diaz Ramos Carvajal~2002! where B = −I , which represents a economy that quickly adjusts its production levels following formation about mismatches in supply and demand:

ẋstd = Axstd + cstd − xstd s44d

Using the classic Leontief I-O model and the results above can extend the IIM to model the industry sectors’ dynamic re ery behaviors and dynamic interactions caused by demand r tion or terrorist attacks on industry sectors. Consider a diag

matrix form of the capital coefficient matrixB:

74 / JOURNAL OF INFRASTRUCTURE SYSTEMS © ASCE / JUNE 2005

J. Infrastruct. Syst., 200

B = diagsbid ; ∀ i = 1,2, . . . ,n s45d

Furthermore, we define aK matrix as follows:

K = diagskid ; ∀ i = 1,2, . . . ,n s46d

We relate Eqs.~45! and ~46! as follows:

K = − B−1 ⇔ ki = 1

bi ; ∀ i = 1,2, . . . ,n s47d

Substituting Eq.~47! for Eq. ~43! and rearranging the terms w yield the following equation:

ẋstd = K fAxstd + cstd − xstdg s48d

Or in discrete form

xsk + 1d − xskd = K fAxskd + cskd − xskdg s49d

Transforming Eqs.~48! and~49! into the normalized inoperabili form will yield the following equations:

q̇std = K fA * qstd + c* std − qstdg s50d

qsk + 1d − qskd = K fA * qskd + c* skd − qskdg s51d

In Eqs.~48! and~49!, matrix A is the Leontief technical coe ficient matrix; vectorcstd is the final demand vector at timet; and vectorxstd represents the total output of sectors at timet. In Eqs. ~50! and ~51!, matrix A* is the normalized interdependency m trix; vector c* std is the normalized final demand vector at timt; and qstd is the inoperability vector at timet. Collectively, Eqs ~48!–~51!, give the formulation for the dynamic IIM.

Matrix K will be referred to as theindustry resilience coeffi cient matrix; each elementki in the matrix measures the resilien of sectori, given an imbalance between supply and deman the case of a terrorist attack or other catastrophic event,ki mea- sures the recovery rate of the industry sectors. In the ca demand reduction,ki measures the production adjustment rat the sector.

The resilience coefficientki can be controlled and manag Each resilience coefficientki in the matrixK is determined by th nature of the individual sector itself as well as the controls o via the risk management policies. Hardening and other risk gation efforts in the industry sectors increaseki during the recov ery. Consequently, economic losses and other adverse impa minimized with shorter recovery times. This would enable p cymakers to assess the return on investments associated wi didate risk management actions for expediting recovery. A ge solution to Eq.~50! is

qstd = e−K sI −A * dtqs0d + E

K e−K sI −A * dst−zdc* szddz s52d

If the final demandc* std is stationary, Eq.~52! can be furthe simplified to

qstd = sI − A * d−1c* + e−K sI −A * dtfqs0d − sI − A * d−1c* g s53d

qstd = q` + e −K sI −A* dtfqs0d − q`g s54d

In the equation above,q` stands for the equilibrium inope ability determined by the final demand vector. The expone

term e−K sI −A * dtfqs0d − q`g is the temporal term that is decay

with time. When Eq.~53! reaches its equilibrium, it becomes

* −1 *

qstd = sI − A d c s55d

5, 11(2): 67-79

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In the equilibrium state, the dynamic IIM reduces to the fo of the static IIM. The dynamic IIM can be viewed as a m general extension of the static IIM, and/or the static model ca viewed as a description of the dynamic model at its equilibr condition.

Assessment of Industry Resilience Coefficient As discussed in the previous section, the industry resilience ficients are the key to modeling the dynamic IIM. The resilie coefficient reflects the output response of each individual ind sector to an imbalance of supply and demand. For a de assessment of the industry resilience coefficients, consid economy consistingn of sectors. It is assumed that initially sec i is attacked by terrorists. Based on the postattack econom sponse, two sets of sectors should be analyzed.

