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

we are faced with the problem of fuzzy number<br />
comparison, because the activity durations<br />
as well as their early and late completions<br />
are also fuzzy numbers. One of the<br />
methods we could use in this case is Kaufmann<br />
and Gupta’s method [4] for fuzzy<br />
number comparison. This is a simple method<br />
of fuzzy number comparison which is performed<br />
in three steps.<br />
Step 1<br />
In this step, we comparet, the „removal”<br />
of fuzzy numbers. It the given fuzzy number<br />
is M = { x, μ ( x),<br />
} so that the x value is<br />
M<br />
within the scope of fuzzy number M and<br />
x ∈ X .<br />
It the lower or upper limit within the X<br />
domain is marked as x ,<br />
−<br />
−<br />
x , respectively<br />
μ ( x ) is the possibility distribution<br />
M<br />
function of the fuzzy number M .<br />
In order to determine „the “removal“<br />
of the fuzzy number M in comparison<br />
with the real value k, we should first define<br />
the terms:<br />
• "left removal", RL( M ,k), and<br />
• "right removal", RD ( M , k).<br />
"Left removal" RL ( M ~ , k) is calculated<br />
as a surface below the curve of the membership<br />
function of the M ~ fuzzy number<br />
between the lower limit of the M fuzzy<br />
number scope and scalar K, which is represented<br />
by the following expressions:<br />
k<br />
RL( M, k) = ∫ μ ( x)<br />
for continuous fuzzy<br />
M<br />
x<br />
−<br />
numbers.<br />
k μ ( xi−1) + μ ( xi)<br />
M M<br />
RL( Mk , ) = ∑<br />
⋅( xi−xi−1) for<br />
i=<br />
2 2<br />
discreet fuzzy numbers.<br />
"Right removal", RD (M ~ , k) is calculated<br />
as a surface below the curve of the<br />
membership function of the M ~ fuzzy<br />
number between the scalar k to the upper<br />
limit of M fuzzy number scope, which is<br />
represented by the following expressions:<br />
No 4, 2011. 162<br />
MINING ENGINEERING<br />
−<br />
x<br />
RD( M , k) = ∫ μ ( x)<br />
for continuous<br />
k<br />
M<br />
fuzzy numbers.<br />
−<br />
x μ ( xi−1) + μ ( xi)<br />
M M<br />
RD ( M, k)<br />
= for<br />
∑<br />
i= k+<br />
1 2<br />
discreet fuzzy numbers.<br />
The total “removal” of M fuzzy number<br />
in comparasion to the real number k is<br />
calculated according to the expression:<br />
RL ( M , k) + RD ( M , k)<br />
R ( M , k)<br />
=<br />
2<br />
In case that M fuzzy number has a<br />
triangular membership function (k=0),<br />
then a half Hamming distance is used:<br />
R ( M , k)<br />
=<br />
x + 2 x′ + x<br />
−<br />
4<br />
where:<br />
−<br />
x , x , x<br />
−<br />
−<br />
' - are abscises of the corre-<br />
sponding vertices of a triangular fuzzy<br />
number.<br />
Now it the two fuzzy numbers are<br />
given be M = { x, μ ( x),<br />
} and<br />
M<br />
N = y, μ ( y)<br />
. It is considered that the<br />
{ } N<br />
fuzzy number M is smaller than the<br />
fuzzy number N if and only if the following<br />
is valid:<br />
R ( M , k) < R ( N, k)<br />
and vice versa.