Review on Energy Resilience

Development of the Multiregional Inoperability Input-Output Model (MRIIM) for Spatial Explicitness in Preparedness of Interdependent Regions Kenneth G. Crowther* and Yacov Y. Haimes

Center for Risk Management of Engineering Systems, University of Virginia, 112A Olsson Hall, Charlottesville, VA 22903 DEVELOPMENT OF MRIIM FOR SPATIAL EXPLICITNESS

Received 27 March 2007; Revised 2 August 2008; Accepted 30 October 2008, after one or more revisions Published online 31 March 2009 in Wiley InterScience (www.interscience.wiley.com) DOI 10.1002/sys.20130

ABSTRACT

Any given region in the US is a complex, interconnected, and interdependent economic system of systems that includes multiple stakeholders, spans multiple sub-regions, and produces a very large number of commodities and services. This paper provides a holistic, methodological framework with which to model the multisectoral and multiregional economic interdependencies inherent in such a large-scale and complex system, factors essential to making strategic preparedness decisions. The framework we present extends the Inoperability Input-Output Model (IIM) [Santos and Haimes, 2004], a model which has been developed to study these large-scale systems and which has been deployed in various studies. The basic IIM yields only average estimates across geography, and thus may provide insufficient information with which to make decisions about preparedness allocation within and across regions. Such average estimates may lead planners and policy makers to overlook geographically concentrated risks and significant cross-regional interdependencies, which are critical in evaluating strategic preparedness options. This paper extends the IIM to model the multiregional interdependencies among the various regions in the US by introducing and developing the Multiregional IIM (MRIIM) and by presenting the spatially explicit concepts of intraregional and multiregional interdependency matrices, A* and T*, respectively. The MRIIM possesses important properties that are derived from the databases that support the model and from the construction of the model itself. These properties guarantee unique solutions when studying the cascade of perturbation across regions and guarantee convergence when computational methods are applied. These properties also lead to a notion of resilience that is gained from cross-regional interdependencies. The major contribution of this paper is to demonstrate the importance of spatial explicitness in interde- pendency analysis through an example case study. © 2009 Wiley Periodicals, Inc. Syst Eng 13: 28–46, 2010

Regular Paper

Contract grant sponsors: Partial support from the National Science Foundation under a grant to the University of Virginia (NSF 0301553: Input-Output Risk Model of Critical Infrastructure Systems); from the Department of Homeland Security through the Institute for Information Infrastructure Protection; and from the Virginia Governor’s Office of Commonwealth Preparedness.

* Author to whom all correspondence should be addressed (e-mail: [email protected]; [email protected]).

Systems Engineering Vol. 13, No. 1, 2010 © 2009 Wiley Periodicals, Inc.

28

Key words: Inoperability Input-Output Modeling (IIM); interdependency analysis; spatial explicitness; multiregional preparedness; infrastructure protection; strategic preparedness

1. INTRODUCTION

In the wake of recent natural and man-made catastrophes around the world, regional planners and regional government policy makers face difficult decisions when endeavoring to improve the competitive advantage of their regions in order to attract business and increase overall local economic secu- rity. Given this situation, many planners are recommending the creation of “risk-based formulas” for the allocation of preparedness management funds (see, for example, the Com- monwealth of Virginia [JLARC, 2005]). Preparedness is the state of readiness that results when investments are made and actions are taken in advance of disruptive events and are designed so as to reduce the probability and/or consequences of disruptive events on the region. These types of recommen- dations for preparedness management stem from and are underscored by the Department of Homeland Security (DHS) directive to “carefully weigh the benefit of each homeland security endeavor and only allocate resources where the bene- fit of reducing risk is worth the amount of additional cost” [DHS, 2006: 64]. Our times require an analytical framework that will enable effective and systemic assessments of the intrinsic value of strategic preparedness. Such a framework should build on existing and available regional databases and be capable of performing quick approximations that will support regional decisions to allocate resources. This paper seeks to develop that framework, and in doing so to contribute a solution to a pressing analytical need.

Preparedness evaluation is complicated by the structure of our society—an interconnected and interdependent network of technologies, businesses, organizations, infrastructures, sociopolitical realities, and regions that require one another for continued efficient operation. Sheffi [2005] provides ex- tensive examples of interdependencies that result from supply networks among global enterprises and describes several methods for reducing enterprise risk by adding resilience. Regional planners and decision makers face a different para- digm in that they do not typically own or control regional operations, they are responsible for regions that are not geo- graphically diversified, and they generally have incomplete knowledge about operations that span a wide range of eco- nomic activities. However, they must be capable of strategi- cally responding to a variety of events that are typically regionally concentrated. Acquiring and consolidating data that represent regional operational entities and their overlap- ping interconnections results in a multiscale and multidimen- sional set of information that can be difficult to utilize effectively. Santos and Haimes [2004] demonstrate how data that is available from the Bureau of Economic Analysis (BEA), US Department of Commerce, can be used to con- struct an interdependency model for pre-disaster mitigation planning. Their proposed Inoperability Input-Output Model (IIM) is based on the basic Leontief [1966] input-output structure and has been used to model the impact of potential

disruptive events on interdependent economic systems. De- tailed literature review of the IIM and related usage can be found in Haimes and Jiang [2001], Haimes [2004], Santos and Haimes [2004], Haimes et al. [2005a,b], Crowther and Haimes [2005], Kujawski [2006], Lian [2006], Lian and Haimes [2006], and Santos [2006], among others.

The main advantages of using the IIM to develop prepar- edness strategies are derived from the large-scale databases that support the model and from the maturity of the field of input-output analysis. However, several limitations have been identified in both the underlying model and the data, and several model extensions and alternatives have been proposed [Brown et al., 2004; Kujawski, 2006; Lian and Haimes, 2006; Santos, 2006.] This paper identifies and illustrates an addi- tional limitation and proposes data sources as well as a model structure that can be adapted to improve the ability of the IIM in strategic preparedness planning activities that are multire- gional in nature. These planning activities include, among other things: (1) the formation of memoranda of under- standing or memoranda of agreement for contingent supply chains within the public sector in order to ensure the contin- ued availability of critical resources in situations where the conventional supply of those resources from the decision making region or another region might be vulnerable, (2) the prioritization of relationships between the public and private sector or between other regions in order to provide the maxi- mum assurance of continuous operation of critical infrastruc- ture, and (3) the development of a value proposition to public and private sector partners that articulates the regional bene- fits of maintaining multiregional continuity of operations.

The IIM, as presented in past publications, suffers from a lack of spatial explicitness and thus produces only average estimates across geography, which may be insufficient for making decisions about preparedness allocations across re- gions. Such average estimates may lead planners and policy makers to overlook geographically concentrated risks or sig- nificant cross-regional interdependencies, which are impor- tant in evaluating relevant preparedness strategies. A model is said to be spatially explicit when it differentiates behaviors and predictions according to spatial location [Goodchild and Janelle, 2004]. The IIM currently considers the nation or a given region of contiguous counties as a whole, and it assumes that this region operates with perfect communication among all acting parties; the resulting effects of a perturbation are thus predicted to be uniform across the entire region. How- ever, spatial explicitness is added when the economy is re- garded as a system of individual regional economies, with a network of encompassing processes connecting the various sub-regions, such as cross-regional transactions of goods and services. Spatial explicitness produces distinct estimates for each region that are determined by that region’s unique char- acteristics as well as its interconnectedness with other regions.

Armington [1969] argues for the need to differentiate commodities and services in an economy according to its

DEVELOPMENT OF MRIIM FOR SPATIAL EXPLICITNESS 29

Systems Engineering DOI 10.1002/sys

region of origin. Worrall [1991: 1] estimates that 80% of all data used by local decision-makers in the UK is geographical or in some way spatially referenced. We feel that similar statements could be made concerning spatially referenced data in many aspects of business and regional operations around the world. Geographic data are traditionally utilized through extensive reports consisting of maps and tables de- scribing myriad geographic dimensions of problems, but these data frequently lack the logical coherence necessary to have an effective impact on choices made by decision-makers, who may have bounded rational capabilities. The need for spatial explicitness in preparedness decisions, juxtaposed with the availability of geospatial data, provides strong moti- vation for pursuing a study of logical models founded on regional location of people, infrastructures, and businesses, models that will aid decision-makers and planners in integrat- ing extensive and varied geographic data.