The first is sectori. After an attack, sectori will start the recovery process~e.g., rebuild the factories, machines, and so! with a recovery rateki, 0ø ki , 1. Depending on the risk mitig tion efforts and the damage, the faster sectori recovers, the large the value ofki will be. The second set of sectors encompasse the others in the economy affected by the attack due to dependence on sectori. To be able to respond efficiently to t attack scenario, the production outputs of these sectors mu immediately adjusted relative to the new level of demand. S immediate adjustments to mismatches in supply and deman respond to the maximum recovery rateskj = 1, j Þ i.

For a special case whereki = 0, and momentarily neglecting t dependence ofi on j , aij

* = 0, ∀ j Þ i, and if final demand stay constant, thei th row in Eq. ~51! will read as follows:

qisk + 1d − qiskd = 0 s56d

from which follows

qisk + 1d = qis0d ; ∀ k = 0,1,2, . . . ,T s57d

In other words, during the period of time under considera sectori has a constant inoperability equal to the initial pertu tion.

In the following discussion, the assessment of the reco rate of the attacked sector, corresponding toki, 0, ki , 1, is ad- dressed and formulated in greater detail.

In Eq. ~52!, if ki . 0, aij * = 0, ∀ j Þ i, and if final demand stay

constant, then the inoperability equation for sectori becomes

1 − qistd = 1 − e −kis1−aii

* dtqis0d s58d

Eq. ~58! is called anindividual sector recovery trajector. Similar to the concept of inoperabilitysistd, the term is 1 −qistd defined as the operability of sectori at time t. From this we conclude that a recovery trajectory that follows an expone curve in temporal space will have a recovery parameterkis1 − aii

* d. The recovery trajectory of a sector can also be written in

following form typically found in reliability literature~note tha the ratiol / t will be clarified in the forthcoming example!:

1 − qistd = 1 − e −sl/tdtqis0d s59d

Comparing Eqs.~58! and ~59! generates the following fo mula, which can be used to estimate the resilience coefficie sectori:

ki = l

ts1 − aii * d

s60d

when aii ! 1, Eq. ~60! can be approximated further, as follows:

JOU

J. Infrastruct. Syst., 200

ki < l

t s61d

This equation, which provides the connection between th silience coefficient~recovery rate! and the recovery paramet justifies the definition ofki in the dynamic IIM as asector resil- ience coefficientor recovery ratesl / td. As an example, the de vation of the recovery rate for the electric power generation supply sector is shown to illustrate the process. Consider a p blackout scenario that follows an exponential recovery such 99% recovery is achieved in 60 days. The resilience coefficie the power sector~denoted by the subscriptp! can be derived a shown below. According to Eq.~59!, the recovery parameter c be calculated as follows:

t =

− lnF qps60d qps0d

G 60

= 0.0768/day

Through the BEA data, we determined for the power se that app

* = 1.2173 10−4. From Eq.~60! we can calculate the reco ery rate to bekp< 0.0768 / day. Therefore the individual recov trajectory for the power sector has the following function, wh is depicted in Fig. 5:

1 − qpstd = 1 − e −kps1−app

* dtqps0d = 1 − e −0.0768tqps0d

Assessment of Economic Loss during Recovery through Dynamic IIM To understand better what the impacts of the attack will be a facilitate the trade-off analysis in risk management decision ing, it is imperative that the economic loss during the reco from each individual industry sector be estimated in quantit dollar amounts for all kinds of possible scenarios. During recovery process, it is important not only to know a sector’s loss compared to the as-planned level, but also, all the ind losses from its interdependent industry sectors should be q fied and taken into account. The national and regional case s both consider the two measures in dollar amounts:~1! economic losses of the attacked sector, and~2! economic losses~direct and indirect! from all sectors. According to the dynamic model in continuous form, the cumulative economic loss for each

Fig. 5. Individual recovery trajectory of power sector

vidual industryi is given by

5, 11(2): 67-79

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Qistd = x̂iE t=0

qistddt s62d

where x̂i = as-planned output rate of industryi ~$/time unit!; qistd = inoperability of industryi at time t; Qistd = cumulative eco nomic loss of industryi by time t; and qistd is subject toqstd = sI − A * d−1c* + e−K sI −A

* dtfqs0d − sI − A * d−1c* g. ThereforeQistd will also be exponential due to the exponential recovery trajecto sectori.

