Page 278 - Sustainable On-Site CHP Systems Design, Construction, and Operations

Managing Risks during CHP Plant Construction 251

probability of a future event occurring, or a condition existing, is generally presented in
the form of a probability density function. If one can obtain some indication of the prob-
ability density function to which a particular price prediction belongs, there is a test of
the likelihood that the estimate is unbiased.
Since it is beyond the scope of this chapter to give a comprehensive description of
each of the above referenced distributions, references have been provided at the end of
this chapter to familiarize one with clear explanations of most, including examples of
their use and application when choosing the best distribution for a given analysis.
Three of the most commonly used distributions applicable to many of the cost issues
related CHP plant construction referenced earlier are the uniform distribution, the trian-
gular distribution, and the normal distributions.
1. In a uniform distribution, all values between the minimum and maximum are
assumed equally likely to occur. For example, if no information about the
existing utilities on the proposed CHP plant site is available, the value for any
of the required connections can be assumed to be equally likely to occur. There
are three conditions that must apply for the uniform distribution to be employed,
namely, both the minimum and maximum values are fixed and all values
between minimum and maximum are equally likely to occur.
2. The triangular distribution can be used to describe a situation where one desires
to estimate the most likely value, where both the minimum and maximum
values are known. However, values near the minimum and maximum value are
less likely to occur than those values near the most likely. The triangular distribution
is widely used in construction estimation due to its ease of use. A common
drawback of the triangular distribution is that it is at best an approximation.
Under some limited circumstances, however, the approximation may be worth
the inherent benefits of using the triangular distribution method.
Triangular distribution construction is relatively simple to describe and a
graphical solution can be found as follows. If one plots the probability density
as the ordinate with the abscissa covering the range of probable values from
minimum to maximum, one can plot a triangle, starting with the minimum
value at the left end of the base of the triangle (also located at the abscissa),
rising in a straight line to the right until an apex results then falling from the
apex in a straight line until it intersects with abscissa at the maximum value
thereby completing the triangle.
The most likely value can then be determined by extending a vertical line
downward from the apex until it intersects the abscissa resulting in two right
angle triangles where the ordinate is common to both. This graphical solution is
based on the location of the intersection of the common ordinate with the most
likely value found where it intersects the abscissa since the smaller area of the
two right angle triangles represents the chance that the price will fall between
the minimum and the most likely value.
3. The normal distribution can be considered the most important distribution in
probability theory. The normal distribution is a family of distributions, each one
shaped like a bell. The bell shape spreads outward and downward but never
quite touches the horizontal scale. The distribution employs two parameters,
the mean and standard deviation. Values are distributed symmetrically about