12- FUZZY ARITMETIC
AND
THE EXTENSION
PRINCIPLE
2
EXTENSION
PRINCIPLE
FIGURE 12-1
A simple single-input, single-output mapping (function).
variable as shown in Figure 12.1. This relationship is a singleinput, single-output process where the transfer function (the
box in Figure 12.1) represents the mapping provided by the
general function f . In the typical case, f is of analytic form, for
example, y = f (x),the input, x, is deterministic, and the
resulting output, y, is also deterministic.
3
Crisp Functions, Mapping, and Relations
Functions (also called transforms), such as the logarithmic
function,
y = log(x), or the linear function y = ax + b, are
mappings from one universe, X, to another universe,Y.
Symbolically, this mapping (function, f ) is sometimes denoted f: X
→ Y . Other terminology calls the mapping y = f (x) the image of x
under f and the inverse mapping, x = f−1(y) , is termed the original
image of y. A mapping can also be expressed by a relation R (as
described in Chapter 3), on the Cartesian space X × Y. Such a
relation (crisp) can be described symbolically as R = (x, y)|y = f
(x), with the characteristic function describing the membership of
specific x, y pairs to the relation R as
4
Crisp Functions, Mapping, and Relations
Now, since we can define transform functions, or mappings, for specific
elements of one universe (x) to specific elements of another universe (y),
we can also do the same thing for collections of elements in X mapped to
collections of elements in Y. Such collections have been referred to in this
text as sets. Presumably, then, all possible sets in the power set of X can
be mapped in some fashion (there may be null mapping formany of the
combinations) to the sets in the power set of Y, that is, f : P(X) → P(Y).
For a set A defined on universe X, its image, set B on the universe Y, is
found fromthe mapping, B = f (A) = y| for all x ∈ A , y = f (x), where B
will be defined by its characteristic valu
5
Crisp Functions, Mapping, and Relations
Example 1.
Suppose we have a crisp set A = 0, 1, or, using Zadeh’s
notation
and a simple mapping
find the resulting crisp set B on an output universe Y using the extension
principle. From the mapping, we can see that the universe Y will be
Y = {2, 6, 10}. The mapping described in Equation will yield the following
calculations for the membership values of each of the elements in universe Y:
6
Crisp Functions, Mapping, and Relations
Notice that there is only one way to get the element 2 in the universe Y, but
there are two ways to get the elements 6 and 10 in Y. Written in Zadeh’s
notation this mapping results in the output
or, alternatively, B = {2, 6}.
Suppose we want to find the image B on universe Y using a relation that
expresses the mapping.
7
Crisp Functions, Mapping, and Relations
The image B can be found through composition (since X and Y are
finite) : that is, B = A ◦ R (we note here that any set, say A, can be
regarded as a one dimensional relation), where, again using Zadeh’s
notation,
and B is found by means of Equation (3.9) to be
or in Zadeh’s notation on Y,
8
Functions of Fuzzy Sets – Extension Principle
The membership functions describing A∼ and B∼ will now be defined on the
universe of a unit interval [0, 1] and for the fuzzy case Equation.
A convenient shorthand for many fuzzy calculations that utilize matrix relations
involves the fuzzy vector. Basically, a fuzzy vector is a vector containing fuzzy
membership values. Suppose the fuzzy set A∼ is defined on n elements in X,
for instance on x1, x2, . . . , xn, and fuzzy set B∼ is defined on m elements in Y,
say on y1, y2, . . . , ym. The array of membership functions for each of the fuzzy
sets A∼and B∼ can then be reduced to fuzzy vectors by the following
substitutions:
9
Functions of Fuzzy Sets – Extension Principle
The image of fuzzy set A∼ can be determined through the use of the
compositionm operation, or B∼=A ∼ ◦ R∼, or when using the fuzzy
vector form, b∼=a∼◦ R∼ where R∼ is an n × m
fuzzy relation
matrix.
More generally, suppose our input universe comprises the Cartesian
product of many universes. Then, the mapping f is defined on the power
sets of this Cartesian input space and the output space, or
Let fuzzy sets A∼1,A∼2, . . . ,A∼n be defined on the universes X1,X2, . . .,Xn.