Similarly, the total economic loss from alln sectors by timeT @denoted byQsTd# is assessed as

QsTd = o i=1

n Sx̂iE 0

qistddtD s63d Comparison of Static IIM and Dynamic IIM This section discusses a connection between the dynamic IIM the static IIM. Static and dynamic models are consistent u equilibrium conditions, and the dynamic model can be tr formed into a static model through the concept ofequivalen static inoperability.

As noted in the previous section, the dynamic IIM takes form of q̇std = K fA * qstd + c* std − qstdg. When equilibrium is reached,q̇std = 0. It follows that A * qstd + c* std − qstd = 0 or qstd = fI − A * qstdgc* std. Therefore, under equilibrium conditions, t dynamic model becomes the static model.

Specialization of Static IIM to Dynamic IIM In the dynamic model, suppose that sectori follows a dynamic inoperability functionqistd from time t = 0 to T. A static inoper ability q̄i exists, defined as follows:

q̄i = 1

T E

t=0

qistddt s64d

where q̄i is called equivalent static inoperability duringf0 ,Tg. Through the equivalent static inoperability, the economic los cumulated during the dynamic recovery can be estimated cally using the following equation derived from Eq.~62!:

Qistd = x̂iE t=0

qistddt = sx̂idsq̄iTd s65d

As depicted in Fig. 6, the scenario where the power sector re ers from 100% inoperability to 1% in 60 days has an equiva

Fig. 6. Dynamic inoperability and equivalent static inoperabilit

constant static inoperability of 22% for the 60-day period.

J. Infrastruct. Syst., 200

Practical Uses of IIM

The IIM provides a computation base for risk-impact analysis utilizes I-O data from the BEA—the agency responsible for d menting the transactions of approximately 500 producing consuming sectors within the U.S. economy. Through our d use of the detailed national I-O tables published by the BEA benefit from their intensive data collection efforts and reso base. In addition, we utilize data available through RIMS II conducting regional-level analysis. This provides a solid fou tion for any analysis, especially one as sensitive to the unkn as the analysis of a terrorist attack.

Given that BEA data provide each producing sector’s req ment or support from other sectors~i.e., production inputs such products and services!, IIM is capable of • Computing the propagating impacts of diverse perturba

scenarios for various regions; • Computing the impact of varying recovery rates for inte

pendent sectors; and • Computing various perspectives of impact, includinginoper-

ability and economic loss, which yield insight into societa consequences and provide a quantitative method for res allocation. As part of using economic-based data for analyzing a ter

attack situation, the IIM application is based upon the assum that the level of economic interdependencies between sect also representative of physical interconnectedness~i.e., in genera two sectors that have a large number of economic transa similarly have a large degree of physical linkage!. Therefore, uti lizing economic interdependencies made accessible to us th the BEA and RIMS II is an efficient and cost-effective alterna for comprehensively accounting for physical linkages betw national sectors.~Otherwise, a similar or even greater special collection effort would be required.! By allowing holistic integra tion of sectors, IIM provides analysts with a tool for systemic prioritizing sectors deemed to be economically critical, in a tion to identifying those sectors whose products are critical du recovery operations. The IIM’s prioritization capability a serves to avoid erroneous assumptions that might otherwise in preselecting most-vulnerable sectors or commodities.

Specifically for a power sector analysis, the IIM would capable of providing the following information essential for assessment and management of the propagating impacts o rorist attack: • Direct economic and power-production impacts of a terr

attack on the power generation and power supply sector • Economic and production capacity impacts to electrical po

users~manufacturing, commerce, household, and others! due to terrorist destruction of vulnerable electronic equipment

• Trade-offs between possible reductions in economic losse the corresponding cost of investment required for carrying various equipment recovery/resource allocation options;

• Labor requirements to support production, delivery, and u as-planned power outputs; and

• Economic and production impacts due to the possible psy logical effects of a terrorist attack.