This paper constructs a spatially explicit multiregional inoperability input-output model (MRIIM) [Crowther, 2007] to expand the interdependency analysis capability of the IIM. It draws on and is built upon multiregional frameworks devel- oped through Riefler and Tiebout [1970], Polenske [1972], Miller and Blair [1985], and Isard et al. [1998]. MRIIM is founded on the multiregional assumption that excess demand

and supply can reasonably be pooled. This assumption im- plies that one commodity produced by one sector is not preferred over the same commodity produced by another sector, but rather only by commodity/service classification and spatial preference. The MRIIM in this paper is built around existing large-scale data collection efforts available through various public and private entities in the US, publish- ed at multiple scales (as discussed later in the text), the nature of which guarantees a unique analytical solution. The major contribution of this paper is to demonstrate the importance of spatial explicitness in interdependency analysis through an example case study.

Sections 2–4 describe the construction of the MRIIM through the selection and adaptation of existing multiregional models. Section 5 describes how the MRIIM can handle complex inputs such as combined supply and demand disrup- tions. Section 6 provides an illustrative example that demon- strates the impact of multiregional interdependencies on the regional analyses used in preparedness decisions. Section 7 summarizes the work. An appendix provides details concern- ing the stability and uniqueness of the model solution that result from the model structure and the character of available data.

Summary of Important Symbols

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2. BASIC IIM CONSTRUCTION AND ASSUMPTIONS

The Inoperability Input-Output Model (IIM) has been applied in various studies for the Commission on High-Altitude Elec- tromagnetic Pulse (attacks against the US) [Haimes et al., 2005a,b], the security division of the Virginia Department of Transportation [Crowther et al., 2004], the demand reduction of air transportation following the terrorist attacks of Septem- ber 11, 2001 [Santos, 2006], the US Northeast blackout of 2003 [Anderson, Santos, and Haimes, 2007], and the US Gulf Coast hurricanes of 2005 [Crowther, Haimes, and Taub, 2007], among others, to study the impact of various risk management policy options on systems of interconnected economic sectors. Derived from Bureau of Economic Analy- sis (BEA) data and a Leontief economic model [Leontief, 1966], the IIM offers a quick and inexpensive method for estimating economic impacts and sector interdependencies, one that describes all economic sectors and most regions of the US.

Santos and Haimes [2004] demonstrate how data available through the BEA, US Department of Commerce, can be used to construct a sector-sector interdependency matrix A*, which is based on the concept of the Leontief technical coefficient matrix A and on a modification of the interdependency matrix proposed in Haimes and Jiang [2001]. This interdependency matrix is the structure that underlies the IIM. The basic formulation of the IIM is shown in Eq. (1), where q is a vector of supply inoperability or percent operational dysfunctional- ity, and f* is a vector of direct perturbation to a sector’s production. This section provides a brief description of each of these model components.

q = A∗q + f∗ or    qi = ∑

j

aij ∗qj + fi

∗   , Wi (1)

Intuitively Eq. (1) is a means of estimating supply inoper- ability from intermediate economic exchanges between sec- tors of the economy. In this basic formulation, the inoperability vector q is a real-valued vector of normalized production loss for each sector in the national economy. The elements of q are ratios of either unrealized (qi > 0) or excessive (qi < 0) production with respect to the nominal production level of the sector. Inoperability is realized either directly from some exogenous restriction (e.g., a bomb or hurricane physically inhibits production) or from the pertur- bation f* of some interconnected sector (e.g., a bomb or hurricane physically inhibits production within one sector, which thereby affects the production of another sector be- cause of typical economic transactions between the two). The perturbation vector f* is a real-valued vector whose elements are the lost final demand with respect to nominal production output for each sector in the nation. As stated by Santos and Haimes [2004], the notion of inoperability, in the IIM-sense, is analogous to unreliability, where a value of zero represents nominal operation and a value of unity represents complete sector production incapability. Typically q is estimated from an input of f*.

The interdependency matrix A* is a matrix whose elements describe sector-sector interdependencies across the nation. Each element a ij∗ describes the amount of inoperability expe- rienced in sector i due to total inoperability in sector j. A* is produced from two large-scale databases that are discussed in Santos and Haimes [2004] and Haimes et al. [2005a], includ- ing: (i) Use matrix and (ii) Make matrix. The Use matrix describes the value (quantity multiplied by producer price) of each commodity consumed by each sector of the national economy. The Make matrix describes the value (quantity multiplied by producer price) of each commodity produced by each sector of the national economy. Note that a vector x of total production output for each sector is equivalent to the row sums of the Make matrix, whose elements are the value (quantity multiplied by producer price) of total annual pro- duction for each sector of the economy. Every 5 years, exten- sive data are collected under the designation of economic census. For most businesses in the US, submission of the economic census report is required by law. (For example, the manufacturing sector is required to submit annually the eco- nomic census under an Act of Congress that resulted in Title 13, United States Code [US Census Bureau, 2002].) Data from the various firms in the economic census are aggregated and published according to sector classifications that are defined according to the primary commodities produced. Santos and Haimes [2004] and Haimes et al. [2005a] provide a detailed summary of these accounts. Detailed accounts are published by the BEA every 5 years, drawn from a complete economic census that is required of all businesses; because of this requirement, the results in the accounts under considera- tion are treated as actual representations of the state of our economy and not as estimates.

Reading Eq. (1) from left to right illustrates the two main assumptions of the IIM. First, supply inoperability q is equal to the sum of cascading inoperability from interconnected sectors A*q and direct perturbation f*. (This is the equilibrium assumption because it assumes that the economic sectors in the nation balance transactions between supply and consump- tion—i.e., all production is consumed.) Second, interdepend- ent sectors receive proportional impact from disruptive sectors; that is, a ij∗ describes the amount of inoperability experienced in sector i due to total inoperability in sector j. The car manufacturing sector clearly illustrates the second assumption in that for every car that is manufactured the sector must consume about four tires, several windows, one engine, etc. Therefore, if one considers the tire sector (i.e., the aggregate of all tire-producing firms) as the sole supplier of the car manufacturing sector (i.e., the aggregate of car manu- facturing firms), then some loss of tire production (e.g., a destroyed tire production plant) will result in a proportional strain on car manufacturing operations. Note that these as- sumptions of the model result in limitations of how it might be used. For example, the highly dynamic nature of impacts immediately following a terrorist attack would violate the equilibrium assumption unless one aggregates the impact across a sufficient span of time. Lian and Haimes [2006] describe model modifications that can help to overcome these challenges for resilience analysis. Moreover, there are several sectors with nonlinear or probabilistic production needs which would violate the proportionality assumption if the

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Systems Engineering DOI 10.1002/sys

sectors and/or regions are not sufficiently aggregated, or if the perturbations have effects on those regions and sectors that are considered too large. Lian and Haimes [2007] and Barker and Haimes [2008] extend the IIM to account for uncertainty and the probabilistic nature of inputs. Crowther [2007] de- scribes spatial and sector aggregations and disruption size limitations under which the proportionality assumption may be correctly incorporated.

