Algebraic characteristics of extended fuzzy numbers

INFORUATION

SCIENCES

Algebraic Characteristics

54,103-130

(1991)

103

of Extended Fuzzy Numbers

RENHONG ZHAO AND RAKESH GOVIND* Department of Chemical Engineering, Mail Location 171, University of Cincinnati, Cincinnati, Ohio 45221

ABSTRACT Extended fuzzy numbers, previously called fuzzy intervals, are discussed by using the resolution identity and the extension principle. The regularity and the spread are defined for describing the algebraic properties of extended fuzzy numbers. Arithmetic operations on a-level set intervals are suggested instead of general set operations in order to reduce the amount of computation. A sufficient and necessary condition for solving A + X= C is derived. Tbe exact solution for A + X = C is obtained. Finally, A - A = 0 (a fuzzification of the crisp O), which is a natural extension from the nonfuzzy field, is proved.

INTRODUCTION Previous

studies

on fuzzy numbers

and mathematical

operations

on them

theory [l, 2, 4, 5, 7, 8, 10, 13-151 are very useful for handling uncertainties in chemical engineering processes. Most chemical processes are complex, and there are uncertainties in some of their parameters, such as mass transfer coefficients, physical properties, plate efficiencies, reaction rate constants, etc., due to imprecise measurement. The problem of imprecisely known parameters is one of the difficulties in process modeling in addition to process understanding and the size and complexity of the model. In practice, chemical engineers often use subjective estimates to approximate real values. The transition membership function in fuzzy set theory is a more precise and effective tool which can be used to represent subjectively evaluated uncertainties. In this paper, extended fuzzy numbers, which were thought of as fuzzy intervals at first, but may have more practical application than generally defined fuzzy numbers, are thoroughly discussed by using the resolution identity and the extension principle. Formulas for arithmetic operations on

based

on fuzzy

set

*To whom correspondence

should be addressed.

OElsevier Science Publishing Go., Inc. 1991 655 Avenue of the Americas, New York, NY 10010

0020-0255/91/$03.50

104

RENHONG

ZHAO AND RAKESH

GOVIND

extended fuzzy numbers with continuous membership functions are derived by using operations on bound points of all a-level set intervals instead of by using general set operations. Based on these formulas, a sufficient and necessary condition for solving the algebraic equation A + X = C is derived. The exact solution of A + X = C is obtained and discussed. Most results derived in this paper can be easily extended to fuzzy numbers as defined in the literature.

1.

FUZZY

NUMBERS

Let x be a continuous real variable restricted by a possibility function &Y) E [O, 11 which satisfies the following assumptions:

distribution

1.1.

A BRIEF

REVIEW

THE DEFINITION

OF PREVIOUSLY

OF FUZZY NUMBERS

DEFINED [2, 141

(1) p(x) is piecewise continuous; (2) p(x) is a convex fuzzy set; (3) there exists exactly one real number

x0 such that &~a) = 1.

A fuzzy set which satisfies above requirements is called a fuzzy number. This definition models a fuzzy number which is expressed linguistically “approximately xc.” x0 is called the mean value of p(x). 1.2.

ARITHMETIC

OPERATIONS

as

ON FUZZY NUMBERS

Using the extension principle given by L. A. Zadeh [14], a binary operation defined in R can be extended to two noninteractive fuzzy numbers A and B by the following equation:

Using the definitions given above, Mizumoto and Tanaka derived algebraic properties of fuzzy numbers [7, 81. One important conclusion following theorem:

some is the

THEOREM. For any j&y number A, there exist no inverse fuzzy numbers B and C for addition and multiplication such that

A+B=O, Here 0 and 1 are crisp numbers.

AxC=l.

EXTENDED 1.3.

