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Fuzzy number theory to obtain conservative results with respect to probability
Computer methods in applied mechanics and engineerlng ELSWIER
Comput.
Methods
Appl. Mech. Engrg.
160 (1998) 205-222
Fuzzy number theory to obtain conservative results with respect to probability Paola Fermi, DISTART-Structural
Engineering,
Faculty
Received
of Engineering, 17 March
Marco Savoia* University
1997; accepted
of Bologna,
Viale Risorgimento
18 September
2, 40136, Bologna,
Italy
1997
Abstract Fuzzy number and possibility theories are used for problems where uncertainties in the definition of input data do not allow for a treatment by means of probabilistic methods. Starting from a scarce/uncertain body of information, fuzzy numbers are used to define possibility distributions as well as upper and lower bounds for a wide class of probability distributions compatible with available data. It is investigated if relations between possibility distributions and probability measures are preserved also when the fuzzy number represents an output variable computed making use of extended fuzzy operations. General real one-to-one and binary operations are considered. Asymptotic expressions (for small/large fractiles) for the membership function of the fuzzy number and for CDFs given by probability theory are obtained. It is shown that fuzzy number theory gives conservative bounds (with respect to probability) for characteristic values corresponding to prescribed occurrence expectations. These results are of special interest for computational applications. In fact, it is easier to define fuzzy variables than random variables when no or few statistical data are available (as in the case of structural design stages). Moreover, extended fuzzy operations are much simpler than analogous operations required in the framework of probability, especially when several variables are involved. 0 1998 Elsevier Science S.A. All rights reserved.
1. Introduction
The advent of more and more powerful computers in the last two decades allowed the engineers for the numerical solution of very complicated problems based on advanced and sophisticated theories. Nevertheless, the intrinsic uncertainties related to the definition of input data, such as loads and resistances in structural engineering, usually set a limit to the validity of the final results of the analysis. For these reasons, evaluation of structural reliability and analysis of sensitivity to variations of input data are typical questions scientists and engineers must answer more and more frequently. Probability theory has been considered for many years as the only way to take into account the uncertainties in the definition of input variables. Nevertheless, it has been recently underlined that probability theory can be used only when variables are random in nature, i.e. when they are disperse but precise information on their fluctuation are available. This is required to define their probability density functions. There are two cases at least where variables cannot be considered as random variables: when the available body of information is small and when their definitions come from subjective judgement. The first case typically arises in structural design, where statistical information on loads and resistances may be scarce or even absent and the definition of probabilistic quantities can be made only on the basis of normative requirements. The second case arises when some variables, described with qualitative words or phrases, must be translated into linguistic variables and, finally, expressed in a numerical way.
* Corresponding
author
00457825/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved. PII: SOO45-7825(97)00301-O
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
206
It has been recently advocated that problems involving uncertain or subjective non-random variables can be treated in a consistent way in the context of theories of possibility and fuzzy sets [l-4]. Possibility of an event is a weaker information than probability, but can be evaluated on the basis of available information only. Then, the confidence on the occurrence of an event can be estimated when its possibility and the possibility of the contrary event (coinciding with the necessity of the event itself) are known. Different from probability, possibility theory is also able to point out those situations where the body of information is so scarce that no useful estimates of the occurrence of the event can be obtained. In reliability estimates or design problems, uncertain or subjective quantities have often been considered as fuzzy parameters. For instance, damage assessment of existing structures requires a large amount of data from experimental testing, inspections and observations. Fuzzy set theory allows for the development of a decisionmaking process merging objective data with subjective information and resulting in simple concluding statements [5-81. In the aseismic design, fuzzy representations of earthquake intensity have been proposed [9,10], starting from the evaluation of basic (zone) intensity, site (local) subsoil conditions and the assignment of structure importance factor and behaviour factor (see for instance [ll]). In the area of geomechanics, fuzzy values have been used to model soil parameters, for which available data are usually limited and imprecise [ 121. Recently, the potentialities of fuzzy logic in the field of numerical analysis have been extensively investigated. In fact, incomplete statistical information or expert knowledge can be easily formalized into fuzzy input data, and the implementation of extended fuzzy operations into finite element codes for static or dynamic analyses appears to be simple [3,13]. Fuzzy logic has been also used for automatic generation of finite element meshes [ 141. Finally, multi-objective structural optimization problems with imprecise data have been formulated with fuzzy variables, and fuzzy goal programming problems have been proposed (see for instance [ 15-191). In all these formulations, a transition stage from absolute acceptance to absolute unacceptance with respect to a constraint condition is considered. In the first part of the present paper, the guidelines of possibility theory and its relations with probability theory are briefly summarized according to Dubois and Prade [20,21]. It is shown that a possibility distribution constructed starting from few statistical data may be used to represent a wide class of probability distributions (compatible with the available information) and to consistently define upper and lower probability distributions. Possibility distribution is represented by means of a normalized fuzzy number. Then, possibility theory is used to obtain conservative estimates of probability characteristic values of uncertain variables (such as loads, resistances, etc.). To the authors’ knowledge, it has never been investigated whether the relations between possibility distributions and probability measures are also preserved when the fuzzy number represents an output variable, computed making use of extended fuzzy operations. This problem has been analyzed in the second part of the paper. For a one-to-one operation, it is shown that probability and fuzzy number theories predict the same characteristic values. The case of a general monotonic (increasing, decreasing or hybrid) real binary operation is then examined. Asymptotic expressions for the membership function of the output variable for small or large possibility fractiles (close to 0 and 1) are derived. Then, analogous expressions are obtained for the probability cumulative distribution function. In both cases, different asymptotic expressions are obtained if input fuzzy numbers present, or not, thin tails at the extremities of their supports. It is shown that fuzzy number theory always gives conservative (i.e. safe) bounds for characteristic values corresponding to prescribed occurrence expectations. These predictions have also been confirmed in two numerical examples reported at the end of the paper.
2. Probability,
possibility,
necessity measures
A confidence measure is a function associating with an event A c f2 (the reference set 0 is the sure event) a number g(A) between 0 and 1, which represents the confidence one has on the occurrence of A. By convention, the extreme values of the confidence are attained by the impossible event 0 and the sure event 0: A =0
=
g(A) = 0,
whereas the converse implications the event is impossible or sure.
A = f2
d
g(A) = 1
are not necessarily
true, i.e. a confidence
(1) equal to 0 or 1 does not mean that
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
201
Several kinds of confidence measures can be defined, depending on the amount and the type of information available for estimating the occurrence of the events. Of curse, estimates based on abundant or scarce data, precise or subjective information should be carried out in different ways. Anyway, any confidence measure must satisfy at least the axiom of monotonicity with respect of the set inclusion: ACB
*
The following
g(A)<@) inequalities
(2) are direct consequences
g(A U B) 3 max]g(A),
g(B)1
g(A fl B) d min[g(A),
g(B)]
of the monotonicity
axiom: (3a)
VA, B C 0
(3b)
Eqs. (l), (3) allow for the definition of a wide class of confidence measures. prescribed assigning, given the confidence on the occurrence of two events and/or A fl B. Probability, possibility and necessity are confidence measures The axiomatic theory of probability is based on the definition of a confidence function) on the elementary events constituting 0. The confidence on the evaluated by introducing the additivity axiom:
For each of them, rules must be A e B, the confidence on A U B belonging to this class. measure (the probability density occurrence of an event is then
P(A U B) = P(A) + P(B) - P(A f7 B)
(4)
It is easy to verify that Eq. (4) satisfies inequalities (3a,b) and, consequently, probability is a confidence measure. Moreover, the probability of A, the complementary event of A in 0, is automatically determined once the probability of A is known, since P(A) + P(i) = 1
(5)
Probability theory requires a full body of precise (even though possibly dispersed) data on the occurrence frequency of elementary events. If this is not the case, i.e. if the pdf is defined through few or imprecise or subjective data, the result of the probabilistic analysis will depend on a set of assumptions. This situation typically occurs during the structural design stages where, in order to use probability approaches, some design variables should be quantitatively defined as random variables before the realization of the structure and, consequently, with no opportunity of directly measuring them. When the data are scarce or imprecise, confidence cannot be assigned with elementary events but with subsets of the sure event at the most. In the case, two important confidence measures, the possibility and necessity measures, can be defined as limiting cases of inequalities (3a,b): I7(A U B) = max[lT(A), II(B)]
(64
N(A fl B) = min[N(A), N(B)]
(6b)
Setting B =A in Eqs. (6a,b) and making max[fl(A),
use of (1) gives
Z7(x)] = 1
(7a)
min[N(A), N(z)] = 0
(7b)
and consequently, n(A) + 17(A) a 1
(84
N(A) + N(x) c 1
(Sb)
Eqs. (8) show that the additivity axiom is not valid for both possibility and necessity measures; hence, the knowledge of the only possibility (or necessity) of an event is not sufficient to estimate the confidence on its occurrence, since the possibility (or the necessity) of the contrary event is unknown. Nevertheless, the following relation can be obtained [21]: U(A) + N(x) = 1 representing
the counterpart
(9) of Eq. (5). It states that, once the possibility
of an event is known, the necessity
of
P. Ferrari, M. Savoia 1 Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
208
the complementary event is determined. Making use of Eqs. (7), (9), the following and necessity of the events A and A can be derived: n(A)<1
ti
N(A)>0
j
N(A)=0
showing that, as is to be expected, possibility Eqs. (6) also suggest that a function r(m), ?T(w) = 17((w))
relations between possibility
=ZJ n(A)=1
(10)
of an event is always greater than or equal to its necessity. called the possibility distribution, can be defined:
w En
(11)
where w’s are the singletons of L? and 0 G r(w) d 1, 3 w(r(w) and necessity of the event A are given by
= 1, so satisfying
Eqs. (1). Hence, possibility
17(A) = sup r(o)
(12a)
OJEA
N(A) = 1 - sup_ r(w) = inf ( 1 - r(w)) OJEA
(12b)
UIEA
Several techniques have been proposed to obtain a possibility distribution starting from an incomplete set of data (see for instance [21,22] for details). It is worth noting that the possibility and the necessity of an event can be evaluated through Eqs. (12a,b) much easier than its probability, for which the integration of pdf is required. As will be shown also in the numerical examples, this circumstance is particularly significant for problems involving several variables. It was previously stated that thesxpectation on the occurrence of the event A is known if its possibility If(A) and that of the contrary event 17(A) are both known. Eq. (9) shows that the same level of information can be obtained from possibility and necessity of the same event A. Moreover, the greater the necessity N(A), the smaller the possibility of the contrary event L!(x) = 1 - N(A) and, consequently, the greater the expectation on the occurrence of A. For instance, if L? denotes the sure event, U(0) = N(R) = 1.
3. Fuzzy numbers to represent possibility measures Dubois and Prade [20] showed that a normalized possibility distribution by means of a fuzzy number Q whose membership function is (Fig. 1):
T(W) can be effectively
&J(w) = r(w)
represented
(13)
The a-cut representation of fuzzy numbers function Q,: [0, l] + R! is defined, such that
is particularly
useful in the framework
Q,(a) = {w E ~/P~+J> s a)
of possibility
theory. A
(14)
NN 1 a
a’
4
Fig. 1. Fuzzy number to represent
a possibility
distribution.
P. Ferrari. M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
4-Y
209
q+s
4
Fig. 2. Membership function of a fuzzy number (- - -); upper F*(x) and lower F,(x) bounds for probability; compatible with the body of information represented through the fuzzy number.