Despite its limitations and assumptions, Rose [2004: 24] maintains that input-output analysis is “well suited to exam- ining how damage in some sectors can ripple through the economy.” Cochrane [2004: 51] finds that the HAZUS ap- proach, a fundamental part of which relies on input-output analysis, provides a “consistent means of evaluating loss, a quality that appears to be lacking in a number of economic impact studies conducted in the wake of the World Trade Center attacks.” Okuyama, Hewings, and Sonis [2004: 78] state that input-output analysis has been employed to “evalu- ate the economic impacts of disasters, mainly because of [its] ability to reflect the structure of regional economy in great detail.” It also offers a means for understanding interdepend- encies as reflected in current operations, and thus provides an analysis of the current state of vulnerability that results from the interconnectivity of the infrastructure and economic sec- tors. For example, Anderson, Santos, and Haimes [2007] use the IIM to produce estimates for the 2003 US Northeast Blackout that are within 4% of other quoted estimates, but that in addition provide details concerning the specific sectors experiencing the losses. For more discussion of economic input-output modeling, see Miller and Blair [1985] or Rose and Miernyk [1989].

3. CONSTRUCTION OF A REGIONAL IIM

The regionalization of the IIM is the first step in adding spatial explicitness to the model. It requires the redefinition of q and f* to account for all regions of interest and a recalculation of A* to reflect the spatially explicit intraregional interdepend- encies which are likely to differ from those of the nation based on the regional mix of production processes. Unfortunately, Use and Make matrices do not exist on the regional level. Instead, regional interdependency coefficients are generally estimated through the use of regional multipliers that are constructed from measures of regional production compared to the nation. Equation (2) describes the formulation of a specific type of regional multiplier called a location quotient, which represents the proportion of demand for sector i in region s that can be satisfied regionally, compared to the ability of the nation to satisfy its own internal demand. Essen- tially, location quotients are ratios representing the relative production of a sector in a region compared to the nation. Empirical tests suggest that regional multipliers can fre- quently be used as surrogates for time-consuming and expen- sive regional surveys in adapting national interdependency data to a region without compromising accuracy [Brucker, Hastings, and Latham, 1990; Rickman and Schwer, 1995; BEA, 1997].

Equation (3) describes the application of location quotients in estimating intraregional technical coefficients. When the location quotient is less than 1, it is assumed that the regional

supply of the sector’s production output is insufficient to meet local demand; otherwise regional supply is sufficient, and no adjustment to the technical coefficient is necessary [BEA, 1997]. Location quotients can be estimated using ratios of total regional earnings of a sector, based upon the assumption that productivity of specific sectors does not significantly vary across the nation.

lqi s =

xi s

/ xs

xi N

/ xN ,

(2)

aij ∗s =

  

lqi s aij

∗,

aij ∗,

lqi

s < 1, lqi

s ≥ 1, (3)

where

lqi s is the proportion of demand for sector i in region s that

is satisfied internally compared to other regions in the nation,

xi s, xi

N is the total production output of sector i in region s or the nation N, respectively (dollars - production quan- tity multiplied by producer price),

xs, xN is the total production output of all sectors in region s or the nation N, respectively (dollars - production quantity multiplied by producer price),

aij ∗s is the amount of inoperability experienced in sector i

due to total inoperability in sector j in the region s, aij

∗ is the amount of inoperability experienced in sector i due to total inoperability in sector j in the nation.

Annual estimates of earnings (used to calculate location quotients) are also based on data that are required by law concerning the total income paid to employees across the US. These data are also considered as benchmarks and are used to test the reliability of advanced and preliminary estimates that are released early based on samples from information submit- ted by employers [e.g., Fixler and Grimm, 2002; Brown, Grimms, and Sacks, 2003].

A spatially explicit interdependency matrix can be formed as a block diagonal matrix in (4), where A*s is a (sub)matrix containing all intraregional technical coefficients for region s calculated using Eq. (3) and the original A* from the IIM in Eq. (1). The rest of the matrix contains zeros.

A* =











A∗1

A∗2

... A∗p











(4)

Note that the dimensions of A* have increased due to the improved degree of spatial explicitness and, moreover, that the individual values have decreased (or remained the same) since each block in the matrix is derived from the same national matrix and from regional multipliers that are strictly between 0 and 1. Intuitively, the values have decreased, be- cause as we account for smaller regions the degree of inter- connectedness between sectors within the individual region

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Systems Engineering DOI 10.1002/sys

typically decreases. This emphasizes the need to account for cross-regional interdependencies, which will be discussed in the following section.

To complete the regionalization of the IIM, redefine q and f*, as shown in (5) and (6), respectively. This redefinition requires that perturbations be estimated on the regional level, but also allows for inoperability to be estimated on the re- gional level. Equation (7) shows that the regional IIM has the same form as the nation, but contains increased spatial explic- itness due to the integration of regional measures of produc- tion into the model.

q =   

 

q1

I

qp

  

  , (5)

f∗ =   

 

f∗1

I

f∗p

  

  , (6)

q = A∗q + f∗ or    qi

r = ∑ j,s

aij ∗sqj

s + fi ∗r

  , Wi, r. (7)

Regionalization, the first step, is critical to the develop- ment of the multiregional model. The definition of the re- gions, including size and location, should be driven by the specific questions to be answered by the model. A large number of independent regions provides a greater amount of spatial explicitness, but also increases the data requirements and computational time. When selecting a regional size, one should consider the nature of the perturbation that is being assessed (try to select a region size that can reasonably repre- sent the direct perturbation); the type of decisions to be made as a result of the model (try to select regions that have jurisdictions that align with the decision-makers and try to make decisions that can be influenced by the analysis); and the potential for interpreting and deriving meaning from the results (a smaller number of regions allows for a much simpler interpretation of results).

Our analysis of the structure of the interdependency matrix A* reveals that this model does not reflect cross-regional interdependencies. In a model that aggregates large regions, these cross-regional interdependencies may not be important, since there may be little incentive to ship over long distances. However, given the increased globalization of our economy and the improved technology that allows services to be per- formed over greater distances, we cannot expect fully to understand interdependencies at a level necessary for prepar- edness without incorporating to some degree the question of cross-regional flow of goods and services. The following section provides this improvement and discusses in detail some of the data sources available for arriving at these esti- mates.

4. CONSTRUCTION OF A BASIC MULTIREGIONAL IIM (MRIIM)

The construction of the Multiregional IIM (MRIIM) builds on the regionalized IIM presented in the previous section by

accounting for cross-regional flows of goods and services that interconnect regions. Taking cross-regional flows into ac- count enables the calculation of multiregional coefficients, which in turn adjust the intraregional interdependency matrix A* in Eq. (7). This section constructs the basic MRIIM; the sections that follow extend the MRIIM to apply more com- plex inputs and provide illustrative examples.

Multiregional coefficients are calculated using commodity and service flow data. These multiregional coefficients de- scribe the way that multiple, smaller (sub)regions are inter- connected as larger regional systems, based upon economic transactions of goods and services across geography. To the decision maker managing large regional systems of intercon- nected regions, these coefficients provide a measure of eco- nomic intraconnections among smaller (sub)regions that will result in either cascades of impacts or sources of resilience following a disaster scenario. To the decision maker respon- sible for smaller (sub)regions they systemically provide: (1) a demand “footprint” describing other regions from which they purchase goods and services and (2) a supply “footprint” describing other regions to which they deliver goods and services. These decision makers of smaller (sub)regions can thus adapt strategic preparedness to mitigate risks against: (1) disaster scenarios that produce supply perturbations in their demand footprint and (2) disaster scenarios that produce demand perturbations in their supply footprint. We will henceforth refer to commodities and services as commodities in this paper. Let zirs be the quantity of commodity i produced in region r and consumed in region s. For each commodity, we form an origin-destination matrix similar to the matrix in Table I for p regions.

The ratio of commodity flow zirs to the total consumed commodities at the final destination sis represents the portion of commodities consumed in region s that arrived from region r.

