A½a ¼ ½a1 (a), a2 (a), B½a ¼ ½b1 (a), b2 (a)

Using the alpha-cut and interval arithmetic method we ¬rst calculate P ¼ A þ B

as P½a ¼ A½a þ B½a ¼ ½a1 (a) þ b1 (a), a2 (a) þ b2 (a). This, of course, gives

alpha-cuts of the sum P. For P ¼ A 2 B, we get P½a ¼ ½a1 (a) À b2 (a), a2 (a) À

b1 (a). In multiplication P ¼ AB we ¬nd P as P½a ¼ A½aB½a. If A . 0 and

B . 0, then P½a ¼ ½a1 (a)b1 (a), a2 (a)b2 (a). When zero does not belong to the

support of B, then P ¼ A 4 B is de¬ned and alpha-cuts of P are calculated as P½a ¼

½a1 (a), a2 (a) Á ½1=b2 (a), 1=b2 (a). You are asked to complete some of these calcu-

lations in the problems for certain fuzzy numbers.

4.5.3 Comparison of Alpha-Cut and Interval Arithmetic Methods

The alpha-cut and interval arithmetic procedure is easily incorporated into computer

programs since we discretize it only computing for say alpha equal to 0. 0, 0.1, . . . ,

0.9, and 1.0. This is the method we will use for fuzzy arithmetic in this book.

It is well known that the two procedures compute the same value for A þ B,

A 2 B, A 4 B and A . B for fuzzy number A and B. However, this is not true for

the evaluation of all fuzzy expressions. For example, for the fuzzy expression

P ¼ (A þ B)/A the two methods can calculate different answers for P. Even so,

we will be using the alpha-cut and interval arithmetic procedure in this book to

do fuzzy arithmetic.

4.6 COMPARISONS BETWEEN FUZZY NUMBERS

4.6.1 Using the Extension Principle

If A and B are two fuzzy numbers, we will need to evaluate the approximate com-

parisons A , B, A ,¼ B, A ¼ B, A .¼ B, A . B and A ,. B, the

fuzzy equivalents of the conventional Boolean numerical comparisons supplied by

most computer languages. The truth value of any one of these comparisons will be a

number in [0, 1]. If either A or B is a single-valued scalar number, we simply replace

it by a singleton fuzzy number. (In FLOPS, at least one of the numbers to be com-

pared must be initially a fuzzy number.)

The simplest comparison to make is to compare A and B for equality. We use the

extension principle in Section 4.5.1, equation (4.40), for this comparison:

tv(A ¼ B) ¼ supx,y {min(M(x), N( y))jx ¼ y} (4:45)

Evaluating (4.45) is simpler than it looks for fuzzy numbers that are ¬rst mono-

tonic upward, then monotonic downward, as the argument increases; we look for the

highest point where the two fuzzy numbers intersect.

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75

4.6 COMPARISONS BETWEEN FUZZY NUMBERS

A fuzzy number A.

Figure 4.10

We now consider approximate fuzzy numerical comparisons such as (A , B).

This implies that we wish to compare fuzzy number A, shown in Figure 4.10, to

another fuzzy number that is less than C. However, the fuzzy literature does not

de¬ne such a fuzzy number; we now proceed to de¬ne a fuzzy number that is

approximately less than C. We ¬rst create a new fuzzy number C such that C is

not equal to A, or C ¼ NOT A, as shown in Figure 4.11.

Next, we create a masking fuzzy number D that is less than A in the Boolean

true“ false sense. Our truth value for D(x) is 1 so long as A(x) is less than its

maximum with x increasing from À1; from there on, as x increases, D(x) is

0. Denote the value of x at the point where A(x) ¬rst achieves its maximum value

by x(Amax). This operation results in D , A shown in Figure 4.12.

We now calculate fuzzy number E that is approximately less than A by demand-

ing that E be (NOT A) AND D, shown in Figure 4.13.

To carry out the comparison, we simply test whether the fuzzy number E, ,A, we

have created equals A using equation (4.45).

To carry out the comparison A .¼ B, we create the fuzzy number F . ¼ B by

ORing A and F , A, as in Figure 4.14.

A fuzzy number C, unequal to A ¼ NOT A.

Figure 4.11

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76 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II

A fuzzy number D , A (in the Boolean sense); D(x) ¼ true if x , x(Amax),

Figure 4.12

else false.

4.6.2 Alternate Method

If A and B are two fuzzy numbers we will need to evaluate, in our fuzzy expert

system, the approximate comparisons

A B, A , B, A ¼ B, A = B, A ! B and A . B

The truth value of any one of these comparisons will be a number in [0, 1]. We

will use the notation v( ) for this value. So, v(A , B) is in [0, 1] and the truth

value associated with A , B is v(A , B); that is, tv(A , B) ¼ v(A , B).

We will ¬rst need to specify some auxiliary fuzzy sets and concepts before we

can de¬ne v( ). This section is based on Klir and Yuan (1995).

For a given fuzzy number A(x), we will construct four other fuzzy sets. Call the

core of A the interval over which the membership of x is one. If the core of A is

[m, m], a single number m, then set d ¼ m. If the core of A is an interval [a, b],

then d ¼ (a þ b)=2. De¬ne the fuzzy set L to have membership value 1 on the

A fuzzy number E , A ¼ ,NOT A AND (E , 30) or E ,¼ A.

Figure 4.13

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77

4.6 COMPARISONS BETWEEN FUZZY NUMBERS

A fuzzy number E ,¼ A ¼ ,NOT A OR b , A, or E ,¼ A.

Figure 4.14

interval (À1, d and zero otherwise. Similarly, fuzzy set R has membership 1 on

½d, 1) and 0 otherwise. Then, de¬ne fuzzy sets

(,A) ¼ Ac > L using T m

( A) ¼ (,A) < A using CL

(.A) ¼ Ac > R for T m

(!A) ¼ (.A) < A for C L

We use CL for OR since (,A) and A, and (.A) and A, are maximally nega-

tively associated (see Section 4.4.1). Fuzzy sets (,A) and ( A) are shown in

Figure 4.15 for triangular fuzzy number A.

Fuzzy numbers A, triangular; , A; and ,¼ A.

Figure 4.15

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78 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II