Assumptions and Limitations of IIM

Several assumptions from the original Leontief structure ar tained in the IIM formulation. Many of the assumptions rem

unchanged because of the desire to capitalize on the vast BEA

5, 11(2): 67-79

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databases designed specifically for the linear, deterministic, librium model for which Leontief won the 1973 Nobel prize economics. It is important to address the underlying mode sumptions for optimal understanding and interpretation of an sis results.

Equilibrium Modeling of Static IIM

It is important to understand the equilibrium assumption of IIM, as perhaps this is the hardest factor to manage in situa where it is highly possible to experience nonequilibrium co tions. Equilibrium implies that industry inputs and outputs find balance with the final consumption of the sectors’ output the long run such a condition is evidently true. Moreover, du a recovery process, equilibrium conditions will also dominat industries are constantly improving their states in an interde dent fashion, as illustrated by the dynamic IIM. However, in short time immediately following scenarios that impose la widespread perturbations, nonequilibrium conditions could d nate and the IIM results would not exactly reflect real reco production rates or economic losses.

Fortunately, a terrorist attack would most likely impact a fined region of our country, while leaving surrounding regi intact. The specific attributes of the attack scenario determin size and location of the impacted region and which regi economies are categorized by equilibrium economic data. W the impacted region is relatively small, the consequences, large within that region, can potentially be dealt with either importing resources from the rest of the country or expor resources or problems out~e.g., hospital patients, or unusa inventory to support increased production in other regions!. These transfers would complement other activities to restore norma erations.

When applying the IIM, we anticipate that the national imp on the economy and production capacity due to a terrorist a is important, but not approaching anything like 100%, even ing the time period immediately following an attack. The sma the fraction~e.g., less than 10%!, the more applicable the resu obtained from the IIM. This is because the overall capacity o country to produce goods plays a significant role, which it d through redistributions that could be feasible without drastic nomic adjustments that would go beyond the IIM model’s sumptions of constant technology and overall economic struc At the regional level, the results from using the IIM would more suspect, since the region under attack would likely very large disruptions. However, the redistribution of nationa sources to the region and the recovery process could quickly, bringing the region into a condition that is within the I boundaries. The initial period of redeployment and reco would not be suitable for IIM analysis of inoperability or e nomic loss, but once within a close fraction of normal~e.g., 10%!, the model could provide results for the remaining periods lea up to full recovery.

For cases encompassing a relatively small region~say, an av erage state!, a brief time period~days to weeks!, or slight inoper ability ~less than 10%!, the bulk of economic losses accumulat the long period of final recovery, where companies operate mally, quarter by quarter, with constantly improving states do nated by equilibrium conditions. Since the bulk of econo losses are accumulated later, the costs in data and time re for building highly accurate transient models are unnecess large. Therefore, although the initial period of redeployment

recovery is not suitable for IIM analysis of inoperability or eco-

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nomic loss, once within a close fraction of normal~e.g., 10%!, the model could provide results for the remaining periods leadin to full recovery. Additionally, it can provide an optimal prior zation strategy for recovery during the uncertain transient pe to minimize the overall losses when the equilibrium conditio reached.

Stability of Technical Coefficient Matrix

At the core of Leontief’s model and the IIM is the techn coefficient matrixA derived from the BEA’s databases. Th equilibrium data define deterministic guidelines for sector in actions based on the assumptions of constant technology an nomic structure. The properties of these data provide the re ing assumptions that lead to model limitations.

The BEA decomposes the U.S. economy into about 500 tors whose outputs contribute to the input of other sectors. element in this matrix gives a constant, deterministic value represents the contributions to one sector, sayj , from any othe sector, sayi, which is proportional to the output of infrastructu j . There are obvious examples where this assumption is exa valid. Moreover, as the number of industry subdivisions creases, a first-order approximation becomes increasingly rate~500 sectors create a matrix defining 250,000 interdepen cies!. For example, if infrastructurei is tire production an infrastructurej is automobile production, then the value of ti used by infrastructurej increases linearly with the value of au mobiles produced.

