Vagueness is embroiled in

several real phenomena. Whether one contemplates vagueness explicitly when

modeling such a phenomenon is one of the modeling choices, the outcome of which

will rest on on the situation. On the other hand, the modeler chooses to reflect

vagueness, he or she will have to select the method for modeling it. Some

experts assert that one theory, e.g. probability theory, is adequate to model

every kind of vagueness. The excellence of the techniques used in a statistical

analysis rest on extremely on the presumed probability model or distribution.

Some physical systems, those complex

ones, are uncompromising to model by an accurate and precise mathematical

procedure or equation due to the complexity of the system structure,

nonlinearity, uncertainty, randomness, etc.

Therefore, approximation

modeling is often necessary for real-world applications. But, the important questions are what kind of

approximation is upright, where the logic of “goodness” has to be first

defined, of course, and how to formulate such a good approximation in modeling

a system such that it is mathematically rigorous and can produce satisfactory

results in both theory and applications.

It is clear that interval

mathematics and fuzzy logic together can provide a promising alternative to

mathematical modeling for many physical systems that are too vague or too

complicated to be described by simple and crisp mathematical formulas or

equations. When interval mathematics and fuzzy logic are employed, the interval

of confidence and the fuzzy membership functions are used as approximation

measures, leading to the so-called fuzzy systems modeling.

Zimmermann H.-J. 134

proposed a definition of uncertainty: Uncertainty implies that in a certain

situation a person does not dispose about information which quantitatively and

qualitatively is appropriate to describe, prescribe or predict

deterministically and numerically a system, its behavior or other

characteristics. Buckley 12 define fuzzy probabilities, which will be fuzzy

numbers, from a set of confidence intervals and use fuzzy numbers for the

parameters in the probability density functions, to produce fuzzy probability

density functions.

Once dealing with the usual

probability concept, an occurrence has its specific limit. For take a

situation, if an event is X= {2, 4, 7, 9, 11}, its margin is sharp and

consequently, it can be characterized as a crisp set. As soon as we deal an

occurrence whose limit is not sharp, it can be measured as a fuzzy set, that

is, a fuzzy event. We can recognize the probability of two behaviors. One is

dealing with the probability of a crisp value (crisp probability) and the other

as a fuzzy set (fuzzy probability).

A hormone activity is a very

complex system, where it secretion processing is narrowly connected to

life-sustaining developments. It is evident now that there are a number of

amino acid combinations in the hormone that keep up its activity. Some details

of hormone activity, like stress or obesity, anxiety are well understood, while

others, like fetal growth, uterus contraction during cesarean and normal

delivery, primary postpartum hemorrhage effects and many others are still

ambiguous.

This means that a reasonable

explanation of the performance of a complex organism similar to a hormone’s

function is expressed in a natural-linguistic method with “sensitive” notations

like great excitability, weak damage, the low anticipation of punishment and so

on, which cannot be through statistically assigned.So, if we need to improve an

adequate tool for the logical theory of a hormone’s effective, in some sense,

we are bound to search for a mathematical tool, which could rightly work with

“perceptions” as with mathematical objects. Such mathematical tool was proposed

by Zadeh 131 and has been further developed during the last decades. It is

called fuzzy logic and fuzzy set theory. Guanrong Chen and Trung Tat Pham 50

give an example for fuzzy rule-based health monitoring expert systems, fuzzy

logic rules to control a focus ring of a camera to allow automatic focusing for

a sharp image, application of fuzzy control to a general class of servo

mechanic systems, the fuzzy controller for the robotic manipulator. The fuzzy

terms and definitions are defined in the following section 67, 135.