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

The equations representing the left and<br />
right side of the fuzzy number C :<br />
α 1 α 14<br />
14 + 3α<br />
= z → α = z −<br />
3 3<br />
α 1 α 20<br />
20 − 3α<br />
= z → α = − z −<br />
3 3<br />
The membership function of the fuzzy<br />
number C is also triangular in shape and<br />
The difference, the product and the<br />
quotient of fuzzy numbers are calculated<br />
similarly.<br />
If the activities Ai, the ti duration of<br />
which is a fuzzy number, are represented<br />
by circles or rectangles (precedence diagram),<br />
then the time of their earliest and<br />
latest completions RZi and RKi are calculated<br />
by the following expressions:<br />
i<br />
p<br />
( RZ t )<br />
RZ = max +<br />
p<br />
( RZ ) = sup{<br />
min[<br />
π ( RZ ) π ( t ) ] }<br />
π ,<br />
i<br />
KZ = min(<br />
KZ − t )<br />
i<br />
n<br />
p<br />
n<br />
π ( KZ ) = sup min[<br />
π ( KZ ) , π ( t ) ]<br />
i<br />
i<br />
i<br />
{ }<br />
i = 1 , 2,...,<br />
m;<br />
p = 1,<br />
2,..,<br />
m −1;<br />
n = 2,<br />
3,...,<br />
m<br />
Fig. 2. Fuzzy numbers A, BC ,<br />
p<br />
n<br />
i<br />
i<br />
it is formally represented by the following<br />
analytical expression:<br />
⎧ 1 14 ⎫<br />
z − , 14≤ z ≤ 17<br />
⎪ 3 3<br />
⎪<br />
μ ( z ) =<br />
C ⎨ ⎬<br />
⎪ 1 20<br />
− z + , 17 ≤ z ≤ 20 ⎪<br />
⎪⎩ 3 3<br />
⎪⎭<br />
A and B fuzzy numbers as well as<br />
their sum C i.e. fuzzy number are presented<br />
in figure 2.<br />
If the times of activity durations are<br />
considered as continuous convex fuzzy<br />
variables, then the beginning and ending<br />
of the activity which is also continuous are<br />
also determined easily.<br />
FUZZY NUMBER COMPARISON<br />
The problem of fuzzy number comparison<br />
is the task which is rather frequent<br />
in decision-making. For instance, when<br />
the alternatives are described by fuzzy<br />
numbers, there is a question of how we<br />
shall determine which alternative is better<br />
than the other. Many works in the literature<br />
are dedicated to this problem [4].<br />
In case of determining the time (deadline)<br />
of project completion for every activity<br />
which is preceded by many other activities,<br />
No 4, 2011. 161<br />
MINING ENGINEERING