Assessment of the potential applicability of fuzzy set theory to accident progression event trees with phenomenological uncertainties

Reliability Engineering and System Safety 37 (1992) 237-252

Assessment of the potential applicability of fuzzy set theory to accident progression event trees with phenomenological uncertainties Moon-Hyun Chun & Kwang-ll Ahn Department of Nuclear Engineering, Korea Advanced Institute of Science and Technology, PO Box 150, Cheongryang, Seoul, South Korea (Received 7 May 1991; accepted 2 December 1991)

Expert opinion is frequently used in the risk analysis of nuclear power plant systems to assess, in particular, the probabilities of rare events. However, this procedure is always accompanied by imprecision and uncertainty that characterise the experts judgment. Since the fuzzy set theory provides a framework for dealing with such judgmental imprecision and uncertainty, the potential applicability of the fuzzy set theory to the uncertainty analysis of accident progression event trees with imprecise and uncertain branch probabilities and/or with a number of phenomenological uncertainty issues are examined as a possible alternative procedure to that used in the current probabilistic risk assessments. The main purpose of this paper is to demonstrate a potential use of fuzzy set theory and provide its formal procedure in the quantification of the uncertainties of accident progression event trees. First, an example application of the fuzzy set theory is made to the simple portion of a given accident progression event tree with typical imprecise and uncertain type of input data, and thereby computational algorithms suitable for application of the fuzzy set theory to the accident progression event tree analysis are identified and illustrated. Secondly, to show the merits of the fuzzy set theory model in real application, the procedure used in the simple example is extended to extremely complex accident progression event trees with a number of phenomenological uncertainty issues, i.e. a typical plant damage state 'SEC' of the Zion nuclear power plant risk assessment, and the results are compared with the one obtained by current probabilistic methods. In addition, discussions and answers for five major questions on its real application are given.

NOTATION

FWA

F,

Ai APET

bj Cij DCH E,(Q)

E*(O)

F

Top event numbers in the event tree Accident progression event tree Branch point for a given issue (top event) Containment matrix Direct containment heating Lower possibilistic mean value of fuzzy quantity Upper possibilistic mean value of fuzzy quantity Frequency of PDS i Fuzzy output function for pathway/" Converse image of z Ranking function suggested by Yager Ranking value of fuzzy outcome ~.

F* LLH m m th min M(S~ n

PDS PRA 237

Fuzzy weighted average Lower distribution function induced by a fuzzy quantity Upper distribution function induced by a fuzzy quantity Limited Latin hypercube Modal value of a fuzzy quantity with unique peak Lower modal value of a fuzzy quantity with flat peak Upper modal value of a fuzzy quantity with fiat peak Minimum operator Mean value of the elements of S~' Number of fuzzy outcomes Plant damage state Probabilistic risk assessment

Moon-Hyun Chun, Kwang-ll Ahn

238

£.

0 r; R Rk sup

so

5. ¢ TFN wi X z

Z~,Zb

Z~ Zj, m

zi, max Zj,mi~

/~,(z~)

Fuzzy input probability (fuzzy variables) Fuzzy probabilities for jth branch point constructed from the LLH issue data Arbitrary fuzzy quantity consequence of type k for containment release category j Real line Risk of consequence type k Supremum operator Fission product source term for release category j of PDS i Fuzzy outcome for pathway j (fuzzy probability) ~-Level set or interval of confidence for Sj Triangular fuzzy number Weighting factor for level i for LLH issue data Event characterised by fuzzy quantity Potential probability for fuzzy outcomes Left and right extreme values of or-level set in a fuzzy set, respectively Potential probability for fuzzy variables Probability for branch point j and for level i Potential probability for fuzzy branch probabilities Mode of the triangular membership function Maximum value of the triangular membership function Minimum value of the triangular membership function t~-Level of membership function Membership function of fuzzy probability Membership function of fuzzy outcome gj

1 INTRODUCTION An important issue faced by contemporary risk analysists of nuclear power plants is how to deal with uncertainties that arise in each phase of risk assessments. In general, assessment of risk from the operation of nuclear power plants is comprised of five principal steps: (1) accident frequency (system) analysis; (2) accident progression, containment loadings, and structural response analysis; (3) radioactive material transport (source term) analysis; (4) offsite consequence analyses; and (5) risk calculations.' There are multiple sources and types of uncertainty in these processes of risk assessment. The major uncertainty addressed here is the one that arises in the second part of the risk analysis which treats the physical processes affecting the core after an initiating event occurs. The type of phenomenon that contributes the most to uncertainty in risk analysis is

the phenomenon that is poorly understood so that there may be several competing models, each incomplete with respect to various aspects of the problem. For example, there is no single accepted scenario for both the high-pressure melt ejection and the subsequent effects leading to direct containment heating (DCH). The physicochemical processes for these phenomena are extremely complex and varied. Major uncertainties involve the impact of DCH on early containment failure, and the impact of core-concrete interactions on both early and late containment failure. In the most recent risk assessment of nuclear power plant systems, 1 an expert opinion polling process was used to assess the probabilities of rare events and the uncertainties related to physical phenomena, in conjunction with the limited Latin hypercube (LLH) sampling approach (a modified Monte Carlo method) for the propagation of input uncertainty. Though it seems that the expert opinion polling as a means of identifying issues and estimating uncertainty is an acceptable part of the current probabilistic risk assessment (PRA) process, a clear disadvantage of this approach is that the sufficient robustness in the final results may not be attained against the ambiguity of the information upon which the experts base their judgment. This is mainly due to the fact that, while expert opinion under an incomplete knowledge and a limited data is inherently imprecise and uncertain, the subjective probabilities estimated by the current PRA process do not adequately reflect such a type of uncertainty which characterises the expert judgment. More efforts are certainly needed to make the current method more effective and more readily acceptable. Also it is desirable to develop and explore some alternative methods or procedures for the assessment of uncertainty in the risk analysis. An alternative approach for modelling the uncertainty under various imprecise and incomplete information has recently been proposed with the use of fuzzy or possibility logic. 2'3 The fuzzy set theory provides a formal framework for dealing with such imprecision and uncertainties inherent in the experts judgment, and therefore it may be used for more effective analysis of accident progression of PRA where experts opinion has been a major means particularly for quantifying some rare event probabilities and uncertainties. The concept of fuzziness or possibility plays a particularly useful role in the management of judgmental uncertainty of experts or in the presentation of meaning of an event. According to this method the total quantity of uncertainty does not have to add up to one. The use of the possibility concept that does not require to normalise such that the total assessed quantity of uncertainty be unity seems to be a great advantage in modelling the imprecision and vagueness inherent in the expert

Applicability of fuzzy set theory to accident progression event trees judgment in risk analyses. This is mainly due to the fact that the probability theory is sometimes too strict a theory to model this type of uncertainty even though it has been somewhat effectively used in the NUREG-1150 risk analyses. Recent advances in the theory of fuzzy sets 2~ make it possible to analyse the uncertainty related to complex physical phenomena, which may occur during a severe accident of nuclear power plants, by means of fuzzy set or possibility concept instead of the probability. To date, however, the use of fuzzy set theory in risk and reliability analyses has been very limited. 5"6 In an effort to explore the full potential of fuzzy set theory as a methodology for dealing with phenomena that are too complex or too uncertain, an example application of the fuzzy set theory is first made to the portion of a simple accident progression event tree (APET) with imprecise and uncertain branch probabilities. Thereby computational algorithms that are suitable for application of the fuzzy set theory to the APET analysis are identified and the formal procedure of this method in the quantification of the uncertainties of APETs is provided. Then, in order to show the merits of the fuzzy set theory model in real applications, the procedure established in the simple example is extended to extremely complex APETs with a number of phenomenological uncertainty issues, i.e. a typical plant damage state 'SEC' of the Zion nuclear power plant risk assessment. The main purpose of this paper is to present the results of an assessment of the potential applicability of the fuzzy set theory to the uncertainty analysis of APETs with imprecise and uncertain branch probabilities and/or with a number of phenomenological uncertainty issues as a possible alternative procedure to the NUREG-1150 methodology. ~ In addition, it is shown here how the fuzzy set theory can be used to represent the imprecision or uncertainty which surrounds the probabilities under certain circumstances, while retaining the structures of a given event tree and consistency which the probability theory provides.

