Fuzzy Importance Measures for Ranking Key Interdependent Sectors Under Uncertainty

42 IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 1, MARCH 2014

Fuzzy Importance Measures for Ranking KeyInterdependent Sectors Under Uncertainty

Gabriele Oliva, Roberto Setola, and Kash Barker

Abstract—In the field of reliability engineering, several ap-proaches have been developed to identify those components thatare important to the operation of the larger interconnected system.We extend the concept of component importance measures to thestudy of industry criticality in a larger system of economically in-terdependent industry sectors that are perturbed when underlyinginfrastructures are disrupted. We provide measures of (i) thoseindustries that are most vulnerable to disruptions and (ii) thoseindustries that are most influential to cause interdependent dis-ruptions. However, difficulties arise in the identification of criticalindustries when uncertainties exist in describing the relationshipsamong sectors. This work adopts fuzzy measures to developcriticality indices, and we offer an approach to rank industries ac-cording to these fuzzy indices. Much like decision makers with theknowledge of the most critical components in a physical system,the identification of these critical industries provides decisionmakers with priorities for resources. We illustrate our approachwith an interdependency model driven by US Bureau of EconomicAnalysis data to describe industry interconnectedness.

Index Terms—Fuzzy numbers, importance measures, inter-dependent sectors.

ACRONYMS AND ABBREVIATIONS

CIM Component Importance Measure

IIM Inoperability Input-Output Model

I-O Input-Output

NOTATION

number of sectors

1 vector of total sector output

1 vector of sector consumer demand

interdependency matrix

the th element of the matrix

1 vector of as-planned total sector output

1 vector of degraded total sector output

Manuscript received March 03, 2013; revised July 25, 2013; acceptedSeptember 09, 2013. Date of publication January 15, 2014; date of currentversion February 27, 2014. Associate Editor: Z. Li.G. Oliva and R. Setola are with the Complex Systems & Security Laboratory,

University Campus Bio-Medico of Rome, Italy, Via A. del Portillo 21, 00128Roma, Italy (e-mail: [email protected]; [email protected]).K. Barker is with the School of Industrial and Systems Engineering, Univer-

sity of Oklahoma, Norman, OK, USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TR.2014.2299113

difference between as-planned and degradedtotal sector output

1 vector of as-planned consumer demand

1 vector of degraded consumer demand

difference between as-planned and degradedconsumer demand

matrix with elements of the inverse ofon its diagonals, zeros elsewhere

1 vector of sector inoperability

1 vector of normalized sector consumerdemand

normalized interdependency matrix

the th element of the matrix

identity matrix

Leontief inverse of the matrix

dependency index for the th infrastructure

influence gain for the th infrastructure

normalized dependency index for the thinfrastructure

normalized influence gain for the thinfrastructure

overall dependency index for the thinfrastructure

overall influence gain for the th infrastructure

normalized overall dependency index for the thinfrastructure

normalized overall influence gain for the thinfrastructure

criticality index for the th infrastructure

normalized criticality index for the thinfrastructure

overall criticality index for the th infrastructure

normalized overall criticality index for the thinfrastructure

interval number with bounds

interval matrix, with boundmatrices

the interval number S dominates the intervalnumber T

0018-9529 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

OLIVA et al.: FUZZY IMPORTANCE MEASURES FOR RANKING KEY INTERDEPENDENT SECTORS UNDER UNCERTAINTY 43

fuzzy number

triangular fuzzy number

fuzzy membership function

-level set of a fuzzy number

support of a fuzzy set

fuzzy matrix

-level set of a fuzzy matrix M

the fuzzy number dominates the fuzzy number

1 vector of fuzzy sector inoperability

normalized fuzzy interdependency matrix

1 vector of normalized sector consumerdemand

fuzzy dependency index for the th infrastructure

fuzzy influence gain for the th infrastructure

fuzzy normalized dependency index for the thinfrastructure

fuzzy normalized influence gain for the thinfrastructure

fuzzy overall dependency index for the thinfrastructure

fuzzy overall influence gain for the thinfrastructure

fuzzy normalized overall dependency index forthe th infrastructure

fuzzy normalized overall influence gain for theth infrastructure

fuzzy criticality index for the th infrastructure

fuzzy normalized criticality index for the thinfrastructure

Leontief inverse of the fuzzy matrix

I. INTRODUCTION

T HE reliable, resilient operation of interdependent criticalinfrastructure systems is “essential to the Nation’s secu-

rity, public health and safety, economic vitality, and way of life”[1], true not just for the US but across the globe. Risk managerspreparing for disruptive events, whether by attack, disaster, ac-cident, or common failure, must plan for the interdependent re-lationships of these infrastructure systems with the industriesthat rely upon them. Rinaldi et al. [3] provide a well-used def-inition of interdependency as a “bidirectional relationship be-tween two infrastructures through which the state of each in-frastructure influences or is correlated to the state of the other,”or more generally, “two infrastructures are interdependent wheneach is dependent on the other.” Understanding the impact thatunderlying infrastructure systems have on the larger economic

system is vital for enabling preparedness and response planningfor a (likely inevitable) disruption. The interdependent nature ofinfrastructure systems has gained considerable government [1],[2] and research attention recently (e.g., [3]–[8]), with less em-phasis placed on the industry impacts of the disruptions to thesesystems (e.g., [9]–[12]).When modeling interdependent systems, and when planning