FUZZY

COMPUTATION

NUMBERS

105

REDUCTION

Computational efficiency is of particular importance for practical application of fuzzy numbers. But the operations given in Equation (1.2-l) require extensive computations except for some operations on a few specific kinds of fuzzy numbers, such as addition and subtraction of triangular fuzzy numbers or normal fuzzy numbers. The computations necessary for some arithmetic operations are considerably simplified on introducing the L-R fuzzy numbers due to Dubois and Prade [l]. Using a triplet (m, (Y,p) for each L-R fuzzy number, exact formulas have been given for addition and subtraction [l]. But for multiplication and division of two L-R numbers, the results are no longer L-R numbers. No exact formulas can be derived. Some approximate expressions under certain assumptions have been suggested [l]. The fuzzy numbers with continuous membership functions are considered to be an extension of the concept of confidence interval by Kaufmann and Gupta [5]. Instead of considering a confidence interval at one unique level, it is considered more generally at all levels from 0 to 1. It is also proved that arithmetic operations on fuzzy numbers based on the extension principle can be decomposed into operations on confidence intervals at all levels from 0 to 1. The problem in the fuzzy field is transformed to a problem in the nonfuzzy field with no information loss. Kaufmann and Gupta simplified the computations for triangular fuzzy numbers and proved results for L-R numbers. They also proved that the lack of additive and multiplicative inverses results from the characteristics of operations on confidence intervals [5].

1.4. SELECTING THE “BEST NONFUZZY MEMBERSHIP FUNCTION

VALUE”

FROM A GIVEN

Even though the information may be uncertain, a definite value is always selected for application. The most possible or the most preferred value of a given fuzzy set can be named its best nonfuzn value. The problem of how to choose a best nonfuzzy value from a given membership function has real significance in engineering application. Various methods have been proposed. Previous studies on fuzzy control for real industrial processes suggested two principal methods for calculating the best nonfuzzy value [6, 111. One is to select a nonfuzzy value which corresponds to the maximum value of the membership function, averaging in some way when there are many such values. This is called the mean-of-maxima (MOM) method. Another method is to choose a value which equally divides the area under the membership function curve. It is called the center-of-area (COA)

106

RENHONG

method. method,

Both of these two methods the problems are:

ZHAO AND RAKESH

have some limitations.

GOVIND

For the MOM

(1) Little information is used. None of the values which correspond to 0 < p(x) < 1 have any effect on choosing the best nonfuzzy value. (2) If the membership function has a flat top region, MOM will fail because there are an infinite number of values which have p(x) = 1. But the MOM method is more suitable method. The reasons are:

for fuzzy numbers

than the COA

(1) From the definition of the membership function, the largest value of p(x) means the highest possibility. If we have to choose one and only one best nonfuzzy value of a fuzzy number “approximately x0,” then x0 is of course the best choice. (2) The COA method may fail unless the membership function of the known fuzzy number is symmetrical. If it is not symmetrical, a value x6 with I) < 1 may be chosen. We have no reason to choose this x6 instead of x0 with p( x0) = 1. We will denote “approximately x0” value. So we have number B which is 1.5.

the best nonfuzzy value of a fuzzy number A which is as A, in this paper. The subscript N means nonfuzzy A, = x0 for the fuzzy number A. Similarly, for a fuzzy “approximately y,,” we have B, = y,.

DIRECT DERIVATION

For any arithmetic result is

OF THE BEST NONFUZZY

operation

VALUE

* on A and B, an intuitive

(A * B)N =

x0

*

yo.

and reasonable

(1.5-l)

For example, when x,, = 4, y, = 5, and * represents addition, Equation (1.5-l) can be expressed linguistically as saying that the most possible result or the best nonfuzzy value for addition of “approximately 4” and “approximately 5” is 4+5=9. Since there is only one value of A of which the value of the membership function is equal to 1.0, for any classical function f of A, in view of the extension principle, we should have

[.W)]N = IThe equations

(1.5-2)

(1.5-l) and (1.5-2) mean that if we have to choose one definite

EXTENDED

FUZZY

value from the result, only operations are required.

2.