2 EXAMPLE APPLICATION OF FUZZY SET THEORY TO AN APET In conventional event tree analysis, 7,s the branch point probabilities have been treated as exact values. As already mentioned, however, for many top event questions of the APETs regarding the phenomena encountered during severe accidents, it is often difficult to assign exact branch probabilities (e.g. 'probabilities concerning the location of induced failure of the reactor coolant system pressure boundary') or parameters (e.g. 'magnitude of pressure loading at vessel breach due to DCH and steam spike'

239

and 'the containment failure pressure') from the current state of knowledge. To examine the potential applicability of the fuzzy set approach to this type of problem, first, a simple portion of a given APET with typical imprecise and uncertain branch probabilities of top events has been analysed by the fuzzy set approach in the following way.

2.1 Formulation of a fuzzy APET analysis problem To illustrate how the fuzzy set theory can be applied to APETs with imprecise and uncertain branch probabilities, suppose one is given a portion of APET as shown in Fig. 1. A typical portion of the given APET has three questions A1, A2, and A 3 ( a s summarised in Table 1), and each question is assumed to have only two branches (success or failure of a system; and/or occurrence or non-occurrence of a phenomenon) for simplicity in presentation. One may notice that the top events are well defined. However, when available data and information for these events are rare, then expert opinion often becomes a major means for quantifying the branch probabilities. In this case, their probabilities might be imprecise and uncertain due mainly to the judgemental uncertainty or subjective bias of the expert. Therefore, let us suppose, because of the nature of the top events considered, it is not possible to assign unique numerical probability values to the branches of three top events shown in Fig. 1. To overcome this difficulty, the concept of fuzzy probability or possibility of probability, 2'3'9 instead of a unique value of probability, can be introduced in Top Events :

Containment sutnls :

A1

A2

A3

I-

Yes

'

r'3

Initiating Event

$3

s4

N° r

1 -P1

P2

P3

$5

S6 P3 ss

Fig. 1. Sample event tree and outcomes.

Moon-Hyun Chun, Kwang-llAhn

240

Table 1. Three top event questions and specified fuzzy variables Top event number

Top event questions

Specified fuzzy variables

A~

Is the reactor cavity dry when a small loss of coolant accident has occurred?

Yes =/5 No = 1 - / 5

A2

What is the probability of occurrence of DCH and steam spike events?

Occurrence =/52 Non-occurrence = 1 - t52

A3

What is the probability of occurrence of hydrogen burn at vessel breach?

Occurrence =/53 Non-occurrence = 1 - p~

the analysis of the event tree. The fuzzy probability is a fuzzy set theoretic or possibilistic representation of an individual's concept of the likelihood of an event, which is described by a fuzzy set or possibility distribution defined in the probability space. 3 Also, it models imprecise and uncertain probability, by means of possibility of particular probabilities which indicates a measure of uncertainty that does not necessarily add up to be unity. Thus, the fuzzy probability which simultaneously models different aspects of uncertainty, probability and possibility seems to be a less arbitrary model and provides more information than the unique subjective probability when the judgment on the branch probability is not clear. By resorting to this fuzzy probability concept, it is possible to allocate a degree of uncertainty to each value of the branch probabilities of top events in the given APET. That is, fuzzy variables can be assigned to the branches of three top events in Fig. 1 as shown in Table 1. Here, a fuzzy variable represents only a variable associated with an unknown quantity whose value is uncertain or fuzzy. The remaining problem is then to transform the fuzzy variables into quantitative fuzzy probabilities or possibility distributions and calculate the possibility distribution of occurrence of accident pathways (i.e. combination of accident sequences and containment events) by means of the axioms of probability and fuzzy propagation logic, given the possibility distribution of occurrence (or success) of top events shown in Fig. 1. In essence, one is dealing with a fuzzy probability on [0, 1], namely, possibility distribution of probability instead of a specific value of probability or probability distribution. A detailed procedure of how to handle this problem is given in the following.

2.2 Fuzzy probabilistic quantification of fuzzy input variables In order to quantify the event tree shown in Fig. 1 using the fuzzy set or possibility approach, quantitative fuzzy probabilities or possibility distributions must be first assigned to the fuzzy input variables. This can be done by introducing the membership function of the fuzzy set theory. The membership function is the central concept of Zadeh's fuzzy set theory 2'1° and it

represents numerically the degree to which an element belongs to a set, small values representing a low degree of membership and high values representing a high degree of membership. This function takes on values between 0 and 1. The membership function of a fuzzy set is often interpreted as a possibility distribution. H In this case, a possibility distribution is defined as being numerically equal to a membership function, then the membership function becomes formally equivalent to the possibility distribution. Thus, a possibility distribution can be obtained from the corresponding membership function and its functional form is represented by the combination of a range of the potential probability and the degree of possibility of particular probabilities belonging to the applicable probability range. In this respect, the actual derivation of the numerical values of the membership function is very important in applying the fuzzy set or possibility_.a__p_Pr0ach_to APETs. It should be noted that the membership or possibility of an element in the fuzzy set is not a statistical quantity as expounded by some authors 2a°'~2 and the assignment of membership or possibility to an event is a matter of subjective opinion. Also, it is not always a yes-or-no matter, but rather a matter of degree. It is basically context dependent. From the viewpoint of membership or possibility, the membership functions or possibility distributions are subjectively determined from expert's experience, intuition and various information including empirical data, just as subjective probabilities are. It seems more important to know the general shape of membership functions to apply the fuzzy set approach, rather than precise numerical values, since the membership function itself is fuzzy. That is, if the membership function has a good shape, it can be generally considered a satisfactory approximation. Most successful applications of the fuzzy set theory have been empirical and the numerical values of membership functions have also been approximately assigned through trial and error. One can assign membership functions with appropriate numerical values for the three fuzzy input variables ~ (i = 1, 2, 3) in Table 1. For simplicity, however, the following linear membership functions are assumed. From these, all the fuzzy inputs in Fig. 1

Applicability o f fuzzy set theory to accident progression event trees are converted into quantitative fuzzy probabilities. Here #~(z;) indicates the membership function corresponding to the potential probability zi of fuzzy variable ~, which is interpreted as the possibility distribution of the proposition that zt is the probability of occurrence of an event.

f 5Zl, 1.0, /~,(zl) = 13.0 _ 5Zl, L0, ~5z2 - 4, ~,,~(z~) = tO, f l.O, #~(z3) = ~ 1.5 - 5z3, 1.0,

if 0 -< zl -< 0.2 if 0.2_< Zl_< 0.4 if 0.4_< Zl_< 0.6 otherwise

(1)

if 0.8_< z2_ 1.0 otherwise

(2)

if 0_< z3__0.1 if 0.1_< z3_< 0.3 otherwise

(3)

The linear membership functions given by eqns (1)-(3) are depicted in Fig. 2. Having established the relationship between the probability and the imprecise notion of likelihood (i.e. possibility) considered here, the fashion in which such imprecision on the probability is propagated is determined completely by the axioms of probability theory and fuzzy logic. 2'13'14 The next step is to calculate the quantitative fuzzy probabilities for each pathway of the example event tree. 2.3 Calculation of fuzzy probabilities for each pathway For the given event tree in Fig. 1 there are eight possible pathways (or outcomes). The fuzzy outcomes ~- (j = 1. . . . . 8) are comprised of the product or subtraction of three fuzzy variables. In the probabilistic framework, the probability z of the second pathway in Fig. 1, for example, is given by z = zl x z2 x (1 - z3). Similarly, in the fuzzy frame1.0

0.5 Probability (zi)

1.0

Fig. 2. Membership functions specified for three fuzzy variables (P~).