for their disruption, it is important to understand which com-ponents of those systems are most influential on the perfor-mance of the whole system, and which are most influenced byother components in the system. This is a well-studied topicin the reliability field, where component importance measures(CIMs) have been introduced to measure the influence of partic-ular components on the overall reliability of a system [13]–[15],or have been used to identify top contributors to cascading fail-ures among a set of tightly interconnected and interdependentsubsystems [16].Specific CIMs include risk reduction worth (RRW), an index

that quantifies the potential damage to the system caused bya particular component, and the reliability achievement worth(RAW) of a component, or the maximum proportion increase insystem reliability generated by that component [17], [18]. SuchCIMs are enabled by reliability block diagrams (RBDs) [19],[20], which provide a depiction of the relationships among com-ponents in physical systems and networks. RBDs can be usedfor risk analysis of critical systems [21], providing feedback forthe implementation of system improvements (e.g., extra compo-nents for redundancy, expediting repair) that may increase ser-vice availability and decrease down time due to a componentfailure.The concept behind CIMs, which primarily have applications

in physical systems, is extended here to describe the reliability(or productive operation) of an interdependent economy whensome of the constituent components (industries) are renderedinoperable by disruptions in underlying infrastructure systems.Namely, rather than finding the components in a physical systemthat impact system reliability the most, we seek those industrysectors that, when their productivity is perturbed by an infra-structure disruption, influence the as-planned performance ofthe interdependent economy the most. System reliability is anal-ogous to interdependent economic operability.In the study of interdependent systems, moreso than an anal-

ysis of CIMs, the evaluation of functional and non-functionalrelationships existing among the different infrastructure and in-dustry sectors is difficult. This difficulty is due to the presenceof complex functionally interdependent relationships that mayresult in higher order widespread impacts, or impacts that do notdepend solely on immediate relations (first order interdependen-cies) but also on ripple effects (higher order interdependencies).Due to this complex, network relationship of interdependent

industries, a mere analysis of top contributors to failures (in-operability) is not sufficient because the impact of the failurethrough first and higher order relationships may be significant.Further, as opposed to many system reliability contexts, themost vulnerable industries and the most influential industriesmay not be obvious due to such higher order relationships. Boththe vulnerability and influence perspectives are indeed valuabletools for a decision maker, especially when presented together.


For example, knowing that the transportation sector is more vul-nerable than the power distribution sector, but when faulted isless able to affect the overall reliability of the other sectors,a preparedness decision maker may choose how to prioritizestrategies aimed at minimizing widespread impacts (e.g., hard-ening of the vulnerable sectors, improving the ability to substi-tute outputs of the influential sectors).Characterizing interdependence among several intercon-

nected critical infrastructures is a non-trivial task, due to theintrinsic complexity of modeling cross-infrastructure relations,and due to the potentially overwhelming dimension of suchsystems. Several models have been used to describe the inter-dependence among infrastructures, along with their disruption,across several infrastructure contexts, including network ap-proaches [22]–[24], agent-based models [25]–[27], amongothers [28]–[31]. A typical approach to face such a complexityis to resort to high-level models, able to capture at a glancethe overall relations among the infrastructures. However, ourdesire here is to focus on industry interdependence. As such, weextend a widely-used input-output approach, the InoperabilityInput-Output Model (IIM) [32], [33] , due to its simplicity andclarity.The IIM is a linear model, extended from the input-output lit-

erature in economics [34], that quantifies how inoperability, theproportional extent to which an industry is not functioning inan as-planned manner, propagates among several industries dueto their interdependent nature. As such, several different perfor-mance measures (e.g., flow capacity, output) could be describedas a proportion on of as-planned performance, making in-operability an appropriate and comparable measure across het-erogeneous industry contexts. The assumption of model lin-earity is overshadowed by the amount of real data describing theinterdependencies among infrastructure sectors, as well as be-tween infrastructures and industries, and among industries, fromthe Bureau of Economic Analysis (BEA) for the US, as well asby similar agencies for over 60 other countries worldwide. TheIIM has been previously deployed to measure the interdepen-dent economic impact experienced across industries due to anunderlying infrastructure disruption, including inland waterwayclosures [35], [36], and electric power outages [37], [38].The IIM, explained in greater detail in Section II, is sub-

ject to several sources of uncertainty [39]–[41], including (i)the behavior of the initiating disruptive event, (ii) how thedisruption initially impacts the infrastructure sectors, (iii) theability of the linear IIM to model potentially nonlinear condi-tions, and (iv) errors in model parameters and underlying data.Uncertainty has been previously addressed in input-outputstudies by treating input coefficients as stochastic variables[42], [43], often using Monte Carlo methods [44]–[46]. Partic-ularly for IIM applications, Santos [47] applied Monte Carlosimulation to study the interdependent outputs of stochasticinput variables, Barker and Rocco [41] address uncertaintyin both input and interdependency parameters with intervalarithmetic, and Pant and Barker [35] apply robust optimizationto infrastructure preparedness decision making with both inputand interdependency uncertainty.In general, it is recognized that resorting to probability distri-

