1 INTRODUCTION
The strong dependence on lifeline and infrastruc- tures is one of the distinctive characteristics of mod- ern societies. Unfortunately, these systems are sub- jected to several hazards such earthquakes causing important physical damage, property loss, dysfunc- tion to urban activities and serious socioeconomic consequences.
Several researchers have proposed different types of interdependencies (Kameda, 2000, Rinaldi et al., 2001, Peerenboom et al., 2001, Tang et al., 2004, Yao et al., 2004):
- Function interaction/ physical, cyber interaction.
- Collocation interaction/ geographic, space, physical damage propagation.
- Substitute interaction/ back-up functions of substi- tute systems.
- Restoration/ recovery interaction.
- Cascade interaction/ functional damage propaga- tion/ system interaction.
- General interaction.
- Logical interaction/ financial markets
The variability of the type of the interactions be- tween the lifelines under investigation and their in- herited complexity makes the assessment of inter- dependent systems’ performance a difficult task and
a very challenging issue for advanced seismic risk management solutions. An “efficient” seismic vul- nerability analysis and a development of an optimum mitigation strategy requires firstly the quantification, in terms of hierarchy importance, of the interdepend- encies between lifeline systems in three different pe- riods (prior, during and after the occurrence of a seismic event) and finally the production of the “sys- temic” fragility curves of interdependent elements.
The “systemic” vulnerability of interacting lifeline elements depends on the vulnerability of individual components, taking into account the way in which the components are connected, and the degree of their interdependency. The “systemic” fragilities can differ significantly from the fragilities of independ- ent lifeline components, as they are based on the vulnerability functions of independent elements and the “cross impact matrix”. Both “systemic” fragility curves and vulnerability curves of individual life- lines’ components are described in terms of the probability of exceeding, in an independent or inter- dependent lifeline component, a specific limit state as a function of ground motion intensity. Moreover, they are described by (cumulative) lognormal distri- bution functions defined by a median value and a standard deviation β (e.g. NIBS, 2004 for independ- ent elements).
Seismic risk and hierarchy importance of interdependent lifelines.
Methodology and important issues.
M. N. Alexoudi, K. G. Kakderi & K. D. Pitilakis
University of Macedonia, Thessaloniki, Greece
Department of Civil Engineering, Aristotle University of Thessaloniki, Greece
11th ICASP International Conference on Application of Statistics and Probability in Civil Engineering, August 1- 4, Zurich, Switzerland
ABSTRACT: The present paper focuses on the proposal of appropriate methodologies for the estimation of seismic risk of “interdependent” lifelines. Moreover, it discusses the quantification of hierarchy importance of these complex systems that are comprised by several subsystems and components, as well as the construction of “systemic” fragility curves of interdependent elements. The latter is performed on the basis of a probabilis- tic approach, as there is lack of real, detailed recorded data of lifelines interaction after earthquakes. Seismic vulnerability of interconnected lifelines is assessed based on the use of appropriate “systemic fragility curves”
that differ from those of individual components. “Systemic fragility curves” are estimated based on vulner- ability functions of independent elements and the “cross impact matrix”. Three different methodologies are used to evaluate the “cross impact matrix”. These include an economic approach (input-output model), an Analytical Hierarchy Process (AHP) and a group decision making approach with linguistic preference rela- tions. The optimum selection between them is depending on available information, quality of data, experience of the experts and the type of lifeline systems involved. Each methodology has it is own restrictions and dis- advantages that must be carefully accounted for and considered during the evaluation.
The aim of the paper is to propose appropriate methodologies for the estimation of the seismic risk of “interdependent” lifelines, to discuss the quantifi- cation of hierarchy importance of these complex sys- tems comprised by several subsystems and to pro- duce “systemic” fragility curves of interdependent elements.
Three different methodologies are proposed herein to evaluate the “cross impact matrix” which represents the degree of corresponding impact of various “elements” (lifelines, infrastructures). The proposed methodologies are based on (i) an eco- nomic approach (input-output model), (ii) a multi- criteria decision making procedure (Analytical Hier- archy Process: AHP) and (iii) a group decision- making approach with linguistic preference rela- tions. For each one of the three methodologies pro- posed, a description is briefly reported together with the basic assumptions, the important issues and the restrictions.