The mapping for these particular in put sets can now be defined as
B∼= f(A∼1,A∼2, . . . ,A∼n), where the membership function of the image B∼
is given by
10
Fuzzy Transform (Mapping)
Formally, let a mapping exist from an element x in universe X(x ∈ X)
to a fuzzyset B∼ in the power set of universe Y, P(Y). Such a mapping
is called a fuzzy mapping, f∼, where the output is no longer a single
element, y, but a fuzzy set B∼, that is, B∼=f∼(x).
If X and Y are finite universes, the fuzzy mapping expressed in
Equation B∼=f∼(x). can be described as a fuzzy relation, R∼ , or, in
matrix form,
11
Fuzzy Transform (Mapping)
For a particular single element of the input universe, say xi , its
fuzzy image, B∼i =f∼(xi ), is given in a general symbolic form as
or, in fuzzy vector notation,
Suppose we now further generalize the situation where a fuzzy input
set, say A∼,maps to a fuzzy output through a fuzzy mapping, or
The extension principle again can be used to find this fuzzy image,
B∼
, by the following expression:
12
Fuzzy Transform (Mapping)
The preceding expression is analogous to a fuzzy composition
performed on fuzzy vectors, or b∼=a∼◦R∼ , or, in vector form,
where b∼j is the j th element of the fuzzy image B∼.
13
Fuzzy Transform (Mapping)
Example 2.
Suppose we have a fuzzy mapping, f∼, given by the following
fuzzy relation,R∼:
which represents a fuzzy mapping between the length and mass of test articles
scheduledfor flight in a space experiment. The mapping is fuzzy because of the
complicated relationship between mass and the cost to send the mass into space,
the constraints on length of the test articles fitted into the cargo section of the
spacecraft, and the scientific value of the experiment. Suppose a particular
experiment is being planned for flight, but specific mass requirements have not
been determined. For planning purposes, the mass (kilograms) is presumed to be
a fuzzy quantity described by the following membership function:
14
Fuzzy Transform (Mapping)
or, as a fuzzy vector, a∼= {0.8, 1, 0.6, 0.2, 0} kg.
The fuzzy image B∼ can be found using the extension principle
(or, equivalently, composition for this fuzzy mapping), b∼=a∼◦R∼
(recall that a set is also a one-dimensional relation). This composition
results in a fuzzy output vector describing the fuzziness in the length of
the experimental object (meters), to be used for planning purposes, or
b∼={0.8, 1, 0.8, 0.6, 0.2} m.
15
Practical Considerations
Suppose there is a mapping between elements, u, of one universe,
U, onto elements, v, of another universe, V, through a function f . Let
this mapping be described by f : u → v. Define A∼ to be a fuzzy set
on universe U; that is, A∼⊂ U. This relation is described bythe
membership functio
The mapping in Equation is said to be one-toone.
16
Practical Considerations
Example 3.
Let a fuzzy set A∼ be defined on the universe U = {1, 2, 3}. We wish to
map elements of this fuzzy set to another universe, V, under the
function
We see that the elements of V are V = {1, 3, 5}. Suppose the fuzzy set A∼is
given as
Then, the fuzzy membership function for v = f (u) = 2u − 1 would be
For cases where this functional mapping f maps products of elements from
two universes, say U1 and U2, to another universe V, and we define A∼ as a
fuzzy set on the Cartesian space U1 × U2, then
17
Practical Considerations
Example 4.
Suppose we have integers 1–10 as the elements of two identical
but different universes. Let
Then, define two fuzzy numbers
U2,respectively:
A∼and B∼ on universe U1 and
The product of (“approximately 2”) × (“approximately 6”) should map to a fuzzy
number “approximately 12,” which is a fuzzy set defined on a universe, say V,
of integers, V = 5, 6, . . . , 18, 21, as determined by the extension principle, or
18
Practical Considerations
The complexity of the extension principle increases when we consider
more than one of the combinations of the input variables, U1 and U2,
mapped to the same variable in the output space, V, that is, the mapping
is not
one-to-one. In this case, we take the maximum membership
grades of the combinations mapping to the same output variable, or, for
the following mapping, we get
19
Practical Considerations
Example 5.