butions to characterize uncertainty is a difficult task when those

distributions are unknown [48], and such is especially true forinput-output approaches [49]. Oliva et al. [50] address uncer-tainty in the IIM with a fuzzy framework, where model param-eters, assessed from infrastructure domain experts, were codi-fied with fuzzy triangular numbers. The approach, unlike othermethodologies, is able to handle uncertain and ambiguous infor-mation about first order interdependencies providing insights onthe structure of the interdependency parameters after an equilib-rium is reached. In fact, even if symmetric fuzzy sets are usedto characterize the ambiguity of first order interdependency re-lationships, higher order interdependencies exhibit a non-sym-metric structure, thus providing further insights on the belief as-sociated to different situations. Even if the formulation providedin [50] imposes a more detailed definition of the coefficients ofthe model, it is possible to find, under quite general hypotheses,a simple closed-form solution to IIM problem.With this paper, we extend the use of component importance

measures to the study of interdependent industries comprising alarger regional economy, ultimately providing a fuzzy approachto prioritization of industries that, when their productivity is per-turbed by an underlying infrastructure disruption, are (i) mostimpacting on the reliability of the larger regional economy, and(ii) most impacted by the reliability of other industries. Such afuzzy approach will account for uncertainty in industry interde-pendence when generating the ranking of importance.The paper is organized as follows. Section II and Section III

provide the methodological background on the IIM, and onfuzzy numbers, respectively. Section IV integrates the two,describing the Fuzzy IIM. Section V illustrates the Fuzzy IIMwith a case study driven by US BEA data and compared withthe interval arithmetic approach [41]. Conclusions follow inSection VI, while a detailed appendix on the mathematicalderivations behind the proposed fuzzy methodology is reportedin the Appendix.

II. INOPERABILITY INPUT OUTPUT MODEL

The traditional input-output (I-O) model is a consolidatedmodel to represent macroeconomic interaction among the eco-nomic sectors of a country or region, often used to describe thegrowth of an economy due to sector expansion or technologyadoption [34], [51]. For interconnected industry sectors (e.g.,transportation, manufacturing, retail trade), vector of length

1 represents the output in monetary units of each sector.Vector , also 1, represents the consumer demand for goods,services, and usage in each sector, while the matrix isthe interdependency matrix whose elements (Leontief tech-nical coefficients) represent the ratio of the input from infra-structure to infrastructure with respect to the overall produc-tion requirements of infrastructure . In essence, (1) describeshow industry output ( ) is a function of intermediate demand( ) among sectors and final demand ( ).

(1)

Equation (1) can be rewritten to more directly relate outputto consumer demand with (2).

(2)


The Input-Output Inoperability Model (IIM) [32], [33]expands (1) and (2) to assess how decreased functionality, orinoperability , in an industry can propagate to a number ofother interconnected sectors. First, using notation from [50],is defined as the as-planned output of the industry, and

as degraded productivity (e.g., production in the manufacturingsector) following a disruptive event, such that is the differ-ence between as-planned and degraded productivity. Further,

is the difference between as-planned final demand ( ) anddegraded final demand ( ).An transformation matrix is introduced [52] in (3).

(3)

Inoperability is computed by applying the transformation in(4) to the reduction of production. As such, the elements ofrepresents the ratio of the change in output to its as-plannedoutput.

(4)

The IIM then has the form of (5), where , and. Vector describes the perturbation in demand re-

sulting from a disruptive event, and is the driver of inoperabilityin the demand-reduction IIM.

(5)

is the interdependency matrix which can be derived fromeconomic data as in [41], or using other sources as in [50]. Al-though it is mathematically possible to have , thecoefficients are strictly less than one [53].In the following, we refer to as the open-loop interde-

pendency matrix, and as the closed-loop in-terdependency matrix. Matrix models the direct effects dueto first-order interdependencies (i.e., represents the fractionof inoperability transmitted by the th infrastructure to the th),or in other terms, how much the operational capability of theth industry is degraded when the th is completely inoperable.On the other side, accounts for the amplification introducedby ripple effects (i.e., second- and higher-order interdependen-cies). Hence, represents the overall effect induced on the thinfrastructure by an infrastructure disruption affecting the thindustry taking into account both direct effects (i.e., those cap-tured by the term) and closed loop amplifications.Note that, for a positive matrix whose eigenvalues are

in modulus less than one, the inverse of matrix isprovided in (6) [54], illustrating an immediate understandingof the cumulative effects of high-order interdependencies inmatrix . The equation defines a procedure for estimatingthe closed-loop interdependency matrix without explicitly com-puting the inverse of matrix .

(6)

The IIM has been deployed in several risk-based studies ofinterdependent impacts, including transportation networks [35],[36], [56], pandemics [57], [58], and electric power outages

[36], [38], among others [27], [59]. Extensions to the IIM par-adigm include a dynamic version of the model [52] as wellas a model expressing multi-regional interdependencies [55].Further, Crowther and Haimes [60] develop a risk-based de-cision making framework for interdependent systems with anIIM-driven optimization problem formulation.

A. Key Sector Metrics for the IIM Model

Analogous to component importance measures (CIMs) in thereliability engineering literature, key sector analysis is used inthe economic input-output literature to identify the criticalitythat certain industry sectors have on the economy, and the pro-ductivity of a country or region. In the literature, two indicesfor expressing the criticality of an infrastructure within a givenscenario have been introduced in [8], [61], based on their eco-nomic input-output counterpart [62], [63]. These two indices arereferred to here as the dependency index, and the influence gain.The dependency index is defined as the sum of the coefficients

of matrix along a row, shown in (7).

