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was introduced to system control by Mamdani (1976) and the reader is
referred to relevant textbooks for more detail on the theory (Höppner,
1999; Yager and Filev, 1994).
The basics of fuzzy system understanding and modeling lie in fuzzy
sets, which can be thought of as classes, where samples of data sets can
have different degrees of membership to a certain class. In a binary
system, a sample either belongs to a class or not (meaning that the degree
of membership is either 1 or 0), whereas in the fuzzy system, a sample can
belong to different classes with different degrees of membership (that all
add up to 1). When represented graphically, fuzzy sets are often
trapezoidal, but can also include triangular, sigmoid, bell-shaped, or
irregular (Cox, 1994). The terms graduation and granulation refer to the
matter of degree of similarity between the data. There are many types of
fuzzy sets, such as type 2, L-fuzzy sets, bipolar fuzzy sets, and intuitive
fuzzy sets (Zadeh, 2007).
It is important to differentiate between probability and membership
degree, since they are sometimes expressed in similar terms. Probability is a
measure of likelihood that a certain event or feature will occur, for example,
there is a 70% chance that patients will respond to the new medication
(70% of patients will respond, whereas 30% will not), whereas the
membership degree would express that patients will have good response to
the new medication (quantifi ed as 70% of the total 100%, or 0.7 degree).
Fuzzy systems are usually described in the form of linguistic rules that
are easily comprehensible and take the form of
if . . . then . . .
sentences.
In this way it is possible to express various system states (e.g. if the drug
solubility is low then the dissolution rate is limited or, if the powder
fl owability is poor then the content uniformity is low, etc.). These fuzzy
rules can be thought of as mathematical relationships mapping inputs to
outputs (Sproule et al., 2002).
One of the most often used algorithms in fuzzy logic modeling was
developed by Sugeno and Yasukawa (1993), whereas other methods are
also available: the min-max composition method (Mamdani, 1976);
improved Sugeno-Yasukawa algorithm (Hadad et al., 2010); GAs
(Ishibuchi et al., 1995), etc. Fuzzy modeling is the process of
if . . . then
. . .
rules defi nition. Development of rules is based upon expert knowledge,
experience, and experimental data. As in any other modeling, the fi rst step
of fuzzy modeling is defi nition and collection of inputs and outputs. Then,
the outputs of the fuzzy system are clustered, using algorithms such as the
fuzzy C-means algorithm (Bezdek, 1981), fuzzy fi ne clustering (Emami
et al., 1998), fuzzy LVQ, fuzzy adaptive resonance theory (Baraldi and
Blonda, 1999), fuzzy C-regression model clustering (Kim et al., 1997), etc.
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