On the other hand there are also, perhaps less obvious, where the linearity assumption may not be valid. For exam consider any process where human innovation is involved production increases proportionally, producers seek to inc resource sharing, among other measures, to consume fewer facturing materials. For example, an increase in the producti cars may not necessarily lead to a proportional increase i requirement for alloy wheels~i.e., basic car models often u simple lug-bolt rims with hubcaps!. Nonlinearity in the use o input requirements may also be observed in the case of s resources~e.g., multiple computers sharing external drives or ripheral devices!. However, in addition to the extreme cost of t added modeling, the database would become overwhelmin probably less meaningful to the individual who is interpreting results of such an analysis.

It is also important to note that the constants that define relationships between industries are time invariant and dete istic. This stems from the fact that the BEA generates data e five years directly from U.S. Census data because of econ momentum that causes values to change very little from ye year. Therefore the variance of commodity flows changes little from year to year and in most industry sectors~this is intui- tive because production procedures change very little over periods of time!. Returning to the car example, four tires for ev one car will certainly vary negligibly over time. However, t obviously will not always be the case. In order to protect ag major flaws, studies have compared various years of econ data. Analysis results have shown only negligible change tween two recent five-year periods.

Summary and Conclusions

The underlying economic data utilized in the IIM provide e

sector’s requirements of support from other sectors~i.e., produc-

5, 11(2): 67-79

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tion inputs such as products and services!. The IIM is capable o calculating the propagating impacts of diverse perturbation narios for various regions. In using economic-based data for lyzing a terrorist attack, application of the IIM is based upon observation that the level of economic interdependency bet sectors is often representative of physical interconnectednes~i.e., in general, two sectors that have a large volume of econ transactions have a similarly large degree of physical linka!. Utilizing interdependencies based on economic data gives th the capability to comprehensively assess the vulnerability o proximately 500 nationwide sectors given an attack to on multiple sectors. By allowing holistic integration of sectors, IIM ~1! provides analysts with a tool for systemically prioritiz sectors deemed economically critical, and~2! identifies those se tors whose continued operability is critical during recovery op tions. The following features and capabilities of the IIM mak particularly useful for conducting an impact analysis of a terr attack: • The IIM considers the numerous sectors of the econom

avoid misleading results that can stem from studying those sectors based on selection criteria related to direc nerabilities without also accounting for indirect ripple effe

• The IIM provides a comprehensive ranking of approxima 500 BEA sectors according to inoperability and economic impact metrics. It does this in a graphic format~e.g., histo grams! that is relevant to terrorist attack risk managemen

• The IIM is capable of modeling workforce recovery, enab the modeling of critical and electronic-vulnerable sectors as the power and health services sectors and identifyin most essential personnel for response to a terrorist attac

• The IIM provides various geographic resolutions~e.g., coun ties, states, and economic regions! that can be customized v RIMS II data to closely approximate different attack stren and intensities; and

• The dynamic IIM is capable of modeling different tempo frames of recovery~e.g., recovery rates for different secto! and establishing various interdependent adjustments to of equilibrium that may occur after a terrorist attack. Application of the IIM to the scenario of a terrorist attack

electric power and power-dependent sectors is described companion paper~Haimes et al. 2004!.

Acknowledgments

We acknowledge the Commission to Assess the Threat t United States from Electromagnetic Pulse Attack~the EMP Com mission! through the Science Applications International Corp tion ~SAIC! for providing partial support for this study. We a preciate the support and input received from Jim Scouras o staff of the EMP Commission. We are grateful for the valua inputs contributed by Mike Frankel, Ira Kohlberg, Rob Mahon Walter Scott, and Paul Spraggs during our regular progress ings, which helped us define the scope and direction for the s We appreciate the contributions of Dr. Stan Kaplan, Dr. Ir Pikus, and Dr. Lester Fink at the University of Virginia. We a highly appreciate the editorial assistance of Grace Zisk an tireless care and support that we received from Della Dirick Manager of the University of Virginia Center for Risk Mana ment of Engineering Systems. The National Science Found under a grant to the University of Virginia Center for Risk M

agement of Engineering Systems provided partial support for this

J. Infrastruct. Syst., 200

study ~NSF 0301553: Input-Output Risk Model of Critical Inf structure Systems, May 2003–January 2006!.