(7)

The dependency index measures the robustness of the th in-frastructure with respect to the inoperability of the other infra-structures, representing ameasure of the residual operational ca-pability of the th infrastructure when every other infrastructureis fully inoperable. The lower the value, the greater the ability ofthe th infrastructure to maintain functionality despite the inop-erability of the other sectors upon which it relies. On the otherside, means that the operability of the th infrastruc-ture might be nullified even if the other infrastructures still havesome residual operational capability.Another index, the influence gain, measures the influence that

one infrastructure exerts over the other infrastructures. It is de-fined as the sum of the coefficients of matrix along a column,as in (8).

(8)

A large influence gain suggests that the inoperability of the-th infrastructure induces significant degradations on the entiresystem. When , the adverse effects in terms of inoper-ability induced by cascading phenomena on the other infrastruc-tures are amplified.Ordering the sectors in decreasing order of provides a

ranking of sectors that are most impacted by inoperability inother sectors, and likewise, a decreasing order of results in aranking of sectors that are most impactful on other sectors. Ina more absolute measure, Hirschmann [64] suggests that keysectors are those sectors for which the normalized version ofindices (7) and (8) are greater than 1.

(9)

The dependency index and influence gain permit quick globalevaluations of the resilience and role of a given infrastructure,


but they are primarily concerned with the nominal, first-order(open-loop) behavior. It is possible to define analogous indicesfor the closed-loopmatrix, namely the overall dependency index( ), and the overall influence gain ( ) [61]. The two indicesare defined as the row-wise sums, and column-wise sums ofmatrix , respectively. The indices and are reportedin (10), together with their normalization and .

(10)

These indices express the influence of a given infrastruc-ture sector considering higher-order interdepedency. The dif-ference in sector rankings for and (likewise comparedto ) is a measure of the amplification due to second-order andhigher-order cascading effects. Ranking the values of , , ,and could provide very different results depending on howfunctionally dependent a sector is on others, how influential asector is on others, and how first order versus higher order in-terdependencies matter.From a systemic point of view, the most critical infrastruc-

tures are those that show remarkable fragility (i.e., high depen-dency index), but have the capability to generate large impact(i.e., high influence gain) in the short term. This property canbe synthesized in the criticality index shown in (11), as well asthe overall criticality index in (12), and in their correspondingnormalizations , , , .

(11)

(12)

These indices can be used to rank the different industries toidentify the most critical, those on which operators and govern-ment decision makers should focus efforts to keep operable tomaintain the larger economy in the face of an infrastructure dis-ruption. Considered alternatively, the ranking provides a quicklist of those industries most impacted for a particular scenario(e.g., electric power outage). Further, across a set of scenarios,decision makers can measure the efficacy of preparedness andrecovery options (e.g., system hardening, redundancies) for un-derlying infrastructures.

B. IIM Uncertainty Analysis With Interval Arithmetic

In the previous section, the degree of interdependency ex-isting among industries and underlying infrastructures were es-timated in terms of reduction of as-planned production. How-ever, these data, especially those quantifying interdependency,may be uncertain but understood with some approximation, dueto the influence of several factors [3], including the peculiar op-erational conditions [8]. To overcome the deterministic nature ofthe IIM, Barker and Rocco [41] deploy interval arithmetic [65],[66] to represent uncertain parameters with upper and lowerbounds within which the nominal (deterministic) value is found.

This form of mathematics uses interval numbers, which are ac-tually an ordered pair of real numbers representing the lowerand upper bound of the parameter range [67].Barker and Rocco [41] explored several occurrences of un-

certainty in (5), including the representation of perturbationfor sector as an interval , and interval represen-tations of the elements of the matrix.In this framework, if several strategies are available to lessen

the impact of a disruption, as manifested in perturbation andsubsequently evaluated with the IIM, there is the need to com-pare those strategies when evaluated with interval arithmetic(e.g., which strategy results in the least inoperability whenstrategies are evaluated with intervals).Several decision rules for comparing intervals S and T for

determining whether S is preferred to T are provided in (13)for the situation where minimization is sought [41]. For singleobjective intervals , and , we will say that

(interval dominates in the minimization case), or inother terms that strategy which generates is preferred to thatwhich generates , if the appropriate criterion in (13) holds.

Best CaseWorst CaseLaplaceHurwiczMin Regret

(13)The appropriate decision rule is chosen based on the decision

maker attitude toward risk. For example, the worst-case crite-rion is consistent with a risk averse attitude. In such a case, thedecision maker would seek to minimize the maximum total eco-nomic losses across all sectors.A limitation of interval arithmetic is that all the values in

the interval have implied the same associated degree of truth,while it might be useful to consider different degrees of beliefassociated with different points in the interval. To this end, apossible choice is to use fuzzy numbers as shown in Fig. 1.

III. FUZZY NUMBERS

With respect to a uniform interval, a fuzzy number [71] pro-vides additional information about the degree of truth associatedwith each point in the interval. Such numbers can be consideredas an extension of interval arithmetic, in that every point in theinterval has an associated degree of truth or belief ranging from0 to 1. Although several choices are possible for the shape of thefuzzy numbers, the most widely adopted are the triangular fuzzynumbers, due to the fact that they can be represented by the ab-scissa of the three vertices. A triangular fuzzy number (TFN)can be defined as the triple of the abscissa ofthe endpoints of the triangle.Formally, a fuzzy number can be defined as a subset of, together with a membership function . Themembership function assigns to each element a degree ofmembership (i.e., a measure of belief that belongs to ).One of the key aspects of fuzzy numbers is that they can be

evaluated as a set of nested intervals; for each , it ispossible to define the -level set of a fuzzy number asthe set of elements with membership grade .The support is the base of the fuzzy number. Note that, for


Fig. 1. Comparison among interval arithmetic (solid line) and triangular fuzzy number (dotted line); for the latter, each value in the interval has its own level ofbelief.