We want to map ordered pairs from input universes X1 = {a, b} and
X2 ={1, 2, 3} to an output universe, Y = {x, y, z}. The mapping is
given by the crisp relation, R,
We note that this relation represents a mapping, and it does not contain
membership values. We define a fuzzy set A∼ on universe X1 and a
fuzzy set B∼on universe X2 as
20
Practical Considerations
We wish to determine the membership function of the output,
C∼= f(A∼,B∼ ), whose relational mapping, f , is described by R. This is
accomplished with the extension principle,
Hence,
21
Practical Considerations
Example 6.
Suppose we have a nonlinear system given by the harmonic
function x∼= cos(ω∼t ), where the frequency of excitation, ω∼, is a
fuzzy variable described by the membership function shown in
Figure 12.2a. The output variable, x∼, will be fuzzy because of the
fuzziness provided in the mapping from the input variable, ω∼. This
function represents a one-to-one mapping in two stages, ω∼→ ω∼t
→ x∼.
The membership function of x∼ will be determined
through the use of the extension principle, which for this example
will take on the following form:
To show the development of this expression, we will take several time
points, such as t = 0, 1, . . . . For t = 0, all values of ω∼ map into a single
point in the ω∼t domain, that is, ω∼t = 0, and into a single point in the x
universe, that is, x = 1. Hence, the membership of
22
Practical Considerations
FIGURE 12.2
Extension principle applied to x∼= cos(ω∼t ), at t =
23
Practical Considerations
x∼is simply a singleton at x = 1, that is,
24
Practical Considerations
FIGURE 12.3
Extension principle applied to x∼= cos(ω∼t ) showing (a) uncertainty in w,
(b) uncertainty in wt, and (c) the overlap in the support of x as t increases.
25
FUZZY
ARITHMETIC
Let I∼ and J∼ be two fuzzy numbers, with I∼ defined on the real line
in universe X andJ∼ defined on the real line in universe Y, and let the
symbol * denote a general arithmetic operation, that is, * ≡ {+, −, ×,
÷}. An arithmetic operation (mapping) between these two number,
denoted I∼∗J∼ , will be defined on universe Z, and can be
accomplished usingthe extension principle, as
26
FUZZY
ARITHMETIC
FIGURE 12.4
Extension principle applied to x∼= cos(ω∼t ) when t causes complete fuzziness
Equation top results in another fuzzy set, the fuzzy number resulting from the
arithmeticoperation on fuzzy numbers I∼and J∼
27
FUZZY
ARITHMETIC
Example 7.
We want to perform a simple addition (∗ ≡ +) of two fuzzy numbers.
Define a fuzzy one by the normal, convex membership function defined
on the integers,
Now, we want to add “ fuzzy one” plus “fuzzy one,” using the extension
principle
28
FUZZY
ARITHMETIC
Note that there are two ways to get the resulting membership value for
a 1 (0 + 1 and 1 +0), three ways to get a 2 (0 + 2, 1 + 1, 2 + 0),
and two ways to get a 3 (1 + 2 and 2 +1). These are accounted for
in the implementation of the extension principle.The support for a fuzzy
number, say I∼ (Chapter 4), is given as
we can find the support of the fuzzy number resulting from the arithmetic
operation,I∼∗J∼, that is,
also valid for general arithmetic operations:
29
INTERVAL ANALYSIS IN ARITHMETIC
When a = b and c = d, these interval numbers degenerate to a scalar real
number. We again define a general arithmetic property with the symbol *,
where * ≡ {+, −, ×, ÷}.
30
INTERVAL ANALYSIS IN ARITHMETIC
Symbolically, the operation
for three intervals, I, J, and K,
31
INTERVAL ANALYSIS IN ARITHMETIC
Example 8.