Isard et al. [1998] prescribe a method for estimating the interregional technical coefficient of a Leontief-based eco- nomic input-output model, given data concerning commodity exchanges across regions. Equation (8) estimates the interre- gional technical coefficient, given the demand-pooling as- sumption.

a ij rs =

  

zi rs

si s

   z ij

•s/xj s = t i

rs zij •s/xj

s = t i rsa ij

•s, (8)

where

t irs = z irs /s is is the proportion of commodity i consumed by region s that originated in region r,

z ij•s is the quantity of production of sector i (from any region) that is consumed by sector j in region s,

xjs is the total production output of sector j in region s, a ij•s is a regional Leontief technical coefficient, which is

the portion of output of sector j in region s that is equivalent to the intermediate consumption by sector j in region s of production from sector i of any region.

Equation (9) defines the spatially explicit interregional proportions matrix T, and Eqs. (10) and (11) define x and f, respectively, for the p-region economy. Note that each block

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matrix Trs in T is a diagonal matrix. The elements in the diagonal of the matrix Trs are the proportions for all commodi- ties i ∈ n consumed by region s that originated in region r.

T =   

 

T11

I

T1p

… ... …

Tp1

I

Tpp

  

  , (9)

x =   

 

x1

I

xp

  

  , (10)

f =   

 

f1

I

f p

  

  . (11)

This construction results in a multiregional Leontief-based model and can be generalized for n sectors and p regions. Eq. (12) shows the multiregional Leontief-based model used to construct the MRIIM. Constructions similar to Eq. (12) can be found in Riefler and Tiebout [1970], Polenske [1972], Miller and Blair [1985], and Isard et al. [1998].

x = TAx + Tf or    xi

r = ∑ js

ti rsaij

•s xj s + ∑

s

ti rs fi

s   , Wi, r.

Reading Eq. (12) from left to right illustrates the three main assumptions of this form of the multiregional model (the first two are already present in the IIM). First, the output of sector i in region r is equal to (in equilibrium with) the sum of intermediate consumption across all regions plus the sum of final consumption across all regions. Second, intermediate consumption of any sector in any region is proportional to its production output; that is, a ijrs = t irsaij•s is the proportion of output of sector j in region s that equates to the amount of intermediate consumption by sector j in region s of sector i production from region r. Third, demand by all sectors in region s for commodity i in region r is pooled to modify the distribution of interme- diate and final consumption; that is, tirs is the proportion of commodity i consumed by region s that originated in region r. The demand pooling assumption results in the multiregional trade coefficient matrix T being formed as shown in Eq. (8). This form also makes it clear that the stability of the intraregional technical coefficient matrix A in (12) does not imply interregional stability or stabil- ity of the multiregional trade coefficient matrix T. (See Riefler and Tiebout [1970] for further details.) Most limitations of this model result from these three funda- mental model assumptions. For further discussion of these limitations, see Crowther and Haimes [2005], Ku- jawski [2006], Santos [2006], and Crowther [2007].

To construct the MRIIM, define x and f in Eq. (12) as spatially explicit vectors of the predisaster level of production output and final demand, respectively. Note that these quan- tities are known from databases on nominal historic produc-

tion and demand. Additionally, define x~ and f ~ as spatially

explicit vectors of postdisaster levels of production output and final demand, as defined in Eq. 13.

x~ =







  

  

x ~

1 1

I

x ~

n 1

x ~

1 2

I

x ~

n p







  

  

and f ~ =







  

  

f~ 1 1

I

f~ n 1

f~ 1 2

I

f~ n p







  

  

. (13)

These two sets of vectors describe predisaster and postdis- aster operating points and can be used to construct two Leon- tief-type multiregional models that represent the state of demand and production before [Eq. (12)] and after [Eq. (14)] a natural or man-made disaster scenario.

x~ = TAx~ + Tf ~ ⇔ 

  x ~

i r = ∑

js

ti rsaij

•s x ~

j s + ∑

s

ti rs f

~ i s   , Wi, r.

Subtracting Eq. (14) from Eq. (12) results in

(x − x~) = TA(x − x~) + T(f − f ~ ). (15)

Define x̂ as a matrix whose diagonal elements are the elements of the vector of production output x with the remain- ing elements zero, as shown in (16). Note that all diagonal elements are positive because economic sectors and regions are defined based on the notion of economic significance [US Census Bureau, 2002]. Thus, x̂ is nonsingular.

x̂ =   

 

x1 1

... xn

p

  

  . (16)

Premultiplying Eq. (15) by x̂−1 and inserting x̂x̂−1 (equiva- lent to the Identity Matrix) results in the MRIIM form of

x̂−1 (x − x~) = (x̂−1Tx̂)(x̂−1Ax̂)x̂−1(x − x~)

+ (x̂−1Tx̂)x̂−1(f − f ~ ) (17)

The substitutions in Eq. (18)–(21) enable a simpler nota- tion of Eq. (22) that is similar to the IIM notation in Santos and Haimes [2004].

(12)

(14)

A∗ = [x̂−1Ax̂] =   

 

[x̂1]−1A•1x̂1 ... [x̂p]−1A•px̂p

  

  , (20)

q = x̂−1(x − x~), (18)

f∗ = x̂−1(f − f ~ ), (19)

34 CROWTHER AND HAIMES

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T∗ = [x̂−1Tx̂] =   

 

[x̂1]−1T11x̂1

I

[x̂p]−1Tp1x̂1

… ... …

[x̂1]−1T1px̂p

I

[x̂p]−1Tppx̂p

  

  .

(21)

Making the substitutions in Eqs. (18)–(21) results in a spatially-explicit MRIIM shown in

It is instructive at this point to consider the additional databases that support multiregional interdependency model- ing and strategic preparedness. The following section dis- cusses the data used to construct the multiregional interdependency matrix T*. Appendix A draws conclusions about the solution to the MRIIM—its existence and unique- ness.

4.1. Multiregional Interdependency Matrix T*

The multiregional interdependency matrix T* contains infor- mation about the cross-regional transactions used in the MRIIM. This section provides details concerning this matrix, its meaning, and the data that are used for its construction in the multiregional modeling presented in this paper.

The product T*A* is an estimate of interregional interde- pendencies, an element of which [T∗A∗] ijrs is the amount of inoperability experienced in sector i in region r due to total inoperability in sector j in the region s. T* is an algebraic transformation of the multiregional trade coefficient matrix as shown in Eq. (23), which is based on commodity flows and cross-regional service transactions. Equation (24) ex- pands this calculation into scalar form. Note the switch in notation in order to specifically communicate the construc- tion of each element in the matrix. By convention, this paper labels rows and columns of T and T* beginning with region 1, commodity 1, moving to region 1, commodity 2, and so on until region p, commodity n. All commodities within the same region are kept together, unless adjust- ments are needed for computational purposes. In (24), i and j are row and column indices in T*, respectively, r is the region of origin, s is the region of destination, and k is the commodity. Equation (24) illustrates the pattern utilized to organize the various data from databases (with the indices k, r, and s) in relation to the matrix T* that is used in the MRIIM calculations.

T∗ = [x̂−1Tx̂] =   

 

[x̂1]−1T11x̂1

I

[x̂p]−1Tp1x̂1

… ... …

[x̂1]−1T1px̂p

I

[x̂p]−1Tppx̂p

  

  ,

(23)

   tij ∗ = t(r−1)n+k,(s−1)n+k

∗ = t k rs

xk s

xk r =

  

zk rs

sk s

   xk

s

xk r

   , Wi, j, k, r, s.

(24)

Equation (24) illustrates the construction of the elements in the multiregional interdependency matrix in terms of the data available from the BEA and other sources described below. Note that there exists a relationship between the total

consumption of commodity k by region s, sks , and the total production output of commodity k by region s, xks. These differ by a factor of net imports to region s, and thus there is a proportional change in the location quotient that reflects a region’s inability to meet its own demand with regional pro- duction when the transformed element of T* becomes large.

The nature of the commodity flow data is such that the intraregional coefficients tend to be much larger than the interregional ones. Figure 1 illustrates this difference using data from BTS [2002] by showing that short-distance flows of commodities between regions dominate longer distances. This means that most production is consumed within the region it has been produced. (In fact, data are more commonly available for regions that are large enough to reflect this attribute of dominating intraregional flows. The smallest re- gions for which data are available are greater metropolitan regions commonly consisting of a city and its surrounding counties.)

