Integration of Fuzzy Functions

7 FUZZY ANALYSIS

7.3 Integration of Fuzzy Functions

a

a

o

r\'

^--

^t

^---

^i'

---

max

f(Xj)

1=1, . . . ,5

Figure 7-4. The maximum of a fuzzy function.

IR

Dubois and Prade [1980a, p. IO1] suggest additional possible interpretations of fuzzy extrema, which might be very appropriate in certain situations, However, we shall not discuss them here and rather shall proceed to consider possible notions of the integral of a fuzzy set or a fuzzy function .

speaking, monotonous functionals), while Lebesque integrals are linear ones"

[Sugeno 1977, p. 92].

We shall focus our attention on approaches along the line of Riemann inte- grals. The main references for the following are Dubois and Prade [1980a, 1982b], Aumann [1965], and Nguyen [1978].

The classical concept of integration of a real-valued function over a closed interval can be generalized in four ways: The function can be a fuzzy function that is to be integrated over a crisp interval, or it can be integrated over a fuzzy interval (that is, an interval with fuzzy foundations) . Alternatively, we may con- sider integrating a fuzzy function as defined in definitions 7-1 or 7-2 over a crisp or a fuzzy interval.

7.3.1 Integration of

a

Fuzzy Function over

a

Crisp Interval

We shall now consider a fuzzy functionj', according to definition 7-2, which shall be integrated over the crisp interval[a, b).The fuzzy functionj'(x) is supposed to be a fuzzy number, that is, a piecewise continuous convexnormalizedfuzzy set onR

We shall further assume that the a-level curves (see definition 2.3)!lj(x)(Y)= a for all a E [0, 1] and a and x as parameters have exactly two continuous solu- tions, y

=

f~(x) and y

=

f-;.(x),for a ;=1 and only one for a

=

^1.f~ and

r:

^are

defined such that

for all a'~a .

The integral of any continuous a-level curve ofjover [a, b)always exists.

One may now define the integral [(a, b) ofj(x) over [a, b) as a fuzzy set in which the degree of membership a is assigned to the integral of any a-level curve ofj(x)over [a, b).

Definition 7-5

Letfix) be a fuzzy function from [a, b) ~ IR to IR such that "IxE = [a, b)j(x) is a fuzzy number andf-;'(x)andf~(x)are a-level curves as defined above. The integralofj(x)over [a,b)is then defined to be the fuzzy set

[(a , b)=

{(J:

^{f ;;(x)dx}

⁺ J: f~(x)

^dx,

an

This definition is consistent with the extension principle according to which

"r

_a!^{(y)= sup inf}gey x e [a,b1^{!l! (x )(g(x», yEIR} y=jgg

where y = {g: [a, b] -t ~Igintegrable} see Dubois and Prade [1980a, p. 107;

1982, p. 5]).

The determination of the integral

1

^(a,b)becomes somewhat easier if the fuzzy function is assumed to be of the LR type (see definition 5-6). We shalI therefore assume thatl(x)=(f(x), sex), t(X»/l1is a fuzzy number in LR representation for alIx E [a,b].f,s,and tare assumed to be positive integrable functions on [a, b].

Dubois and Prade [1980a, p. 109] have shown that under these conditions lea,b)=

(J:

^f(x)dx,

J:

^s(x)dx,

J:

^t(x)dxt

It is then sufficient to integrate the mean value and the spread functions oflex) over [a,b], and the result will again be an LR fuzzy number.

Example 7-5

Consider the fuzzy function lex) = (f(x), sex), t(X»LRwith the mean function f(x)

= r,

the spread functions sex)

=

x/4, and

t(x)=-x 2 L(x)=--1

1+x²

R(x)= ₁₊₂1

I I

_x

Determine the integral froma= 1 tob=4, that is, compute

r ^f.

According to the above formula, we compute I

J:

^f(x)dx

=f

^x2dx

=

²¹

f

^abs(x)dx

^{= 5,4}

^{I -dxx}4

⁼

^1.875

f

^abt(x)dx

^{= 5,4}

^{I -dxX2}

⁼

^3.75

This yields the fuzzy number lea, b) = (21, 1.875, 3.75)LR as the value of the fuzzy integral.

Some Properties of Integrals of Fuzzy Functions. LetAa be the a-level set of the fuzzy set

A.

The support

SeA)

^of

A

^{is then}

SeA)

^{= U}

An.

The fuzzy set

A

can now be written as ae[O ,I]

where

A

=

_aE[O,I]^{U aAu}

=

_aE[O,1]^{U {(x,/-laAa(x)lxEAu)}}

(x ) ={a forxEAu /-laA"

°

^clor xli:F'<J.^>1

(see Nguyen [1978, p, 369]).

Let

A

represent a fuzzy integral, that is,

A= f)

then

J ^{] =}

^U

a(J])

I a E[O,I] I 0.

= U_aE[O,I]

a(f]a)

I

Definition 7-6 [Dubois and Prade 1982a, p, 6]

L]

satisfies the commutativity condition

iffVaE[0,

l](f]) =f]a

_I

0. I

Dubois and Prade [1982a, p. 6] have proved the following properties of fuzzy integrals, which are partly a straightforward analogy of crisp analysis,

Theorem 7-1

Let] be a fuzzy function; then

f -

_I

f= f=- f ^fb-

_a

r.

