Introduction to Water Infrastructure Asset Management - Global Drinking Water Management and Conservation - page 163

t 2 0 ½

; 1

This is the age of the asset. The utility must decide to replace or continue to use
the asset as a result of this information. The action variable is:

x 2 keep, replace ½

These are the choices that the utility has. The state transition function is:

t þ 1 x ¼ keep

;

gt ðÞ ¼

;

1 x ¼ replace

;

'

s
age is 1; if the utility decides to keep the asset, the age increases by 1 unit. The
reward (present cost) function is

If the utility decides to replace the asset, the age is reset to zero and next year

M t ; x ¼ keep

ft ðÞ ¼ ;

I x ¼ replace

;

The reward will be either the maintenance cost or the investment cost. The value
(present and present discounted future cost) of the asset is given by the Bellman
equation:

Vt þ 1 ð Þ V ðÞ

1 þ d

V ðÞ ¼MIN M t þ I þ

;

ð 1 þ d Þ ð Þ

The goal of the utility is to minimize V(t). The utility must simulate the decision
process at each and every future time period to determine when it will choose to
replace the asset. The solution to this decision process will minimize the total
present and future cost given in Eq. 5.1 .

7.4.2.3 The DSS in Continuous Time

Equation 7.1 can be converted to continuous time as follows:

Z

t 1

M ðÞ e xd dx þ Ie td þ e dt þ 1

ð Þ C ðÞ

C ðÞ ¼ ð 7 : 3 Þ

1