241

work the fuzzy outcome ~ for the second pathway is defined by 2'9 Sz =/51 x P2 x (1 -/53)

(4)

The calculus of fuzzy sets is based on three specific operators of set complement, union, and intersection. 2 More specifically, the intersection of fuzzy sets is modelled by the min-operator and the union by the max-operator. The operators of fuzzy sets are less sensitive to variations in membership grades than the corresponding probabilistic system. 2 Also, a basic principle that allows the generalisation of crisp mathematical concepts to the fuzzy framework is known as the 'extension principle' of Zadeh. TM That is, to calculate the fuzzy probability for each pathway via the input membership functions #~,(zi) (i = 1, 2, 3), these relationships have been used in the following way. 14 Let ~ be a real function of three variables (/1, P2, and 153) and let /51,/52, and /53 be three fuzzy sets defined in the probability space (i.e. unit interval [0, 1]). The 'extension principle' allows one to define the image of/51,/52, and /53 through ~ =fj(P1, P2,/3) whose membership function is #s,.(z)--

sup

Z1,Z2,Z3E IJ~I(z)

{min[#~,(zl), #&(z2), #~(z3)])

(5)

where f}-l(z) = {(Zl, Z2, Z3) ER Ill(z1, 7.2, z 3 ) = z } This principle can be interpreted as follows: the possibility for the quantity (Pl,/52,/53) to be represented by zl, z2, z3 is #~1×~×~(Zl, z2, z3) = min[#~,(zl), #~(z20), #~(z3)]

(6)

Here, /51 x t52 x 153 denotes the Cartesian product or three fuzzy variables P1,/52, and /53. Then, the possibility for ~. =fj(P1, P2, P3) to be represented by z is the greatest possibility value for the quantity (zl, z:, z3) in the converse image of z, f71(z), to be in P1 x/52 >( /53" Note that whenever f f l ( z ) = O, #~j(z) = 0. Thus, fuzzy numbers can be processed in this manner similar to the non-fuzzy case, and the operations are sometimes called the 'extended operations' (extended addition, extended subtraction, etc.). 2,15 In practice, however, the implementation of the solution procedure is not trivial, although the solution of the various extended operation is defined by the extension principle of eqn (5). The reason is that the solution procedure corresponds to a non-linear programming problem which is very complex except for the simplest mapping functions. In the present work, the extension principle of eqn (5) has been implemented by the 'fuzzy weighted average' (FWA) algorithm proposed by Dong and Wong. 15 The computational algorithm of the FWA operation is

Moon-Hyun Chun, Kwang-II Ahn

242 1.0 0.9 0.8 0.7 0.6

~( z )

gj

0.5 0.4 0.3 0.2 0.1 0.0 0 u...)uu~

Probability (z)

U.8-"

Fig. 3. Output fuzzy probability distributions (4) obtained by the FWA algorithm.

based on ideas from the o~-level representation of fuzzy sets which indicates any membership grade, 2 non-linear programming implementation of the extension principle, and interval analysis. This method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner. 15 For example, to obtain the simple product of three fuzzy probabilities ;~1 = P~ x P2 x P3 by means of the FWA computational algorithm, it requires the following steps: (1) Select a particular o~-level value (shown on the /~(z)-coordinate in Fig. 3) where 0 -< tr _ 1.0. (2) Find the interval(s) in fuzzy probabilities of #1,/52, and/53 which correspond to tr (these are the ~-levels o f / 1 , P2, and/53). (3) Using interval operations, compute the interval(s) in $1 which correspond to those of /51 X /52 X /53 (the results are the o:-levels of $1). The above steps are repeated for as many values of o~ as needed to refine the solution. The eight fuzzy outcomes Sy (j = 1. . . . . 8), i.e. possibilities of occurrence of accident pathways as a fuzzy set given the possibilities of occurrence of top events ~ (i = 1, 2, 3) shown in Table 1, have been obtained following the procedures outlined above and they are shown in Fig. 3. The output fuzzy probability distributions ~ (j = 1 , . . . , 8) shown in Fig. 3 are the result of repeating the above steps for 100 values of re.

2.4 Interpretation and comparison of output fuzzy probabilities So far it has been shown how one might assess the likelihood of certain branch points (or phenomena) in terms of a series of possibility distributions. In addition, possibility distributions for a set of accident pathways have been obtained for a postulated event tree using a fuzzy logic. The next step of the fuzzy set approach is the interpretation and comparison of the output functions (i.e. the eight fuzzy outcomes ~, j = 1 , . . . , 8) obtained in the previous step and shown in Fig. 3. This step is equivalent to the problem of decision making and it is basically context dependent. From the viewpoint of decision making, it is very important to identify numerical quantities which characterise the fuzzy outcomes. For the purpose of the present work, four representative quantities, i.e. representative values, modal values, upper and lower possibilistic mean values, and intervals of confidence are computed to compare and to enhance the robustness in the results of the present analysis. Basically, these quantities are obtained by defuzzifying the fuzzy outcomes into single numerical values or a scale of intervals. Brief definitions of the four quantities are given here, whereas the actual numerical values obtained are summarised in Table 2.

2. 4.1 Representative value The representative value of a fuzzy quantity (i.e. a fuzzy set of real numbers) can be obtained from the

Applicability of fuzzy set theory to accident progression event trees

243

Table 2. Four quantifies characterising eight fuzzy outcomes S~ Representative valuesa j=1 2 3 4 5 6 7 8

0.052 0.292 0.006 0.027 0-092 0.640 0.011 0-047

2 0 4 0 6 1 2 2

Modal values

Mean values [E., E*] b

Intervals of confidence c

I0.0, 0-041 [0-18, 0.4] 0.0 0.0 [0-0, 0.08] [0.54, 0.8] 0,0 0.0

[0-0, 0.103 31 [0.078, 0-51 [0.0, 0-012 7] [0-0, 0.053 31 [0.0, 0.183 3] [0.367 3, 0-90] [0-0, 0-022] [0.0, 0.093 3]

[0.0, 0.050 41 [0-155 2, 0.421 [0.0, 0.001] [0-0, 0.008 41 [0.0, 0-098 4] [0-500 2, 0.82] [0.0, 0.002] [0-0, 0.016 4]