The a-level The following (A)
cut Q,(a) properties
V a E (0, l] ,
(B) Q,(l,=Q, (C)
is an interval collecting hold:
Va,
dE(0,
Q,(a)
is a bounded
points with membership
and closed interval
function
a general
greater than or equal to a.
;
(154
={d; 11,
CDF F(X)
(15b) cy?a’
3
Q,
(15c)
Eq. (15a) states that Q is a convex fuzzy number. Moreover, the peak Q, is a single-value the modal value (see Eq. (15b)). The subset Q, is obtained by considering the strong inequality
interval, and q is (>O) in Eq. ( 14)
(strong a-cut) and represents the support of Q. Finally, property (15~) states that Q, is monotonic with respect to the set-inclusion. This condition is necessary in order to satisfy the axiom of monotonicity (2). It has been shown [21] that the fuzzy number Q can be used to select a class B of probability measures P, such that: 9 = {P/V A C 0, N(A) d P(A) G n(A)}
(16)
i.e. for which possibility and necessity represent upper and lower bounds for probability of each event A. Since the estimate of characteristic values is the main subject of the paper, only events such as (-co, X] or [x, +“) will be considered in the following. If A = (-co, x], Eq. (16) provides for the definition of two upper and lower distribution functions (see Fig. 2): F*(x) = n((-?
xl),
F*(x) = NC--~, xl)
such that the general CDF F(x) corresponding relation:
to a probability
(17) measure P belonging
to the class .!??satisfies the
F*(x) G F(x) s F*(X)
(18)
Making use of Eqs. (12), (13), upper and lower bounds for probability terms of the membership function of the fuzzy number Q as F*(X) = supk&WJ
G4 ,
F*(x) = inf{ 1 - ~~(w)/w
F*(x) and F,(x)
may be rewritten
> x}
In the following, probability measures whose CDFs satisfy Eq. (18) will be called ‘compatible’ number b and consequently, with the body of information represented through it.
in
(19) with the fuzzy
4. Bounds for characteristic values of uncertain variables through fuzzy number theory Possibility theory is very useful to obtain conservative estimates of probability fractiles of uncertain variables. Consider, for instance, the problem of defining a design variable representing an external action without
P. Ferrari, M. Savoiu I Comput. Methods Appl. Mech. Engrg. 160 (1998) 20.5222
210
having enough information to model it in terms of probability distribution. The available body of information is interpreted in the context of possibility and fuzzy number theories and represented by means of the LR fuzzy considered number 0 = (9, y, a),,. In this way, a full class of probability distributions can be simultaneously (see Eq. (16)) compatible with the available data. Moreover, upper and lower bounds for probability can be derived from Eqs. (19). These bounds are useful to obtain estimates of probabilistic parameters, such as the characteristic value qk corresponding to a given k (high) fractile (i.e. 0.95). Using possibility theory, a lower bound for the characteristic value qk is given by the value ik such that the possibility of the event A = (Gk, +m) is (see Fig. 3(a)): II((@,, +“))
= 1 -k
(20)
In fact, from Eqs. (lo), (16), the probability of the event measured through every probability measure compatible with the fuzzy number is 0 S P(A) S 1 - k. Then, & is a conservative estimate for qk, i.e. & 3 qk. The same result can be obtained by selecting the value 4, such that the necessity of the event A = (--cc, @,I is N(-m,
4J) = F*(&J = k
(21)
i.e. such that its probability is k S P(x) G 1. Both conditions membership function of the fuzzy number 0 as r((qk) = ~~(4~) = 1 - k
with
(20)
and (21) can be written
qk E [q, q + 61
in terms
of
(22)
An analogous problem is to determine the characteristic value mk of a design variable, such as a resistance, corresponding to a k (low) fractile (i.e. 0.05), starting from the fuzzy number k = (m,7,S),,.A (conservative) lower bound for mk is given by the value S, such as (see Fig. 3(b)): ZI(A)=F*(&,)=k
or
N(x)=
1 -k
(23)
m,
4
m ,4=(lii,, +=-)
Fig. 3. (a) Upper and (b) lower bounds for characteristic
____.. values obtained
through
fuzzy number theory.
P. Ferrari,
M. Savoia
/ Comput. Methods Appl. Mech. Engrg.
160 (1998)
205-222
111
where A = (-cc), fik]. In the two cases of Eq. (23) the probability of the events A and 2 are set 0 d P(A) G k and 1 - k s P(x) s 1, respectively. Then, Kz, is a conservative estimate of mk, i.e. Kz, d mk. In terms of the fuzzy number membership, both Eqs. (23) yield r(fik)
= ,uQ(&) = k
with
rii, E
[q -
(24)
y, q]
5. Fractiles of variables obtained from probability and fuzzy number theories In the previous sections it has been shown that the membership function of a fuzzy number representing a possibility distribution can be used to obtain upper and lower bounds for probability distributions compatible with the available body of information. An important question is: are these properties preserved when the fuzzy number represents an output variable computed through extended fuzzy operations? Consider a set of uncorrelated variables X, , . . , X,, whose nature is essentially random even though the body of information is so scarce that their distribution functions cannot be safely defined. These variables are represented through the fuzzy numbers X, and membership functions pL(xj) represent their possibility distributions and, correspondingly, select the classes 9(X,) of probability measures P, compatible with the available data. According to Eq. (18), F,,(x,) and FT(x,) give lower and upper bounds for all these probability measures. Moreover, consider the variable Y, which is obtained as an explicit function of the input variables, i.e. X,). Of course, not a single probability measure, but a whole class !?(Y) should be obtained, Y=Y(X,,..., collecting all the probability measures obtained starting from the classes 9(X,). It will be shown in the following that fuzzy number theory gives conservative (i.e. safe) estimates of characteristic values corresponding to small and large (i.e. close to 0 and 1) fractiles of Y. Hence, properties described in Section 4 hold also when the fuzzy number is an output variable obtained from extended fuzzy operations. The cases of monotonic real one-to-one and binary operations will be analyzed separately in the following. The reader is referred to textbooks on fuzzy algebra for details on the extension of ordinary operations to operations between fuzzy numbers [23,24]. 5.1. One-to-one
operations
The monotonic real one-to-one operation y = u(x) is considered here. The input variable is the fuzzy number r? = (xc, xc. - XL,XR - ~c)~~. Making use of Eq. (19) upper and lower bounds for probability are given by
F;(x) =
0
x
0 ,+(x) 1 1
xL 6.x 4.~ x,
F,,(x)
=
x
1 - /J&) 1 1
xc s x s xR XR
By virtue of extension principle [23], the extended operation output fuzzy variable Y, whose membership is
(25)
Y = u(X) associates with the fuzzy number X the
&J(Y) = sup k(X) x y=U(li) Since u(x) is a monotonic
(26)
function,
Eq. (26) reduces to
&(Y) = &0-‘(Y)) Correspondingly,
upper and lower bounds 0
F;(Y) =
(27)
,+K’(Y)) { 1
for probability 0
Y
are obtained
F,,(Y) =
1 -,+&-I(Y)) 1 1
where Y, = U(Q), Y, = u(xc), y, = u(xR) if y = u(x) is an increasing
from Eqs. (19) in the form: Y
(28)
Y,
whereas y, = u(x,), y, = u(x,),
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
212
yR = u(x,) if it is a decreasing operation. Comparison for probability of X and Y are related as Incr.) Deer.)
‘( y))
F*,(y) = F;(u F;(y)
of Eqs. (25) and (28) shows that upper and lower bounds
(29a)
F,*(Y) = Fx*(u_l(Y))
= 1 -F,.&-‘(y))
(29b)
F,*(Y) = 1 - W-‘(y))
if y = u(x) is a monotonic increasing or decreasing operation, respectively. Probabilistic analysis is an alternative approach to obtain the class L?‘(Y) of probability measures for the output variable. In this case, two CDFs F,,(x) and F,,(x) are considered, coinciding with upper and lower bounds for probability of the input variable X, i.e. F,,(x)
= G(x)
Since, making Incr.) Deer.)
?
FXR(4 = F,*(x)
use of probability
theory,
F,(Y) = F&‘(Y))
(31a)
F,(y) = 1 - F,(L’(y))
is the CDF of the output variable prove that
(3 lb) Y obtained
Qx, FmW~FxW~FxL(4 where F,(y), Incr.) Deer.)