Consider the following example of subdistributivity. For I = [1, 2], J = [2,
3], K = [1, 4],
32
APPROXIMATE METHODS OF EXTENSION
A serious disadvantage of the discretized form of the extension principle in propagating
fuzziness for continuous-valued mappings is the irregular and erroneous membership
functions determined for the output variable if the membership functions of the input
variables are discretized for numerical convenience. The reason for this anomaly is that
the solution to the extension principle, is really a nonlinear programming problem for
continuous-valued functions. It is well known that, in any optimization process,
discretization of any variables can lead to an erroneous optimum solution because
portions of the solution space are omitted in the calculations. For example, try to plot a
10th-order curve with a series of equally spaced points; some local minimum and
maximum points on the curve are going to be missed if the discretization is not small
enough. Again, these problems do not arise because of any inherent problems in the
extension principle itself; they arise when continuous-valued functions are discretized,
then allowed to propagate from the input domain to the output domain using the
extension principle. Other methods have been proposed to ease the computational
burden in implementing the extension principle for continuous-valued functions and
mappings. Among the alternative methods proposed in the literature to avoid this
disadvantage for continuous fuzzy variables are three approaches that are summarized
here along with illustrative numerical examples. All of these approximate methods make
use of intervals, at various λ-cut levels, in defining membership functions.
33
Vertex Method
The algorithm works as follows. Any continuous membership function can
be represented by a continuous sweep of λ-cut intervals from λ = 0+
to λ = 1. Figure 12.5 shows a typical membership function with an interval
associated with a specific value of λ . Suppose we have a single-input
mapping given by y = f (x) that is to be extended for fuzzy sets, or B∼= f
(A∼), and we want to decompose A∼ into a series of λ-cut intervals, say Iλ.
When the function f (x) is continuous and monotonic on Iλ = [a, b], the
interval representing B∼at a particular value of λ , say Bλ, can be obtained
as
34
Vertex Method
FIGURE 12.5
Interval corresponding to a λ-cut level on fuzzy set A∼.
35
Vertex Method
FIGURE 12.6
Three-dimensional Cartesian region involving intervals for three input variables, x1, x2, and
x3.
Figure 12.6. Each of the input variables can be described by an interval, say Iiλ, at a
specific λ-cut, where
36
Vertex Method
The value of the interval function for a particular λ-cut can be
obtained as
where cj is the coordinate of the j th vertex representing the n-dimensional
Cartesian region.
where j = 1, 2, . . . , N and k = 1, 2, . . . , m for m extreme points in the region
37
Vertex Method
Example 9.
We wish to determine the fuzziness in the output of a simple nonlinear
mapping given by the expression y = f (x) = x(2 − x), seen in Figure 12.7a,
where the fuzzy input variable, x, has the membership function shown in Figure
12.7b. We shall solve this problem using the fuzzy vertex method at three λ-cut
levels, for
λ = 0+, 0.5, 1. As seen in Figure 12.7b, the intervals
orresponding to these λ-cuts are I0+ =[0.5, 2], I.5 = [0.75, 1.5], I1 = [1, 1]
(a single point). Since the problem is one dimensional, the vertices, cj , are
described by a single coordinate; there are
N = 21 = 2
vertices (j = 1,2). In addition, an extreme point does exist within the region of the
membership function and is determined using a derivative of the function,
df
(x)/dx = 2− 2x = 0, x0 = E1 = 1 (Ek, where k = 1). This extreme point is within each
of the three λ-cut intervals, so will beinvolved in all the following calculations for
Bλ:
FIGURE 12.7
Nonlinear function and fuzzy input membership.
38
Vertex Method
39
Vertex Method
Figure 12.8 provides a plot of the intervals B0+,B0.5, and B1 to form
the fuzzy output, y.
FIGURE 12.8
Fuzzy membership function for the output to y = x(2 −
x).
40
DSW Algorithm
The Dong, Shah, and Wong (DSW) algorithm (Dong, follShah, and
Wong, 1985) also makes use of the λ-cut representation of fuzzy sets,
but, unlike the vertex method, it uses the full λ-cut intervals in a
standard interval analysis. The DSW algorithm consists of the owing
steps:
1.
