EXISTING MATHEMATICAL METHODS FOR UNCERTAINTY ASSESSMENT - Modelling Uncertainty in Flood Forecasting Systems - page 31

Figure 3.1. Membership function (arbitrary) of an input fuzzy variable X i with
its lower and upper bounds for a given ! -cut level.
Figure 3.2. Membership function (arbitrary) of the output Y with the upper and
lower bounds for a given ! -cut level. The value of ! is identical to
that in Figure 3.1.
It can be seen from Equations (3.12) and (3.13) that finding the membership function of
an output from the Extension Principle by the ! -cut method amounts to finding the
minimum and maximum of a function with constraints on the bounds in the input
variables defined for the given ! -cut. Various algorithms for finding minimum and
maximum of a function are available, which range from linear, nonlinear to global. The
suitability of the applications of these algorithms depends on the nature of the problem
among other things. In the present study, global optimisation methods called genetic
algorithms are used. More information on the algorithms is given in Section 5.3.
Recent applications of the Extension Principle in uncertainty analysis include
groundwater flow problems (Schulz and Huwe, 1997 & 1999), chemical equilibrium
problems (Schulz et al, 1999), environmental risk assessment (Guyonnet et al., 1999),
and pipe networks analysis (Revelli and Ridolfi, 2002).
Extension Principle for monotonic function
If the given function f is strictly monotonic with respect to each input variable within the
computation domain of the variables, the problem can be simplified as given in Equations
(3.15) through (3.18).
(3.15)