Computing a Model for Asset Management with Risk - Global Drinking Water Management and Conservation - page 184

8.4.4.2 Determination of F(x)

To determine the frequency of failure in City C, frequency was regressed on age.
This initial regression ( 6.4.2 a) provided signi cant results. Again after correcting

for heteroscedasticity, a second regression, utilizing White ' s matrix also demon-

strated signi cant results. Since non-normality was not an issue in this second

regression, these results were used to obtain Eq. 8.19 , which is the frequency
function for City C:

F ðÞ¼ 13 : 587 þ 0 : 4987

2 : 341

Age i ð 8 : 19 Þ

ð Þ

8.4.4.3 Solution of the DSS

The DSS parameters in City C are: Eqs. 8.18 and 8.19 , d = 0.0423 and
I = 152169.31. Average values were assumed for depth, size, ACAGE and AC,
resulting in Eq. 8.13 , a modi ed cost of failure function:

S ðÞ¼ 209 : 83 þ 87 : 05 Age i ð 8 : 20 Þ

These parameters yield Eq. 8.21 , a DSS for City C:

R

t 1

Þ e 0 : 0423x dx þ 152169 : 31e 0 : 0423t

ð 209 : 83 þ 87 : 05x Þ 13 : 587 þ 0 : 4987x ð

1

C ðÞ¼

1 e 0 : 0423 t þ 1

ð Þ

ð 8 : 21 Þ

Iterated graphical minimization yields an interesting solution in this case as
demonstrated in Figs. 8.18 and 8.19 .
Figure 8.19 seems to indicate that the net service cost can be negative if the asset
is renewed every 30.2 years. Continuous optimization con rms this result. Of

course, a negative cost does not make sense. This result demonstrates the limita-
tions of linear analysis. A simple examination of the parameters demonstrates that
the number of failures in the early years of the asset is actually negative, which
results in a negative expected failure cost and this obviously does not make any
sense.

8.4.4.4 Nonlinearity

Nonparametric regressions for City C demonstrate that the frequency of failure
follows a similar pattern to Cities A and B. However, the cost of failure does not
seem to behave as in the other two municipalities. Rather than resemble a logistic
S function, the cost of failure in City C seems to resemble a sine function. The
results of the nonparametric regressions are shown in Figs. 8.20 and 8.21 .