Fig. 2. Examples of -levels of a triangular fuzzy number.

, the alpha level collapses into a singleton. Fig. 2 showsan example of -levels for a triangular fuzzy number.The -level can be extended also to a matrix containing

fuzzy numbers, and in that case the -level matrix is aninterval matrix. Such an interval matrix can be described by twomatrices of real numbers and , where each entry of

, and are respectively the left, and right endpoints ofthe -level of the corresponding entry of . Again, in the caseof triangular entries, , hence collapses into asingle real matrix.The extension of arithmetic operations to triangular fuzzy

numbers is relatively straightforward for the sum and scalarproduct, by either summing the endpoints or scaling them by ascaling factor. Product and division are much more complex inthat only a bound can be given for the result. For two non-neg-ative fuzzy numbers and , aconvenient means to calculate a bound for their product is with(14).

(14)

While for the division, the bound is calculated with (15).

(15)

In the case of the fuzzy product of a matrix for a vector(both assumed non-negative), the bound is given by (16),

where (17) provides each element of the triplet.

(16)

where

(17)

Detailed information on the mathematical derivations behind(14)–(17) are collected in the appendix.As we are ultimately providing a means with which to rank

sectors according to several dependency and influence indices,and as those indices will be measured with fuzzy numbers, weconclude the section introducing an approach to compare fuzzynumbers. The proposed approach is an extension of the intervaldecision rule illustrated in (13). Suppose we have to comparetwo fuzzy numbers and , and decide if . For each, we obtain two intervals and that can be comparedusing (13). Combining the information from the comparison ofthe -levels, we can provide a measure of preference by sam-pling and counting how many times .Specifically, let a function be equal to 1 ifaccording to (13) (based on one of the criteria defined

there), and 0 otherwise. Define a function as

(18)

A fuzzy number is preferred to iff , wherewould typically be equal to , and of course for any practicalcomputation it is possible to pick an approximation with .

IV. FUZZY IIM

We apply the arithmetic of fuzzy numbers to the IIM to ac-count for uncertainty in the elements of the matrix. Of pri-mary interest here is a means to measure and rank the key sec-tors, taking into account also the associated uncertainty.As such, the IIM is rewritten in (19), where is a vector offuzzy numbers, each representing the inoperability of a given

infrastructure, while , also a vector of fuzzy numbers, is


Fig. 3. Comparison between the open and closed loop indices according to [41] and [50] for sector n.1.

the perturbation in demand resulting from a disruptive event.Finally, the entries of the matrix are fuzzy numbers.

(19)

As shown in the appendix, the solution of the above fuzzyequation has to be computed by means of a difference inclu-sions framework, looking for the set of composed by allthe possible solutions where the parameters assume any valueamong those specified by their own interval.As shown in the appendix, exploiting peculiarities of trian-

gular fuzzy numbers, this calculation can be simplified, and an-level of the bound can be calculated by (20) and (21).

(20)

(21)

A. Open-Loop Key Sector Metrics for the Fuzzy IIM

We extend the open loop key sector metrics , and tothe fuzzy case, i.e., , and . In general, the indices canbe defined for a particular level of . The fuzzy coefficients ofmatrix are represented as a collection of intervals for each

. Such intervals are collected for each in two crispmatrices and , and the dependency index and influencegain are computed in the usual way. For each infrastructure , theresult is an interval for the dependency index,for the influence gain, and for the criticality index.

(22)

The above results hold true for any fuzzy number. However,assuming the coefficients are triangular fuzzy numbers,it is possible to provide a bound for the indices, which are alsotriangular fuzzy numbers. Such a bound is completely deter-mined by three numbers (i.e., the index evaluated for the leftand right matrix endpoints for , and the index evaluatedfor ), where and coincide.

(23)

B. Closed-Loop Key Sector Metrics for the Fuzzy IIM

The fuzzy closed-loop dependency index , influence gain, and criticality index are more complex as it is non-

trivial to compute the fuzzy matrix due to the ma-trix inverse calculation. However if the elements are non-negative, then the following bound can be provided [50].

(24)

Consequently, it is sufficient to consider the bounds of thelevel sets 0 and 1 to characterize the fuzzy inverse. Clearly, theinterval matrix for collapses into a single crisp matrix,while the level set for expresses the maximum level ofuncertainty of the inverse.The elements of the fuzzy matrix are

defined in (25), where .

(25)


Fig. 4. Comparison between the open and closed loop indices according to [41] and [50] for sector n.3.

Note that the approach suggested by (25) is simple and compact,requiring the inversion of three crisp matrices. Also, it yieldsresults that are numerically identical to those obtained by usingthe fuzzy version of the series of (6).Based on the above results, it is possible to define the generic-level of the closed loop dependency index , of the closedloop influence gain , and of the closed loop criticality index

as follows.