EXTENDED

107

NUMBERS

FUZZY

on the mean value of each fuzzy number

NUMBERS

From the above brief review we can conclude that fuzzy set theory can be used to solve real problems relating to subjectively evaluated numbers. But there are still some unanswered questions. Equations (1.5-l) and (1.5-2) are simple formulas which can be used for deriving the “best nonfuzzy values” from mathematical operations on general fuzzy numbers. But in evaluating an uncertain parameter, it is often hard to specify only one most possible value. Indeed, if we can do so, we should use this value directly with no uncertainty. In real cases it is more practical and convenient to specify a set of ranges with different possibilities than only one most possible value. This means that the so-called fuzzy inter& [2] may have more practical applications for engineering. Some papers about fuzzy intervals have already been published. All of these papers are based on set operations. The computational difficulty of directly using set operations for the extension principle has been mentioned. The motivation of this paper is to propose some simple methods for application of the extension principle. In this paper it is shown that fuzzy numbers can be expressed as special cases of fuzzy intervals. It is reasonable to interpret the fuzzy intervals as extended jkzy numbers. The fuzzy numbers defined in the literature will be referred to as general fuzzy numbers in this paper.

2.1.

THE DEFINITION OF EXTENDED FUZZY NUMBERS

Let x be a continuous real variable restricted function p(x) E [O, 11, and suppose that: (1) p(x) is piecewise continuous. (2) &) is a convex fuzzy set. (3) There exists a flat top region /L(x) = 1.

for p(x).

by a possibility

This region

distribution

corresponds

to

A fuzzy set which satisfies the above requirements is called an extended fuzzy number. In general, Equations (1.5-1) and (1.5-2) do not hold for extended fuzzy numbers.

108

RENHONG

v(x)

GOVIND

A AL”’

Fig. 1.

2.2.

ZHAO AND RAKESH

MEMBERSHIP

Bound

FUNCTION

AR”’

A

points for mlevel

EXPRESSION

set interval

of p,Jx).

FOR (Y-LEVEL

From previous studies by Kaufmann and Gupta [5], it is known that the computation of operations on fuzzy numbers based on the extension principle can be reduced by decomposing the membership functions into a-levels and conducting the mathematical operations directly on confidence intervals. But in general, the membership function is written as p(x), which cannot be used directly for a-level calculations. For any fuzzy number A (either a general fuzzy number or an extended fuzzy number) which has membership function pA(x), an interval bounded by two points at each a-level (0 < (Y< 1) can be obtained by using the a-cut method. The symbols A(LU)and A%) are used in this paper to represent the left end point and the right end point of this interval respectively, as shown in Figure 1. When At) and At) are considered for all cu-levels, because of the monotonicity and normality of the membership function for A, the quantities A(L) and A(i) are just two inverse functions of kCLa(x)in two ranges. So we can express a general or extended fuzzy number A by using the following form: ‘4 + [ AP’, A$)],

O
This is the resolution identity defined by Zadeh [14]. Using the expression for the membership function at the a-level instead of the usually used p(x), algebraic properties of extended fuzzy numbers can be analyzed much more conveniently.

EXTENDED

FUZZY

109

NUMBERS

2.3. MATHEMATICAL EXPRESSIONS OF EXTENDED FUZZY NUMBERS

There extended

are two important fuzzy numbers:

FOR SOME IMPORTANT PROPERTIES

points

about the discussion

of the properties

of

(1) The COA method rather than the MOM method is used for selecting the best nonfuzzy value of an extended fuzzy number. Therefore the shape of membership function will affect this choice. This is not the case for selecting the best nonfuzzy value of general fuzzy numbers by using the MOM method. (2) By using the resolution identity, we avoid being restricted to some special kind of fuzzy numbers. Common features of the extended fuzzy numbers can be studied. 2.3.1.

Normality

For an extended fuzzy number A, at (Y= 1 level, all x-values located in the interval [A t), A(A)] satisfy pa(x) = 1, and we have /Q’ < A(‘) R(see Figure 1). If A is a general

fuzzy number, ‘@’ = A’;‘.

(2.3.1-1) then (2.3.1-2)

REMARK 2.3.1. A general fuzzy number is a special extended fuzzy number with its membership function interval at (Y= 1 level degenerating into a point.

2.3.2.

Positivity and Negativity

If Af’) > 0, A is said to be positive. If A$) < 0, A is said to be negative. 2.3.3.