A major concern for the quantification of lifeline elements’ interactions is the description of the typol- ogy and the functioning of systems involved, the na- ture of the reciprocal influence when the specific synergy is evolved (normal, co-seismic or restora- tion/recovery period) and the importance of the link (slight / strong) between components and systems.
An example of the application of the three meth- odologies in terms of hierarchy importance for the
“inter-dependent” lifelines of port system facilities can be found in a companion paper in this confer- ence (Kakderi et al., 2011).
2 METHODOLOGY
The seismic risk (SR) of interdependent lifeline sys- tems is given by the expression (Equation 1):
{S.R.interdependent}={S.R.independent}*{Interaction function} (1) The risk of failure or deviation from normal oper- ating conditions in one infrastructure (or part of it) can affect the risk in another infrastructure, if the two are interdependent. In case of a strong earth- quake, malfunction of a component can result in cas- cading effects, within the same system and to other connected and interacting systems. The nature of the identified interactions, as well as the degree of inter- connectedness (type and degree of coupling), is the determinant parameter of the interdependent sys- tems’ seismic behavior. To develop an interaction function different approaches can be used: eco- nomic, fuzzy logic, decision-making or combination of different approaches. Three different methodolo- gies are proposed herein.
The Input-Output model comprises a linear, de- terministic, equilibrium approach and a framework capable to describe the degree of interconnectedness.
It can be used in national or regional level and with
some restrictions in city level if economical data is available. It gives better results in large systems, if coefficients are considered to be constant for a fixed unit of time (static approach); however it should be carefully used in “close systems”. Kakderi et al.
(2007) give an application of the proposed method- ology through an illustrative example.
The Analytical Hierarchy Process (AHP) is listed among the most popular multi-criteria decision mak- ing procedures, as with a theoretical robustness it employs simplicity, accuracy, can handle both intan- gible and tangible criteria and has the capability to directly measure the inconsistency of the respon- dent’s judgments. AHP combines subjective and ob- jective alternatives into a single measure in a hierar- chical framework. On the other hand, there is a restriction of a maximum number of criteria that can be used and number of lifelines or lifeline elements that can be involved. An early version of the meth- odology combined with an illustrative example is provided in Alexoudi et al. (2008) research.
The third methodology is based on linguistic (hy- brid) geometric averaging operator, for group deci- sion making with linguistic preference relations. It can be used in cases where the linguistic preference information provided by the experts does not take the form of precise linguistic variables. Value ranges can be obtained due to the experts’ vague knowledge about the preference degrees of one alternative over another. This methodology exploits the opinion of every expert without losing information and preci- sion. Using this methodology, the uncertainty of the opinion of the experts is transported in the estimated
“systemic” fragility curve. The developed approach is described through a numerical example in Alexoudi et al. (2008, 2009) studies.
The optimum selection between the different ap- proaches depends on available information, quality of data, experience of the experts and the type of lifeline systems involved. Each methodology has its own restrictions and disadvantages that must be carefully accounted for and considered during the evaluation.
Adequate interdependency indices can be esti- mated to measure the degree of connection between different lifelines and resulted perturbations to one system from induced malfunctions to the other. The concept of systemic vulnerability or “vulnerability of interdependent elements” is introduced through the definition of a propagated inoperability matrix and a methodology is proposed for the vulnerability as- sessment of interdependent systems using “interde- pendent” or “systemic” fragility curves. The latter are derived based on the use of independent compo- nents’ fragility curves (e.g. NIBS, 2004) and the
“cross impact matrix”, which represents the degree of probabilistic contribution (functional dependence) of one system compared to another. The importance
of each lifeline component is assigned through ade- quate weight coefficients.
