DESARROLLO DE LAS SESIONES

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 Lotfi 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 P¼_{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

M4Nwe assume N(0)¼0. The membership function for P is determined using the extension principle as follows:

P(z)¼sup_x,y{min(M(x),N(y))jxy¼z} (4:40)

Suppose we wish to add a fuzzy M and a fuzzy N to yield fuzzy number P. Let us first consider evaluating P(z¼4). We consider all pairs of values of x and y that sum to 4. For each suchx2ypair, 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 defining this second method, we discuss the relationship between the two procedures.

4.5.2 Alpha-Cut and Interval Arithmetic

First, we will discuss alpha-cuts and then interval arithmetic. Ifa[(0, 1, the alpha- cut of A, a fuzzy subset of universal set X, writtenA½a, is defined 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 define 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 infinite. Practically, if we assume A(x) is effectively 0 for xa3s and for xaþ3s, then A½0 ¼ ½a3s,aþ3s.

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 writeA½a ¼ ½a1(a),a2(a)for fuzzy number A, where theai(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 x¼2, and straight line segments for its sides, then we seeA½a ¼ ½1þa, 42afor alpha in [0, 1]. If

A½0 ¼ ½a1,a2, then we write: (1) A.0 ifa1.0; (2)A0for a10; (3) A,0 meansa2 ,0; and (4)A0,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 ½c,d ¼{xyjx in½a,b,y in ½c,d} (4:41) It follows that ½a, b þ ½c, d ¼ ½aþc, bþd (4:42) ½a, b ½c, d ¼ ½ad, bc (4:43) ½a,b4½c,d ¼ ½a,b 1 d, 1 c (4:44)

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

½a,b ½c,d ¼ ½k,n, for

k¼min{ac,ad,bc,bd}

n¼max{ac,ad,bc,bd}

Multiplication and division may be simplified 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

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

½a, b½c, d ¼ ½ad, bc

Now we may return to fuzzy arithmetic. For fuzzy numbers A and B let

A½a ¼ ½a1(a),a2(a),B½a ¼ ½b1(a),b2(a)

Using the alpha-cut and interval arithmetic method we first calculate P¼AþB as P½a ¼A½a þB½a ¼ ½a1(a)þb1(a),a2(a)þb2(a). This, of course, gives alpha-cuts of the sum P. For P¼A2B, we get P½a ¼ ½a1(a)b2(a),a2(a) b1(a). In multiplication P¼AB we find P as P½a ¼A½aB½a. 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, thenP¼A4Bis defined and alpha-cuts of P are calculated asP½a ¼ ½a1(a),a2(a) 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 AþB,

A2B,A4B 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 P¼(AþB)/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.