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

For every α intersection the fuzzy<br />
numbers are represented by :<br />
α α α α<br />
A = ⎡<br />
⎣xL, x ⎤ D ⎦ and B = ⎡<br />
⎣ yL , y ⎤ D ⎦<br />
Then the operations with triangular<br />
fuzzy numbers are defined by the following<br />
expressions [5].<br />
{ , μ ( ) , }<br />
C<br />
C = A+ B= z z z∈ R<br />
⎡ α α α α⎤<br />
z= ⎢xL + yL , xR + yR⎥; μ ( z)<br />
= α; α = 0,1<br />
⎣ ⎦ C<br />
{ , μ ( ) , }<br />
C<br />
C = A− B= z z z∈R [ ]<br />
⎡ α α α α⎤ z= ⎢xL − yR, xR − yL⎥; μ ( z)<br />
= α; α = 0,1<br />
⎣ ⎦ C<br />
{ , μ ( ) , }<br />
C<br />
C = A⋅ B= z z z∈R [ ]<br />
⎡ α α α α⎤<br />
z= ⎢xL ⋅yL , xR ⋅ yR⎥; μ ( z)<br />
= α; α = 0,1<br />
⎣ ⎦ C<br />
{ μ }<br />
C<br />
C = A: B= z, ( z) z∈ R,<br />
[ ]<br />
⎡ α α α α⎤<br />
z= ⎢xL / yR, xR / yL⎥; μ ( z)<br />
= α; α = 0,1<br />
⎣ ⎦ C<br />
Example:<br />
[ ]<br />
There are strictly positive fuzzy numbers<br />
given A = x, μ ( x) x∈<br />
[ 6,10]<br />
and<br />
{ }<br />
A<br />
{ , μ ( ) [ 8,10 ] } .<br />
A<br />
B = y y y ∈<br />
μ ( x)<br />
=<br />
A<br />
⎧ 1<br />
⎫<br />
x −3, 6 ≤ x ≤8 ⎪ 2<br />
⎪<br />
⎨ ⎬<br />
⎪ 1<br />
− x + 5, 8 ≤ x ≤10⎪<br />
⎩⎪ 2<br />
⎭⎪<br />
( x)<br />
=<br />
⎧ y − 8, 8 ≤ y ≤ 9 ⎫<br />
⎨ ⎬<br />
⎩− y + 10, 9 ≤ x ≤ 10 ⎭<br />
No 4, 2011. 160<br />
MINING ENGINEERING<br />
μ<br />
B<br />
Determine: C = A+ B<br />
In order to make the marked operations<br />
over the given triangular fuzzy numbers,<br />
it is necessary to determine the left<br />
and right confidence limit for every confidence<br />
level α ∈[<br />
0,<br />
1]<br />
. In other words, it is<br />
necessary to determine the extreme values<br />
in α intersection of A and B fuzzy numbers.<br />
The left limit value of A fuzzy number<br />
for the α ∈[<br />
0,<br />
1]<br />
confidence level α x L<br />
is obtained according to the expression:<br />
1 α<br />
α<br />
xL<br />
− 3 = α → xL<br />
= 6 + 2α<br />
2<br />
The right limit value of A fuzzy number<br />
for the α ∈[<br />
0,<br />
1]<br />
confidence level α x R is<br />
obtained according to the expression:<br />
1<br />
− xα<br />
5 α α<br />
R + = → xR<br />
= 10 − 2α<br />
2<br />
α intersection of a fuzzy number is:<br />
[ α , α ] = [ 6 + 2α<br />
, 10 − 2α<br />
]<br />
x<br />
1<br />
α = L R<br />
2<br />
x A<br />
Analogue to the previous expression<br />
we obtain for α intersection of B fuzzy<br />
number:<br />
[ α α<br />
y ] = [ 8 + α,<br />
−α<br />
]<br />
B α = L , yR<br />
10<br />
Let us calculate the sum of A and B<br />
fuzzy numbers which is marked as C . C<br />
fuzzy number is formally written down as<br />
{ , μ ( ) } .<br />
C<br />
C = z z<br />
According to the rule of addition of<br />
triangular fuzzy numbers, it follows that:<br />
[ (6 2 ) (8 ),(10 - ) (10 2 ]<br />
[ 14 3 α,203α] C = + α + + α α + − α =<br />
= + −