ECONOMIC APPROACH
FUZZY LOGIC
DECISION MAKING
COMPOSITE APPROACH
ESTIMATION OF LIFELINE SYSTEMS’
INTERDEPENDENCY INDICES ESTIMATION OF LIFELINE SYSTEMS’
PROPAGATED INOPERABILITY MATRIX INDEPENDENT LIFELINE
COMPONENTS’
VULNERABILITY CURVES
INDEPENDENT LIFELINE COMPONENTS’
WEIGHT COEFFICIENTS ESTIMATION OF INTERDEPENDENT LIFELINE
COMPONENTS’ VULNERABILITY CURVES
Figure 1. Flowchart of the general methodology.
The proposed methodology for the evaluation and quantification of the interactions of different lifelines (components or elements) is illustrated in Figure 1.
Description of the methodologies, along with spe- cific guidelines for the evaluation of the degree of corresponding impact of “inter-dependent” lifeline systems to the whole system through the evaluation of the “cross impact matrix” and the estimation of
“systemic” fragility curves of interdependent ele- ments is given in the next paragraphs.
An example of the application of the appropriate methodologies to encounter all inter and intra de- pendencies and the global performance of port com- ponents in Thessaloniki (Greece) under the effect of an earthquake event is given in Kakderi et al. (2011) research.
3 ECONOMIC APPROACH
Input-Output model (I-O) offers a macrolevel, linear, deterministic, and equilibrium modeling of interde- pendencies among economic infrastructures. The original Leontief model can be extended to the Inop- erability Input-Output Model (IIM) in order to spec- ify systems’ interactions and quantify the degree of coupling. In the case of lifelines, the model was ini- tially developed in the form of a physical-based model (Haimes and Jiang, 2001, Jiang, 2003,Jiang and Haimes, 2004) and later as a demand reduction model (Santos and Haimes, 2004, Haimes et al., 2005 a, b). Recently, supply- and output-side exten- sions to the IIM for interdependent infrastructures have been proposed (Leung et al., 2007).
3.1 Methodology
Input-Output model (I-O) is a representation of eco- nomic activity among several sectors that includes inter-sector flows. These parameters are recorded in
tables (Input-Output tables – Table 1) which are the core of the I-O analysis.
Table 1. Form of Input-Output table.
Sectors
1 2 … n
Final demand
Total output
1 X11 X12 … X1n c1 X1
2 X21 X22 … X2n c2 X2
. . .
. . .
. . .
. . .
. . .
. . .
. . .
Sectors
3 Xn1 Xn2 … Xnn cn Xn
Value
added z1 z2 … zn
Total
supply r1 r2 … rn
In its most basic form, the I-O model consists of a system of linear equations, each one of which de- scribes the distribution of a sector’s production throughout the economy (Equation 1).
i c x a X c Ax X
j
i j ij
i ∀
= +
⇔ +
=
∑
(1)where i,j = 1,2,………,n are the system’s interacting sectors, X is the total output matrix, c is the final demand matrix and A is the technical coefficient ma- trix.
Making the basic assumption that the level of economic dependency is the same as the level of physical dependency, the original Leontief model can be used to specify systems’ interactions and quantify the degree of coupling. A different interpre- tation of the model parameters adequate for lifeline systems is incorporated as follows:
- The interacting sectors i=1, 2, …., n are different lifeline systems.
- The c and X matrices represent lifeline commodi- ties measured in monetary terms. In detail:
- xij is the input commodity (in monetary terms) of lifeline system i to the production of the commod- ity of lifeline system j (intermediate consumption).
- Xi is the total output commodity (in monetary terms) of lifeline system i.
- ci is the final demand (or final consumption) for the ith lifeline system – the portion of the total com- modity output (in monetary terms) of lifeline sys- tem i for final consumption by end-users.
- rij is the amount of the ith lifeline system (resource) input commodity in the production of the jth com- modity. Total inputs are referred to as value-added.
- ri is the total supply of the ith lifeline system input commodity.
- The A matrix describes the degree of dependence between infrastructures and it is determined on the basis of the physical connections that exist among infrastructures. αij is the proportion of the commod-
ity input of lifeline system i to j, with respect to to- tal production requirements of lifeline system j. It is referred to as the Leontief technical coefficient.