(30)
F,,(y)
F,(Y)
are obtained
*
from a general F,(x) through the operation
Q Y 3 FY,AY)~FY(Y)~FYL(Y)
from FxL(x), F,,(x)
= F,,(u-‘(Y))
(32)
as
F,,(Y) = F,&-‘(y))
F,,(Y) = 1 - F,,&-‘(Y))
y = U(X), it is easy to
(33a)
F,,(Y) = 1 - Fx,(u-l(Y))
(33b)
F,, and F, are then the two upper (left) and lower (right) CDFs bounding all CDFs F,(y) obtained from F,(x) compatible with the body of information represented through the fuzzy number X. Finally, from Eqs. (29) (30), (33), it can be easily verified that, for one-to-one operations, fuzzy number algebra and probability give the same lower and upper bounds, i.e.:
F,*(Y) = F&Y)
7
(34)
F:(Y) = F,(Y)
5.2. Binary operations In this section, pl (xl) and &(x2) denote the membership functions of two fuzzy numbers X,, _f2. Moreover, X is a real binary operation associating the output variable y = y(x,, x2) = xl X xJ to the ordered pair (xl, x2). The corresponding extended operation between fuzzy numbers is denoted by Y = Xl @X2 and &uy(y) is the membership function of the output variable I? For each membership function, the left (increasing) part and the right (decreasing) part are denoted by L and R, respectively. Making use of fuzzy operations, the left and the right parts of Y are obtained from ,+(x1), &(x2) in different ways, depending on the nature of the operation X. Increasing, decreasing or hybrid operations are defined as Increasing
Decreasing
operation:
operation:
and
Q.x~,~Q.~~>x~~
*
x10
xX2,>%b
Q-cuQ~lh>~lo and
Qx~~,Qx~,,
3
Xla
xx2u
Qx’~,Qx,,,>x,~
xx2h
Hybrid operation:
Xx2,
(35)
213
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
Making use of extension principle [23], it can be shown that the part H = L, R of h(y) parts J = L, R of I, and K = L, R of I, according to the following rules: (A) Increasing
operation
(B) Decreasing
(C) Hybrid operation
H#J=K
* *
from the
H = J = K
=+
operation
is obtained
(36)
H = J # K
In the following, the notation introduced in Eq. (36) will be used to consider simultaneously both parts (L and R) of fuzzy numbers. Conversely, when the two parts are considered separately, U, V denote the two parts giving the part L of Y and -U, -V those giving R. Of course, U = V= L and -V = -V = R for increasing operations, U = V= R and -tF = -V = L for decreasing operations, U = -V = L and tF = V= R for hybrid operations. Once the membership function of Y is obtained, upper and lower bounds for probability are derived from equations analogous to (25), and bounds for characteristic values corresponding to given fractiles can be found according to the procedures outlined in Section 4. Fractiles close to 0 and 1 are of special interest: they will be determined in the following through asymptotic approximations of F*y close to yL and of Fy* close to yR, where YT,= [yL, yR] is the support of Y. First of all, yL and yR are obtained from the pairs of extreme values of r?, and x,, i.e. xIL, .x,~ and xZL, xZR as (H = L, R): YH
=
xI J
’
(37)
X2K
where subscripts J and K follow the rules (36). Then, Taylor’ ,ur(y) in the neighbourhood of yH gives P?(y)=
T
:
T
2K (Y - YH) + WY
series expansion
of the membership
function
- YH>‘l
IJ
where (il = 1J, 2K): Ti, = C,,x,:, and (i=
(39) lJ,2K):
1,2;iZ=
(40) As is to be expected, Eq. (38) gives positive values of ,+(y) the signs of coefficients reported in Eq. (40) are (i = 1,2):
C,,>O I=H; x,!, 2 0
C,,
I= L ;
in the neighbourhood
of both yL and yR. In fact,
I#H (41)
I=R
Hence, from Eq. (39) (il = 1J, 2K): T,,aO
T,,
H=L;
H=R
(42)
where, of course, H, J, K follow the rules in Eq. (36). Substituting (42) in (38), the validity of the asymptotic expansion is proved. Finally, Eq. (38) gives the following upper and lower bounds for probability:
F;(Y)=
T
111
F,*(Y)
= 1 -
:T2” (Y T,,,
1
+ T2+
YL) + WY
- Yd21 (43)
(y - yp.) + o[(y
-ye>‘]
which are valid for y -+ yL from above and for y + yR from below, respectively. Eqs. (43) show that the leading terms in the expansions of bounds for probability distributions are linear functions of the distance of y from the extremities of the support y, and yR. Eqs. (43) break down in two cases: (a) xi; = 0 (i = 1,2); (b) x,!, +w (I= J, K; i = 1 and/or 2). Case (a) means
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
214
that pdf corresponding to upper and lower probability distributions associated with Xi takes an infinite value at xi,: this case is meaningless in practical applications. On the contrary, case (b) is particularly significant: at least one variable Xj has thin tails at xi,, i.e. a,ui/&x, = 0. This case will be discussed in details at the end of the section. The results from fuzzy number theory are now compared with those obtained through probability theory. In particular, upper and lower CDFs of the output variable Y are computed, starting from the limit distributions defined through fuzzy numbers X, and 2,. The membership F&-Q
functions
of Xi, r?, are used to define probability
distributions
F,,(x,),
F,,+(x,) and FZL(x2),
as
‘jL(‘i) = FT(xj)
FiR(XI) = Fi*(xi)
9
(i = 1,2)
(44)
where FT(xi), F,,(xi) are upper and lower bounds defined from Xi through equations functions are used to define upper and lower pdf s as (i = 1,2) (see Fig. 4): 0 PiL(‘i)
_I
=
Xl <
d,u d-T
x rC
0 If X,, X2 are uncorrelated
PYL(Y) =
random
variables,
.O
Y
P,&I)P2”(~*)
YL GY CY,
-0
Yc
with (25). These
xx
cxj sxi,
(45)
XiR< x;
two important
PyR(Y) =
analogous
pdf’s for the output variable
0
Y
PI&,)P2&)
Yc d Y s YR
0
YR
1
Y are given by
(46)
where yL and yR are defined in Eq. (37) and yc = xIc X xZc is the modal value of I? For the sake of brevity, pdf’s in Eq. (46) will be rewritten in the following as P,(Y)
= P&I
with H, J, K following
)P&Z)
the rules reported in Eq. (36). Integration
(47) of Eq. (47) gives upper and lower CDFs for Y
(H = L, R):
Fig. 4. Increasing operation: upper and lower CDFs F,,, F,R (i = 1.2) and corresponding to obtain upper (FYL) and lower CDFs of the output variable I’.
pdf’s p,,., P,~. Domains of integration
for Eq. (48)
215
P. Ferrari, M. Savoia / Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
FY"(Y)
=Iy?
P,(Y)
(48)
dY
I
It can be proved that the following VXl > F;,(x,l
G FLX,)
property *
=s (Lb,)
holds (see Appendix
v y >
Fyx(Y)
SFY(Y)
A): (49)
sG F,,(Y)
representing the counterpart of Eq. (32) for binary operations. Eq. (49) states that F,,,(y) and F,,(y) in Eq. (48) are lower and upper bounds for all CDFs of the output variable Y obtained starting from CDFs F,(x,) of input variables X,. Making use of (47), Eq. (48) is rewritten as (50) where 8(J) = + 1 if J = L or 6(J) = - 1 if J = R (J = U, V). Moreover, the argument
of the integral can be given
the form:
4% 4%
P&I khK(X2)= S(J)S(K)dw, z and, correspondingly,
Fu,(y)
(51)
upper (H = L) and lower (H = R) CDFs in Eqs. (48) can be computed from the equations:
=
(52)
for all the cases, i.e. for increasing, decreasing or hybrid operations according to relations (36). Asymptotic expressions for CDFs in Eq. (52) will be computed, for y close to yL and yR. For this purpose, Eq. (52) is rewritten as FyL(y)
(53)
=
By expanding (JK = UV, WI+):
the membership
function
A::,k -4’
+ A&
+
CL, in the neighbourhood
(x1 - x,~)(x~ - xZK) +
of x,,, straightforward
.4::2(x, - 4’
+ *. .
algebra
yields
(54)
where
(55)
The leading term Ai in the expansion (54) vanishes if one or both derivatives of the membership functions a,ui/ax, are zero. This singular case, analogous with that already singled out in the computation through fuzzy numbers, will be discussed later in the section. The double-integration in Eqs. (53) is performed by writing, for a given y, the right-hand points of the integration intervals in the form (Fig. 4): x2 = X2(% x1)
x1 =x,(Y,x,,)
K=V,aL
(56)
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
216
Taylor’ series expansion obtaining
of function
y(x,, x2) is performed
in the neighbourhood
of yH up to linear terms, so
(57) where yH and C,,, C,, are defined endpoints (56) are given by
x,(yJ,,)=x,,+~(Y
in Eqs. (37),
_Y”)
II
Finally, performing the integrations obtained, when V J, K, A? # 0:
x2
=
(40a). Hence,
linearized
expressions
+ (Y - YH) - CL/(x, - XiJ) C 2K
X2K
in Eqs. (53), the following
equations
for the integration
(58) for upper and lower CDFs are
CJV
AlI
FYL(Y)
=
___ (y - YLj2 + O[(Y - YL131 2c, “C,, (59)
e-v
F,(Y)
=
l-
All
(y - Y,j2 + O[(Y - Y,j31
2c,,c,,
for y + y, from above and y + yR from below, respectively. It is easy to verify that, as is to be expected, the two coefficients multiplying the quadratic terms are positive. In fact, from Eqs. (41) and (55a), C, JC2K > 0, A: 2 0 if J = K, and C, JC2K < 0, A: C 0 if J # K. Fiqs. (59) can now be compared with the corresponding upper and lower bounds for probability obtained through fuzzy numbers, see Eq. (43). Both expansions are valid when, V J, K, A: # 0, i.e. api / axi (i = 1,2) are different from zero. These equations show that, in the neighbourhood of the extremities of the support of r (y, and yR), i.e. for small or large values of fractiles of the output variable, asymptotic expressions for upper and lower bounds from fuzzy number and probability theories are FUZZY) F;(Y)
WY>
YLI
F*,(Y)
FYL(y) = 0[( y - Y,)~]
Probability) respectively.
= 01~ -
=
1 - O[Y - YKI
F&Y)
= 1 - W(Y - YR)~I
(604 (60b)
Hence, for y close to y, and yR: > F,(Y)
F,*(Y)
(61)
< Fw.(y)
i.e. fuzzy number theory gives conservative estimates of characteristic values given by probability theory, for small (or large) fractiles. As has been previously announced, peculiar situations may occur if the membership function of at least one variable fi has thin tails at xi1 (il = 1U and/or 2V), i.e. ap,/axi = 0. In this case, the leading term in the expansion of upper bound for probability vanishes, and Eq. (43a) takes the form
WY)
1 d2fi = y dq’2
This situation
yI, (Y - YLj2 + WY
may occur in three different
(a)
xi, = fm
(b) (c)
xi, E (-a, xIU = km
circumstances
xiv E (-m,
+a) - (0)
(62)
- YL)31
’ = fm x2v I = km x2v
t-m) -
(see Eqs. (39), (40)):
(0) (63)
which will be analyzed separately in the following. It will be shown that, in all three cases, the upper CDF obtained through probability theory is smaller than that of Eq. (62) and, consequently, fuzzy number theory gives conservative results with respect to probability.
P. Ferrari, M. Savoia / Comput. Methods Appl. Mech. Engrg. 160 (1998)
217
205-222
Case a):
The coefficient
of the leading
term in Eq. (54) is AyxI; by performing
the integration
of Eq. (53a) yields
UV
(y - y,13+ O[(Y - Yd41 F,,(Y) = 6ctix;
(65a)
zv
IU
In passing, it can be verified that the coefficient of the leading term is always positive; in fact A yxy 2 0 and C,, > 0 for V = L, whereas A yx: s 0 and C,, < 0 for V= R. Comparison of Eqs. (62) and (65a) shows that Eq. (61a) holds for y-y,. Cuse b): 1 ,=O
‘ZO, XI I/
3
A;“=AK=O
xzv
In this case the coefficient
of the leading term in Eq. (54) is AyXI. Integration
of Eq. (53a) yields
UV
A Ix2 F,(Y)
=
6c
(Y - YJ3 + O[(Y - Yd41
c2
II/
2v
and Eq. (61a) is verified. Case c):
( 64~) Hence, integration A
F,,(Y) =
of Eq. (53a) gives L/V
24c2;‘;2 (Y - YL14 + WY - Yd51 I11
(65~)
2v
and Eq. (61a) is verified also in this case. Analogous results can be obtained if lower distributions FYI and F, are considered. Hence, Eqs. (6 1) are verified also when input variables i, have thin tails at the extremities of the fuzzy support.
6. Two numerical examples The theoretical results previously obtained concerning characteristic values obtained from probability and fuzzy numbers theories are confirmed in this section through two numerical examples. The operation y(xl, x2) = xl .x2 (multiplication) is considered, with xl, x2 ranging over real positive values. In the first case, fuzzy input variables have triangular membership functions whereas, in the second case, membership functions have thin tails at the extremities of the support. 6.1. Triangular fuzzy
numbers
The input variables are represented by means membership function is given by (i = 1,2):
of two fuzzy numbers
zi = (xc, xc - xL, xR - x~)~~, whose
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
218
xi -
XjL
xic &,(Xi)
xiR
=
‘iR
-
-
XiL
c
xi
s
xic
s
xi G
xic
X,L
‘i
(66)
X,R
xiC
0
otherwise
where [xiL, xiR] is the support and xic is the modal value. The membership obtained from the fuzzy extended operation r = x, Or?, is
+-
-b, -b,
of the output variable
aLcL(y))“*
(b:
YLCYGYC
aL Pruy(Y)=
function
- (bi - aRcR(y))“*
(67) YCCY GY,
aR 0
otherwise
where (H = L, R): aH =
kc
-
+fDZC
-
X2H)
9
b,
=
[@,C
-
X,H)XZH
+
62C
-
X*H)%fl/2
c&) =X,f,X2H -Y (68)
Since multiplication is an increasing operation, the left (right) parts of J?,, 2,. Probability theory requires the integration of (H = L, R) and pIH = Unif[x,,, xiB] is the uniform A = C, B = R if H = R. Lower and upper CDFs
FYH=
according
to Eq. (36) the left (right) part of p is obtained from
Eq. (48) where, according to (47) and (36), pyH = pIH . pZH pdf over the interval [xiA, xiB], with A = L, B = C if H = L, and can be given the form
sAB cx,B
-
X,A)(X2B
-
%A)
where
forY,,
s,, = y[Log(y/&) -
GY GY,
for y,GyYyy, Or
‘A,
=
and
YIJ=x,lex2J
-Y
(704
+ X,A%A ;
SAB=~[Lo~(~/x;?A)-Lo~(~~x2B)l-x,A(x2R-~A)
YILog(x,B)
for YWGY
L”g(x,,)l
-
sAB
(1,
J
=
A,
L”g(x,A)
=
B)3
if
=
YAB~YBA
YBAs YAB;
y[Log(x,,)
y,
if
-
minbAB?
