The basic theory behind fuzzy arithmetic is Zadeh's extension principle. That
is, to evaluate C = A o B, where A, B and C are fuzzy numbers and o is some
operator, we go about it this way. Let x be a member of A, and y be a member of
B. We want to calculate the membership of z = x o y in C = A + B. If we do
this for all possible z values, we will have the membership function for C.
To do this, we select a value for z. Now for all x and y with x o y = z, we
calculate the MAXIMUM of all the MINIMUMs of the memberships of x in A and y in
B. This "max of all the mins" is Zadeh's famous extension principle.
For example, we will take a fuzzy 2 and a fuzzy 5, and add them.
1.0 - a b
.8 - a a b b
.6 - a a b b
.4 - a a b b
.2 - a a b b
0.0 ---------a--------------a---------------------b------------------------
0 1 2 3 4 5 6 7
8 9
Now we take every pair of numbers x and y which add to some value z = x + y. We
calculate the membership of x in A, a(x); of y in B, b(y); and take the MINIMUM
of these. This is a "trial" value for the membership of z = x + y in C . We do
this for every x-y pair for which x + y = z, and take the MAXIMUM of all these
trial values as the membership of x + y in C. = Say (to start) z = 7. For a sum
of 7, say x is 3 and y is 4. The membership of x in A is zero, and the
membership of y in B is .5. We take the minimum of these, which is zero, and
note it. Now we try x = 2..5 and y = 4.5.The membership of x in A is 1, and the
membership of y in B is also 1. The minimum of these is 1. This is bigger than
our previous value of .5, so we discard the .5 and keep it. If we keep on going
we will get values which are smaller than 1, so the membership of 7 in C is 1.
Of course, we will do this analytically if we can. For triangular membership
functions this is easy to do and we will end up with this for C = A + B;
1.0 - a b c
.8 - a a b b c c
.6 - a a b b c
c
.4 - a a b c b
c
.2 - a a b c b
c
0.0 ---------a--------------a-------c-------------b-----------------------c-
0 1 2 3 4 5 6 7
8 9
Hope this helps some - William Siler
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