Values obtained by ranking function F proposed by Yager. b Lower and upper possibilistic mean values, respectively. c o~= 0.9. concept of 'ranking function' proposed by Yager 16 or 'removal number' proposed by Kaufmann and Gupta. 17 The ranking function is defined as the integral of the mean of the level sets associated with a fuzzy quantity, which was originally suggested to help in the ordering or comparing of fuzzy quantities of the unit interval. The ranking function used here is the following. 16 If "¢7 is the re-level set of k a n d if M ( S ~ is the mean value of the elements of S~, then F(~) =

f0flcmaxM(g~) dcr

(7)

where tr = [0, 11. Consider, for example, a fuzzy probability ~qt with the membership grade shown in Fig. 3, where for each grade of membership the dashed line represents the average value of the elements having at least that grade of membership. Then F(~q~) is equal to the area between the dashed line and the membership axis. Also, this ranking function is the same as the removal number, relative to k = 0 recently suggested as an ordinary representative of a fuzzy number by Kaufmann and Gupta.17 Because of this feature of the ranking function, the value obtained from the ranking function can be regarded as a representative value of all the elements of a fuzzy quantity. Therefore, the rank value is selected as a representative value of each fuzzy outcome in the context of the present paper. The results of application of this method to the fuzzy outcomes ~ (j = 1. . . . ,8) obtained for the postulated event tree are shown in the second column of Table 2.

2. 4. 2 Modal value When a fuzzy quantity is normalised, 2 any element of its peak is called a modal value of the fuzzy quantity, lS If the fuzzy quantity has a flat peak, the modal value becomes an interval [_m, fit] whose possibility is one. However, if the fuzzy quantity has a unique peak

number m, the modal value becomes the unique value m itself.18 In this case, both m and fit of the fiat peak are identical to m. In general, the height of a fuzzy quantity, i.e. possibility of an element, can be viewed as a measure of its reliability. 2 In this respect, the modal value can be regarded as an element with the highest reliability in the fuzzy quantity. The modal values of the fuzzy outcomes ~ are shown in the third column of Table 2.

2. 4. 3 Upper and lower possibilistic mean value The upper and lower possibilistic mean value of a fuzzy quantity ls'19 is defined from a probabilistic interpretation of the fuzzy quantity, thus relating it to the probability distribution while keeping the intrinsic characteristics of the fuzzy quantity. This value is essentially based on the upper and lower probabilities in the sense of Dempster. 2° That is, the possibility grades of an event X quantified by a fuzzy quantity and one minus the complement of the event can be interpreted, respectively, as upper and lower bounds on grades of probability P ( X ) , in the sense of Dempster. 2° Under this interpretation of a fuzzy quantity, a set of upper and lower distribution function F*(x), F.(x) is defined by F,(x)=[ItO(x) (1 F,(x) =

if x-< _m otherwise

l - lim f~o(r) if x -> fit r~x 0 otherwise

f

(8) (9)

Then, the upper and lower mean values of Q can be defined, i.e. E*(Q) and E , ( Q ) , respectively, such thatlSA 9 E*(Q) = f__+fr dF,(r)

(10)

E , ( Q ) = f_+~ r dF*(r)

(11)

Moon-Hyun Chun, Kwang-ll Ahn

244

The mean value of a fuzzy quantity Q is the interval [E,(~)), E*(~))] containing the mean values of all random variables compatible with C). This mean value is different from the representative value which is represented by a single value. The mean values of the fuzzy outcomes ~ obtained by using this concept are shown in the fourth column of Table 2.

in the fifth column of Table 2. In Fig. 3, in particular, notice that the range of $6 values that corresponds to 0~= 0.9 level is indicated by dashed lines.

3 APPLICATION OF FUZZY SET THEORY TO ZION ACCIDENT PROGRESSION ANALYSIS

2. 4. 4 Interval of confidence The interval of confidence of a fuzzy quantity ~2"~7 is defined as an interval that contains all the elements of the fuzzy quantity corresponding to any specific possibility, which represents a type of uncertainty. Freeling ~2 has made an initial attempt to provide an axiomatic basis where the 'possibilities' are to be interpreted as 'degrees of confidence'. Then the specific possibility indicates a degree of belief that a true value of the fuzzy quantity may exist within the interval. For example, an interval of confidence corresponding to a possibility 0.9 can be assumed to be an interval with the possibility where a true value may lie. The endpoints of this interval are represented by the extremes at (r-level of the membership functions. Thus, this interval is slightly different from the concept of confidence limit which has been used in the probability theory. However, the interval of confidence can be used to enhance the interpretation of fuzzy outcomes. The symbolic representation of an 'interval of confidence' at o<-level is usually written

To show the merits of the fuzzy set theory in real applications, an application of the procedure used in the foregoing example is made to a typical plant damage state (PDS) 'SEC' of the Zion. 7 The representative accident sequence for the PDS 'SEC' is a small loss of coolant accident, emergency corecooling system failure on injection, and with operational containment sprays but without operational fan coolers. 7 That is, for the purpose of comparison between the two approaches, one by the NUREG-1150 methodology ~ and the other by the fuzzy set theory approach, Zion APET for 'SEC' has been evaluated by the two methods. For convenience in discussion, a brief summary of (1) the methodology of Zion accident progression analysis, (2) Zion APET analysis of 'SEC' by statistical methods, and (3) Zion APET analysis of 'SEC' by fuzzy set theory is presented successively in the following.

a s ~7

3.1 Methodology of Zion accident progression analysis

S'~ = [Za, zbl

(12)

Figure 4 shows the 'intervals of confidence' at tr = 0.9 level in the fuzzy outcome ~. The numerical values of ~,=o.9_~q~,=o.9 that correspond to Fig. 4 can be obtained directly from Fig. 3 and they are also listed

The risk from a nuclear power plant can be defined by ~ Rk = E fi E Cijr~(Sij) (13)

i

j

where Pmbabfli~

R ~ is the risk of type k (associated with consequence k), f~ is the frequency of P D S / , Cij is the conditional probability of containment release category j given PDS i (i.e. containment matrix), S0 is the fission product source term for containment release category j of PDS i, and r~' is the consequence of type k, given fission product source term Sit, for release category j.

1.0 0.9

g6

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

I

52

I

g5

t

r

g8

g7

~j 0=i .....8) Fig. 4. Intervals of confidence at o~= 0.9 level in the fuzzy outcomes.

The first part of the risk analysis ('accident frequencies' f~) represents the estimation of the frequencies of accident sequences leading to core damage. In this part of the analysis, combinations of potential accident-initiating events and system-failure frequencies are calculated. The major concern in the present work is the second part of the risk analysis (accident progression, containment loading, and structural response analysis to obtain C0) which deals with the progression of the accident after the core has begun to degrade. For each

