>Please be a little more specific. I just bought the book and it seems
>improbable to me that anyone could figure out how he is doing his
>fuzzy arithmetic if at all. Either re-post your correction or mail it to me
>for evaluation.
>Do they use and/or operators which are not t-norms or t-conorms? I'm
>curious what kind of fallicies are being taught out there.
The example in Table 2.1 on p. 30: Let fuzzy numbers A and B be
A = [-2, 3, 8]
B = [-1, 2, 7]
where [l, m, r] denotes the leftmost, modal, and rightmost values of a
triangular fuzzy number (TFN). By the method outlined in McNeill/Thro, the
results of the following operations are:
Addition: [-4,5,14]
Subtraction: [-8,1,10]
Multiplication: [-3,6,15]
Division: [-7.5,1.5,10.5]
The *correct* results for addition and subtraction are:
Addition: [-3,5,15]
Subtraction: [-9,1,9]
The correct multiplication and division results are more difficult due
to the fact that both fuzzy numbers contain 0 in their support and the
fact that the result is *not* a TFN. However, the support for the
multiplication result is definitely [-14,56] since those are the min and
max possible values of the product. The support for the division operation
is the entire real line since B contains 0 and therefore the result can
approach both positive and negative infinity.
This is how they explain their addition operation:
1. Add the two modal numbers to get the result modal: 3 + 2 = 5
2. The base range of A is 8 - (-2) = 10. The base range of B is
7 - (-1) = 8. Therefore the average base range is 9.
3. The leftmost value is then 5 - 9 = -4 and the rightmost value is
5 + 9 = 14.
It seems that subtraction, multiplication, and division are done the
same way except that the operators are changed.
Note what they say on p. 28: "For simplicity, FuzNum Calc presents the
results as a symmetrical triangle. A more sophisticated computer would
be able to represent results as asymmetrical triangles, as well."
---That's funny...the *same* computer that I have FuzNum Calc running on
seems to be able to do the *correct* fuzzy arithmetic using my code
..hmmm...I guess I don't need a "more sophisticated computer", only
the correct algorithm!!
Sarcasm aside, it would have been *very* easy to show the correct way to
do fuzzy addition and subtraction---the correct method is easier than their
method. And it would have been more instructive to introduce some of
the difficulties with fuzzy multiplication and division (especially with
TFNs) to new readers, rather than ignoring the problems altogether and
giving the impression that it is very simple.
*********
For those that aren't familiar with fuzzy arithmetic, here is a
brief review:
Dubois and Prade defined the operations on fuzzy numbers back in 1978 (see
Dubois and Prade, "Operations on fuzzy numbers." International Journal
of Systems Science, 9:612-626, 1978) but McNeill and Ehro must not have
known this. Some people have said that the Dubois and Prade
definitions are not the *only* way, and that you can define your own...to
that I agree, BUT the definitions have to be consistent and reasonable.
If we are adding the two numbers above, and A can be as low as -2 and B
can be as low as -1, then HOW can you say that the result will be any
lower than -3 ?
DUBOIS AND PRADE DEFINITION OF THE ADDITION OPERATION:
The membership function for the sum of two fuzzy numbers, M1 + M2 is:
mu_{M1+M2}(x) = sup{min{mu_{M1}(x-y), mu_{M2}(y)} | y in R}
which, for TFNs, simplifies to M1+M2 = [l1+l2, m1+m2, r1+r2].
Subtraction is defined similarly and reduces to M1-M2 = [l1-r2, m1-m2,
r1-l2]. Multiplication and division are more difficult to simplify due
to their peculiar nature around 0.
If anyone has any comments or gripes, please feel free to contact me.
Warren Hearnes
Graduate Research Assistant
Georgia Institute of Technology
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