L”i%bh2,)l
YBA), Y,
G’Ob)
(7Oc) -%A&R
= max(YA8,
-
‘,A)
+
X2BX,A
+
Y
(704
YBA).
The fuzzy numbers J?, = (2,0.2,0.2),, and J?, = (3,0.3,0.3),, have been considered in the numerical example. Fig. 5 shows the comparison between the upper and lower bounds for probability F*y, Fy* obtained from fuzzy extended operation (dashed line) and limit CDFs F,, F, from probability theory (solid lines). In the
Fig. 5. Triangular input fuzzy numbers: Upper (F:) (F,,~, F,,), a general CDF (F,) satisfying Eq. (47).
and lower (F,,) bounds from extended fuzzy operations,
limit CDFs from probability
P. Ferruri, M. Savoia Table 1 Fuzzy number with triangular
membership
219
I Compur. Methods Appl. Mech. Engrg. 160 (1998) 205-222
function: characteristic
values obtained through probability
theory and conservative
bounds given
by fuzzy number theory
Probability F!,. Fuzzy theory F: Probability F!, Fuzzy theory F,.
0.05
0.10
5.0318 4.9142
5.1035 4.9686
0.90
0.95
6.9668 7.1286
7.0523 7.1942
same figure, CDF F, obtained from two compatible pdf’s for Xj, i.e. X, = Unif[xiL, xiR] is also reported. Of course, F, and F, are upper and lower bounds for Fy, i.e. Eq. (49) is satisfied. The figure confirms the theoretical prediction of Section 5, that is fuzzy number theory gives conservative estimates of characteristic values with respect to probability. Even if the analysis in Section 5 was confined to low and high fractiles (close to 0 and l), the figure shows that fuzzy operation gives a conservative lower bound for F, - ~0.5 and an upper bound for F, - >0.5. Finally, Table 1 shows the characteristic values for various fractiles, obtained from probability and compared with lower and upper bounds given by fuzzy number theory. 6.2. FUZZY numbers
with thin tails
; 1
In this example, functions:
the input variables
x, x
1 -cos
[
/+x,(X,)
=
1 2
X
1 - cos rR
r
_X,L 7T
IC
(
are two fuzzy numbers
X 1R (
‘1. -X.
lr Xi(
0
11 XiL
>I
s
x;
s
b, = (x,, xc - xL, xR - -x,=)~~ with memberships
x,c
(71)
xjc G xi =Gx,R otherwise
Both variables have thin tails at x,~ and .K,~. By performing output variable p is found in the form: -b,
+ (bt - aLcL(Y))“*
?r
fuzzy operation,
the membership
function
of the
Y,SYYYy,
QL -b,
- (bi - aRcR( Y))“2
>I
VT aR
lo where a”, b,,
(72)
Y,GYGY,
otherwise
cH (H = L, R) are the same defined
5
in Eqs. (68).
5.5
Fig. 6. Input fuzzy numbers with member function of Eq. (69): upper (F;) and lower (F,.) bounds from fuzzy operations, probability (F,,,, F,,).
limit CDFs from
220
P. Ferrari, M. Savoia / Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
Table 2 Fuzzy number with membership given by fuzzy number theory Probability F,, Fuzzy theory FT Probability F,, Fuzzy theory F,.
0.05 5.1278 5.0163
Table 3 Fuzzy number with membership number theories
Probability FuL Fuzzy theory Ft Probability FIR Fuzzy theory F,.
function
0.05 5.1144 5.0137
of FQ. (71): characteristic
0.10 5.1855 5.0837
function
0.90
0.95
6.8718 6.9921
6.9392 7.0717
of Eq. (71): characteristic
0.10 5.1626 5.0774
values obtained
through
probability
values obtained from asymptotic
0.90
0.95
6.9574 6.9943
7.0056 7.0721
theory and conservative
expressions
for probability
bounds
and fuzzy
In this case, limit CDFs from probability theory cannot be given an analytical form as in the previous case and have been computed numerically. The fuzzy numbers 2, = (2,0.2,0.2),, and J?Z = (3,0.3,0.3),, have been considered, and results obtained from fuzzy numbers and probability theory are shown in Fig. 6. The corresponding characteristic values are reported in Table 2. It is easily verified that, in this case also, fuzzy number theory gives conservative values with respect to probability. Finally, the approximate values obtained from asymptotic expressions in Eqs. (62) and (65~) are shown in Table 3. Comparison of Tables 2 and 3 shows the effectiveness of asymptotic approximations, the maximum error with respect to exact values being 1.25 and 0.31% for probability and fuzzy number theory, respectively.
7. Conclusions The usefulness of fuzzy numbers and possibility theories for problems with uncertainties associated with input data are discussed. Following Dubois and Prade [21], fuzzy numbers are used to represent possibility distributions and upper and lower bounds for probability. It has been investigated whether relations between possibility and probability measures are preserved when fuzzy numbers are output variables computed through extended fuzzy operations. For one-to-one operations, it has been shown that fuzzy number theory gives the same characteristic values as probability. For monotonic (increasing, decreasing or hybrid) real binary operations, asymptotic expressions (for small and large fractiles) of membership function (fuzzy number theory) and CDF (probability theory) have been derived. It has been shown that fuzzy number theory gives conservative bounds, with respect to probability, for characteristic values corresponding to prescribed occurrence expectations. These results are of special interest in the framework of computational analysis. In fact, it is well known that, when incomplete statistical information is only available, it is easier to define fuzzy variables than random variables. Moreover, extended fuzzy operations are simpler than analogous operations required in the framework of probability, and the implementation into finite element codes is simple [3].
Acknowledgment The financial support of the (Italian) Ministry of University and Scientific and Technological (MURST - 40%, ‘Modelling of Solids and Structures and Experimental Investigations’) is gratefully edged.
Research acknowl-
P. Ferrari.
Appendix
M. Savoia
I Comput.
Methods
Appl.
Mech.
Engrg.
I60 (1998)
205-222
221
A
It is proved here that CDFs F,,(y), F,,(y) defined in Eq. (46) are upper and lower bounds for all CDFs of the output variable Y obtained starting from general CDFs F,(x,) satisfying F,,(x,) G F,(x,) d FtL(x,) (see Eq. (47)). Starting from two general pdf p,(x,), p2(x2) defined over the intervals [x,~,x,~], [.xZL,xZR],respectively, the CDF for the output variable Y can be obtained as (see Eq. (48)): F,(Y) = 8(U) S(V) Jr~;:“‘i” 6:fl’“’ Integration
of Eq. (A.l)
P,(X,)P,(X,) d.X, &,
(A.1)
with respect to x2 gives
‘;,(?..C~“) F,(y) = WJ 1 which, integrated
According
I *1u
by-parts
P,@,)lW,)
(A.21
- ~*(%“)I h,
with respect to xi is finally rewritten
to Eqs. (44) and (46), CDFs F,(y)
or F,,(y)
as
can be obtained from Eq. (A.3) by replacing
F,(x,),
F2(x2) with F,,,(x,), F2v(xz) or F,,(x,), F2+(Q, respectively. An additional CDF is required for the proof, i.e. F,,(y) (S = V, -V), where the only FT(x2) is replaced with F&x,) in Eqs. (A.2), (A.3). Making use of Eq. (A.2) it can be easily verified that F,,(Y)
s F,(Y)
(A.4)
%,(Y)
In fact, F&x2) G F2(x2) SF,,(x,) and, if V= L, F2&cZL) = Fz(xzL) = F,,(x,,) = 0. Then, lF2R(~2)F2,&)I s IF&,) - F&V)( d /F2L(x2) - F,,(q,)(. On the contrary, if V= R FZH(~ZR)= F2h) = F,,(.%,) = 1
and, consequently, Moreover,
In
\F,,(x,)
making
- F,,(x,,)/
G IF&d
- Fz(.+)l
use of Eq. (A.3), the following
&&9
= F,,(Y)
fact,
F,,(x,)~F,(x,)~F,,(x,)
3
F,,(Y)
c IF,,&,)
inequalities
- F2,(~2v)l.
are proved: (A.5)
+,,(Y)
and,
F,,(x,.)latF,(x,)-F,(x,.)l~lF,,(x,)-F,,(x,.)l. (A.3) is positive and, since F,(x,) c F,,(x,),
F,R(~,L)=F,(~,L)=F,L(~,L)=O. Then, IF,&,)M oreover, the second term at the right-hand side of Eq. analogous inequality holds for their integrals. On the contrary, if if lJ=L,
U = R, F,,GIR) = F,(x,~) = F,,(x,,) = 1 and IF,,@,) - F,,(xrU)I 6 (F,(x,) - F,(x,,,)\ d IF,&,) - FIR(xIU)I. Moreover, the second term at the R.H.S. of Eq. (A.3) is negative and, since F,,(x,) G F,(x,), the opposite inequality holds for their integrals. Making use of Eqs. (A.4) and (AS), Eq. (47) is immediately proved, i.e. F,,_(y) and F,,(y) of Eq. (46) are upper and lower bounds for CDFs of the output variable Y.
References [I] C.B. Brown and J.T.P. Yao, Fuzzy sets and structural engineering, J. Struct. Engrg. ASCE 109 (1983) 121 l-1225. and H. Furuta, Probabilistic treatment of fuzzy events in civil engineering, Prob. Engrg. Mech. I (1986) 58-64. [31 S. Valliappan and T.D. Pham, Fuzzy logic applied to numerical modelling of engineering problems, Comput. Mech. Adv. 2 (1995) 213-281. I41 P Ferrari and M. Savoia, Fuzzy set theory for dynamic analysis with uncertain design data, in: G.D. Manolis, D.E. Beskos and C.A. Brebbia eds.. Earthquake Resistant Engineering Structures (Comput. Mech. Publications, Southampton, 1996) 233-252. PI J.T.P. Yao, Damage assessment of existing structures, J. Engrg. Mech. ASCE 106 (1980) 785-799. [61 J.T.P. Yao, Safety and Reliability of Existing Structures (Pitman Adv. Publ. Progr., London, 1985). approach to reliability of existing structures, J. Struct. Engrg. ASCE I19 ( 1993) 3276-3290. r71 K.C. Chou and J. Yuan, Fuzzy-Bayesian fuzzy logic integrated neural network approach, VI A.S. Ramu and V.T. Johnson, Damage assessment of composite structures-A Comput. Struct. 57 (1995) 491-502.