Applicability of fuzzy set theory to accident progression event trees

245

Table 3. Six types of APET top events Dependency upon prior events

Type of inputs

(1) Branch point probabilities only (2) Branch point probabilities and parameter values (3) A set of parameters to be summed and compared to reference parameters to obtain branch point probabilities general type of accident, defined by the PDSs, the analysis considers the important characteristics of the core-melting process, the challenges to the containment building, and the response of the building to those challenges. Event trees were used to organise and quantify the large amounts of information used in this analysis. The event trees combined information from many sources, e.g. detailed computer accident simulations and panels of experts providing interpretations of available data. The principal steps of the 'accident progression analysis' are (1) development of APETs, (2) probabilistic quantification of event trees, and (3) grouping of event tree outcomes into a smaller set of 'accident progression bins'. In the Zion study, 7 the APET in the form of computer codes (such as EVNTRE and EVNTREISS) provided the necessary framework for quantification of the likelihood of various failure modes. The structure of the Zion APET is based on 59 top events, many of which have multiple outcomes or branches. The list of top event questions for the Zion APET can be found in Ref. 7. Depending on the type of input, there are six different types of top events as shown in Table 3. 3.2 Zion APET analysis of 'SEC' by statistical methods The uncertainty analysis of Zion APET relies on the selection of key uncertainty issues that can have a significant impact on the estimated risk at Zion. The approach used in the selection and evaluation of key uncertainty issues for Zion APET is essentially the same as that used for Surry. 21 Uncertainties in the estimates of containment loading and performance were treated through a stratified Monte Carlo sampling procedure called LLH. The elicitation of expert judgments was necessary to develop the probability distributions for some individual parameters in this uncertainty analysis. For certain key issues in the uncertainty analysis, panels of experts were convened to discuss and help develop the needed probability distributions. 1,7 For statistical quantification of the Zion APET, the EVNTREISS code 7 is used with 'issue' and 'sample' data along with the three input data required in the

Independent

Dependent

Type 1 Type 3 Type 5

Type 2 Type 4 Type 6

EVNTRE code (i.e. data for 'binning', 'branch-point probability', and 'dependency'). The product of the accident progression and containment loading analysis is a set of accident progression bins. Each bin consists of a group of postulated accidents (with associated probabilities for each PDS) that have similar outcomes with respect to the subsequent portion of the risk analysis (i.e. analysis of radioactive material transport). Quantitatively, the product consists of a matrix of conditional probabilities, with rows and columns defined by the sets of PDSs and accident progression bins, respectively. 1 The results of statistical APET analysis for the plant damage state 'SEC' (selected from 14 PDSs of Zion) is shown in Table 4 to provide a direct comparison with the results obtained by the fuzzy set theory approach. These LLH results were obtained by application of the EVNTREISS computer code that has incorporated the LLH sampling technique for the key uncertainty issues of the containment loading and performance. The number of LLH samples was limited to 100. 3.3 Zion APET analysis of 'SEC' by fuzzy set theory approach In order to analyze the Zion APET for the plant damage state 'SEC' by fuzzy set theory and compare directly with the results obtained by the LLH procedure, the probabilistic input data used in the Zion APET must first be converted into fuzzy input data by the appropriate assumption and adjustment. For this purpose, the following steps are taken. Except for the eight containment loading and performance issues, all other input data for the Zion APET are kept unchanged (sample input data for the Zion APET can be found in Ref. 7). Then, all the other input data can be regarded as point values with possibility one. For the key eight issues included in the Zion APET uncertainty analysis by the LLH procedure, 7 the 'issue data' for the EVNTREISS code are replaced by fuzzy inputs prepared for each issue. In addition, the EVNTREISS code has been modified to treat the fuzzy set theory procedure used in the example~ application.

246

Moon-Hyun

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Kwang-ll

Ahn

Table 4. Percentile values for lt~ conditional probabilities obtained by LLH approach for PDS 'SEC' Bin no.

Min

5th

Median

95th

Max

Mean

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.0 0.0 1.000 x 10 4 0-0 0-0 0.0 6.000 × 10 6 0.0 0.0 0.0 0-0 0.0 0-0 0.0 1.826 × 10 ~ 0.0 0.0 0.0 0.0

0.0 0.0 1.000 x 10 4 0-0 0.0 0.0 1-050 × 10 3 0-0 0.0 0.0 0.0 0.0 0.0 0.0 2.216 × 10 ~ 0.0 0.0 0.0 0.0

0.0 0.0 1.000 x 10 ~ 0.0 0.0 0-0 2-475 × 10 3 2.613 × 10 ~ 2 . 8 0 4 × 1 0 `2 0-0 0.0 0.0 0.0 0.0 9-383 x 10 ~ 0.0 0.0 0.0 0.0

0.0 0.0 1.008 x 10 2 5.300 x 10 4 0-0 0.0 2.498 × 10 ~ 4.958 × 10 ~ 1.444×10 ~ 3.886×10 ~ 0.0 0.0 0.0 0.0 9.965 × 10 ~ 0.0 0.0 2.282 × 10 -2 6.847 X 10 -2

0.0 0-0 1.051 x 10 2 3.204 x 10 3 0.0 0-0 2.500 × 10 3 5-796 × 10 ~ 2.285×10 ~ 4.469×10 -~ 0.0 0.0 0.0 0.0 9-974 × 10 ' 0.0 0.0 1.728 × 10 I 5.184 x 10 ~

0.0 0.0 3.908 x 10 3 1.24(I x 10 4 0.0 0.0 2.302 × 10- 3 6.908 × 10 2 3.150×10 2 8.507×10 ,3 0-0 0-0 0.0 0.0 8.664 × 10 0.0 0.0 4-447 × 10 3 1-373 X 10 -2

In the Z i o n A P E T analysis, 7 the d i f f e r e n t sets of values for b r a n c h e s o f e a c h issue a r e c a l l e d ' l e v e l ' . T h e levels m a y r e p r e s e n t different o p i n i o n s a b o u t t h e severities o f a c e r t a i n physical phenomenon. H o w e v e r , the p r o b a b i l i t y o f o c c u r r e n c e o f e a c h ' l e v e l ' can be different f r o m e a c h o t h e r . T h e ' w e i g h t i n g f a c t o r s ' a r e u s e d for this s u b j e c t i v e p r o b a b i l i t y o f o c c u r r e n c e of e a c h level. T o p r e p a r e fuzzy i n p u t s f r o m t h e issue d a t a (i.e. c o m b i n a t i o n o f t h e b r a n c h p r o b a b i l i t i e s a n d t h e c o r r e s p o n d i n g w e i g h t i n g factors), the c o n c e p t o f ' t r i a n g u l a r fuzzy n u m b e r s ( T E N ) '17'18 o r ' t r i a n g u l a r t y p e o f fuzzy p r o b a b i l i t i e s ' has b e e n used. T h u s , w e i g h t i n g f a c t o r s a n d b r a n c h p o i n t p r o b a b i l i t i e s for e a c h level u s e d for e a c h issue in the u n c e r t a i n t y analysis o f t h e Z i o n A P E T h a v e b e e n t r a n s f o r m e d into t h e T F N as s h o w n in Fig. 5: A ' T F N ' can be d e f i n e d by a t r i p l e t (Gmi,, Zj,m, Zj . . . . )" A l t h o u g h o n e can a s s u m e v a r i o u s t y p e s of m e m b e r s h i p f u n c t i o n s , fuzzy n u m b e r s o f this t y p e a r e very simple to c o n s t r u c t a n d m a n i p u l a t e . F o r a given d e p e n d e n c y case in a given issue, t h e ' m i n i m u m ' a n d the ' m a x i m u m ' values s h o w n in Fig. 5 c o r r e s p o n d to the m i n i m u m a n d t h e m a x i m u m p r o b a b i l i t i e s o f all

levels given for a b r a n c h p o i n t in the given d e p e n d e n c y case. T h e ' m o d e ' for a b r a n c h p o i n t s h o w n in Fig. 5, on the o t h e r h a n d , is t h e s u m m a t i o n o f the p r o d u c t s of t h e w e i g h t i n g factors a n d t h e b r a n c h p o i n t p r o b a b i l i t i e s for e a c h level. T h e s e q u a n t i t i e s can b e m o r e concisely e x p r e s s e d in m a t h e m a t i c a l forms. S u p p o s e that T a b l e 5 is the given L L H issue d a t a , t h e n t h e m i n i m u m , m a x i m u m , a n d m o d e s of t r i a n g u l a r fuzzy n u m b e r s t h a t c o r r e s p o n d to the values s h o w n in T a b l e 5 can be e x p r e s s e d as 4

Zj, m : E ZijWi'

Modes:

j = 1, 2, 3

i--1

M i n i m u m value:

Zj,min = min[zlj, z2j, z3j, z4i]

M a x i m u m value:

zj . . . . = max[zlj, z2, z3j, z4j]

The TFNs are then p o n e n t s as follows:

constructed

from

these

(14)

com-

~,~,(zj) =

(Zj--Zjmin)/(Zjm--Zj, min), ifZj,min<-Zi<~Zj,m ,, (zj max

Zj)/(Zj . . . . -- Zj,m),

0, '

if Zj,,,, <-- Zj <-- Zj . . . .