Select a λ value where 0 ≤ λ ≤ 1.
2.
Find the interval(s) in the input membership function(s) that
correspond to this λ .
3.
Using standard binary interval operations, compute the
interval for the output membership function for the selected λ-cut
level.
4.
Repeat steps 1–3 for different values of λ to complete a λ-cut
representation of the solution.
41
DSW Algorithm
Example 10.
Let us consider a nonlinear, 1D expression similar to the previous
example, or y = x(2 + x) = 2x + x2, where we again use the fuzzy
input variable shown in Figure 12.7b. The new function is shown in
Figure 12.9a, along with the fuzzy input in Figure 12.9b. Again, if we
decompose the membership function for the input into three λ-cut
intervals, for
λ = 0+, 0.5, and 1, we get the intervals I0+ = [0.5, 2]
, I0.5 = [0.75, 1.5], and
FIGURE 12.9
Nonlinear function and fuzzy input membership.
42
DSW Algorithm
I1 = [1, 1] (a single point). In terms of binary interval operations, the functional
mapping on the intervals would take place as follows for each λ-cut level:
FIGURE 12.10
Fuzzy membership function for the output to y = x(2 + x).
43
Restricted DSW Algorithm
This method, proposed by Givens and Tahani (1987), is a slight
restriction of the original DSW algorithm. Suppose we have two
interval numbers, I = [a, b] and J = [c, d]. For the special case where
neither of these intervals contains negative numbers, that is, a, b, c, d
≥0, and none of the calculations using these intervals involves
subtraction, the definitions of interval multiplication, and interval
division, can be simplified as follows:
44
Restricted DSW Algorithm
Example 11.
Let us consider the function in Example 10, y = x(2 + x) and another
nonlinear, 1D expression of the form y = x/(2 + x), where we again
use the fuzzy input variable shown in Figure 12.7b in both functions.
In interval calculations, we can represent the scalar value 2 by the
interval [2, 2]. The λ-cut interval calculations using the restricted DSW
calculations are now as follows:
Note that these three intervals for the output B∼ are identical to those in the
previous example
45
Restricted DSW Algorithm
46
Comparisons
It will be useful at this point to compare the three methods
discussed so far – the extension principle, the vertex
method, and the DSW algorithm – by applying them to the
same problem.This comparison will illustrate the problems
faced with using the extension principle on discretized
membership functions, as compared to the other two methods
47
Comparisons
Example 12.
We define fuzzy sets X∼and Y∼ with the membership functions as shown in
Figure 12.13. We will use the following methods to compute X∼∗Y∼ and to
demonstrate the similarity of results:
• the extension principle
• the vertex method
• the DSW algorithm.
FIGURE 12.13
Fuzzy sets X∼and Y∼.
48
Comparisons
the extension principle:
and
Their product would then give us
The result of the operation X∼×Y∼ for a discretization level of
seven points is plotted in Figure 12.14a.
49
Comparisons
Vertex method: I0+: Support for X is the interval [1, 7] and support for Y
is the interval [2,8].
Therefore, min = 2, max = 56, and B0+ = [2, 56]. I0.33 : X[2, 6], Y[3,
7].
Therefore, min = 6, max = 42,
6].
and B0.33 = [6, 42].
Therefore, min = 12, max = 30, and B0.66 = [12, 30].
5].
I0.66 : X[3, 5],Y[4,
I1.0 : X[4, 4], Y[5,
Therefore, min = 20, max = 20, and B1.0 = [20, 20].
50
Comparisons
FIGURE 12.14
X∼×Y∼ for increasing discretization of both X and Y (both variables are discretized
for the same number of points): (a) 7 points; (b) 13 points; (c) 23 points; (d)
63 points.
51
Comparisons
DSW method:
FIGURE 12.15
Output profile of X∼×Y∼determined using the vertex method.
FIGURE 12.16
Output profile of X∼×Y∼ determined using the DSW algorithm.
52