121J.T.P. Yao
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[9] G.Y. Wang and W.Q. Wang, Fuzzy optimum design of aseismic structures, Earth. Engrg. Str. Dyn. 13 (1985) 827-837. [lo] S. Alim and D.L. Smith, Fuzzy set-theoretic models for interpretation of seismic design codes, Fuzzy Sets Syst. 29 (1989) 277-291. [ II] Eurocode 8, Structures in Seismic Regions-Design-Part I, General and Building, 1993. [12] S. Valliappan and T.D. Pham, Fuzzy finite-element analysis of a foundation on an elastic soil medium, bit. J. Numer. Anal. Methods Geomech. 17 (1993) 771-789. [ 131 S. Valliappan and T.D. Pham, Computational mechanics and fuzzy logic-Marriage a la mode, 3rd World Congress on Computational Mechanics, Chiba, Japan (1994) 1500-1505. [14] S. Valliappan, T.D. Pham, V. Murti and J. Franklin, Fuzzy logic for automatic finite element mesh generator, Microcomputers Civil Engrg. 8 (1993) l-8. [IS] J. Munro and P.H. Chuang, Optimal plastic design with imprecise data, J. Engrg. Mech. ASCE 112 (1986) 888-903. [16] S.S. Rao, Multiobjective optimization of fuzzy structural systems, bit. J. Numer. Methods Engrg. 24 (1987) 1157-l 171. [17] S.U. Mohandas, T.A. Phelps and K.M. Ragsdell, Structural optimization using a fuzzy goal programming approach, Comput. Struct. 37 (1990) 1-8. [18] Y.C. Yeh and D.S. Hsu, Structural optimization with fuzzy parameters, Comput. Struct. 37 (1990) 917-924. [19] C.J. Shih and T.K. Lai, Fuzzy weighting optimization with several objective functions in structural design, Comput. Struct. 52 (1994) 917-924. [20] D. Dubois and H. Prade, The mean value of a fuzzy number, Fuzzy Sets Syst. 24 (1987) 279-300. (211 D. Dubois and H. Prade, Possibility Theory. An Approach to Computerized Processing of Uncertainty (Plenum Press, New York, 1988). [22] S. Valliappan and T.D. Pham, Constructing the membership function of a fuzzy set with objective and subjective information, Microcomputers Civil Engrg. 8 (1993) 75-82. [23] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications. Mathematics in Science and Engineering Inc., San Diego, 1980). [24] A. Kaufmann and M.M. Gupta, Introduction to Fuzzy Arithmetic (Van Nostrand, New York, 1991).
(Academic
Press
Comput.
Methods
Appl. Mech. Engrg.
160 (1998) 205-222
Fuzzy number theory to obtain conservative results with respect to probability Paola Fermi, DISTART-Structural
Engineering,
Faculty
Received
of Engineering, 17 March
Marco Savoia* University
1997; accepted
of Bologna,
Viale Risorgimento
18 September
2, 40136, Bologna,
Italy
1997
Abstract Fuzzy number and possibility theories are used for problems where uncertainties in the definition of input data do not allow for a treatment by means of probabilistic methods. Starting from a scarce/uncertain body of information, fuzzy numbers are used to define possibility distributions as well as upper and lower bounds for a wide class of probability distributions compatible with available data. It is investigated if relations between possibility distributions and probability measures are preserved also when the fuzzy number represents an output variable computed making use of extended fuzzy operations. General real one-to-one and binary operations are considered. Asymptotic expressions (for small/large fractiles) for the membership function of the fuzzy number and for CDFs given by probability theory are obtained. It is shown that fuzzy number theory gives conservative bounds (with respect to probability) for characteristic values corresponding to prescribed occurrence expectations. These results are of special interest for computational applications. In fact, it is easier to define fuzzy variables than random variables when no or few statistical data are available (as in the case of structural design stages). Moreover, extended fuzzy operations are much simpler than analogous operations required in the framework of probability, especially when several variables are involved. 0 1998 Elsevier Science S.A. All rights reserved.
1. Introduction
The advent of more and more powerful computers in the last two decades allowed the engineers for the numerical solution of very complicated problems based on advanced and sophisticated theories. Nevertheless, the intrinsic uncertainties related to the definition of input data, such as loads and resistances in structural engineering, usually set a limit to the validity of the final results of the analysis. For these reasons, evaluation of structural reliability and analysis of sensitivity to variations of input data are typical questions scientists and engineers must answer more and more frequently. Probability theory has been considered for many years as the only way to take into account the uncertainties in the definition of input variables. Nevertheless, it has been recently underlined that probability theory can be used only when variables are random in nature, i.e. when they are disperse but precise information on their fluctuation are available. This is required to define their probability density functions. There are two cases at least where variables cannot be considered as random variables: when the available body of information is small and when their definitions come from subjective judgement. The first case typically arises in structural design, where statistical information on loads and resistances may be scarce or even absent and the definition of probabilistic quantities can be made only on the basis of normative requirements. The second case arises when some variables, described with qualitative words or phrases, must be translated into linguistic variables and, finally, expressed in a numerical way.
* Corresponding
author
00457825/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved. PII: SOO45-7825(97)00301-O
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
206
It has been recently advocated that problems involving uncertain or subjective non-random variables can be treated in a consistent way in the context of theories of possibility and fuzzy sets [l-4]. Possibility of an event is a weaker information than probability, but can be evaluated on the basis of available information only. Then, the confidence on the occurrence of an event can be estimated when its possibility and the possibility of the contrary event (coinciding with the necessity of the event itself) are known. Different from probability, possibility theory is also able to point out those situations where the body of information is so scarce that no useful estimates of the occurrence of the event can be obtained. In reliability estimates or design problems, uncertain or subjective quantities have often been considered as fuzzy parameters. For instance, damage assessment of existing structures requires a large amount of data from experimental testing, inspections and observations. Fuzzy set theory allows for the development of a decisionmaking process merging objective data with subjective information and resulting in simple concluding statements [5-81. In the aseismic design, fuzzy representations of earthquake intensity have been proposed [9,10], starting from the evaluation of basic (zone) intensity, site (local) subsoil conditions and the assignment of structure importance factor and behaviour factor (see for instance [ll]). In the area of geomechanics, fuzzy values have been used to model soil parameters, for which available data are usually limited and imprecise [ 121. Recently, the potentialities of fuzzy logic in the field of numerical analysis have been extensively investigated. In fact, incomplete statistical information or expert knowledge can be easily formalized into fuzzy input data, and the implementation of extended fuzzy operations into finite element codes for static or dynamic analyses appears to be simple [3,13]. Fuzzy logic has been also used for automatic generation of finite element meshes [ 141. Finally, multi-objective structural optimization problems with imprecise data have been formulated with fuzzy variables, and fuzzy goal programming problems have been proposed (see for instance [ 15-191). In all these formulations, a transition stage from absolute acceptance to absolute unacceptance with respect to a constraint condition is considered. In the first part of the present paper, the guidelines of possibility theory and its relations with probability theory are briefly summarized according to Dubois and Prade [20,21]. It is shown that a possibility distribution constructed starting from few statistical data may be used to represent a wide class of probability distributions (compatible with the available information) and to consistently define upper and lower probability distributions. Possibility distribution is represented by means of a normalized fuzzy number. Then, possibility theory is used to obtain conservative estimates of probability characteristic values of uncertain variables (such as loads, resistances, etc.). To the authors’ knowledge, it has never been investigated whether the relations between possibility distributions and probability measures are also preserved when the fuzzy number represents an output variable, computed making use of extended fuzzy operations. This problem has been analyzed in the second part of the paper. For a one-to-one operation, it is shown that probability and fuzzy number theories predict the same characteristic values. The case of a general monotonic (increasing, decreasing or hybrid) real binary operation is then examined. Asymptotic expressions for the membership function of the output variable for small or large possibility fractiles (close to 0 and 1) are derived. Then, analogous expressions are obtained for the probability cumulative distribution function. In both cases, different asymptotic expressions are obtained if input fuzzy numbers present, or not, thin tails at the extremities of their supports. It is shown that fuzzy number theory always gives conservative (i.e. safe) bounds for characteristic values corresponding to prescribed occurrence expectations. These predictions have also been confirmed in two numerical examples reported at the end of the paper.
2. Probability,
possibility,
necessity measures
A confidence measure is a function associating with an event A c f2 (the reference set 0 is the sure event) a number g(A) between 0 and 1, which represents the confidence one has on the occurrence of A. By convention, the extreme values of the confidence are attained by the impossible event 0 and the sure event 0: A =0
=
g(A) = 0,
whereas the converse implications the event is impossible or sure.
A = f2
d
g(A) = 1
are not necessarily
true, i.e. a confidence
(1) equal to 0 or 1 does not mean that
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
201
Several kinds of confidence measures can be defined, depending on the amount and the type of information available for estimating the occurrence of the events. Of curse, estimates based on abundant or scarce data, precise or subjective information should be carried out in different ways. Anyway, any confidence measure must satisfy at least the axiom of monotonicity with respect of the set inclusion: ACB
*
The following
g(A)<@) inequalities
(2) are direct consequences
g(A U B) 3 max]g(A),
g(B)1
g(A fl B) d min[g(A),
g(B)]
of the monotonicity
axiom: (3a)
VA, B C 0
(3b)
Eqs. (l), (3) allow for the definition of a wide class of confidence measures. prescribed assigning, given the confidence on the occurrence of two events and/or A fl B. Probability, possibility and necessity are confidence measures The axiomatic theory of probability is based on the definition of a confidence function) on the elementary events constituting 0. The confidence on the evaluated by introducing the additivity axiom:
For each of them, rules must be A e B, the confidence on A U B belonging to this class. measure (the probability density occurrence of an event is then
P(A U B) = P(A) + P(B) - P(A f7 B)
(4)
It is easy to verify that Eq. (4) satisfies inequalities (3a,b) and, consequently, probability is a confidence measure. Moreover, the probability of A, the complementary event of A in 0, is automatically determined once the probability of A is known, since P(A) + P(i) = 1
(5)
Probability theory requires a full body of precise (even though possibly dispersed) data on the occurrence frequency of elementary events. If this is not the case, i.e. if the pdf is defined through few or imprecise or subjective data, the result of the probabilistic analysis will depend on a set of assumptions. This situation typically occurs during the structural design stages where, in order to use probability approaches, some design variables should be quantitatively defined as random variables before the realization of the structure and, consequently, with no opportunity of directly measuring them. When the data are scarce or imprecise, confidence cannot be assigned with elementary events but with subsets of the sure event at the most. In the case, two important confidence measures, the possibility and necessity measures, can be defined as limiting cases of inequalities (3a,b): I7(A U B) = max[lT(A), II(B)]
(64
N(A fl B) = min[N(A), N(B)]
(6b)
Setting B =A in Eqs. (6a,b) and making max[fl(A),
use of (1) gives
Z7(x)] = 1
(7a)
min[N(A), N(z)] = 0
(7b)
and consequently, n(A) + 17(A) a 1
(84
N(A) + N(x) c 1
(Sb)
Eqs. (8) show that the additivity axiom is not valid for both possibility and necessity measures; hence, the knowledge of the only possibility (or necessity) of an event is not sufficient to estimate the confidence on its occurrence, since the possibility (or the necessity) of the contrary event is unknown. Nevertheless, the following relation can be obtained [21]: U(A) + N(x) = 1 representing
the counterpart
(9) of Eq. (5). It states that, once the possibility
of an event is known, the necessity
of
P. Ferrari, M. Savoia 1 Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
208
the complementary event is determined. Making use of Eqs. (7), (9), the following and necessity of the events A and A can be derived: n(A)<1
ti
N(A)>0
j
N(A)=0
showing that, as is to be expected, possibility Eqs. (6) also suggest that a function r(m), ?T(w) = 17((w))
relations between possibility
=ZJ n(A)=1
(10)
of an event is always greater than or equal to its necessity. called the possibility distribution, can be defined:
w En
(11)
where w’s are the singletons of L? and 0 G r(w) d 1, 3 w(r(w) and necessity of the event A are given by
= 1, so satisfying
Eqs. (1). Hence, possibility
17(A) = sup r(o)
(12a)
OJEA
N(A) = 1 - sup_ r(w) = inf ( 1 - r(w)) OJEA
(12b)
UIEA
Several techniques have been proposed to obtain a possibility distribution starting from an incomplete set of data (see for instance [21,22] for details). It is worth noting that the possibility and the necessity of an event can be evaluated through Eqs. (12a,b) much easier than its probability, for which the integration of pdf is required. As will be shown also in the numerical examples, this circumstance is particularly significant for problems involving several variables. It was previously stated that thesxpectation on the occurrence of the event A is known if its possibility If(A) and that of the contrary event 17(A) are both known. Eq. (9) shows that the same level of information can be obtained from possibility and necessity of the same event A. Moreover, the greater the necessity N(A), the smaller the possibility of the contrary event L!(x) = 1 - N(A) and, consequently, the greater the expectation on the occurrence of A. For instance, if L? denotes the sure event, U(0) = N(R) = 1.