(15)

otherwise

Table 5. Representation of a typical LLH issue data"

1.01

Level no.

1

ZII

2

z21

3

z31 z4~

4

Parameter Zj,min

Zj,m

Zj,max

Fig. 5. Fuzzy input constructed from LLH issue data.

Branch point . . . . . . . . bl b2 b3 ZI2 Z22 z32 Z42

Z13 Z23 z33 Z43

Weighting factors

WI

lAY2 w~ 14)4

"Constraints: ~=L wl = ~)~=zzo = 1.0 where bj is the branch point for a given issue (top event), zq is the probability for branch point j and for level i, and wi is the weighting factor for level i.

Applicability of fuzzy set theory to accident progression event trees

(1) Fuzzy probabilisic quantification of fuzzy input variables by means of membership functions or possibility distributions. (2) Calculation of quantitative fuzzy probabilities for each pathway using the extension principle. Then the extension principle can be implemented by the computational algorithm of FWA operation. (3) Sufficient information can be deduced from two representative quantities, i.e. the representative and modal values. In addition, two quantities such as 'the upper and lower mean' and 'the interval of confidence' may be used to properly interpret and support the fuzzy outcomes.

These fuzzy inputs are obtained based on the assumption that the mode (Zi,,n) is the most possible value (possibility approaches one) in a given range and the extremes (Zj.min and zj. . . . ) are the least possible values (possibility approaches zero). Then, these fuzzy inputs cover the original probability ranges for each branch of the L L H issue data. They are not strictly based on the opinions of actual experts, but rather they are evaluated based on the subjective judgment of the problem considered. The rest of the procedure to obtain the final fuzzy probabilities for each pathway on the A P E T is essentially the same as the procedure used in the example application. The results of the Zion A P E T analysis for 'SEC' obtained by the fuzzy set theory approach are shown in Table 6. To allow for meaningful comparisons with the L L H results, the four quantities whose definitions are already given, i.e. representative values, modal values, upper and lower possibilistic mean values, and intervals of confidence of the fuzzy outcomes are presented in Table 6.

4.2 Comparison of fuzzy outcomes and LLH results In the preceding sections, an application of the fuzzy set theory has been made to APETs with phenomenological uncertainties and the two results of Zion A P E T analysis of 'SEC', one obtained by statistical methods and the other by fuzzy set approaches, are presented in Tables 4 and 6. Because of the inherent differences of the two approaches it is not proper to compare every value shown in Tables 4 and 6 directly with one another. Rather, from the standpoint of comparison one should identify particular fuzzy quantities that are meaningful to compare. To provide some guide and insight for a proper interpretation and comparison, the following additional information is given regarding the nature of the four quantities shown in Table 6.

4 RESULTS AND DISCUSSION 4.1 A brief outline of the computational procedure of the fuzzy set theory application In the example of real application, it is shown that the major computational steps of the fuzzy set theory application to the analysis of APETs with imprecise and uncertain input data are as follows:

Table 6. Four quantifies characterising fuzzy outcomes obtained by fuzzy set approach for PDS 'SEC' Bin no,

Representative valuesa

Modal values

1 2

0.0

0-0

o.o

o.o

3 4

6.413 × 10-3 2.842 × 10-4

3.862 x 10-3 0.0

5

0.0

0.0

6

0.0

0.0

7

3.233 x 10 -3

2.487 x 10 -3

8 9 10 11 12 13 14 15 16

3.304 x 10-2 5.178 x 10-2 3.597 X 10 -2 0.0 0.0 0-0 0-0 6.485 x 10-1 0.0

3-055 x 2.059 x 9.406 x 0.0 0.0 0-0 0-0 9.331 x 0.0

17

0.0

0.0

18 19

3.760 x 10-3 9.522 × 10 -3

6.000 X 10 -6 1-800 X 10-3

10-2 10-2 10 -3

10-1

247

Mean values [E,, E*] b

[0.0, 0.0] [o.o, O.Ol [1.287x 10-3, 1.146 x 10 2] [0.0, 5.498 x 10-4] [0.0, 0.0l [0.0, 0-0] [8.016 × 10 -4, 5.646 x 10 -3] [9-734X 10 -3, 5-622 x 10-2] [5.337× 10 -3, 9.722 x 10-2] [1-854 X 10 -3, 6.926 x 10-2] [0-0, 0.0] [0.0, 0.0] [0.0, 0.0] [0.0, 0-0] [3.025x 10-I, 9"989 x 10-~] [0.0, 0-01 [0.0, 0.0] [1.616x 10 -7, 7-273 x 10 -3] [1-212× 10 -6, 1.843 x 10-2]

a Values obtained by ranking function F proposed by Yager. 16 b Lower and upper possibilistic mean values, respectively. ~tr = 0.9.

Intervals of confidence ~

[0-0, 0.0] [o.o, o.o] [3.185 x 10-3, 4.959 × 10-3] [0.0, 0.0]

[0.0, 0.01 [0.0, 0.0] [1.982 x 10 -3, 3.054 x 10 -3]

[2-424 x 10-2, 3.493 x 10-2] [1.492 x 10-2, 2.765 x 10-2] [6-087 X 10 -3, 1.440 x 10-2] [0-0, 0-01 [0.0, 0-0l [0.0, 0.0] [0-0, 0-0] [7.449 x 10-1, 1-0] [0-0, 0.0]

I0.0, 0.0] [1-000 X 10 -6, 1"900 x 10-5] [2-000 x 10-6, 5.200 x 10-5]

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Moon-Hyun Chun, Kwang-ll Ahn

(1) 'Representative values' are similar to the concept of the mean value in the probability distribution. Therefore, this value can be directly compared with the 'mean values' obtained by the LLH approach. (2) Since a 'modal value' of the fuzzy quantity indicates the most possible value of parameters considered in the fuzzy quantity, 18 this value shown in the second column of Table 6 can be compared with 'median values' shown in Table 4. (3) The 'upper and lower possibilistic mean value' of a fuzzy quantity shows the mean interval in which a true value may lie. However, it is not quite proper to compare directly with any of the statistical LLH results. (4) The 'interval of confidence' shows the range for the output at o~-level of confidence. Also, this quantity can not easily be compared with any of the LLH results shown in Table 4. Thus, of the four quantities shown in Table 6, 'representative' and 'modal' values are two proper values that can be compared directly with 'mean' and 'median' values of LLH results given in Table 4. The remaining two quantities given in Table 6, on the other hand, can be used to support and properly interpret the other fuzzy quantities. From the results shown in Tables 4 and 6 the following observations can be made. (1) Representative values (in Table 6) are reasonably close to the mean values (in Table 4). (2) The order of magnitude of the modal values of fuzzy outcomes and the medium values of LLH results is generally close except for the bin 10.