The interdependency matrix A* indicates the de- gree of coupling between the different infrastruc- tures. It is given from the following equation:
[
( )] [ ] [
( )]
A* = diag X −1⋅ A ⋅ diag X ,
=
i j
ij X
a X
*
aij (2) Estimated interdependency indices vary between 0
and 1, with higher values referring to higher degree of interconnectedness.
It should be noted that using this approach, only first order interdependencies (direct dependent ef- fects) are simulated.
Briefly, the I-O model includes two steps for the estimation of the interactions between lifelines or in- frastructures. Firstly, the Leontief technical coeffi- cient matrix A is estimated. Τhen the interdepen- dency matrix A* is calculated based on intermediate consumptions, the total output in monetary terms and the Leontief technical coefficient matrix. The in- terdependency matrix A* is the “cross impact ma- trix”.
3.2 Basic assumptions The Input-Output Model:
- Represents the economic activity among several sectors at equilibrium state.
- Studies the interdependency of an economy’s pro- ducing and consuming units based on their cross transactions.
- Uses observed economic data for a specific geo- graphic region.
- Describes inter-sector flows measured for a par- ticular time period in monetary terms.
- Assumes that the level of economic dependency is the same as the level of physical dependency.
3.3 Important issues
Some of the important issues of the I-O analysis are described below:
- A key parameter is the initial perturbation. Often there is no direct translation of physical infrastruc- tures disruption to perturbations while in some cases the issue is compounded by the fact that the infrastructure is not included in the defined sectors of the available I-O table.
- Haimes and Jiang (2001) recognized that there is a lack of data availability and specific information needed for establishing the probability of inopera- bility due to interconnectedness of the infrastruc- tures.
- IIM is more useful as a guideline to the potential cascading effects rather than a forecasting model (Sadoulet and de Janvry, 1995).
3.4 Restrictions
The main restrictions of the method are based on the assumptions of the Leontief I-O model. Haimes et al.
(2005a, b) define the main limitations in their stud- ies. Such limitations are refereed to:
- Equilibrium modeling and as a static model only equilibrium values can be determined.
- Difficulties in the modelling of a more complex economic environment, including substitutions among inputs and economies in scale.
- The impact of region and the time period.
- The lack of data.
- The perturbation amount. The smaller the perturba- tion, the more applicable and stable the results.
4 ANALYTICAL HIERARCHY PROCESS (AHP) The Analytical Hierarchy Process (AHP) (Saaty, 1977, 1980) is a mathematical theory for deriving ra- tio scale priority vectors from positive reciprocal matrices with entries established after comparisons.
Moreover, AHP is listed in the most popular multi- criteria decision making procedures (Hwang and Yoon, 1981) which elicits preferences of decision makers/ experts by means of pair-wise comparisons.
AHP yields unique answer for the interactions be- tween different lifelines or elements inside the same systems and resulted perturbations to one system or element from induced malfunctions to the other.
4.1 Methodology
The AHP combines subjective and objective alterna- tives into a single measure in a hierarchical or net- work framework. The assessment is based on a ratio scale and pair-wise comparisons for the element in interest (eg. inter and intra dependencies between elements or systems) either verbally or numerically.
In verbal comparisons decision makers select one phrase out of a list of nine phrases that best repre- sents their opinion.
Afterwards, a table proposed by Saaty (1989) con- vert the phrases into an 1- 9 scale number (Table 2).
For each expert, the derived pair-wise compari- sons of relative importance aij= wi/wj for all decision elements and their reciprocals aji= 1/aij are inserted into a “reciprocal square matrix” A = {aij} as shown in Equation 3.
A=
n n n
n
n n
w w w
w w w
w w w
w w w
w w w
w w w
/ ...
/ /
...
...
...
...
/ ...
/ /
/ ...