3. Fuzzy numbers to represent possibility measures Dubois and Prade [20] showed that a normalized possibility distribution by means of a fuzzy number Q whose membership function is (Fig. 1):
T(W) can be effectively
&J(w) = r(w)
represented
(13)
The a-cut representation of fuzzy numbers function Q,: [0, l] + R! is defined, such that
is particularly
useful in the framework
Q,(a) = {w E ~/P~+J> s a)
of possibility
theory. A
(14)
NN 1 a
a’
4
Fig. 1. Fuzzy number to represent
a possibility
distribution.
P. Ferrari. M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
4-Y
209
q+s
4
Fig. 2. Membership function of a fuzzy number (- - -); upper F*(x) and lower F,(x) bounds for probability; compatible with the body of information represented through the fuzzy number.
The a-level The following (A)
cut Q,(a) properties
V a E (0, l] ,
(B) Q,(l,=Q, (C)
is an interval collecting hold:
Va,
dE(0,
Q,(a)
is a bounded
points with membership
and closed interval
function
a general
greater than or equal to a.
;
(154
={d; 11,
CDF F(X)
(15b) cy?a’
3
Q,
(15c)
Eq. (15a) states that Q is a convex fuzzy number. Moreover, the peak Q, is a single-value the modal value (see Eq. (15b)). The subset Q, is obtained by considering the strong inequality
interval, and q is (>O) in Eq. ( 14)
(strong a-cut) and represents the support of Q. Finally, property (15~) states that Q, is monotonic with respect to the set-inclusion. This condition is necessary in order to satisfy the axiom of monotonicity (2). It has been shown [21] that the fuzzy number Q can be used to select a class B of probability measures P, such that: 9 = {P/V A C 0, N(A) d P(A) G n(A)}
(16)
i.e. for which possibility and necessity represent upper and lower bounds for probability of each event A. Since the estimate of characteristic values is the main subject of the paper, only events such as (-co, X] or [x, +“) will be considered in the following. If A = (-co, x], Eq. (16) provides for the definition of two upper and lower distribution functions (see Fig. 2): F*(x) = n((-?
xl),
F*(x) = NC--~, xl)
such that the general CDF F(x) corresponding relation:
to a probability
(17) measure P belonging
to the class .!??satisfies the
F*(x) G F(x) s F*(X)
(18)
Making use of Eqs. (12), (13), upper and lower bounds for probability terms of the membership function of the fuzzy number Q as F*(X) = supk&WJ
G4 ,
F*(x) = inf{ 1 - ~~(w)/w
F*(x) and F,(x)
may be rewritten
> x}
In the following, probability measures whose CDFs satisfy Eq. (18) will be called ‘compatible’ number b and consequently, with the body of information represented through it.
in
(19) with the fuzzy
4. Bounds for characteristic values of uncertain variables through fuzzy number theory Possibility theory is very useful to obtain conservative estimates of probability fractiles of uncertain variables. Consider, for instance, the problem of defining a design variable representing an external action without
P. Ferrari, M. Savoiu I Comput. Methods Appl. Mech. Engrg. 160 (1998) 20.5222
210
having enough information to model it in terms of probability distribution. The available body of information is interpreted in the context of possibility and fuzzy number theories and represented by means of the LR fuzzy considered number 0 = (9, y, a),,. In this way, a full class of probability distributions can be simultaneously (see Eq. (16)) compatible with the available data. Moreover, upper and lower bounds for probability can be derived from Eqs. (19). These bounds are useful to obtain estimates of probabilistic parameters, such as the characteristic value qk corresponding to a given k (high) fractile (i.e. 0.95). Using possibility theory, a lower bound for the characteristic value qk is given by the value ik such that the possibility of the event A = (Gk, +m) is (see Fig. 3(a)): II((@,, +“))
= 1 -k
(20)
In fact, from Eqs. (lo), (16), the probability of the event measured through every probability measure compatible with the fuzzy number is 0 S P(A) S 1 - k. Then, & is a conservative estimate for qk, i.e. & 3 qk. The same result can be obtained by selecting the value 4, such that the necessity of the event A = (--cc, @,I is N(-m,
4J) = F*(&J = k
(21)
i.e. such that its probability is k S P(x) G 1. Both conditions membership function of the fuzzy number 0 as r((qk) = ~~(4~) = 1 - k
with
(20)
and (21) can be written
qk E [q, q + 61
in terms
of
(22)
An analogous problem is to determine the characteristic value mk of a design variable, such as a resistance, corresponding to a k (low) fractile (i.e. 0.05), starting from the fuzzy number k = (m,7,S),,.A (conservative) lower bound for mk is given by the value S, such as (see Fig. 3(b)): ZI(A)=F*(&,)=k
or
N(x)=
1 -k
(23)
m,
4
m ,4=(lii,, +=-)
Fig. 3. (a) Upper and (b) lower bounds for characteristic
____.. values obtained
through
fuzzy number theory.
P. Ferrari,
M. Savoia
/ Comput. Methods Appl. Mech. Engrg.
160 (1998)
205-222
111
where A = (-cc), fik]. In the two cases of Eq. (23) the probability of the events A and 2 are set 0 d P(A) G k and 1 - k s P(x) s 1, respectively. Then, Kz, is a conservative estimate of mk, i.e. Kz, d mk. In terms of the fuzzy number membership, both Eqs. (23) yield r(fik)
= ,uQ(&) = k
with
rii, E
[q -
(24)
y, q]
5. Fractiles of variables obtained from probability and fuzzy number theories In the previous sections it has been shown that the membership function of a fuzzy number representing a possibility distribution can be used to obtain upper and lower bounds for probability distributions compatible with the available body of information. An important question is: are these properties preserved when the fuzzy number represents an output variable computed through extended fuzzy operations? Consider a set of uncorrelated variables X, , . . , X,, whose nature is essentially random even though the body of information is so scarce that their distribution functions cannot be safely defined. These variables are represented through the fuzzy numbers X, and membership functions pL(xj) represent their possibility distributions and, correspondingly, select the classes 9(X,) of probability measures P, compatible with the available data. According to Eq. (18), F,,(x,) and FT(x,) give lower and upper bounds for all these probability measures. Moreover, consider the variable Y, which is obtained as an explicit function of the input variables, i.e. X,). Of course, not a single probability measure, but a whole class !?(Y) should be obtained, Y=Y(X,,..., collecting all the probability measures obtained starting from the classes 9(X,). It will be shown in the following that fuzzy number theory gives conservative (i.e. safe) estimates of characteristic values corresponding to small and large (i.e. close to 0 and 1) fractiles of Y. Hence, properties described in Section 4 hold also when the fuzzy number is an output variable obtained from extended fuzzy operations. The cases of monotonic real one-to-one and binary operations will be analyzed separately in the following. The reader is referred to textbooks on fuzzy algebra for details on the extension of ordinary operations to operations between fuzzy numbers [23,24]. 5.1. One-to-one
operations
The monotonic real one-to-one operation y = u(x) is considered here. The input variable is the fuzzy number r? = (xc, xc. - XL,XR - ~c)~~. Making use of Eq. (19) upper and lower bounds for probability are given by
F;(x) =
0
x
0 ,+(x) 1 1
xL 6.x 4.~ x,
F,,(x)
=
x
1 - /J&) 1 1
xc s x s xR XR
By virtue of extension principle [23], the extended operation output fuzzy variable Y, whose membership is
(25)
Y = u(X) associates with the fuzzy number X the
&J(Y) = sup k(X) x y=U(li) Since u(x) is a monotonic
(26)
function,
Eq. (26) reduces to
&(Y) = &0-‘(Y)) Correspondingly,
upper and lower bounds 0
F;(Y) =
(27)
,+K’(Y)) { 1
for probability 0
Y
are obtained
F,,(Y) =
1 -,+&-I(Y)) 1 1
where Y, = U(Q), Y, = u(xc), y, = u(xR) if y = u(x) is an increasing
from Eqs. (19) in the form: Y
(28)
Y,
whereas y, = u(x,), y, = u(x,),
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
212
yR = u(x,) if it is a decreasing operation. Comparison for probability of X and Y are related as Incr.) Deer.)
‘( y))
F*,(y) = F;(u F;(y)
of Eqs. (25) and (28) shows that upper and lower bounds
(29a)
F,*(Y) = Fx*(u_l(Y))
= 1 -F,.&-‘(y))
(29b)
F,*(Y) = 1 - W-‘(y))
if y = u(x) is a monotonic increasing or decreasing operation, respectively. Probabilistic analysis is an alternative approach to obtain the class L?‘(Y) of probability measures for the output variable. In this case, two CDFs F,,(x) and F,,(x) are considered, coinciding with upper and lower bounds for probability of the input variable X, i.e. F,,(x)
= G(x)
Since, making Incr.) Deer.)
?
FXR(4 = F,*(x)
use of probability
theory,
F,(Y) = F&‘(Y))
(31a)
F,(y) = 1 - F,(L’(y))
is the CDF of the output variable prove that
(3 lb) Y obtained
Qx, FmW~FxW~FxL(4 where F,(y), Incr.) Deer.)