(26)

Again, assuming that the coefficients are triangularfuzzy number, the above indices represent a triangular bound,and thus are completely determined by three values (i.e., theindex evaluated for the left, and right matrix endpoints for, and the index evaluated for ), where

for all .

(27)

Let , and be the sum of all the entries of , and, respectively; and let , and be the sum of all the

entries of , and , respectively. Also,let

(28)

According to Theorem 1 (see appendix), the normalized de-pendency index, influence gain, and criticality (both open andclosed loop) indices are defined as follows.

(29)

In this case, the graphical representation, especially of thecriticality index and ( and ), provides an easy


Fig. 5. Graphical representation of the fuzzy open loop dependency index and influence gain.

vision of the different quantities, as illustrated in Figs. 5 and 6(in Figs. 7 and 8 for the normalized values).In the figures, the values assumed by the different quantities

for (e.g., the most credible value) are represented as ablack point. On the other side, the different levels of belief arerepresented by a gray scale. In this way, the relevance of each in-frastructure is reported as a fuzzy ellipse in the dependency-in-fluence plane. Note that the ellipse’s width is ameasure of the as-sociated uncertainty, which can be different for the dependencyand influence.Analogous to the crisp case, it could be that ranking the above

fuzzy indices according to the fuzzy operator could providevery different results due to how a sector depends on the others,on how influential a sector is on others, and how first orderversus higher order dependencies matter.

V. CASE STUDY

We have applied the proposed approach to the 15 15 sectorcase study in [40], [41] from the 2005 BEA annual IO tables[68]. The 15 sectors representing all industry and infrastructuresectors in the US are given in Table I.The matrix for the 15-sector example is provided in

Table VII in the Appendix, while Table VIII presents the totaloutput, vector , of these sectors. The calculations provided inthis paper assume a two-week disruption; therefore, the annualtotal output values in Table VIII are divided by 26.As in [41], we assume that each entry of is affected by

an uncertainty of with respect to the nominal value. Toavoid a uniform characterization of the ambiguity, we hypothe-size that, using the fuzzy framework, the coefficients are mod-eled as symmetric triangular fuzzy numbers: the peak value is incorrespondence of the nominal value, and the width of the sup-port is equal to the amplitude of the uncertainty. Note that the

TABLE IINDUSTRY AND INFRASTRUCTURE SECTORS CONSIDERED IN

THE CASE STUDY (SOURCE: [41])

choice of a triangular shape for the entries of has been madefor the sake of clarity, but the extension to a more sophisticatedrepresentation of the uncertainties is straightforward.Let us describe the fuzzy interdependency indices obtained.

Table II reports the open loop dependency index, the influencegain, and the criticality index, while Table III reports the nor-malized ones. Analogously, Table IV reports the closed loop in-dices, and Table V the normalized ones.Figs. 3 and 4 depict the open- and closed-loop indices for two

industries (the Agriculture and Utilities industries). To performa comparison, we report also the intervals obtained in [41], andthe crisp value when no uncertainty is considered.Plots 3 and 4 show that, for the open-loop indices, the interval

representation coincides with the support of the fuzzy numbers(i.e., ). Closed loop indices, instead, are very differentin terms of ambiguity. This difference is because in the fuzzyframeworkwe are able to effectively compute the inverse of ,


Fig. 6. Graphical representation of the fuzzy closed loop dependency index and influence gain.

Fig. 7. Graphical representation of the fuzzy normalized open loop dependencyindex and influence gain.

while in [41] the intervals are obtained by adding to thenominal inverse matrix. For example, in Fig. 3, the fuzzy

TABLE IIFUZZY OPEN LOOP INDICES

has much more ambiguity (wider support) compared to the in-terval approach, while the opposite is true for the index . Asshown by Fig. 4, other sectors show different results in terms ofambiguity. This difference means that, even assuming uniformuncertainty, the fuzzy framework allows us to accurately calcu-late the effect of second and higher order dependencies.It is evident that industry n.2 (Mining) depends largely on the

services provided by other infrastructures (it has ).On the other side, the industry most able to negatively influ-ence the operability or reliability of the others is industry n.5(Manufacturing), with a (i.e., it is able to producean amplification effect of about the 250%). Such a large value,together with a non-marginal degree of dependency, makes thissector also the most critical, with . Note that Man-ufacturing is roughly three times more critical than the second


Fig. 8. Graphical representation of the fuzzy normalized closed loop dependency index and influence gain.

most critical industry n.11 (Professional and Business sector)which has .Conversely, the less critical sector is n.12 (Educational ser-

vices, health care, and social assistance), with a ,because it is at the same time resilient (i.e., ),and able to exert limited effects ( ). The least im-pacting sectors are n.14 (Other services, except government,

), n.6 (Wholesale trade, ), and n.2(Mining, ).These aspects are emphasized in Table III, where values

greater than 1 imply that the corresponding sector has aboveaverage vulnerability, influence, and criticality.The first aspect to consider is that, in accordance with (6), all

the terms are biased by one. The criticality of sector n.5 (Manu-facturing) is considerably increased due to the effect of secondand higher order dependencies ( ). Analogously,the high order dependencies increase the criticality of sectorsn.2 (Mining, ), n.11 (Professional and business ser-vices, ), and n.1 (Agriculture, forestry, fishing andhunting, ).The above considerations have been done considering the

most believable value of the fuzzy numbers describing the in-dices (i.e., the value assumed for ). This work representsa simplification because it considers only implicitly the uncer-tainties associated with the different quantities.To address this issue, we propose both an ordering of the in-

dices according to (13), and the plots in Figs. 5–8.