References

Blanc Diaz, M., and Ramos Carvajal, C.~2002!. The foundations of d namic input-output revisited: Does dynamic input-output belon growth theory? ^http://www19.uniovi.es/econo/DocumentosTrab 2002/258-02.pdf& ~April 6, 2004!.

Brucker, S. M., Hastings, S. E., and Latham III, W. R.~1990!. “The variation of estimated impacts from five regional input-output m els.” Int. Region. Sci. Rev., 13, 113–139.

Converse, A. O.~1971!. “On the extension of input-output analysis account for environmental externalities.”Am. Econ. Rev., 61, 197– 198.

Embrechts, P., Kluppelberg, C., and Mikosch, T.~1997!. Modeling ex tremal events for insurance and finance, Springer, New York.

Ernst and Young.~2002!. Manhattan lodging forecast, Ernst and Youn Real Estate Advisory Group, New York.

Federal Aviation Administration~FAA. ~2002!. Aviation industry over view fiscal year 2001, Office of Aviation Policy and Plans, Washin ton, D.C.

Galea, S., et al.~2002!. “Psychological sequelae of the Septembe terrorist attacks in New York City.”N. Engl. J. Med., 346~13!, 982– 987.

Griffin, J. ~1976!. Energy input-output modeling, Electric Power Resear Institute, Palo Alto, Calif.

Guo, J., Lawson, A. M., and Planting, M. A.~2002!. From make-use t symmetric I-O tables: An assessment of alternative technolog sumptions, 14th Int. Conf. on Input-Output Techniques, Bureau o Economic Affairs, U.S. Dept. of Commerce, Washington, D.C.

Haimes, Y. Y.~1977!. Hierarchical analyses of water resources syste Modeling and optimization of large-scale systems, McGraw-Hill, New York.

Haimes, Y. Y.~2002!. “Roadmap for modeling risks of terrorism to t homeland.”J. Infrastruct. Syst.8~2!, 35–41.

Haimes, Y. Y.~2004!. Risk modeling, assessment, and managemen, 2nd Ed., Wiley, New York.

Haimes, Y. Y., and Horowitz, B. M.~2004!. “Adaptive two-player hier archical holographic modeling game for counterterrorism intellig analysis.” J. Homeland Security Emergency Manage., 1~3!, Article 302, 1–24.

Haimes, Y. Y., and Jiang, P.~2001!. “Leontief-based model of risk complex interconnected infrastructures.”J. Infrastruct. Syst. 7~1!, 1–12.

Haimes, Y. Y., and Nainis, W. S.~1974!. “Coordination of regional wate resource supply and demand planning models.”Water Resour. Res, 10~6!, 1051–1059.

Haimes, Y. Y., Horowitz, B. M., Lambert, J. H., Santos, J. R., Crow K. G., and Lian, C.~2005!. “Inoperability input-output model fo interdependent infrastructure sectors: Case study.”J. Infrastruct. Syst., 11~2!, 80–92.

Haimes, Y. Y., et al.~2004!. “Applying inoperability input-output mode ~IIM ! to the impact of high-altitude electromagnetic pulse~HEMP! on interdependent infrastructure sectors.”Rep. to the Commission to A sess the Threat to the United States from Electromagnetic Puls tack, Center for Risk Management of Engineering Systems, Uni Virginia, Charlottesville, Va.

Isard, W. ~1960!. Methods of regional analysis: An introduction to gional science, MIT Press, Cambridge, Mass.

Jiang, P.~2003!. “Input-output inoperability risk model and beyond.” P dissertation, Dept. of Systems and Information Engineering, Un Virginia, Charlottesville, Va.

Jiang, P., and Haimes, Y. Y.~2004!. “Risk management for Leontie based interdependent systems.”Risk Anal, 24~5!, 1215–1229.

Krause, U. ~1992!. “Path stability of prices in a nonlinear Leontief

5, 11(2): 67-79

e .”

e y.”

ith

19–

d- -

man-

eet

ood

and lec- eer-

n in-

t t.

IMS ern-

t s, , D.

U.S.