(30)
F,,(y)
F,(Y)
are obtained
*
from a general F,(x) through the operation
Q Y 3 FY,AY)~FY(Y)~FYL(Y)
from FxL(x), F,,(x)
= F,,(u-‘(Y))
(32)
as
F,,(Y) = F,&-‘(y))
F,,(Y) = 1 - F,,&-‘(Y))
y = U(X), it is easy to
(33a)
F,,(Y) = 1 - Fx,(u-l(Y))
(33b)
F,, and F, are then the two upper (left) and lower (right) CDFs bounding all CDFs F,(y) obtained from F,(x) compatible with the body of information represented through the fuzzy number X. Finally, from Eqs. (29) (30), (33), it can be easily verified that, for one-to-one operations, fuzzy number algebra and probability give the same lower and upper bounds, i.e.:
F,*(Y) = F&Y)
7
(34)
F:(Y) = F,(Y)
5.2. Binary operations In this section, pl (xl) and &(x2) denote the membership functions of two fuzzy numbers X,, _f2. Moreover, X is a real binary operation associating the output variable y = y(x,, x2) = xl X xJ to the ordered pair (xl, x2). The corresponding extended operation between fuzzy numbers is denoted by Y = Xl @X2 and &uy(y) is the membership function of the output variable I? For each membership function, the left (increasing) part and the right (decreasing) part are denoted by L and R, respectively. Making use of fuzzy operations, the left and the right parts of Y are obtained from ,+(x1), &(x2) in different ways, depending on the nature of the operation X. Increasing, decreasing or hybrid operations are defined as Increasing
Decreasing
operation:
operation:
and
Q.x~,~Q.~~>x~~
*
x10
xX2,>%b
Q-cuQ~lh>~lo and
Qx~~,Qx~,,
3
Xla
xx2u
Qx’~,Qx,,,>x,~
xx2h
Hybrid operation:
Xx2,
(35)
213
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
Making use of extension principle [23], it can be shown that the part H = L, R of h(y) parts J = L, R of I, and K = L, R of I, according to the following rules: (A) Increasing
operation
(B) Decreasing
(C) Hybrid operation
H#J=K
* *
from the
H = J = K
=+
operation
is obtained
(36)
H = J # K
In the following, the notation introduced in Eq. (36) will be used to consider simultaneously both parts (L and R) of fuzzy numbers. Conversely, when the two parts are considered separately, U, V denote the two parts giving the part L of Y and -U, -V those giving R. Of course, U = V= L and -V = -V = R for increasing operations, U = V= R and -tF = -V = L for decreasing operations, U = -V = L and tF = V= R for hybrid operations. Once the membership function of Y is obtained, upper and lower bounds for probability are derived from equations analogous to (25), and bounds for characteristic values corresponding to given fractiles can be found according to the procedures outlined in Section 4. Fractiles close to 0 and 1 are of special interest: they will be determined in the following through asymptotic approximations of F*y close to yL and of Fy* close to yR, where YT,= [yL, yR] is the support of Y. First of all, yL and yR are obtained from the pairs of extreme values of r?, and x,, i.e. xIL, .x,~ and xZL, xZR as (H = L, R): YH
=
xI J
’
(37)
X2K
where subscripts J and K follow the rules (36). Then, Taylor’ ,ur(y) in the neighbourhood of yH gives P?(y)=
T
:
T
2K (Y - YH) + WY
series expansion
of the membership
function
- YH>‘l
IJ
where (il = 1J, 2K): Ti, = C,,x,:, and (i=
(39) lJ,2K):
1,2;iZ=
(40) As is to be expected, Eq. (38) gives positive values of ,+(y) the signs of coefficients reported in Eq. (40) are (i = 1,2):
C,,>O I=H; x,!, 2 0
C,,
I= L ;
in the neighbourhood
of both yL and yR. In fact,
I#H (41)
I=R
Hence, from Eq. (39) (il = 1J, 2K): T,,aO
T,,
H=L;
H=R
(42)
where, of course, H, J, K follow the rules in Eq. (36). Substituting (42) in (38), the validity of the asymptotic expansion is proved. Finally, Eq. (38) gives the following upper and lower bounds for probability:
F;(Y)=
T
111
F,*(Y)
= 1 -
:T2” (Y T,,,
1
+ T2+
YL) + WY
- Yd21 (43)
(y - yp.) + o[(y
-ye>‘]
which are valid for y -+ yL from above and for y + yR from below, respectively. Eqs. (43) show that the leading terms in the expansions of bounds for probability distributions are linear functions of the distance of y from the extremities of the support y, and yR. Eqs. (43) break down in two cases: (a) xi; = 0 (i = 1,2); (b) x,!, +w (I= J, K; i = 1 and/or 2). Case (a) means
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
214
that pdf corresponding to upper and lower probability distributions associated with Xi takes an infinite value at xi,: this case is meaningless in practical applications. On the contrary, case (b) is particularly significant: at least one variable Xj has thin tails at xi,, i.e. a,ui/&x, = 0. This case will be discussed in details at the end of the section. The results from fuzzy number theory are now compared with those obtained through probability theory. In particular, upper and lower CDFs of the output variable Y are computed, starting from the limit distributions defined through fuzzy numbers X, and 2,. The membership F&-Q
functions
of Xi, r?, are used to define probability
distributions
F,,(x,),
F,,+(x,) and FZL(x2),
as
‘jL(‘i) = FT(xj)
FiR(XI) = Fi*(xi)
9
(i = 1,2)
(44)
where FT(xi), F,,(xi) are upper and lower bounds defined from Xi through equations functions are used to define upper and lower pdf s as (i = 1,2) (see Fig. 4): 0 PiL(‘i)
_I
=
Xl <
d,u d-T
x rC
0 If X,, X2 are uncorrelated
PYL(Y) =
random
variables,
.O
Y
P,&I)P2”(~*)
YL GY CY,
-0
Yc
with (25). These
xx
cxj sxi,
(45)
XiR< x;
two important
PyR(Y) =
analogous
pdf’s for the output variable
0
Y
PI&,)P2&)
Yc d Y s YR
0
YR
1
Y are given by
(46)
where yL and yR are defined in Eq. (37) and yc = xIc X xZc is the modal value of I? For the sake of brevity, pdf’s in Eq. (46) will be rewritten in the following as P,(Y)
= P&I
with H, J, K following
)P&Z)
the rules reported in Eq. (36). Integration
(47) of Eq. (47) gives upper and lower CDFs for Y
(H = L, R):
Fig. 4. Increasing operation: upper and lower CDFs F,,, F,R (i = 1.2) and corresponding to obtain upper (FYL) and lower CDFs of the output variable I’.
pdf’s p,,., P,~. Domains of integration
for Eq. (48)
215
P. Ferrari, M. Savoia / Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
FY"(Y)
=Iy?
P,(Y)
(48)
dY
I
It can be proved that the following VXl > F;,(x,l
G FLX,)
property *
=s (Lb,)
holds (see Appendix
v y >
Fyx(Y)
SFY(Y)
A): (49)
sG F,,(Y)
representing the counterpart of Eq. (32) for binary operations. Eq. (49) states that F,,,(y) and F,,(y) in Eq. (48) are lower and upper bounds for all CDFs of the output variable Y obtained starting from CDFs F,(x,) of input variables X,. Making use of (47), Eq. (48) is rewritten as (50) where 8(J) = + 1 if J = L or 6(J) = - 1 if J = R (J = U, V). Moreover, the argument
of the integral can be given
the form:
4% 4%
P&I khK(X2)= S(J)S(K)dw, z and, correspondingly,
Fu,(y)
(51)
upper (H = L) and lower (H = R) CDFs in Eqs. (48) can be computed from the equations:
=
(52)
for all the cases, i.e. for increasing, decreasing or hybrid operations according to relations (36). Asymptotic expressions for CDFs in Eq. (52) will be computed, for y close to yL and yR. For this purpose, Eq. (52) is rewritten as FyL(y)
(53)
=
By expanding (JK = UV, WI+):
the membership
function
A::,k -4’
+ A&
+
CL, in the neighbourhood
(x1 - x,~)(x~ - xZK) +
of x,,, straightforward
.4::2(x, - 4’
+ *. .
algebra
yields
(54)
where
(55)
The leading term Ai in the expansion (54) vanishes if one or both derivatives of the membership functions a,ui/ax, are zero. This singular case, analogous with that already singled out in the computation through fuzzy numbers, will be discussed later in the section. The double-integration in Eqs. (53) is performed by writing, for a given y, the right-hand points of the integration intervals in the form (Fig. 4): x2 = X2(% x1)
x1 =x,(Y,x,,)
K=V,aL
(56)
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
216
Taylor’ series expansion obtaining
of function
y(x,, x2) is performed
in the neighbourhood
of yH up to linear terms, so
(57) where yH and C,,, C,, are defined endpoints (56) are given by
x,(yJ,,)=x,,+~(Y
in Eqs. (37),
_Y”)
II
Finally, performing the integrations obtained, when V J, K, A? # 0:
x2
=
(40a). Hence,
linearized
expressions
+ (Y - YH) - CL/(x, - XiJ) C 2K
X2K
in Eqs. (53), the following
equations
for the integration
(58) for upper and lower CDFs are
CJV
AlI
FYL(Y)
=
___ (y - YLj2 + O[(Y - YL131 2c, “C,, (59)
e-v
F,(Y)
=
l-
All
(y - Y,j2 + O[(Y - Y,j31
2c,,c,,
for y + y, from above and y + yR from below, respectively. It is easy to verify that, as is to be expected, the two coefficients multiplying the quadratic terms are positive. In fact, from Eqs. (41) and (55a), C, JC2K > 0, A: 2 0 if J = K, and C, JC2K < 0, A: C 0 if J # K. Fiqs. (59) can now be compared with the corresponding upper and lower bounds for probability obtained through fuzzy numbers, see Eq. (43). Both expansions are valid when, V J, K, A: # 0, i.e. api / axi (i = 1,2) are different from zero. These equations show that, in the neighbourhood of the extremities of the support of r (y, and yR), i.e. for small or large values of fractiles of the output variable, asymptotic expressions for upper and lower bounds from fuzzy number and probability theories are FUZZY) F;(Y)
WY>
YLI
F*,(Y)
FYL(y) = 0[( y - Y,)~]
Probability) respectively.
= 01~ -
=
1 - O[Y - YKI
F&Y)
= 1 - W(Y - YR)~I
(604 (60b)
Hence, for y close to y, and yR: > F,(Y)
F,*(Y)
(61)
< Fw.(y)
i.e. fuzzy number theory gives conservative estimates of characteristic values given by probability theory, for small (or large) fractiles. As has been previously announced, peculiar situations may occur if the membership function of at least one variable fi has thin tails at xi1 (il = 1U and/or 2V), i.e. ap,/axi = 0. In this case, the leading term in the expansion of upper bound for probability vanishes, and Eq. (43a) takes the form
WY)
1 d2fi = y dq’2
This situation
yI, (Y - YLj2 + WY
may occur in three different
(a)
xi, = fm
(b) (c)
xi, E (-a, xIU = km
circumstances
xiv E (-m,
+a) - (0)
(62)
- YL)31
’ = fm x2v I = km x2v
t-m) -
(see Eqs. (39), (40)):
(0) (63)
which will be analyzed separately in the following. It will be shown that, in all three cases, the upper CDF obtained through probability theory is smaller than that of Eq. (62) and, consequently, fuzzy number theory gives conservative results with respect to probability.
P. Ferrari, M. Savoia / Comput. Methods Appl. Mech. Engrg. 160 (1998)
217
205-222
Case a):
The coefficient
of the leading
term in Eq. (54) is AyxI; by performing
the integration
of Eq. (53a) yields
UV
(y - y,13+ O[(Y - Yd41 F,,(Y) = 6ctix;
(65a)
zv
IU
In passing, it can be verified that the coefficient of the leading term is always positive; in fact A yxy 2 0 and C,, > 0 for V = L, whereas A yx: s 0 and C,, < 0 for V= R. Comparison of Eqs. (62) and (65a) shows that Eq. (61a) holds for y-y,. Cuse b): 1 ,=O
‘ZO, XI I/
3
A;“=AK=O
xzv
In this case the coefficient
of the leading term in Eq. (54) is AyXI. Integration
of Eq. (53a) yields
UV
A Ix2 F,(Y)
=
6c
(Y - YJ3 + O[(Y - Yd41
c2
II/
2v
and Eq. (61a) is verified. Case c):
( 64~) Hence, integration A
F,,(Y) =
of Eq. (53a) gives L/V
24c2;‘;2 (Y - YL14 + WY - Yd51 I11
(65~)
2v
and Eq. (61a) is verified also in this case. Analogous results can be obtained if lower distributions FYI and F, are considered. Hence, Eqs. (6 1) are verified also when input variables i, have thin tails at the extremities of the fuzzy support.
