RESULTADOS Y VALORACIÓN GLOBAL DE LA PROPUESTA

3.4.1 Matrices of Truth Values

We will have several situations in which we deal with arrays of truth values. For example, if we have digitized a membership function by sampling it at discrete values of its numeric argument, we have created a vector of truth values. In the fol- lowing section on fuzzy relations, we will deal with matrices of truth values. We will call these fuzzy matrices. We now define an important operation we can employ on

Figure 3.5 Membership functions of a linguistic variable speed.

such arrays. Roughly speaking, the operation of addition in ordinary matrices is analogous to the fuzzy logical OR, and the operation of multiplication on ordinary matrices is analogous to the fuzzy logical AND.

The most important operation on matrices of truth values is called composition, and is analogous to matrix multiplication. Suppose I have two fuzzy matrices A and B. To compose these matrices, they must meet the same size compatibility restric- tions as in ordinary matrix multiplication. Let A have l rows and m columns, and let B have m rows and n columns. They are compatible, since the number of columns of A equals the number of rows of B. The composition of A and B is denoted AWB, and

will produce a matrix C having l rows and n columns.

In ordinary matrix multiplication, we would obtain cijby summing the product aikbkjover k. We could write this as

ci,j¼ai,1b1,jþai,2b2,jþ þai,nbn,j

In fuzzy matrix composition, we obtain cijby repeatedly ORing (aikAND bkj) over all values of k, using min – max logic by default. This procedure gives:

cij¼(ai,1AND b1,j)OR(ai,2AND b2,j)OR OR(ai,nAND bn,j) (3:22)

The operation of composing matrices A and B is writtenC¼AWB. For example,

suppose we have these two fuzzy matrices to be composed:

A¼ 0:2 0:4 0:6 0:3 0:6 0:9 B¼ 0:5 1 0:7 0:5 1 0 2 6 4 3 7 5 (3:23)

We first compute c1,1, and apply (3.22) to the first row of A and the first column of B. The minimum of 0.2 and 0.5 is 0.2; min(0.4, 0.7) is 0.4; and min(0.6, 1) is 0.6; and the maximum of these minima is 0.6. Continuing this procedure, we obtain C as:

C¼ 0:6 0:4 0:9 0:5

(3:24)

3.4.2 Relations Between Sets

We have two universal sets X and Y. ByXYwe mean the set of all ordered pairs (x, y) for x in X andyin Y. A fuzzy relation R onXYis a fuzzy subset ofXY; that is, for each (x, y) pair we have a number ranging from 0 to 1, a measure of the relationship between x and y.

As a simple example of a fuzzy relation let X¼{John,Jim,Bill} and

Y ¼{Fred,Mike,Sam}. The fuzzy relation R between X and Y, which we will call “resemblance”, might be as shown in Table 3.3.

Fuzzy relations may be given as matrices if the sets involved are discrete, or ana- lytically if the sets are continuous, usually numbers from the real line. The members of X and Y are often the members of fuzzy sets A and B; the fuzzy relation between their members is often a function of their grades of membership in A and in B.

Theoretical fuzzy logical inference involves an important application of fuzzy relations, in which the fuzzy relation is usually an implication, as discussed in Section 3.1. (While this theory is important to fuzzy logicians, it is much less so to builders of fuzzy expert systems, as we shall see below.) First, we must choose an implication operator valid for classical logic. Suppose we picked the one given in equation (3. 8). Next let A and B be two fuzzy numbers with membership func- tions A(x) and B(x), respectively, shown in Figure 3.6. (Here X¼Y¼the set of real numbers.) The fuzzy relation, from equation (3.8) is

R(x,y)¼min(1, 1A(x)þB(y)) (3:25)

for x and y any real numbers. Then tv(A!B)¼R(x, y) in fuzzy logic. If A and B are the two fuzzy numbers shown in Figure 3.6, the quadrant (x0, y0) of the fuzzy relation R(x, y) is shown in Figure 3.7. (Other quadrants are not shown, since they would obscure the graph.) R(x, y) is symmetric about the (y, R) plane, and is everywhere 1 outside the region (24, x, 4).

In fuzzy inference, we will need to compose fuzzy relations. If the fuzzy relation exists in matrix form, the procedure in Section 3.4 may be followed. However, the fuzzy relation may be given as a continuous function. Let R be a fuzzy relation on

XY and S another fuzzy relation onYZ. Then R(x, y) is a number in [0, 1] for all x in X and allyin Y, and S(y, z) has its values in [0, 1] for all y in Y and all z in Z. We compose R and S to get T, a fuzzy relation on XZ. This is written asRWS¼T. We compute T as follows:

T(x,z)¼supy{min{R(x,y),S(y,z)}} (3:26)

In equation (3.26), “sup” stands for supremum, which must be used in place of max for many infinite sets. For example, the sup of x in [0, 1)¼1, but this interval has no max. On the other hand, sup of x in [0, 1]¼max of x in [0, 1]¼1. Other AND type operators [eqs. (3.2) – (3.3)] may be used in place of min in equation (3.26).

TABLE 3.3 A Fuzzy Relation

Fred Mike Sam

John 0.2 0.8 0.5

Jim 0.9 0.3 0.0

Bill 0.6 0.4 0.7

As an example of composition, suppose the universe X has three members, and the universe Y has two members. Let the relation between X and Y be given in (3.27) as matrix R: R¼ 0:4 0:8 0:2 0:9 1 0 2 4 3 5 (3:27)

Figure 3.6 Membership functions of two fuzzy numbers A and B.

Figure 3.7 Part of one quadrant of fuzzy relation A!B between fuzzy numbers A(x) and

Now suppose that a fuzzy set A, a fuzzy subset of X, has grades of membership

A¼(0:5 0:8 0:2) (3:28)

and we wish to compose fuzzy set A with R to get the grades of membership in fuzzy set B, a fuzzy subset of Y. Then

Bj¼max(min(A1,Ri,j), min(A2,Ri,j). . . (3:29)

Applying (3.29) to the data in (3.27) and (3.28), we obtain

B¼(0:4 0:8) (3:30)