Figs. 5 and 6 show the open loop, and closed loop indices inthe , and planes, respectively; (Figs. 7 and 8 reportthe corresponding normalized values). Specifically, a grayscalehas been used to depict: values with higher belief are repre-sented in black, while values with lower belief fade to white.By means of such plots, it becomes immediate to capture theambiguity and uncertainty associated with the different indices;this immediacy can be seen as a useful visual tool for decisionmaking.In Figs. 7 and 8, the axes intersect at , hence an infra-

structure in the upper right quadrant is vulnerable and influen-tial, both above the average. An industry in the upper left quad-rant has above average influence, and an industry in the lowerright quadrant has above average vulnerability. Note that, fromopen- to close-loop analyses, the shape of the ellipses changes(i.e, the uncertainty varies in different ways along and ),and the criticality changes. Obviously, the uncertainty tends togrow while moving from an open- to a closed-loop analysis.Table VI provides an ordering of the industries according to

the preference operator with respect to , , , , , and. The ranking results in Table VI were the same for all of thedecision rules in (13). According to fuzzy , the Mining sector(sector n.2) is most functionally dependent upon the other 14sectors.The Manufacturing sector (sector n.5) tops the fuzzy rank-

ings, having a large capability to influence other sectors. As


TABLE IIIFUZZY OPEN LOOP NORMALIZED INDICES

TABLE IVFUZZY CLOSED LOOP INDICES

mentioned before, it is also the most critical sector both in anopen- and closed loop perspective.The ranking of the closed-loop versions of the fuzzy de-

pendency index, , influence gain, , and criticality index ,provide a similar picture of dependency and influence for themost relevant sectors, but the relevance of the others changesstressing how the higher order dependencies can alter theperceived relevance. Backup strategies are typically basedonly upon first order dependencies; considering also higherorder contributions allows us to better estimate the effectiverelevance and criticality of the different sectors, as well asbetter planning resource allocation.From a decision making perspective, the rankings of these

two indices can lead to areas of emphasis for preparedness in-vestments. For example, ensuring the continued operation ofmanufacturing facilities will significantly contribute to the re-liable function of the remaining sectors.Notice that the ranking of Table VI takes into account only

indirectly the different uncertainties associated to the fuzzy in-dices, even if the ranking methodology operates on the fuzzynumbers and not only on the most believable value. More ex-plicit information can be gathered from Figs. 7, 8, 5 and 6, which

provide an immediate feedback about the uncertainties associ-ated with the different quantities (e.g., the width of the base ofthe triangles).

VI. CONCLUDING REMARKS

Component importance measures identify the components ofa physical system that influence its reliable performance, thusprompting decisionmakers to alter the system in someway (e.g.,redundant components, more frequent inspection and mainte-nance cycles). We analyzed a system of a different sort: an eco-nomic system of different component industries that are im-pacted by disruptions in underlying infrastructures. Such disrup-tions, whether by attack, disaster, accident, or common failure,can impact across several industries both directly and indirectly.And, much like the ranked components resulting from a CIManalysis, a prioritized list of industries can focus decision makerattention and resources more effectively.To address the identification of important components in the

interdependent economic system, we developed two measures:(i) the dependency index measures the extent to which an in-dustry is impacted by those other industries to which it is in-terconnected, and (ii) the influence gain measures the extent towhich an industry impacts those industries to which it is in-terconnected. These measures were defined for open-loop (firstorder), and closed-loop (higher order) impact analyses.Much like physical systems, the study of the importance of

industry sectors can be fraught with uncertainty. Hattis and Bur-master [69] motivate analyses that do not involve point esti-mates in various parameters in risk assessments as they couldhave substantial effects on risk assessment outcomes if they areincorrect. Point estimates used for parameters in an interdepen-dency model, in particular, can “exaggerate the certainty of theanalysis and will almost assuredly be incorrect” [70]. As such,the indices developed here were extended to the situation wherethe interdependent nature of industry interaction is quantifiedby fuzzy numbers, thus representing the uncertainty in sectorinterconnectedness. Further, we developed a ranking approachfor prioritizing industries according to their fuzzy indices.We are fundamentally motivated by the CIMs of the re-

liability engineering literature. And though we apply the


TABLE VFUZZY CLOSED LOOP NORMALIZED INDICES

TABLE VISECTORS CONSIDERED IN THE CASE STUDY (SOURCE: [41]) WITH THE SAMEORDERING OBTAINED FOR EVERY STRATEGY (BEST CASE, WORST CASE,

LAPLACE, HURWICZ, MIN REGRET)

approaches developed here to interdependent industries sec-tors, parallels can be drawn to applications in physical systemswhere direct and indirect impacts (via feedback loops) arepresent.Future work efforts include the definition of effective strate-

gies for the composition of multiple information sources (e.g.,economic input-output data, the knowledge of experts, techni-cians, and stakeholders) in a unified framework. Further workincludes the application of the methodology to more complexand nonlinear models, considering how the interdependenciesamong the different sectors change over time to model and pre-dict adverse situations, as well as a comparison of critical infra-structures of different nations.