_b

where the fuzzy integrals are fuzzy sets with the membership functions /-l_{- a}fb_f(u)=

"r

_af^(-u)Vu

Theorem 7-2

LetI andl'be two adjacent intervalsI

=

^{[a, b], I'}

=

^[b,c] and a fuzzy function

1:

^[a,c] ~ P(~).Then

whereEBdenotes the extended addition of fuzzy sets, which is defined in analogy to the subtraction of fuzzy numbers (see chapter 5).

Let

j

and

g

be fuzzy functions. Then

j

EB

g

is pointwise defined by

(] EB g)(u)

⁼

j(u) ^EB g(u),

^UEX

(This is a straightforward application of the extension principle from chapter 5.1.)

Theorem -7-3

Let

j

and

g

be fuzzy functions whose supports are bounded. Then

f/j ^EB g)

^d

f,J ^EB Lg L(]EBg)= f,JEB Lg

iff the

commutativity condition

is satisfied for

fj

and

L ^g.

7.3.2 Integration of

a

(Crisp) Real-Valued Function over

a

Fuzzy Interval

(7.1) (7.2)

We now consider a case for which Dubois and Prade [1982a, p. 106]proposed a quite interesting solution:Afuzzy domain ;g; of the real line ~is assumed to be bounded by two normalized convex fuzzy sets, the membership functions of which are J.la(x)and J.lb(X), respectively. (See figure 7-5.) J.la(x) andJ.lb(X)can be interpreted as the degrees (of confidence) to which x can be considered a lower or upper bound of;g; .If

ao

andb_oare the lower/upper limits of the supports ofii orb, then ao or bo are related to each other by

ao

=infS(ii) ~sup S(b)=boo

Definition 7-7

Letfbe a real-valued function that is integrable in the intervalJ = [ao, bo]; then according to the extension principle, the membership function of the integral

f:9'f

is given by

J.lWt(z)=supmintu,(x),J.lb(y»

x ,ye J

z= ff

l+---::ll"'r---,.,.---

IR

O...

_~---Jl---~"---~-

Figure 7-5. Fuzzily bounded interval.

Let F(x)=

r

^{fey) dy,C E J (F}is the antiderivative of f). Then, using the exten- sion principle again, the membership function ofF(ii), ii E Pc~),is given by

~f(a)(Z)= sup _~a(x)

x:z=F(x l

Proposition 7-1 [Dubois and Prade 1982b, p. 106]

J2iJ

f

^{= F(b)}

e

^F(ii)

where

e

denotes the extended subtraction of fuzzy sets.

Proofs of proposition 7-1 and of the following propositions can be found in Dubois and Prade [1982b, pp. 107-109].

A possible interpretation of proposition 7-1 is as follows: If ii and bare normalized convex fuzzy sets, then J2iJj is the interval between "worst" and

"best" values for different levels of confidence indicated by the respective degrees of membership (see also Dubois and Prade [1988a, pp. 34-36]).

Example 7-6 Let

ii= {(4, .8), (5, I), (6,A)}

b

={(6, .7), (7, I), (8, .2)}

f(x)=2, xE[ao,bo]=[4,8]

Then

f~

^{f(x) dx= f: 2dx= 2x}

I~

The detailed computational results are:

(a, b) J:2dx min(~x(a), ~x(b))

(4,6) 4 .7

(4,7) 6 .8

(4,8) 8 .2

(5,6) 2 .7

(5, 7) 4 1.0

(5, 8) 6 .2

(6,6) 0 .4

(6, 7) 2 .4

(6,8) 4 .2

Hence choosing the maximum of the membership values for each value of the integral yields f~f=^{{(a, .4),} (2, .7), (4, 1), (6, .8), (8, .2)}.

Some properties of the integral discussed above are listed in propositions 7-2 to 7-4 below. Their proofs, as well as descriptions of other approaches to "fuzzy integration," can again be found in Dubois and Prade [1982a, pp. 107-108].

Proposition 7-2

Let

f

and g be two functionsf, g:I -) Iffi,integrable onI.Then f:(f+g)

c

f: f EB f: g

where EB denotes the extended addition (see chapter 5).

Example 7-7 Let

f(x)=2x-3 g(x)=-2x+3

a

={(I,.8), (2,1), (3,.4)}

fj =({3, .7), (4, I), (5, .3)}

So

f

^{a f(x)dxb =[X2- 3xtb}

f

^{a g(x)dxb}

⁼

^[_X2+5xtb

f

^{a f(x)b +g(x)dx=[2x]ab}

In analogy to example 7-6, we obtain

f: ^f

⁼{(O, .4), (2, .7), (4, .4), (6,1),(10, .3), (12, .3)}

f:

^{g={(-6, .3),(-4,.3), (-2,}.1), (0, .8), (2, .7)}

Applying the formula for the extended addition according to the extension prin- ciple (see section 5.3) yield s

f: ^f

⁺

f:

^g

⁼

^{{(-6, .3),(-4,}.3), (-2, .4), (0, .7), (2, .7), (4,.1),(6, .8), (8, .7), (10, .3), (I2, .3), (I4,.3)}

Similarly to example7-6,we compute

f:

^(j+g)={(O, .4), (2, .7), (4, 1), (6, .8), (8, .3) Now we can easily verify that

f: ^{f EB} f:

^g

_~ f:

^(j+g)

Proposition 7-3

Iff, g:I ~I?'"orf, g:I ~K, then equality holds:

Proposition 7-4

Let~

=

(ii, b), ~'

=

(ii, C),and ~"

=

(c,b).Then the following relationship s hold:

f~J ^s f~ ^{fi EB} f~.Ii

f~J

⁼

f~ ^fi

^EB

f~J2

^{iff CEIR}

(7.3) (7.4)