6. Two numerical examples The theoretical results previously obtained concerning characteristic values obtained from probability and fuzzy numbers theories are confirmed in this section through two numerical examples. The operation y(xl, x2) = xl .x2 (multiplication) is considered, with xl, x2 ranging over real positive values. In the first case, fuzzy input variables have triangular membership functions whereas, in the second case, membership functions have thin tails at the extremities of the support. 6.1. Triangular fuzzy
numbers
The input variables are represented by means membership function is given by (i = 1,2):
of two fuzzy numbers
zi = (xc, xc - xL, xR - x~)~~, whose
P. Ferrari, M. Savoia I Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
218
xi -
XjL
xic &,(Xi)
xiR
=
‘iR
-
-
XiL
c
xi
s
xic
s
xi G
xic
X,L
‘i
(66)
X,R
xiC
0
otherwise
where [xiL, xiR] is the support and xic is the modal value. The membership obtained from the fuzzy extended operation r = x, Or?, is
+-
-b, -b,
of the output variable
aLcL(y))“*
(b:
YLCYGYC
aL Pruy(Y)=
function
- (bi - aRcR(y))“*
(67) YCCY GY,
aR 0
otherwise
where (H = L, R): aH =
kc
-
+fDZC
-
X2H)
9
b,
=
[@,C
-
X,H)XZH
+
62C
-
X*H)%fl/2
c&) =X,f,X2H -Y (68)
Since multiplication is an increasing operation, the left (right) parts of J?,, 2,. Probability theory requires the integration of (H = L, R) and pIH = Unif[x,,, xiB] is the uniform A = C, B = R if H = R. Lower and upper CDFs
FYH=
according
to Eq. (36) the left (right) part of p is obtained from
Eq. (48) where, according to (47) and (36), pyH = pIH . pZH pdf over the interval [xiA, xiB], with A = L, B = C if H = L, and can be given the form
sAB cx,B
-
X,A)(X2B
-
%A)
where
forY,,
s,, = y[Log(y/&) -
GY GY,
for y,GyYyy, Or
‘A,
=
and
YIJ=x,lex2J
-Y
(704
+ X,A%A ;
SAB=~[Lo~(~/x;?A)-Lo~(~~x2B)l-x,A(x2R-~A)
YILog(x,B)
for YWGY
L”g(x,,)l
-
sAB
(1,
J
=
A,
L”g(x,A)
=
B)3
if
=
YAB~YBA
YBAs YAB;
y[Log(x,,)
y,
if
-
minbAB?
L”i%bh2,)l
YBA), Y,
G’Ob)
(7Oc) -%A&R
= max(YA8,
-
‘,A)
+
X2BX,A
+
Y
(704
YBA).
The fuzzy numbers J?, = (2,0.2,0.2),, and J?, = (3,0.3,0.3),, have been considered in the numerical example. Fig. 5 shows the comparison between the upper and lower bounds for probability F*y, Fy* obtained from fuzzy extended operation (dashed line) and limit CDFs F,, F, from probability theory (solid lines). In the
Fig. 5. Triangular input fuzzy numbers: Upper (F:) (F,,~, F,,), a general CDF (F,) satisfying Eq. (47).
and lower (F,,) bounds from extended fuzzy operations,
limit CDFs from probability
P. Ferruri, M. Savoia Table 1 Fuzzy number with triangular
membership
219
I Compur. Methods Appl. Mech. Engrg. 160 (1998) 205-222
function: characteristic
values obtained through probability
theory and conservative
bounds given
by fuzzy number theory
Probability F!,. Fuzzy theory F: Probability F!, Fuzzy theory F,.
0.05
0.10
5.0318 4.9142
5.1035 4.9686
0.90
0.95
6.9668 7.1286
7.0523 7.1942
same figure, CDF F, obtained from two compatible pdf’s for Xj, i.e. X, = Unif[xiL, xiR] is also reported. Of course, F, and F, are upper and lower bounds for Fy, i.e. Eq. (49) is satisfied. The figure confirms the theoretical prediction of Section 5, that is fuzzy number theory gives conservative estimates of characteristic values with respect to probability. Even if the analysis in Section 5 was confined to low and high fractiles (close to 0 and l), the figure shows that fuzzy operation gives a conservative lower bound for F, - ~0.5 and an upper bound for F, - >0.5. Finally, Table 1 shows the characteristic values for various fractiles, obtained from probability and compared with lower and upper bounds given by fuzzy number theory. 6.2. FUZZY numbers
with thin tails
; 1
In this example, functions:
the input variables
x, x
1 -cos
[
/+x,(X,)
=
1 2
X
1 - cos rR
r
_X,L 7T
IC
(
are two fuzzy numbers
X 1R (
‘1. -X.
lr Xi(
0
11 XiL
>I
s
x;
s
b, = (x,, xc - xL, xR - -x,=)~~ with memberships
x,c
(71)
xjc G xi =Gx,R otherwise
Both variables have thin tails at x,~ and .K,~. By performing output variable p is found in the form: -b,
+ (bt - aLcL(Y))“*
?r
fuzzy operation,
the membership
function
of the
Y,SYYYy,
QL -b,
- (bi - aRcR( Y))“2
>I
VT aR
lo where a”, b,,
(72)
Y,GYGY,
otherwise
cH (H = L, R) are the same defined
5
in Eqs. (68).
5.5
Fig. 6. Input fuzzy numbers with member function of Eq. (69): upper (F;) and lower (F,.) bounds from fuzzy operations, probability (F,,,, F,,).
limit CDFs from
220
P. Ferrari, M. Savoia / Comput. Methods Appl. Mech. Engrg. 160 (1998) 205-222
Table 2 Fuzzy number with membership given by fuzzy number theory Probability F,, Fuzzy theory FT Probability F,, Fuzzy theory F,.
0.05 5.1278 5.0163
Table 3 Fuzzy number with membership number theories
Probability FuL Fuzzy theory Ft Probability FIR Fuzzy theory F,.
function
0.05 5.1144 5.0137
of FQ. (71): characteristic
0.10 5.1855 5.0837
function
0.90
0.95
6.8718 6.9921
6.9392 7.0717
of Eq. (71): characteristic
0.10 5.1626 5.0774
values obtained
through
probability
values obtained from asymptotic
0.90
0.95
6.9574 6.9943
7.0056 7.0721
theory and conservative
expressions
for probability
bounds
and fuzzy
In this case, limit CDFs from probability theory cannot be given an analytical form as in the previous case and have been computed numerically. The fuzzy numbers 2, = (2,0.2,0.2),, and J?Z = (3,0.3,0.3),, have been considered, and results obtained from fuzzy numbers and probability theory are shown in Fig. 6. The corresponding characteristic values are reported in Table 2. It is easily verified that, in this case also, fuzzy number theory gives conservative values with respect to probability. Finally, the approximate values obtained from asymptotic expressions in Eqs. (62) and (65~) are shown in Table 3. Comparison of Tables 2 and 3 shows the effectiveness of asymptotic approximations, the maximum error with respect to exact values being 1.25 and 0.31% for probability and fuzzy number theory, respectively.
7. Conclusions The usefulness of fuzzy numbers and possibility theories for problems with uncertainties associated with input data are discussed. Following Dubois and Prade [21], fuzzy numbers are used to represent possibility distributions and upper and lower bounds for probability. It has been investigated whether relations between possibility and probability measures are preserved when fuzzy numbers are output variables computed through extended fuzzy operations. For one-to-one operations, it has been shown that fuzzy number theory gives the same characteristic values as probability. For monotonic (increasing, decreasing or hybrid) real binary operations, asymptotic expressions (for small and large fractiles) of membership function (fuzzy number theory) and CDF (probability theory) have been derived. It has been shown that fuzzy number theory gives conservative bounds, with respect to probability, for characteristic values corresponding to prescribed occurrence expectations. These results are of special interest in the framework of computational analysis. In fact, it is well known that, when incomplete statistical information is only available, it is easier to define fuzzy variables than random variables. Moreover, extended fuzzy operations are simpler than analogous operations required in the framework of probability, and the implementation into finite element codes is simple [3].
Acknowledgment The financial support of the (Italian) Ministry of University and Scientific and Technological (MURST - 40%, ‘Modelling of Solids and Structures and Experimental Investigations’) is gratefully edged.
Research acknowl-
P. Ferrari.
Appendix
M. Savoia
I Comput.
Methods
Appl.
Mech.
Engrg.
I60 (1998)
205-222
221
A
It is proved here that CDFs F,,(y), F,,(y) defined in Eq. (46) are upper and lower bounds for all CDFs of the output variable Y obtained starting from general CDFs F,(x,) satisfying F,,(x,) G F,(x,) d FtL(x,) (see Eq. (47)). Starting from two general pdf p,(x,), p2(x2) defined over the intervals [x,~,x,~], [.xZL,xZR],respectively, the CDF for the output variable Y can be obtained as (see Eq. (48)): F,(Y) = 8(U) S(V) Jr~;:“‘i” 6:fl’“’ Integration
of Eq. (A.l)
P,(X,)P,(X,) d.X, &,
(A.1)
with respect to x2 gives
‘;,(?..C~“) F,(y) = WJ 1 which, integrated
According
I *1u
by-parts
P,@,)lW,)
(A.21
- ~*(%“)I h,
with respect to xi is finally rewritten
to Eqs. (44) and (46), CDFs F,(y)
or F,,(y)
as
can be obtained from Eq. (A.3) by replacing
F,(x,),
F2(x2) with F,,,(x,), F2v(xz) or F,,(x,), F2+(Q, respectively. An additional CDF is required for the proof, i.e. F,,(y) (S = V, -V), where the only FT(x2) is replaced with F&x,) in Eqs. (A.2), (A.3). Making use of Eq. (A.2) it can be easily verified that F,,(Y)
s F,(Y)
(A.4)
%,(Y)
In fact, F&x2) G F2(x2) SF,,(x,) and, if V= L, F2&cZL) = Fz(xzL) = F,,(x,,) = 0. Then, lF2R(~2)F2,&)I s IF&,) - F&V)( d /F2L(x2) - F,,(q,)(. On the contrary, if V= R FZH(~ZR)= F2h) = F,,(.%,) = 1
and, consequently, Moreover,
In
\F,,(x,)
making
- F,,(x,,)/
G IF&d
- Fz(.+)l
use of Eq. (A.3), the following
&&9
= F,,(Y)
fact,
F,,(x,)~F,(x,)~F,,(x,)
3
F,,(Y)
c IF,,&,)
inequalities
- F2,(~2v)l.
are proved: (A.5)
+,,(Y)
and,
F,,(x,.)latF,(x,)-F,(x,.)l~lF,,(x,)-F,,(x,.)l. (A.3) is positive and, since F,(x,) c F,,(x,),
F,R(~,L)=F,(~,L)=F,L(~,L)=O. Then, IF,&,)M oreover, the second term at the right-hand side of Eq. analogous inequality holds for their integrals. On the contrary, if if lJ=L,
U = R, F,,GIR) = F,(x,~) = F,,(x,,) = 1 and IF,,@,) - F,,(xrU)I 6 (F,(x,) - F,(x,,,)\ d IF,&,) - FIR(xIU)I. Moreover, the second term at the R.H.S. of Eq. (A.3) is negative and, since F,,(x,) G F,(x,), the opposite inequality holds for their integrals. Making use of Eqs. (A.4) and (AS), Eq. (47) is immediately proved, i.e. F,,_(y) and F,,(y) of Eq. (46) are upper and lower bounds for CDFs of the output variable Y.
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Press