The Commodity Flow Survey (CFS) reports data on the movement of shipped commodities, their values, weights, and modes of transport, as well as their origins and destinations [BTS, 2002]. This information is publicly available in various aggregate summaries that protect the anonymity of the indi- vidual sampled companies. The 2002 CFS presents shipping data from business establishments in the US classified accord- ing to the 1997 North American Industry Classification Sys- tem (NAICS). Even though surveys are constructed based on NAICS classifications, numbers are reported according the Standard Classification of Transported Goods (SCTG).

Figure 1 shows a snapshot of the data published in the 2002 CFS. The figure was generated from Tables 3a and B-3a of BTS [2002] summarizing the distribution of total value of commodities shipped in the US by distance of shipment. In addition, it presents the estimated errors associated with this report. The bars in Figure 1 show the 90% confidence inter- vals for the estimates reported in Table 3a of BTS [2002].

It is interesting to note that the error bars remain relatively the same size even as the quantities of shipped commodities decrease. This shows that estimates for shorter to medium size transports are more accurate. The dip in the 50–99-mile category probably reflects the fact that most production that can be satisfied with short-haul traffic is either produced or otherwise stored within 50 miles of the destination, and where this is not possible longer trips are more likely. When the distance intervals are divided into equal intervals (as opposed to what is shown), the value of shipment decays exponentially with the distance traveled. This suggests that cross-regional spillovers will commonly cascade through neighboring re- gions before impacting more distant regions. This phenome- non can be explained by the amplified costs associated with total long-distance shipments.

Figure 2 shows a snapshot of the data from Tables 8 and B-8 of the 2002 CFS report. Here estimates are disaggregated by commodity type and distances between origins and desti- nations. Again, the error bars show the 90% confidence inter- vals.

The Freight Modeling Improvement Plan (FMIP) is a forum in which practitioners and researchers can discover how to develop local understanding of freight through local data collection and modeling built on the national picture

q = T*A*q + T*f*. (22)

DEVELOPMENT OF MRIIM FOR SPATIAL EXPLICITNESS 35

Systems Engineering DOI 10.1002/sys

Figure 2. Total 2002 value ($billions) of commodities shipped by STCG classification [BTS, 2002].

Figure 1. Total 2002 value ($billion) of commodities shipped over various distances [BTS, 2002].

36 CROWTHER AND HAIMES

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provided by the Freight Analysis Framework [FMIP, 2005]. The Freight Analysis Framework (FAF) is a federally funded initiative that integrates data from multiple major commodity flow data sources and then complement it with economic production data to provide the best possible picture of freight flow across the US [FHWA, 2005]. However, no reliability information is released through reports associated with the FAF 2002 database. The goal is to begin to decentralize the data gathering efforts and to provide all regions with the analytical tools to construct high-resolution commodity flow tables within their jurisdictions and then to share that infor- mation with the US Census Bureau and other jurisdictions that may be interested. Figure 3 shows a snapshot of FAF 2002 data through a geodatabase implementation of the MRIIM [Crowther, 2008]. It shows the value of commodities (in millions of US dollars) consumed by the Greater Richmond, VA region. Each region shown in the diagram is one of 114 regions defined by FAF for which data are published reporting origin-destination flows by commodity type for each mode of transportation.

A third source of cross-regional data is published by Global Insight, a company that sells a database of detailed county-to-county commodity flows for approximately 600 commodity types; the Virginia Department of Transportation purchases this database for use in developing critical infra- structure and transportation projects, among others. The data collection and analysis methodology is proprietary and no reliability information has been released concerning its per- formance. However, it is based on the openly available CFS estimates that are supplemented by a proprietary survey that is twice the size of the CFS sample. The underlying assump- tion is that with a sample three times that of the CFS, the data is more representative of actual cross-regional flows.

Service flows are not specifically characterized by any one database, but many recent studies attest their significance. Harrington, MacPherson, and Lombard [1991] review a large number of studies of producer- and intermediate-service ex- ports from local regions; they conclude that most services are

established to serve local economies, but that those services in localities with substantive physical and communication infrastructures and specialized labor forces can have wide- spread clients. They also conclude that there are exceptions where businesses derive large portions of revenue from exter- nal localities; these generally include businesses that are very specialized or that are parts of large enterprise structures. Persky and Wiewel [1994] discover that improved transport and communications technologies have surprisingly resulted in producer services simultaneously becoming more local and more global, resulting both in shrinking multiregional trade and in spatial relationships in middle distance regions. Esparza and Krmenec [1996] explain this phenomenon in their presentation of a general hierarchical structure for spatial economies in which smaller cities interact with nearby larger cities that are sufficiently substantive and specialized. Yet, according to a recent survey of American households by the Bureau of Transportation Statistics [2003], about one-fifth of all long distance US travel is for work or work-related activi- ties. Currently, this model does not account for multiregional transactions of services, but it is easily extended to capture such transactions as they become available. The large geo- graphic size of the regions used in this paper, combined with the estimates from literature showing that large percentages of service-sectors’ revenues are derived locally, lead us to conclude that the current model we present offers a reasonable improvement over existing nonspatially explicit models.

5. MRIIM WITH PARTIALLY EXOGENOUS SUPPLY INOPERABILITY

Inoperability is largely a measure of lost supply value. A supply-driven model can be derived that is similar to this model, but it is driven by value-added inputs and prices rather than final demand. There has been a thorough debate ques- tioning the usefulness (and appropriateness) of the supply- driven model (originally conceived for planned economies such as the Soviet Union), because of its assumptions about

Figure 3. Regions that ship commodities to the Greater Richmond, VA region. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

DEVELOPMENT OF MRIIM FOR SPATIAL EXPLICITNESS 37

Systems Engineering DOI 10.1002/sys

the nature of demand always exceeding supply. Oosterhaven [1988, 1989, 1996], Gruver [1989], Rose and Allison [1989], Dietzenbacher [1997], and Kujawski [2006] provide exten- sive discussions concerning the utility of this modeling para- digm, taking into account its inherent assumptions and model framework. In general, however, these discussions discount the direct use of the supply-driven model.

Miller and Blair [1985] consider a conceptually different approach to supply constrained problems. Using as an exam- ple a 3-sector Leontief input-output economy, they partially constrain production output (i.e., supply) and balance the demand-driven input-output equilibrium model using the tra- ditional demand-driven modeling framework of the basic Leontief equation. This approach enables the analysis of a supply-constrained state under the more generally accepted assumptions of the demand-driven model. It is still possible to derive an analytic solution and prove its existence and uniqueness given the data. This section applies this concept to derive a solution to the MRIIM for a problem with partially exogenous inoperability. This method is applied in cases when supply is rendered inoperable and exogenously speci- fied instead of a final-use perturbation. For example, if a power plant incurs damage and is unable to supply power, then the inoperability, qpower, should be exogenously specified, rather than a final use perturbation, fpower. This section ex- pands the MRIIM under conditions where both final use perturbation and exogenous inoperability exist simultane- ously following a disaster scenario.

Rearrange the indexing of the matrices and vectors in Eq. (23) such that all the sectors for which the initial inoperability is known (e.g., sector inoperability is directly caused by a terrorist attack or natural disaster) are grouped together in the q _ partition of the inoperability vector. The elements of all

other matrices are rearranged to account for this restructuring. The resulting MRIIM is shown in Eq. (25) without loss of generality.