Convexity

At) and A$) can each be decomposed into one or more subfunctions of LY. Each of these subfunctions is continuously differentiable with respect to (Y in the interior of its domain. The convexity of pLA(x) can be easily expressed as follows: for each continuous differentiable subfunction of A?), (2.3.3-l)

110

RENHONG

For each continuous

differentiable

ZHAO AND RAKESH

subfunction

of Ag),

dAk”‘
2.3.4.

(2.3.3-2)

.

da

GOVIND

Regularity

Using the COA method, there exists only one nonfuzzy number A, which divides the area under the curve of the membership function of A into two equal parts. However, three cases will occur. These three cases are shown in Figures 2, 3, 4 and can be expressed mathematically as: (1) At’ < A, < Ay’, (2) At) < A, < A’#, (3) A$) < A, < Ag). For extended fuzzy numbers which correspond to case (1) or (3), 114(x = AN) < 1; they are called irregular extended jkzy numbers in this paper. Extended fuzzy numbers which correspond to case (2) are called regular extended fuzzy numbers. It is obvious that the best nonfuzzy value chosen for a regular

AL’lZ

1.0

____

______

&I’

______

a

0.0

)

t

t

t

AL’“’

AN = ALi”)

AR””

O
Af’ < A,

< All’.

x

EXTENDED

FUZZY

111

NUMBERS

ALd

AN

Fig. 3.

At) d A,

AR””

Q Ag’.

extended fuzzy number has a more reasonable explanation than that chosen for an irregular extended fuzzy number. If A is a regular extended fuzzy number, from the definition of the best nonfuzzy value A, based on the COA, we have

(2.3.4-l)

AL@,

AN = AR(“) OCa
Fig. 4.

Ag’ < A,

< A$).

112

RENHONG

ZHAO AND RAKESH

GOVIND

2.3.4.1. Formula for Deriving A, for a Regular Extended Fuzzy Number A From Equation (2.3.4-l), another useful equation can be derived: A, = ; /,‘( At”) + Ak”)) da. Equation (2.3.4.1-1) is the formula for deriving function of a regular extended fuzzy number A.

A,

(2.3.4.1-1)

from the membership

2.3.4.2. Criterion for Regularity of an Extended Fuzzy Number regular extended fuzzy number, from Section 2.3.4, we have

Substitute

A,

by using Equation

(2.3.4.1-1). After rearrangement

2Av)$(AL”‘+

Ak”))da d 2Ag).

If A is a

we have (2.3.4.2-l)

Equation (2.3.4.2-l) is the criterion for regularity of an extended fuzzy number. By analogy with the linguistic expressions for general fuzzy numbers, we can express a regular extended fuzzy number A linguistically as “approximately A,.” Here A, is derived by using the COA. 2.3.5.

Symmetry

An extended fuzzy number function satisfies At)+ nA is a real nonfuzzy By using Equation

A is said to be symmetrical

A($)= nA,

(crisp) constant (2.3.1-l),

O
(2.3.5-l)

for all cy-levels.

2 Ay) < Ay) + Ati’ < 2 A$‘. From Equation

if its membership

(2.3.5-2)

At) + At) = At) + Ag) = n,; then

(2.3.5-l),

jb(A’:)+A~))dol=ll(A~)+A~))da=jlnlda=Afl)+AC’. 0

0

(2.3.5-3)

EXTENDED Substituting

FUZZY Equation

NUMBERS (2.3.5-3) into Equation

(2.3.5-2), we have

(a) + At))

da < 2k#.

(2.3.5-4)

(2.3.5-4) with Equation

(2.3.4.2-l),

the following conclu-

2A~kjo’(A,

Comparing Equation sion can be drawn.

113

fuzzynumber

REMARK2.3.5. Any symmetrical extended 2.3.5.1.

Formula

for Deriving

A,

for

Symmetrical

A From Equations that for a symmetrical extended

(2.3.4.1-1) and (2.3.5-3), it can be derived fuzzy number A, 1

/ 20

1

(A?)

must be regular.

+ A’$)) da = A,

= in,

(2.3.5.1-1)

and A,

= inA.

(2.3.5.1-2)

This formula can be also used for general symmetrical general fuzzy number,

2.3.6.

A(La)+ Ag))

A,=;(

x0=

fuzzy numbers.

When

A is a

= in,.