APPENDIX

As discussed above, the sum of two fuzzy numbers is a linearoperation, and the resulting fuzzy number is also triangular[72]. Specifically, the sum of two triangular fuzzy numbers

and is given in (30).

(30)

Equation (31) shows the multiplication of a triangular fuzzynumber by a real scalar value .

(31)

Note that, while fuzzy numbers, specifically triangular fuzzynumbers, are closed with respect to the sum and scalar productoperations, the product of two triangular fuzzy numbers is, ingeneral, not triangular.However, using the theory of inclusions [73]–[75], it is

possible to overcome such a problem and provide a triangularbound for the product. Let us first provide a simple example.Consider the scalar equation provided in (32), where andare triangular fuzzy numbers.

(32)

The product is performed level-wise by considering a productof intervals for each -level, and the resulting collection of-levels is a fuzzy number and a bound for the product. Foreach -level, we have to take into account the product of twointervals. That is, , and . Asa consequence, the solution for is no more a singlevalue, but becomes a set of values such that (33) holds.

(33)

Note that in (33) the equality symbol “ ” has been replacedwith “ ,” because (33) represents the set of all the feasible solu-tions of the , where , and ,i.e., any realization of and within the intervals and ,respectively.The extension of the product to the vectori case is straightfor-

ward. Consider a fuzzy linear system of equations of the formof (34), where is an fuzzy-valued matrix, i.e., arefuzzy numbers, and are vectors of fuzzy numbers.

(34)

Analogous to the scalar case, we have the following.

(35)


TABLE VIIMATRIX FOR THE 15 SECTOR EXAMPLE, WITH VALUES ROUNDED TO FOUR DECIMAL PLACES (SOURCE [40])

TABLE VIIITOTAL OUTPUT FOR THE 15 SECTOR EXAMPLE (IN MILLIONS OF US$)

(SOURCE: [40])

Note that, for each , the set is indeed an in-terval vector , which is characterized by two real vectors, and representing the extremal values of each compo-

nent (i.e., )In general, the evaluation of such bounds is not straightfor-

ward, but in the case of non-negative coefficients [50], [75], [76]the bounds are given by

(36)

Hence, in this case, it is sufficient to evaluate for eachthe two endpoints to reconstruct the solution. If the fuzzy

numbers are triangular, the -levels 0 and 1 are sufficient tocompletely describe a bound for the product [50], and we justneed to know , where .A similar result can be obtained for the division of two trian-

gular fuzzy numbers.Theorem 1: Let , and be a non-negative fuzzy number,

and a positive fuzzy number respectively (i.e., ).Then the generic -level of a bound for the fuzzy numberis given by

(37)

Moreover, if , and are trian-gular fuzzy numbers, then a bound for is given by

(38)

Proof: is the set of values such that

(39)

That is,

(40)

Because the maximum interval is , the state-ment is proved. Equation (38) follows as a corollary, because theleft and right endpoints of the triangular fuzzy numbers coincidefor .Let us report a definition of the inverse of a fuzzy matrix. The

inverse of a fuzzy matrix can be defined in several ways [77].In the following, we employ the level-wise approach knownas Rohn’s Scheme [78]. For each -level, the inverse of theinterval matrix is defined as the smallest interval matrix

that contains the following set.

(41)

This interval matrix satisfies (42) and (43).

(42)

(43)

Clearly, the inverse of a fuzzy matrix exists only if the eachmatrix is invertible for each -level.

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Gabriele Oliva received a master degree in Computer Science and Automa-tion in 2008, and a Ph.D degree in Computer Science and Automation in 2012,both at University Roma Tre or Rome, Italy. He currently holds a post-doctorateresearch position at University Campus Bio-Medico of Rome, Italy. His re-search interests include critical infrastructure interdependency modeling, fuzzysystems, distributed systems, and indoor sensor network localization. He is in-volved in several national and European projects on Critical Infrastructures In-terdependency Modeling and Networked Control.

Roberto Setola received a Laurea degree in Electronic Engineering (1992), anda PhD (1996) from the University Federico II of Naples . From 1999 to 2004,he served at the Italian Prime Minister’s Office. He is currently Associate Pro-fessor of Automatic Control at the University CAMPUS Bio-Medico and Headof the Complex System and Security Laboratory. He was responsible for theItalian Government Working Group on CIIP, and a member of the G8 SeniorExperts’ group for CIP-CIIP. He wrote three books on the simulation of dy-namic systems, and more than 100 scientific papers related to the modellingand control of complex systems (electro-mechanical, biological and social) andcritical infrastructures.

Kash Barker is an Assistant Professor in the School of Industrial and Sys-tems Engineering at the University of Oklahoma. He earned his Ph.D. degreein Systems Engineering in 2008 from the University of Virginia, where hewas a graduate assistant in the Center for Risk Management of EngineeringSystems, following B.S. and M.S. degrees in Industrial Engineering fromthe University of Oklahoma. His research interests have dealt primarily with(i) analyzing risk in interdependent industry and infrastructure sectors, and(ii) enhancing data-driven decision making for large-scale system sustain-ment. Research projects have been funded by the National Science Foun-dation, Federal Highway Administration, Army Research Office, and NavalPostgraduate School. He is an Associate Editor for IIE Transactions (Gov-ernment, Policy, and Society department), and on the Editorial Board of RiskAnalysis.

Fuzzy Importance Measures for Ranking Key Interdependent Sectors Under Uncertainty

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