[I − T∗A∗]   

q

q _    = [T∗]

  

f _ ∗

f∗    . (25)

Without loss of generality, reindex the matrices such that each sector in each region is treated as a unique sector with m = np total region-sectors, and suppose that the inoperability for sectors k+1, …, m is exogenously specified. Then partition the matrix [I – T*A*] as shown in

[I − T∗A∗]  

D1 (−D2) D3 (−D4)

   , (26)

where D1, D2, D3, and D4 are defined as follows:

D1: the k × k matrix containing the elements from the first k rows and first k columns of [I – T*A*],

D2: the k × (m – k) matrix of elements containing the first k rows and last (m – k) columns of (–[I – T*A*]),

D3: the (m – k) × k matrix containing the elements from the last (m – k) rows and first k columns of [I – T*A*],

D4: the (m – k) × (m – k) matrix of elements containing the first (m – k) rows and last (m – k) columns of (– [I – T*A*]).

Furthermore, adjust the definitions of perturbation and inoperability vectors to account for the incorporation of mul- tiple regions, as follows:

q : unknown inoperability vector for sectors 1, …, k, f*: unknown spatially explicit perturbation vector for

sectors k + 1, …, m, q _

: exogenous inoperability vector for sectors k + 1, …, m,

f _

∗: exogenous spatially explicit perturbation vector for

sectors 1, …, k.

Finally, for notational simplicity define the partitions of the matrix T* as follows:

T∗ :   

T1∗ (−T2∗) T3∗ (−T4∗)

   ,

(27)

where the partitions of T* are defined as follows:

T1*: the k × k matrix containing the elements from the first k rows and first k columns of T*,

T2*: the k × (m – k) matrix of elements containing the first k rows and last (m – k) columns of (–T*),

T3*: the (m – k) × k matrix containing the elements from the last (m – k) rows and first k columns of T*,

T4*: the (m – k) × (m – k) matrix of elements contain- ing the first (m – k) rows and last (m – k) columns of (–T*).

With these substitutions, the MRIIM in Eq. (25) becomes Eq. (28), which can be reorganized into Eq. (29).

  

D1 (−D2) D3 (−D3)

     

q

q _    =

  

T1∗ (−T2∗) T3∗ (−T4∗)

     

f _ ∗

f∗   

(28)

  

D1 T2∗

D3 T4∗      

q f∗    =

  

T1∗ D2 T3∗ D4

      f _ ∗

q _ 

(29)

Appendix B derives the analytical solution to this system of equations. The calculation of the necessary inverse is instructive to show if a unique solution exists for the MRIIM with partially exogenous inoperability.

6. ILLUSTRATIVE MRIIM CASE STUDY

This section explores results from the MRIIM compared to the traditional IIM by considering the economic impacts resulting from the loss of a crude-oil terminal in the Gulf of Mexico. Turk et al. [1989] describe the potential physical impacts to the US associated with physical pipeline interde- pendencies. These impacts are used as inputs to illustrate some basic properties of MRIIM results. This discussion adopts the perspective of two decision makers: (1) a national decision-maker interested in the extent and distribution of the impact of a crude terminal outage on the US economy and (2)

38 CROWTHER AND HAIMES

Systems Engineering DOI 10.1002/sys

representatives from the Gulf Coast states, considered as if they were in a consortium capable of making collective deci- sions, interested in taking action designed to decrease total impact to the region.

In the Gulf of Mexico, large tankers arrive and attach themselves to pipeline terminals so that their crude oil can be transported to refineries on land. Incapacitating these termi- nals would result in decreased pipeline distribution and refin- ery capacity. The Gulf Coast states must make planning decisions that will encourage the construction of new refiner- ies in their region, which will in turn increase local revenues, but such construction will also increase the vulnerability of the terminals—and thus the overall economic viability of the region—to physical impacts. The national government must make decisions that will incentivize interconnectivity in such a way that the impact of a terminal outage is spread propor- tionally among users, thereby minimizing its overall effects. The direct result of a 10-week loss of a crude-oil terminal and its associated physical cascade, discussed by Turk et al. [1989], can be adapted as inputs to each of the Petroleum Administration and Defense Districts (PADD) regions by translating the reduced capacity and available physical sub- stitute capacity into a percentage of lost capacity. The follow- ing data on physical impacts resulting from the disruption and its cascade through physical interdependencies is taken from

Turk et al. [1989]. Tables II–IV detail the resulting physical perturbation for three PADD regions receiving a direct or indirect physical impact; the other two PADD regions do not receive a cascading impact from physical pipeline intercon- nectivity. This is an expected result from the Turk et al. [1989] model, since at the time of the study there were no physical interdependencies linking the Gulf Coast to the West Coast.

We use the MRIIM with partially exogenous inoperability to estimate how the impacts will propagate under two differ- ent scenarios to illustrate the extent to which model output differs when accounting for cross-regional economic interde- pendencies. In the first scenario we assume each PADD is only physically interconnected across regional boundaries; this disregards any cross-regional economic transactions that might exist to assess each region independently (i.e., we use the model from Sections 3 and 5 without the details from Section 4). In the second scenario we assume that the regions are economically interconnected as assumed in the MRIIM (i.e., we use the model derived in the paper from Sections 3, 4, and 5). Figure 4 shows the results of the top ten sectors to receive impact in the first scenario, not accounting for cross- regional economic transactions between regions. Figure 5 shows the top ten sectors’ impacts resulting from the same perturbation, but using the MRIIM with partially-exogenous inoperability including cross-regional transactions. A com- parison of the results of the two scenarios will indicate whether the accounting for cross-regional transactions is criti- cal and if it will alter decisions made in strategic preparedness. Appendix C contains descriptions of the sector abbreviations used throughout this section.

Figure 4 shows that the economic losses are nonhomo- geneous across regions, despite the fact that perturbations are similarly structured. (See Appendix C for a summary of sector abbreviations.) This is simply a reflection of the structure of perturbations and the intraregional interdependencies result- ing from the calculated location quotients. This is evident by examining the relative size of economic losses to specific sectors and by examining the ranking of economic losses to sectors. For example, Rental and leasing services (RENT) sector receives the tenth largest economic loss in PADD I,

Table I. Multiregion Origin-Destination Table for Commodity i

Table II. Perturbation to PADD I (East Coast) from 10-Week Crude Terminal Outage

Table III. Perturbation to PADD II (Midwest) from 10-Week Crude Terminal Outage

DEVELOPMENT OF MRIIM FOR SPATIAL EXPLICITNESS 39

Systems Engineering DOI 10.1002/sys

whereas it is the sixth largest for PADDs II and III. These results illustrate heterogeneity of internal interdependency structures, which vary by region and result in different re- gional impact structures that support correspondingly differ- ence preparedness decisions for regions.

Figure 5 uses the same perturbation shown in Table II–IV, but uses the MRIIM to capture multiregional economic inter- dependencies existing between regions. The total economic losses are the same (at least they differ by an amount smaller than the margin of error in the databases). Although the total economic losses are the same, the results differ strongly from those in Figure 4 in several ways. First, PADDs IV and V now experience an impact that results from their inability to export or import commodities because of disrupted economic opera- tions in the perturbed regions. Second, values and rankings of sectors have changed. Figure 6 compares these impacts from Figures 5 and 4.

Figure 6 shows the difference between Figures 5 and 4. The most obvious change is found in the impacts to PADDs IV and V, for which Figure 4 shows no impact in the first scenario. Another illustrative example is the Chemical manu- facturing sector (CHEM), which exhibits very clear changes from Figure 4 to Figure 5. When the MRIIM is used, the impacts to PADDs I, IV, and V are increased, while the impacts to PADD II and III are decreased compared to the scenario in which only regionalization is used. The MRIIM can help to make regional decision makers aware of interre- gional adjustments that a catastrophic event in an intercon- nected region would necessitate. Graphical displays of these interdependencies will enhance the capacity of decision mak-

ers to arrive at appropriate decisions concerning multiregional substitutions.

Table V reports the differences among the multiregional and regional impact models as a percentage of the multire- gional impact calculated in the second scenario. It shows that the adjustments for cross-regional interdependencies are sig- nificant for regional preparedness decision-makers to under- stand the economic impacts of disasters to the specific regions.