/ /
2 1
2 2
2 1 2
1 2
1 1 1
(3)
The priority vectors that describe the importance of criteria are derived using Eigenvalue Method (EM) (Saaty, 1994; 1988) or Logarithmic Least Squares Method (LLSM) (Barzilai et al, 1987; Craw-
ford, 1987) or Least Squares Methods (LSM). Ac- cording to the eigenvalue b method the normalized right eigenvector (W = {w1, w2,…,wn}T) associated with the largest eigenvalue (λmax) of the square ma- trix A provides the weighting values for all decision elements.
A * W= λmax * W (4)
Table 2. Conversion table used in AHP to translate verbal pref- erences into numbers (Saaty, 1988).
Verbal judgment Numerical judgment
Equally preferred 1
Equally to moderately 2 Moderately preferred 3 Moderately to strongly 4
Strongly preferred 5
Strongly to very strongly 6 Very strongly preferred 7 Very strongly to extremely 8 Extremely preferred 9
The analytical solution of Equation 4 then pro- vides the relative weights for each decision ele- ments. Because judgements are frequently inconsis- tent, a Consistency Index (CI) is used to measure the degree of inconsistency in the square matrix A. Saaty (1980) compared the estimated CI with the same in- dex derived from a randomly generated square ma- trix, called the Random Consistency Index (RCI).
The ratio of CI to RCI for the same order matrix is called the Consistency Ratio (CR). The reliability of the experts and their judgmental consistency is cal- culating using the Consistency Ratio of individual experts. Generally, a CR of 0.10 or less is considered acceptable; otherwise the “reciprocal square matrix”
A is revised to improve the judgmental consistency.
In case of several experts, inconsistent experts should be removed and Geometric Mean Method can be employed to aggregate the different judgments from the consistent experts. A new square matrix of the group experts (Agroup) can be produced. Weight coefficients can be assigned to all infrastructure ele- ments as estimated from (Agroup) matrix. Group Con- sistency Index (GCI) and Group Consistency Ratio (GCR) are calculated in the same way as the typical CR value (GCR= GCI/RCI). The group judgment is considered consistent if the GCR is less than 0.10.
The relative importance of each element com- pared to another as derived from the normalized Agroup matrix is then used in the “systemic vulner- ability”. The lifeline interactions are described as cross impact indices that are parts of the Group Re- ciprocal square matrix and represent the degree of probabilistic contribution (functional dependence) of one system compared to another. The degree of cor- responding impact of “inter-dependent” lifeline in the operability of lifelines during the study period is defined summing all the elements in each line of
“cross impact matrix” for the study period.
4.2 Basic assumptions
The basic assumptions of AHP are the following:
- Experts can use verbal expressions or numerical values in order to give their opinion.
- Experts should give their opinion using pair-wise comparisons.
- Verbal expressions should be converted into num- bers using Saaty (1989) table.
- Saaty (1989) conversion table is used also in each set of pair-wise comparisons of (sub)-criteria or al- ternatives.
4.3 Important issues
The triggering issue of this approach is the:
- Production of hierarchy structuring of simple con- nection without indirect links for the complex rela- tions between inter and intra dependencies for life- lines’ elements.
- Translation through verbal expressions or numbers of the importance of each lifelines’ element for the functioning of others or for the perturbation of the functionality of the system itself. The “importance”
of lifelines’ element can measure the interdepend- encies.
- Careful selection of the group of experts according to their experience to lifelines and earthquake engi- neering.
Some advantages of AHP method are given be- low:
- The verbal or the numerical modes itself do not in- fluence the quality of the AHP results. Both of them are equally representing the opinion of deci- sion makers for the prediction of the alternative and the ranking chosen from a set of alternatives or the preferences (scores) for the same set. Moreover, both modes can provoke results in the same degree of consistency for a set of judgments.
- Verbal expressions can have the numerical mean- ing as given by Saaty (1989) convert table and ver- bal expression can have the same interpretation be- tween the sender of information and the receiver.
- The pair-wise comparisons don’t influence the quality of the AHP analysis.
- The result into the priority vectors is independent from the method used (p.g Eigenvalue).
4.4 Restrictions
Some of the important restrictions that must be taken into consideration are given below:
- There is no direct translation of physical infrastruc- tures disruption, especially when applying the method to estimate lifeline interdependencies.
