Environmental Engineering Reference
In-Depth Information
a function of the age of the asset, whereas the investment is a constant. The
rationale for these assumptions is intuitive. Including maintenance in the investment
cost does not change the model; it merely moves that period
s maintenance into
investment. However, the mathematical representation becomes less convoluted as
a result
'
the model is simpler to understand. Representing maintenance cost as a
function of age simply allows for more
—
flexibility in the model as the expense may
vary with age.
The DSS can be formulated in discrete and continuous time and the following
sections are a mathematical statement of both.
7.4.2.1 The DSS in Discrete Time
In discrete time, the DSS can be described as a recursive function that includes the
sum of the maintenance up to the period before replacement, the replacement cost
itself, and future service cost, all discounted accordingly:
C
t
¼
X
t
1
M
i
1
þ
d
I
1
þ
d
C
t
1
þ
d
i
þ
t
þ
ð
7
:
1
Þ
t
þ
1
ð
Þ
ð
Þ
ð
Þ
i¼1
where
M
i
maintenance in period i
I
investment
C
t
aggregate cost of service over an in
nite horizon for a given length between
renewals (t)
d
discount rate employed by utility for cost benefit analysis
Isolating C
t
yields:
t
þ
1
X
t
1
ð
1
þ
d
Þ
I 1
þ
d
ð
Þ
M
i
1
þ
d
C
t
¼
i
þ
ð
7
:
2
Þ
t
þ
1
t
þ
1
ð
Þ
ð
1
þ
d
Þ
1
ð
1
þ
d
Þ
1
i¼1
The objective is to choose the time period, t*, that minimizes C
t
. This t* can be
found through dynamic programming or via a transformation to continuous time.
7.4.2.2 Dynamic Programming
The above objective can be solved using a deterministic Markov method; the utility
must simulate the decisions that it will be faced with over an in
nite time horizon.
The state variable is:
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