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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