- Uncertainties can’t be accounted and although, there is no need of data availability, it is extremely time-consuming process especially when expert’s opinions are inconsistent.
- The size of the group of the experts should be “ef- ficiently” selected in order to increase the accuracy and minimize the management of communication problems. An optimum group size is consisted of about 5 members
- No more than seven indicators should be used to describe a problem.
- In case that some alternatives appear to be very close with each other, managers and practitioners might wish to consider a correction to avoid some errors that traditional AHP cannot detect.
5 GROUP DECISION MAKING WITH UNCERTAIN LINGUISTIC PREFERENCE RELATION METHOD
The stochastic nature of the decision process, the un- certainty and the imprecise numeric values of deci- sion data (including incomplete information and/or unknown condition) and of humans are incorporated with the flexibility of fuzzy set theory (Chen and Hwang, 1992, Chen and Tzeng, 2004) and with methodologies that exploit the advantages of uncer- tain additive linguistic preference relations.
Xu, (2004 a, b) developed a method, based on lin- guistic geometric averaging operator and linguistic hybrid geometric averaging operator, for group deci- sion making with linguistic preference relations. In addition, Xu (2006) proposed a methodology that combines a formula based on possibility measure for comparing two uncertain linguistic preference values and the uncertain additive linguistic preference rela- tions as given by the group experts, without loss of information.
5.1 Methodology
Xu (2006) approach is proposed for the quantifica- tion of the hierarchy importance of interconnected lifelines and infrastructures and estimation of the
“systemic” fragility curves of interdependent ele- ments.
The methodology encounters four- steps until the estimation of the “cross-impact matrix”. Firstly, a fi- nite set of alternatives X = {x1, x2,..., xn} and experts E = {e1, e2, ………, em} are considered. The weight vector of the decision makers is:
ω = (ω1, ω2, …, ωm}T, where ωi ≥ 0 and 1
1
∑
== m
i
ωi (5)
The experts provide their preferenceswith respect to the importance of interaction between the systems by using the linguistic terms in the set. Using the sets, uncertain additive linguistic preference rela- tions are produced R~(k)
=( r~
ij(k)
)nxn according to the number of experts.
Secondly, the Uncertain Linguistic Weighted Av- eraging (ULWA) operator is utilized to aggregate all the uncertain additive linguistic preference relations provided by the experts to obtain the collective un- certain additive linguistic preference relation
R~
=( r~
ij)nxn, where ~rij =ULWAw
(
~rij(1),~rij(2),...,~rij(m))
(6) for all i, j = 1, 2, …, n and ULWA operator is defined as:n n
ULWAω(µ~1,µ~2,...,µ~n)=ω1µ~1 ⊕ω2µ~2 ⊕...⊕ω µ~
[ω = (ω1, ω2, …, ωn}T is the weighting vector of the µ~i
with ωi є [0,1] and 1
1
∑
== n
i
ωi ].
In the third step, the Uncertain Linguistic Averag- ing (ULA) operator zi ULA
(
r~i ,r~i ,..., r~in)
2
= 1 , for all
i = 1, 2, …, n with:
~ ) ...
~ (~ ) 1 ,...,~ ,~
(~1 2 n 1 2 n
ULA µ µ µ = n µ ⊕µ ⊕ ⊕µ (7)
is estimated to aggregate the preference information r~ij (j = 1, 2, …, n) in the ith line of theR~
in order to get the global preference degree z~iof the ith alternative over all the other preferences. Finally, the complementary matrix is developed:
( )
pij nxnP= , where pij ≥0,pij + pji =1, pii =1/2(8) for all i, j = 1, 2, …, n. All elements in each line of ma- trix P are summed up to obtain:
∑
== n
j ij
i p
p
1
, i = 1, 2, …, n. (9) The z~i(i = 1, 2, …, n) are then ranked in descending order in accordance with the values of pi(i = 1, 2, …, n).
Complementary matrix is the “cross impact matrix”
of interconnected lifelines that can be used to evalu- ate the “systemic vulnerability” of interconnected lifelines and infrastructures.
