Rudarski radovi br 4 2011 - Institut za rudarstvo i metalurgiju Bor

A large number of factors influence that<br />
planning process must take into accountuncertainty<br />
or imprecision. This is why deterministic<br />
model, which is used most frequently<br />
(CPM) becomes quite inappropriate<br />
and inapplicable. The classical approach with<br />
quantitative treatment of uncertainty uses<br />
mathematical tools of Probability Theory (the<br />
relationship of favourable and total number of<br />
possible outcomes). The Probability Theory<br />
approach is of limited application possibilities<br />
and there are certain weaknesses in considering<br />
the problem where types of uncertainties<br />
are different from those calculated as a ratio<br />
of the number of favourable and the number<br />
of possible outcomes.<br />
The most significant alternative approach<br />
was developed by the Californian<br />
Professor L. Zadeh [1]. This approach<br />
allows for the description of fuzzy events<br />
and knowledge, operations over them and<br />
drawing conclusions and also represents<br />
the basis for the development of the fuzzy<br />
set mathematical theory. Fuzzy management<br />
is useful in cases when technological<br />
processes are complex for the analysis by<br />
means of the known methods or when the<br />
information available are interpreted<br />
qualitatively but uncertainly [2].<br />
FUZZY SET THEORY AND THE<br />
NOTION OF POSSIBILITY<br />
Fuzzy set theory provides for the formal<br />
system of representation and comprehension<br />
of situations when unsecure, subjective and<br />
imprecise information appear. In classical<br />
modelling the relations are expressed by<br />
mathematical functions. As the systems become<br />
more complex, it becomes more difficult<br />
to apply mathematical modelling, and<br />
therefore fuzzy models are used in these<br />
situations.<br />
For any given set T whose elements<br />
are t real numbers there is a sub-set<br />
Ti ∈ T , whose elements are assigned the<br />
values of a function μ( ti)<br />
the values of<br />
which are real numbers in the interval (0,<br />
1). The sub-set Ti represents a fuzzy set or<br />
the so-called fuzzy restriction on the set T.<br />
The function μ( ti)<br />
is called the membership<br />
function of ti elements on the set Ti.<br />
The membership function shape can be<br />
entirely arbitrary. Figure 1. represents<br />
standard membership functions. Based on<br />
the experimental research, the conclusion<br />
has been drawn that the standard membership<br />
functions can be used to solve the<br />
majority of tasks<br />
Fig. 1. Standard membership functions of a fuzzy set<br />
No 4, 2011. 158<br />
MINING ENGINEERING