4.3 Comparisons between the fuzzy set theory and the NUREG-1150 methodology for uncertainty analysis of APETs with phenomenological uncertainties The subjective probability distribution used in the uncertainty analysis of the most recent APETs (i.e. NUREG-1150) 1 has been constructed by the combination of some set of possible probability estimates (which represent the uncertainties of models or hypothesis called as a 'level') for each branch point and weighting factors (that represent experts' degree of belief for each set of the probability estimates), in conjunction with an expert opinion polling process. Therefore, a subjective probability distribution is given to the discrete variable associated with the uncertainties of various models or hypothesis. The propagation of this type of probability distribution depends on Monte Carlo sampling method which is a direct approach to the problem of uncertainty propagation. This approach is a short-cut to the

propagation of modelling and input parameter uncertainties, assessing the approximate distribution of an output directly. From the conceptual and operational viewpoint, the major drawback of this method can be summarised as follows. (1) The sufficient robustness in the final results may not be attained against the ambiguity of the information upon which the experts base their judgment, since the probability itself estimated by the existing approach does not adequately model the imprecision and vagueness that characterise the judgmental uncertainty of the experts. (2) A full elicitation of probability and probability distributions is always time-consuming and sometimes extremely difficult since probability distribution may be too strict for modelling the uncertainty under extremely complex and uncertain environments. (3) The probability distribution constructed by the above approach is a substantially discrete distribution and the number of hypothesis or models which can be treated in the APET is limited to small numbers because of the sampling mechanism for propagating the discrete distribution to obtain outputs. On the other hand, the use of fuzzy probability by means of possibility distribution gives a clear advantage over the probabilistic techniques used in the NUREG-1150. Mainly the advantage comes from the peculiar formal properties of fuzzy or possibility logic which can take account of experts' judgmental behaviour in the face of imprecision of events. In terms of operation and the basic concept, the fuzzy probabilistic modelling of accident progression presented in this paper provides several definite advantages over the probabilistic method of NUREG1150, and the outcomes obtained by the fuzzy method give more information than the values computed by the statistical LLH method of NUREG-1150 particularly in the following respects. (1) The fuzzy set theory method enhances the robustness in the final results of APET uncertainty analysis against the ambiguity involved in the expert judgment for phenomena or events that are too complex or too ill-defined to be susceptible to analysis by conventional approaches since this approach models basically the judgmental uncertainty of experts. (2) The modelling of uncertainty analysis for the APET branch points (events) by this method is far more convenient since the total quantity of uncertainty by means of possibility measures does not have to add up to one.

Applicability of fuzzy set theory to accident progression event trees

(3)

When the fuzzy logic is used, the propagation of the fuzzy probability in the APET can easily be made directly or analytically to obtain the output without going through a tedious sampling procedure. In addition, all the ranges of the potential probabilities included in the fuzzy probability can be propagated to obtain the output.

4.4 On the remaining questions regarding the real application of the fuzzy set theory to APETs with phenomenologicai uncertainties Even though the main purpose of this work is to demonstrate a potential use of the fuzzy set theory (or rather, theory of possibility) and provide a formal procedure in the quantification of uncertainties of APETs, one may still have the following questions. (1) Does the present work (or theory of possibility) give any insight that is beyond the existing knowledge on the phenomenological uncertainties of the accident progression? (2) Does the procedure provided in the present paper give any better instrument for eliciting the subjective measure (i.e. probability in PRA) or mechanism for combining expert opinions, which seems to be two of the major concerns in current PRA research? Wu et al. ,22 in particular, expressed some concern regarding the future use of fuzzy set theory in PRA based on their observation that a rational rule for information combination (e.g. the combination of expert opinions in the framework of non-probabilistic theories) similar to Bayes' theorem in the theory of probability is not well-developed for non-probabilistic theories. In addition, how are the fuzzy probability distributions going to be obtained? (3) How are the fuzzy measures going to be combined with standard probabilistic quantities, like equipment failure rates that are widely used? (4) Does the new measure (possibility) proposed in the present paper suggest an easier way for the communication of the concept of uncertainty and risk in the framework of accident progression? (5) Does the paper provide a simplified model with which a satisfactory justification is stood by its side? Although the above questions (1)-(5) go beyond the actual scope of the present work, a brief answer for each question is provided here to give further insight on the merits of the present method in real application.

249

Answer for question no. (1) The theory of possibility gives more insight than the existing method (probability) in the sense that the concept of theory of possibility simultaneously models the phenomenological uncertainties by the subjective probabilities as well as the judgmental uncertainties associated with expert's subjectivity or bias by the possibility. In the conventional PRA, subjective probabilities are the only means for expressing the phenomenological uncertainties of the accident progression. Since the subjective probability indicates a degree of belief for the occurrence of an event it is one way of representing the uncertainty of an event. When the opinion of a group of experts are sought to quantify the phenomenological uncertainties, the current PRA method for expressing their competing opinions is to employ the concept of weighting factors. In this case, the weighting factors reflect the importance of each expert's opinion. In the case of NUREG-1150, the weighting factors are assigned to each level of key uncertainty issues as already shown in Table 5 and these weighting factors may be regarded as the uncertainty for each level. The main drawback of the existing method, however, is that each expert has to express his uncertain opinion in terms of a unique probability or weighting factor even when the expert can not make a clear judgment (e.g. because the phenomena or events are too complex or too ill-defined). Compared to the existing method, each expert can express his opinion (i.e. probability) with more ease and confidence when the possibility distribution is used because it models not only the phenomenological uncertainties by the subjective probabilities but also the judgmental uncertainties associated with expert's subjectivity or bias by the possibility.

Answer for question no. (2) The judgmental uncertainty associated with expert's subjective bias makes it very difficult to correctly elicit the subjective measure (i.e. probability in PRA). As already noted in the answer for Question No. 1, the difficulty is mainly because the expert has to describe his uncertain opinion in terms of a specific value (probability) even when his judgment is uncertain and imprecise. This difficulty in eliciting the subjective measure can be greatly reduced by using the membership function or possibility distribution of probability since it can simultaneously model the probability and its degree of possibility. In this respect, the present procedure can be viewed as a better instrument for eliciting the subjective measure. While subjective probabilities have a firm axiomatic base, at present, these aspects are somewhat lacking in fuzzy set or possibility theory. Because of this reason, it is not always clear how to construct

