Computing a Model for Asset Management with Risk - Global Drinking Water Management and Conservation - page 170

that early data is missing, the mean observation in first time period would be

considered the intercept. Upon plotting these points, one would be able to detect
nonlinearity in the model. If nonlinearity is present, then an appropriate functional
form will have to be chosen that

fits the data. In the case studies that follow, two
functional forms are used: logistic S curves and sine curves.

8.2 Towards Solving the DSS

Upon determining the parameters and functions involved in the model, an
approximate solution can be derived. This solution will be useful in a practical
application since the solution will only be

in the sense that at some
point the decimals must be rounded. However, rounding the fourth decimal of a
number identifying the year at which to replace an asset does not sacri

“ approximate ”

ce the
quality of the analysis, as scheduling and other factors will not permit a utility to be
much more precise in terms of renewal.
There are two methods by which one would derive this approximate solution:
graphically and numerically. A graphical solution involves plotting the cost function
over time and visually identifying the minimum present value of service cost as a
function of the renewal length of the asset. Adjusting the range of time to re ect only

the area to either side of the solution will assist in deriving the more appropriate
numerical solution. This process can be iterated until a solution of the desired pre-
cision is obtained. The numerical solution involves the use of computerized mathe-
matical packages to solve numerically for a minimum. Since the cost function
involves integrals, the computerized program may be required to derive a numerical
solution where it is not reasonable to do so algebraically. A mathematical software
package can be used to differentiate the cost function with respect to time. This

rst

derivative would be set to zero to find the minimum. Upon discovery of the minimum,

the second derivative would be checked to ensure that the cost function is convex (i.e.
the second derivative is negative). The next section utilizes an example to demonstrate
both of these techniques.

8.3 Application of Risk into the DSS

This section makes a number of reasonable assumptions as to the parameters of the
model in order to demonstrate the application of the risk incorporated DSS. Con-
sider parameters as follows:

M ðÞ¼ x 2 ð 8 : 1 Þ

S ðÞ¼ x 3 ð 8 : 2 Þ