D ow

nl oa

de d

fr om

a sc

el ib

ra ry

.o rg

b y

P en

ns yl

va ni

a S

ta te

U ni

ve rs

it y

on 0

8/ 11

/2 0.

C op

yr ig

ht A

S C

E . F

or p

er so

na l

us e

on ly

; al

l ri

gh ts

r es

er ve

Model.” Ann. Operat. Res., 37~1!, 141–148. Lahr, M. L., and Dietzenbacher, E., eds.~2001!. Input-output analysis

Frontiers and extensions, Palgrave, New York. Lahr, M. L., and Stevens, B. H.~2002!. “A study of regionalization in th

generation of aggregation error in regional input-output modelsJ. Regional Sci., 42~3!, 477–507.

Lambert, J. H., and Patterson, C. E.~2002!. “Prioritization of schedul dependencies in hurricane recovery of a transportation agencJ. Infrastruct. Syst.8~3!, 103–111.

Lee, K. S.~1982!. “A generalized input-output model of an economy w environmental protection.”Rev. Econ. Stat., 64~3!, 466–73.

Leontief, W. W. ~1951a!. “Input-output economics.”Sci. Am., October 15–21.

Leontief, W. W.~1951b!. The structure of the American economy, 19 1939, 2nd Ed., Oxford University Press, New York.

Leontief, W. W. ~1966!. Input-output economics, Oxford University Press, New York.

Liew, C. J. ~2000!. “The dynamic variable input-output model: An a vancement from the Leontief dynamic input-output.”Annals of Re gional Science, 34~4!, 591–614.

Miller, R. E., and Blair, P. D.~1985!. Input-output analysis: Foundation and extensions, Prentice-Hall, Englewood Cliffs, N.J.

Miller, R. E., Polenske, K. R., and Rose, A. Z.~1989!. Frontiers of input-output analysis, Oxford Univ. Press, New York.

Norris, F. H., Byrne, C. M., Diaz, E., and Kaniasty, K.~2002!. “The range, magnitude, and duration of effects of natural and hu caused disasters: A review of the empirical literature.”A National Center for Post-Traumatic Stress Disorder (PTSD) fact sh, NCPTSD, White River Junction, Vt.

JOU

J. Infrastruct. Syst., 200

Olsen, J. R., Beling, P. A., Lambert, J. H., and Haimes, Y. Y.~1997!. “Leontief input-output model applied to optimal deployment of fl protection.”J. Water Resour. Plan. Manage.124~5!, 237–245.

Proops, J. L.~1984!. “Modeling the energy-output ratio.”Energy Econ., 6~1!, 47–51.

Santos, J. R.~2003!. “Interdependency analysis: Extensions to dem reduction inoperability input-output modeling and portfolio se tion.” PhD dissertation, Dept. of Systems and Information Engin ing, Univ. of Virginia, Charlottesville, Va.

Santos, J. R., and Haimes, Y. Y.~2004!. “Modeling the demand reductio input-output~I-O! inoperability due to terrorism of interconnected frastructures.”Risk Anal, 24~6!, 1437–1451.

Susser, E. S., Herman, D. B., and Aaron, B.~2002!. “Combating the terror of terrorism.”Sci. Am., 287~2!, 70–77.

Tsang, J. L., Lambert, J. H., and Patev, R. C.~2002!. “Extreme even scenarios for planning of infrastructure projects.”J. Infrastruct. Sys 8~2!, 42–48.

United States Department of Commerce.~1997!. Regional multipliers: A user handbook for the regional input-output modeling system (R II) , Bureau of Economic Analysis, Washington, D.C., U.S. Gov ment Printing Office, Washington, D.C.

United States Department of Commerce.~1998!. Benchmark input-outpu accounts of the United States, 1992, Bureau of Economic Analysi Washington D.C., U.S. Government Printing Office, Washington C.

United States Department of Commerce.~2003!. Digital Economy 2003, Economics and Statistics Administration, Washington, D.C., Government Printing Office, Washington, D.C.,^http:// www.esa.doc.gov/DigitalEconomy2003.cfm&.

5, 11(2): 67-79