However drastic the differences between sector impacts might be, the total impacts are the same. For the regional case the economic losses total $28.17 billion for the nation. The national losses for the multiregional case total $27.91 billion, a difference of 0.9% of the multiregional value. This suggests that multiregional calculation is simply a redistribution of impacts and is expected since it is derived by decomposing cross-regional imports and exports from the final demand and accounting for them explicitly in the model as multiregional interconnections. Moreover, the independent regional models can be calculated in the MRIIM framework by replacing the multiregional interdependency matrix T* with the identity matrix of correct dimensions. The identity matrix then has the interpretation that all production is consumed internally, which is logically represented in an independent regions model.

The differences between the two scenarios depict how cross-regional interdependencies can create both vulnerabili- ties and resilience to individual regions through their inter- connections. Figure 5 illustrates this concept when it is interpreted as the change in impact given cross-regional inter- dependency; the absence of such cross-regional interdepen-

Figure 4. Economic losses of top ten sectors for PADDs I – V (considered as independent regions). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Table IV. Perturbation to PADD III (Gulf Coast) from 10-Week Crude Terminal Outage

40 CROWTHER AND HAIMES

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dency would result in the impacts shown in Figure 4. Figure 6 is a comparison of these results and provides an initial method for understanding how cross-regional transactions of commodities can impact strategic preparedness through the ability to provide resilience through interdependencies. Table V can then be interpreted as setting forth the decreased level of resilience resulting from cross-regional transactions with a vulnerable region. In the model, some of the exports of PADD III are implicitly combined with the region’s final demand for the analysis of scenario one (independent regions model). However, this fact fails to capture the reality that these exports actually externalize some of the regions’ losses to other re- gions that rely on various supplies that are imported from PADD III, exports which are explicit in the analysis of sce- nario two (multiregional model). The Gulf Coast decision- makers would utilize this pattern of analysis to understand the

tradeoffs between (1) dominating petroleum-based produc- tion and consumption and thus owning the liability of such activity and (2) assuring that the externalization of risk is sufficient to balance regional revenue streams. The national decision-maker would utilize this pattern of analysis to un- derstand the cascading and potentially amplifying impact of a terminal outage. Using this pattern allows the national as well as the Gulf Coast decision-makers to understand how changes in both the economic landscape and interdependency among the regions can change the distribution of impact and perhaps motivate regulation and incentives to fairly distribute the risk of a terminal outage.

7. CONCLUSIONS

This paper explores the impact of spatial explicitness in the estimation of disruptions to interdependent regional eco- nomic systems for strategic preparedness planning. We exam- ine this impact first through the development of the Multiregional Inoperability Input-Output Model (MRIIM)

Table V. Percent (%) Increase in Impact Estimates between the Independent Regional Impact Model and the MRIIM

Figure 5. Economic losses of top ten sectors for PADDs I – V (considered as interdependent regions). [Color figure can be viewed i n t h e o n l i n e i s s u e , w h i c h i s ava i l a b l e a t w w w. in te r- science.wiley.com.]

Figure 6. Increased calculated losses for PADDs I through V, when considering cross-regional trade flows in the calculation. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

DEVELOPMENT OF MRIIM FOR SPATIAL EXPLICITNESS 41

Systems Engineering DOI 10.1002/sys

and then through a case study that compares the results from independent and interconnected multiregional models. The IIM in previous publications considers the nation or some region of contiguous counties as a whole, operating with perfect communication among all acting parties; in such a scenario, the resulting impacts of a perturbation are predicted to be uniform across the entire region. However, spatial ex- plicitness is added when the economy is regarded as a system of regional economies with processes such as economic trans- actions of goods and services that link the various sub-regions. Adding spatial explicitness results in a model that produces distinct estimates for each region determined by its regional characteristics and its interconnectedness with other regions.

Spatial explicitness is added to the IIM through the selec- tion and application of multiregional input-output modeling principles from the literature, which are then adjusted accord- ing to available databases. Moreover, the existence of the solution existence and the uniqueness of the MRIIM are explored through an evaluation both of the structure of the model and of the character of the data supporting the model. An application of the MRIIM with inputs from another early interdependency work [i.e., Turk et al., 1989] illustrates the change in impact estimates that result from adding spatial explicitness to a multiregional analysis. Although the total impact remains unchanged, the impacts to specific sectors across the analysis regions change significantly. These signifi- cant changes would greatly impact a regional strategic prepar- edness decision as described in the introduction to this paper. Moreover, they illustrate the regional resilience that can result from multiregional interdependencies, an important aspect of regional preparedness.

The MRIIM overcomes the lack of spatial explicitness and thus provides an example method to produce more than only average estimates across geography to better support deci- sions about preparedness allocations across regions. Through the MRIIM, decision-makers are more likely to identify geo- graphically concentrated risks and significant cross-regional interdependencies, which are important in evaluating relevant preparedness strategies. To the decision-maker of large re- gional systems of interconnected regions, the MRIIM pro- vides a measure of economic intraconnections between smaller (sub)regions that may result in either cascades of impacts or sources of resilience following a disaster scenario. To the decision-maker of smaller (sub)regions they systemi- cally provide a means to investigate risks mitigation against: (1) disaster scenarios that produce supply perturbations in their demand footprint (the regions from which their eco- nomic sectors purchase goods and services) and (2) disaster scenarios that produce demand perturbations in their supply footprint (the regions that purchase their economic output). Moreover, this paper has characterized how multiregional interdependency can provide resilience to a region by allow- ing externalization of portions of supply or demand risks. It is the opinion of the authors that multiregional considerations be included by regional analysts and planners at all levels of strategic preparedness and regional planning.

APPENDIX A: DISCUSSION OF THE MRIIM SOLUTION

The mathematical solution to the MRIIM can be obtained by linear algebra and is shown in

q = [I – T*A*]–1 T*f*. (30)

The analyst of strategic preparedness is able to relate some set of perturbations f* to a set of economic inoperabilities q. This section shows that the spatial explicit structure of this model and the character of the data is such that it results in a guaranteed solution that is unique.

A solution to Eq. (30) exists and is unique whenever the matrix [I – T*A*] is nonsingular, or equivalently is full rank or has no eigenvalues of zero. However, it is easier to charac- terize the matrix T*A*, since it contains the data that drives the solution.

Consider the following lemma: If the matrix T*A* is full rank and has no eigenvalues equal to 1, the matrix [I – T*A*] is also nonsingular, and Eq. (30) has a unique solution. To illustrate this sufficient condition, define σ(D) as the spectrum of the matrix D. The spectrum of a matrix is the set of all eigenvalues λ.

Let the matrix D be a positive, real-valued, np × np matrix (just like T*A*) whose spectrum does not include 0 or 1. Furthermore, consider the nonsingular matrix S, such that the matrix [S-1DS] is the Jordan form of D, a diagonal matrix with all eigenvalues as the elements of the diagonal. The eigenval- ues µ of the matrix [I – D] can be calculated through the following relationship:

(I – D)x = µx

⇒ S–1(I – D)xS = S–1µxS

⇒ (I – S–1DS)S–1xS = µS–1xS

⇒











1 − λ1

1 − λ2 ...

1 − λnp









 S–1xS = µS–1xS.

(31)

It is easy to see from Eq. (31) that σ(I – D) can contain a 0 only if σ(D) contains a 1. In other words, if the matrix D is full rank and has no eigenvalues of 1, it is sufficient to assume that [I – D] is nonsingular. We now show that this assumption is met by the MRIIM by characterizing how the data is used for constructing the model.

Return to the scalar data representations of TA shown in Eq. (12), where T is the multiregional technical coefficient matrix and A is the intraregional technical coefficient matrix. The eigenvalues of [TA] and [TA]T are the same, since both matrices are characterized by the same characteristic polyno- mial, the roots of which are the eigenvalues. As such we can show that the matrix [TA] has eigenvalues strictly less that unity.