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Moon-Hyun Chun, Kwang-II Ahn

reasonable membership functions or possibility distributions (especially fuzzy probability distributions in the context of this paper). In the real application, therefore, the elicitation of detailed shape of the possibility distributions has been largely empirical, heuristic, and context dependent. Many guidelines on elicitating the possibility distributions 2 have been proposed including the use of statistical data, 23-25 the composition of simpler functions, 23 and the psychophysical judgement and measurement. 6 When available data is limited, the concept of modal value and confidence level used in this paper can be used as an appropriate numerical representation or shape of possibility distribution. It should be noted, however, that no completely general approach for obtaining the reasonable possibility seems to be available yet. With regard to the mechanism for combining expert opinions, in the present work, the TFN is used as a key mechanism for combining expert opinions given in the form of a set of branch probabilities and weighting factors. The TFN (which is a type of possibility distribution) has been represented by three parameters (i.e. minimum, modal, and maximum values). The modal value indicates an element with the highest possibility whereas other values indicate a tolerance limit of elements considered by experts. These parameters can be estimated from available information and expert's experience along with the subjective judgement of the problem considered. Thus, the TFN gives a rational approximation with appropriate bound for an uncertain probability. In addition, the TFN satisfies several properties of rational distribution: (a) its form is simple and therefore easy to manipulate, (b) it has relatively small number of parameters for estimation, (c) it can appropriately model the given information and (d) it is easy to communicate between experts. These properties of the TFN provide a good basis for combining the information when available information (such as empirical data or expert opinions) is limited to a small number. In some respects, it is true that a rational rule for combining expert opinions in the framework of non-probabilistic theories is not well-developed as Wu et al. 22 pointed out. It should be noted, however, that the observation of Wu et al. 22 is based on their assessment of only a special case where uncertainty measures are represented by dual concepts of possibility and necessity rather than a membership function or possibility distributions which are used in the context of the present paper. When expert opinions on the occurrence of an event are represented by membership functions or possibility distributions as shown in this paper, the different opinions expressed in the form of membership function or possibility distribution can easily be manipulated by the combination rules of fuzzy sets

since the expression of a possibility distribution can be viewed as a fuzzy set or membership function. 2"~ In the fuzzy framework, several methods of combination operation have been suggested that have formal axiomatic justifications, z'4'26'27 The existing methods of combination operation include such as pessimistic combination by intersection operation, optimistic combination by union operation, and compensatory combination by generalised mean operations that cover the entire interval between min and max operations. Basically, they combine different possibility grades at a fixed value of parameter rather than the values of parameter at a fixed possibility level. When the combined possibility distribution is unnormalised, it can be normalised through the normalising process. 26 Depending on the context in a given application, of course, one can use other combination operations. 2,27 In actual applications, the choice of the method of combination operations will be dependent on the given specific situation and the model used. The NUREG-1150 approach may provide a useful insight for selecting a method of combination operation in the risk assessment. Although the NUREG-1150 does not explicitly give any theoretical basis on the combination operation, it uses the principle of simple arithmetic averaging of probabilities to combine all probability distributions obtained from multiple experts. 1,28This averaging has the same concept of the arithmetic mean operation 2"4'27 in the fuzzy framework. This operation satisfies two characteristics of rational combination: (a) a small variation in any possibility distribution does not produce a noticeable change in the combined possibility distribution; and (b) when the experts are not equally weighted, it can also include weights that contain the relative importance of one expert to another. 24 However, it should be noted that this method is not the only possible operation in the risk assessment. Answer for question no. (3) The fuzzy measures can easily be combined with standard probabilistic quantities by introducing the concept of possibilistic uncertainty on the probabilistic quantities. The conventional methods of obtaining probabilistic quantities like equipment failure rates are based on the statistical information. When the statistical information associated with a certain quantity is rare and therefore there is a large uncertainty, such probabilistic quantities can be estimated in the form of fuzzy probabilistic quantities. The fuzzy probabilistic quantities contain the uncertainty which is associated with an expert's subjectivity that may arise during the process of modelling and evaluation of the probabilistic failure rates. In the framework of fuzzy set or possibility theory, the standard probabilistic quantities can be interpreted as

Applicability of fuzzy set theory to accident progression event trees a special case of fuzzy probabilistic quantities with possibility one. Therefore, the fuzzy probabilistic quantities include the standard probabilistic failure rates. In the field of reliability analysis, Kaufmann and Gupta 17 pointed out that the standard probabilistic failure rates can be replaced by the possibilistic failure rates. For this, they showed that the reliability function based on the concept of possibilistic failure rates can be well manipulated by the concept of intervals of confidence or lower and upper bounds of the failure rates corresponding to any degree of possibility. One can see that the above approaches are applicable to the other standard probabilistic quantities, e.g. frequencies or probabilities of component failures in the fault tree analysis. 5'29 Answer for question no. (4) The present measure (i.e. possibility) proposed here can be judged to be an easier way for the communication of the concept of uncertainty and risk in the framework of accident progression in the following respects: When the phenomenological uncertainties of accident progression are given in terms of probabilities and weighting factors, the only way of handling the uncertainties and risk is to use some statistical sampling such as a stratified Monte Carlo. In this case, the number of uncertainty issues that can be treated in the APET to examine their impact on the risk is limited to small numbers as pointed out already. In addition, the magnitude of risk obtained by this method may vary depending on the sampling size and its mechanism. Therefore, with this method it is not always clear how one can correctly determine the impact of uncertainties on the risk. The extension principle of Zadeh used in the present work, on the other hand, has several advantages in the sense that (a) its operation is relatively easier than the statistical sampling procedure, (b) it gives always invariant results, and (c) the easy operation and the robustness in the final outcome make it simpler to determine the impact of the uncertainties on the risk. Answer for question no. (5) A satisfactory justification for the present model can be found from the following features. (a) The fuzzy phenomena or judgmental uncertainty that arises in the probabilistic quantification of the APET top events can be more clearly expressed in terms of the possibility distribution which carries more information than a unqiue probability. Therefore, better approximations of real phenomena are feasible. (b) The fuzzy APET model quantified by possibility distributions can be easily operated by means of a well-defined formal logic, i.e. the extension principle. The fuzzy APET model operated by the extension

251

principle gives always exact fuzzy outcomes. Also, the fuzzy outcomes or the four quantities characterising the fuzzy outcomes used in the present procedure provide at least as much information as the existing methods based on the probabilistic approach.

5 CONCLUDING REMARKS In conclusion, an assessment of the potential applicability of the fuzzy set theory to the APET with imprecise and uncertain branch probabilities has been made as a complement or an alternative to the methods currently used in the risk assessment of nuclear power plants where experts opinion is a major means for quantifying some event probabilities and uncertainties. This paper provides a formal procedure for applying the fuzzy set theory to APETs with imprecise and uncertain branch probabilities and/or with a number of physical uncertainty issues. In addition, the computational algorithms considered to be acceptable for application of the fuzzy set theory to the APET analysis for each step are identified and used for real application. It is clear, however, that a lot more work must be done to improve the present method before it can easily be used for PRA purposes. That is, although the current fuzzy set theory has not reached fully satisfactory level for easy application, it has certainly a great potential for application in the PRA. In the future use of fuzzy set theory in PRAs, however, considerable care should be exercised particularly in the elicitation of fuzzy probability distributions and the combination of expert opinions. In addition, more comparative studies of fuzzy and probabilistic approaches to the APET analysis are needed. The results presented in this paper should be viewed as a first attempt to construct a formal procedure and to assess the potential for practical applications of the fuzzy set theory to the analysis of APETs with imprecise and phenomenological uncertainties.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of the Korea Science and Engineering Foundation. Also, the authors are deeply grateful to the anonymous reviewer whose thoughtful comments contributed to make a better presentation of the paper.

REFERENCES 1. Anon., Severe Accident Risks: An Assessment for Five U.S. Nuclear Power Plants, NUREG-1150, Vols 1 and

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2. 3.

4. 5. 6. 7.

8. 9. 10. 11. 12.

13. 14.

15.

Moon-Hyun Chun, Kwang-ll Ahn

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