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71

4.4 HEDGES

A membership function modi¬ed by hedges.

Figure 4.8

shown graphically as a fuzzy 5 in Figure 4.7. We assume that the fuzzy number is

symmetrical, with a single value of the argument (the central value) at which the

membership is one. The shape of the resulting fuzzy number is assumed separately

speci¬ed, and may be linear, s-shaped (piecewise quadratic) or normal.

The second type of hedge is applied to truth values. “very Small” reduces the

truth value of Small; “somewhat Small” increases the truth value. Usually, the orig-

inal truth value is raised to a power greater than 1 for terms that reduce truth values,

and less than 1 for terms that increase truth values. Table 4.2 de¬nes hedges for

modi¬cation of truth values, and Figure 4.8 gives a sample membership function

modi¬ed by hedges. Again, the hedges employed by FLOPS are similar to those

used by Cox.

As shown in Figure 4.8, hedges can operate on membership functions producing

modi¬ed membership functions, and can be used to modify clauses in fuzzy

propositions. Consider the fuzzy proposition

speed is Fast

Figure 4.9 Membership function Very_Small created without hedges.

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

This can be modi¬ed by substituting very Fast, somewhat fast, and so on. If the grade

of membership of Fast in speed is 0.5, the truth value of (somewhat Fast) would be

0.51/2 ¼ 0.707, of (slightly Fast) 0.51/3 ¼ 0.794, of (very Fast) 0.52 ¼ 0.25, and of

(extremely fast) 0.53 ¼ 0.125.

However, we have found that in practice such hedges applied to membership

functions can be confusing and in¬‚exible. It is possible to use separate linguistic

terms such as Slow and Very_slow, each with its own membership function,

rather than use hedges applied to membership functions. For example, the member-

ship function for Very_Small in Figure 4.9 cannot be created by using the usual

power-based hedges just described.

4.5 FUZZY ARITHMETIC

Fuzzy arithmetic is concerned with the addition, subtraction, multiplication, and

division of fuzzy numbers. There are two methods of performing fuzzy arith-

metic: (1) from the extension principle; and (2) using alpha-cuts and interval

arithmetic.

4.5.1 Extension Principle

The extension principle, due like so much of fuzzy theory to Lot¬ Zadeh, is a power-

ful and very general tool used to fuzzify crisp operations, crisp equations, crisp

functions, and so on. To fuzzify arithmetic, let * denote addition, subtraction,

multiplication, or division of real numbers. We wish to compute P ¼ M * N for

fuzzy number M and N producing fuzzy number P. If * is division we need to

assume that zero does not have positive membership in the divisor. So, for

M 4 N we assume N(0) ¼ 0. The membership function for P is determined using

the extension principle as follows:

P(z) ¼ supx,y {min(M(x), N( y))jx Ã y ¼ z} (4:40)

Suppose we wish to add a fuzzy M and a fuzzy N to yield fuzzy number P. Let us ¬rst

consider evaluating P(z ¼ 4). We consider all pairs of values of x and y that sum to

4. For each such x 2 y pair, we calculate M(x) and N( y). We take the minimum of

M(x) and N( y). We now take the maximum of all these minima, and that is the value

of P(4).

If we can evaluate (4.40) analytically, we have a very general method for general-

izing crisp operators to fuzzy operators. However, unless this can be done analytic-

ally, the procedure is computationally unfriendly, involving two nested loops. So, let

us now present a second procedure that is more easily incorporated into computer

programs. After de¬ning this second method, we discuss the relationship between

the two procedures.

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73

4.5 FUZZY ARITHMETIC

4.5.2 Alpha-Cut and Interval Arithmetic

First, we will discuss alpha-cuts and then interval arithmetic. If a [ (0, 1, the alpha-

cut of A, a fuzzy subset of universal set X, written A½a, is de¬ned to be the crisp set

{xjA(x) ! a}. This set is the collection of all the x in X whose membership value is

at least alpha. We must separately de¬ne A[0], because otherwise it will be all of

X. A[0] will be simply the base of the fuzzy number. For the triangular fuzzy

number A in Figure 3.1, A[0] ¼ [a, c]. For the s-shape fuzzy number in Figure 3.2

A[0] ¼ [a, d ]; and for the trapezoidal fuzzy number in Figure 3.4 A[0] ¼ [a, c].

For the normal fuzzy number in Figure 3.3, the support is in¬nite. Practically,

if we assume A(x) is effectively 0 for x a À 3s and for x ! a þ 3s, then

A½0 ¼ ½a À 3s, a þ 3s.

The core of a fuzzy number is the set A[1] and the support is the interval A[0]. A

fuzzy set is normal if the core is nonempty. All our fuzzy number will be normal.

The alpha-cut of a fuzzy number is always a closed, bounded, interval. We will

assume that A[0], for normal fuzzy numbers, is the interval given above. So we

will write A½a ¼ ½a1 (a), a2 (a) for fuzzy number A, where the ai (a) give the end

points of the interval that are, in general, functions of alpha. For example, if A is

a triangular fuzzy number with base the interval [1, 4], vertex at x ¼ 2, and straight

line segments for its sides, then we see A½a ¼ ½1 þ a, 4 À 2a for alpha in [0, 1]. If

A½0 ¼ ½a1 , a2 , then we write: (1) A . 0 if a1 . 0; (2) A ! 0 for a1 ! 0; (3) A , 0

means a2 , 0; and (4) A 0, whenever a2 0.

Next we need to review the basic ideas of interval arithmetic. Let [a, b] and [c, d]

be two closed, bounded, intervals. If * denotes addition, subtraction, multiplication

or division of real numbers, then we extend it to interval as follows:

½a, b Ã ½c, d ¼ {x Ã yjx in ½a, b, y in ½c, d } (4:41)

It follows that

½a, b þ ½c, d ¼ ½a þ c, b þ d (4:42)

½a, b À ½c, d ¼ ½a À d, b À c (4:43)

11

½a, b 4 ½c, d ¼ ½a, b Á , (4:44)

dc

if zero does not belong to [c, d ], and

½a, b Á ½c, d ¼ ½k, n, for

k ¼ min{ac, ad, bc, bd}

n ¼ max{ac, ad, bc, bd}

Multiplication and division may be simpli¬ed if we know a . 0, c . 0 or

b , 0, c . 0, and so on. For example, if a . 0 and c . 0, then

½a, b½c, d ¼ ½ac, bd

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

but if b , 0 and c . 0 we see that

½a, b½c, d ¼ ½ad, bc