Introduction to Water Infrastructure Asset Management - Global Drinking Water Management and Conservation - page 165

C t ¼ X

t 1

M i þ S i F i
1 þ d

I
1 þ d

C t
1 þ d

i þ

t þ

ð 7 : 5 Þ

t þ 1

ð Þ

ð Þ ð Þ

i¼1

The cost function still represents the aggregate flow of costs, discounted to the

present value. However, this flow now distinguishes between regular maintenance,

M, and expected accident costs, SF. Again, the goal is to minimize this eternal cost
function. To do so, C must first be isolated:

1 X

t þ 1 t 1

ð 1 þ d Þ M i þ S i F i

1 þ d

I 1 þ d ð Þ

C t ¼

i þ

ð 7 : 6 Þ

t þ 1 t þ 1

ð 1 þ d Þ ð Þ ð 1 þ d Þ 1

i¼1

flow, C, can now be minimized in two ways: via dynamic pro-
gramming, as outlined in above, or through the transformation to continuous time
that follows.

The eternal

7.4.3.2 Continuous Model

In continuous terms SF becomes S(x)F(x) allowing the transformation of Eq. 7.5 to
continuous time as follows:

Z

t 1

Þ e xd dx þ Ie td þ e dt þ 1

ð Þ C ðÞ

C ðÞ ¼ ð M ðÞþ S ðÞ F ðÞ ð 7 : 7 Þ

1

Isolating C yields:

t 1

R

Þ e xd dx þ Ie td

ð M ðÞþ S ðÞ F ðÞ

1

C ðÞ ¼ ð 7 : 8 Þ

1 e dt þ 1

ð Þ

As in Sect. 7.4.3.1 , it is desirable to minimize eternal cost with respect to t. For a
practical application this can be accomplished through continuous optimization or
numerical approximation. A numerical approximation may be a more reasonable
approach considering the dif

of precision required
for practical applications. Asset management necessitates rounding of time at some
point as it is not reasonable that utilities can measure time in in

culty of isolating t and the ' lack '

nitely small
amounts. Nevertheless, if greater precision is desired, continuous optimization may
provide a precise, meaningful and practical solution.
The above discussion on a solution to the DSS focused on practical solutions
since, as identi ed in Sect. 7.4.2.3 , a generalized solution to this problem is nearly

impossible to derive.