5.2 Basic assumptions
Some of the basic assumptions of the methodology are given below:
- An introduction of linguistic terms scale that col- laborate the numerical value with verbal expres- sions is needed.
- All experts should use the same linguistic terms scale.
- Experts should give their opinion in intervals after the use of pair-wise comparisons as in AHP.
- The choice between decision alternatives necessi- tates the ranking of uncertain linguistic variables.
5.3 Important issues
The important issues of this methodology are given briefly below:
- Flexibility to represent and analyse the impre- cise/vague information resulting from the lack of knowledge and information.
- Relative ease with which it handles multiple per- ception-based judgment intervals instead of provid- ing deterministic preferences.
- No limit of indicators or criteria or size of the group of the experts used to describe a problem.
Nevertheless, in order to minimize communication problems, an optimum group size is consisted of about 5 members.
- Experts can express their preferences and their rat- ings using qualitative criteria as considered in lin- guistic or numerical terms using uncertain additive linguistic preference relation.
5.4 Restrictions
Below are given some important remarks and restric- tions.
- A series of papers (Triantaphyllou et al., 1994, Tri- antaphyllou and Mann, 1994, Triantaphyllou and Mann, 1995) report that one of the problems of us- ing pairwise comparisons to characterize fuzzy membership functions was potentially very high failure rates.
- The discrete set of the Saaty’s scale used for pair- wise comparisons does not work well when the ac- tual pairwise comparisons are assumed to be con- tinuous.
- As all opinions of the group of experts are equally used, a careful selection is needed according to their subject- research domain.
6 INTERDEPENDENT FRAGILITY CURVES Fragility curves of the interdependent components are estimated based on vulnerability functions of in- dependent elements (e.g. NIBS, 2004) and the “cross impact matrix”. The concept of “systemic vulner- ability” or “vulnerability of interdependent ele- ments” can be written as a probability of the interde- pendent event EN for a three systems’
interconnectivity:
P(EN)= P(E1)+ (1-P(E1))P(E2)*a12N + (1 P(E1))P(E3)a23N (10)
where: P(E1) denotes the probability of event inside system 1 (system 1 independency), P(E2) denotes the probability of event 2 inside system 2 (system 2 in- dependency) and P(E3) denotes the probability of event inside system 3 (system 3 independency), a12N
denotes a cross impact factor representing the degree of probabilistic contribution (functional dependence) of system 1 and 2 to the lifeline element of node N
and a23N denotes a cross impact factor representing the degree of probabilistic contribution of system 2 and 3 to the lifeline element of node N.
This method inevitably involves complex interac- tions among events and high uncertainty. Four main steps can be accounted for: (1) construction of a structural model of interrelationship of all systems under consideration, (2) quantification of expert opinions/ or economical activities for the importance of each lifeline system or element compared to an- other (3) conversion of the opinions/ inter- sector flows into the form of a “cross impact matrix” com- posed of elements representing degree of corre- sponding impact and (4) modification of probability of an event occurrence using the “cross impact ma- trix”.
7 CONCLUSIONS
Interdependencies among infrastructures create new challenges that need to be addressed on several deci- sion-making levels involving multiple stakeholders, and modeled from various perspectives; therefore, it is impossible for a single model to address all con- cerns of infrastructure interdependencies. The main objective of this effort is to give decision-makers the means and capacities to formulate effective risk management, based on the quantification (hierarchy importance) of the interdependencies according to the available data, time and expertise.
In the present paper three different methodologies with different theoretical background, assumptions and implementation fields are presented briefly for the quantification (hierarchy importance) of the in- terdependencies between lifelines and infrastruc- tures.
Moreover, as a next step, the seismic risk of “in- ter-dependent” lifeline systems can be performed on the basis of a probabilistic approach using “systemic fragility curves”. The development of appropriate methodologies to estimate the seismic risk of “inter- dependent” lifeline systems depends on available in- formation and resources as well as the desirable level of reliability of the performed analysis. “Sys- temic” fragility curves can be produced for each method, for all the elements at risk and for the three study periods according to available information.
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