Consider the following eigenpair (λ, y) of D = [TA]T = ATTT, a positive, real-valued np by np matrix. Furthermore,

42 CROWTHER AND HAIMES

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assume y has been normalized such that ||y||∞ = 1. If yq is an element of y such that |yq| = 1, then

λyq = [λy]q = [Dy]q = ∑ j=1

np

dqj yj

⇒ |λyq| = |λ||yq| =   

  ∑ j=1

np

dqj yj   

  ≤ ∑

j=1

np

dqj |yj| ≤ ∑ j=1

np

dqj |yq|

⇒ |λ| ≤ ∑ j=1

np

dqj < ∑ q=1

np

ajq ≤ 1.

This argument is plausible due to the fact that the matrix A is normalized by column sums and the elements of T are all strictly less than unity when there are positive cross-re- gional flows of commodities or services. When there are 0 cross-regional flows, then the corresponding element of T is 1, the location quotient is 1, and the value of the row will be strictly less than 1, due to the fact that there must be positive final demand in the region. Moreover, the interregional tech- nical coefficient matrix [TA] is composed of elements that are the product of a single element of T multiplied by a single element of A.

This result can be applied to the solution of the MRIIM by the understanding that a similarity transformation does not impact the eigenvalues of the matrix being transformed. Eq. (32) shows the relationship between T*A*and TA as a simi- larity transformation of a positive definite diagonal matrix, whose diagonal values are the levels of production output for each of n sectors in each of p regions.

T*A* = [x̂–1Tx̂][x̂–1Ax̂] = [x̂–1TAx̂] = [TA]*. (32)

Finally, given these properties of the matrix combination there exists a unique solution for the MRIIM shown in Eq. (30) due to the fact that all eigenvalues of T*A* are between 0 and 1. This provides a useful property for assessing the cascading impacts amongst economically interconnected re- gions for strategic preparedness. The use of this model in strategic preparedness is illustrated in Section 6.

APPENDIX B: SOLUTION FOR THE MRIIM WITH PARTIALLY EXOGENOUS INOPERABILITY

Equations (33)–(39) show the derivation of a partitioned inverse, similar to those in Eq. (29). Premultiplying and postmultiplying by block triangular matrices results in a block diagonal matrix shown on the right-hand side of Eq. (33). Equation (34) shows the result of taking the inverse of both sides and distributing the inverse on the left-hand side. Note that the matrix D1 is full rank (i.e., nonsingular) due to the nature of the economic data that are used to construct it.

  

I −D3[D1]−1

0 I      

D1 T2∗

D3 T4∗      I 0

−[D1] −1T2∗

I   

=   

DI 0

0

−D3[D1]−1T2∗+T4∗    ,

(33)

  

I 0

−[D1]−1T2∗

I

  

−1   

D1 T2∗

D3 T4∗   

−1   

I −D3[D1]−1

0 I   

−1

=   

DI 0

0

−D3[D1]−1T2∗+T4∗    . (34)

Equation (35) shows the isolation of the desired inverse matrix on the left-hand side, and the reduction of the expres- sion on the right-hand side to a single matrix expression is shown in Eqs. (36) and (37).

  

D1 T2∗

D3 T4∗   

−1

=

  

I 0

−[D1]−1T2∗

I

      [D1]−1

0

0 [T4∗ − D3[D1]−1T2∗]−1

  

  

I −D3[D1]−1

0 I

  

(35)

=    [D1]−1

0

−[D1]−1T2∗[T4∗ − D3[D1]−1T2∗]−1

[T4∗ − D3[D1]−1T2∗]−1   

  

I −D3[D1]−1

0 I

   (36)

=   

[D1]−1 + [D1]−1T2∗[T4∗ − D3[D1]−1T2∗]−1D3[D1]−1

−[T4∗ − D3[D1]−1T2∗]−1D3[D1]−1

  

−[D1]−1T2∗[T4∗ − D3[D1]−1T2∗]−1

[T4∗ − D3[D1]−1T2∗]−1    .

(37) For notational simplicity make the substitution defined in

Eq. (38) to result in a final relationship in Eq. (39) that shows the partitioned inverse relationship.

F = [T4* – D3[D1]–1T2*]–1, (38)

  

D1 T2∗

D3 T4∗   

−1

=   

[D1]−1 + [D1]−1T2∗FD3[D1]−1

[−FD3[D1]−1 −[D1]

−1T2∗F] F

  

(39) Inserting Eq. (39) into Eq. (29) results in

  

q f∗    =

  

[D1]−1 + [D1]−1T2∗FD3[D1]−1

−FD3[D1]−1 −[D1]

−1T2∗F] F

  

  

T1∗ D2 T3∗ D4

  

   f _ ∗

q _  . (40)

DEVELOPMENT OF MRIIM FOR SPATIAL EXPLICITNESS 43

Systems Engineering DOI 10.1002/sys

APPENDIX C: SECTOR ABBREVIATIONS AND DESCRIPTIONS

44 CROWTHER AND HAIMES

Systems Engineering DOI 10.1002/sys

Equation (40) is algebraically equivalent to Eq. (25) or Eq. (22), whose solution in Eq. (30) was shown to be unique. Algebraic equivalence is sufficient to establish that the MRIIM with partially exogenous inoperability in Eq. (40) also has a unique solution. Equation (40) also provides the set of multipliers that show an impact of mixed supply/demand perturbations will result in a cascade of impact across sectors and regions of the economy, given the preparedness strategy and disaster scenario under consideration.

ACKNOWLEDGMENTS

This research is supported by the National Science Founda- tion under a grant to the University of Virginia Center for Risk Management of Engineering Systems (NSF 0301553: Input- Output Risk Model of Critical Infrastructure Systems). Addi- tionally, this research is supported by the Department of Homeland Security through the Institute for Information In- frastructure Protection (I3P), by Virginia Governor’s Office of Commonwealth Preparedness. We highly appreciate the contributions from Jim Lambert, Barry Horowitz, and Joost Santos.

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Kenneth Crowther is a Research Fellow with the Institute for Information Infrastructure Protection, is a Research Assistant Professor with the Department of Systems and Information Engineering at the University of Virginia, and has been a member of the Center for Risk Management of Engineering Systems since 2007. He is active with several professional engineering societies and has served as Chair of the Engineering and Infrastructure Specialty Group in the Society of Risk Analysis. His interests include large-scale empirical problems related to data integration, regional/mul- tiregional analysis, emergency resource management, and strategic regional preparedness. Results from his work have benefited sponsors, such as US Department of Homeland Security, the Virginia Governor’s Office of Commonwealth Preparedness, and various localities, and have benefited from his ability to integrate a variety of data sources for risk-based decision problems and appropriately interpret the meaning and limitation of the results for the region of interest.

Yacov Haimes holds the Lawrence R. Quarles Professorship at the School of Engineering and Applied Science, University of Virginia, Charlottesville, and is a member of the Systems and Information Engineering faculty. He is the Founding Director (1987) of the university-wide Center for Risk Management of Engineering Systems, University of Virginia. On the faculty of Case Western Reserve University, Cleveland, OH, for 17 years, he was the Chair of the Systems Engineering Department, and Director of the University-wide Center for Large-Scale Systems and Policy Analysis. During the 1977–1978 sabbatical year, he was an AAAS/American Geophysical Union Congressional Science Fellow, joining the staff of the Executive Office of President Jimmy Carter, and later the staff of the House Science and Technology Committee. Over the years he has held several high offices as president and chair of boards of directors of professional and public service organizations. In 1998, he served as the President of the Society for Risk Analysis. He has published more than 250 articles and technical papers, more than 150 of which are in archival journals and encyclopedias. He has edited or co-edited 21 volumes, and authored or co-authored six books. His most recent book is Risk Modeling, Assessment, and Management (Wiley, New York, 1998). The third edition will be released in December 2008.

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Systems Engineering DOI 10.1002/sys