Absolute Spread and Relative Spread

Membership functions can be also used to represent the transition from an extended fuzzy number through a general fuzzy number to a related nonfuzzy number. Two methods can be employed for this purpose. (1) If A is an extended

fuzzy number,

Ak”’ - At”’ > 0,

If A is a general

fuzzy number,

then

&-A?)>0

for O
and

A(g)-

O<(Y
A(f)=0

(2.3.6.0-l)

for (~=l.

(2.3.6.0-2)

114

RENHONG

If A degenerates

into a nonfuzzy

ZHAO AND RAKESH

value,

Ak”’ - At'

= 0,

(2) If A is a positive extended

O
fuzzy number,

(2.3.6.0-3)

we have

O
If A is a positive general Ak”’ > 1 A(La) If A degenerates

GOVIND

(2.3.6.0-4)

fuzzy number,

for 0
into a nonfuzzy

Ak’ -=1 At”’

and

for a=l.

(2.3.6.0-5)

number,

Ak”’

-=

1

O
’

A?’

Based on above analysis, the definitions relative spread RSD can be given in follows.

(2.3.6.0-6)

of the absolute

spread

ASD

and

2.3.6.1. Absolute Spread ASD The absolute spread ASD of a fuzzy number A at a-level is defined as the distance between the two end points of its membership function at that level. The mathematical expression of this definition is Asoy) = Ak”’ - A?). If A is an extended

fuzzy number, ASD

%$If A is a general ASD~)>

If A is a nonfuzzy

fuzzy number, 0

then O
0,

(2.3.6.1-2)

then

for O
number,

(2.3.6.1-1)

and

ASD~)=O

for cr=l.

(2.3.6.1-3)

then ASD$+=O,

O
(2.3.6.1-4)

EXTENDED

FUZZY

115

NUMBERS

2.3.6.2. Relative Spread RSD The relative spread RSD of a fuzzy number A at a-level is defined as a ratio of the two end points of its membership function at that level. The mathematical expression of this definition is: If A is positive, (2.3.6.2-l) If A is negative, RSDw

=

A

At”’

(2.3.6.2-2)

Ak”’ *

No definition of RSD is given in this paper for an A which is not confined the positive or the negative region. If A is an extended fuzzy number, whether positive or negative, RSDy)>

If A is a general

(2.3.6.2-3)

fuzzy number,

RSD~) >

If A is a nonfuzzy

O=Gcr
1,

to

1

for 0 d LY< 1

and

RSD~) =

1

for (r = 1. (2.3.6.2-4)

number, RSD$+=

1,

O
(2.3.6.2-5)

Absolute spread and relative spread are two useful criteria for solving algebraic equations containing known and unknown fuzzy numbers. 3.

ARITHMETIC

OPERATIONS

ON EXTENDED

FUZZY

NUMBERS

The arithmetic operations on two extended fuzzy numbers can be represented more compactly by using the resolution identity. The properties of the results can be analyzed more conveniently based on interval arithmetic [91. 3.1.

a-LEVEL EXPRESSIONS FOR THE RESULTING MEMBERSHIP FUNCTION

Using the confidence interval idea, arithmetic operations on two extended fuzzy numbers A and B give the following expressions, which are similar to

RENHONG ZHAO AND RAKESH GOVIND

116 that defined

for interval

arithmetic:

C=A+B+[Cp,ck*‘]

=[At”‘+Bp,Ap+Bp],

(3.1-1)

C=A-B-,[C~‘,C~‘]

=[AP)-B~),A~)-BP)],

(3.1-2)

C=AXB+[Ct”‘,Ck*‘] = [min( At*) X BP), A$) X BP), &) max( &)

X I#$), Ag) X BP)),

X BP), A$) x B p’, A?) x BP), A’$ x BP’)] , (3.1-3)

C = A + B + [ Ct”‘, Cg’] = [A~),A%)]

x [l/~g),l/~f)]

if

o e [BP), ~g)]. (3.1-4)

3.2.1.

Normality of C

PROPOSITION 3.2.1. Under any arithmetic operation on two extended fuzzy numbers which are either positive or negatiue, normality is preserved. This can be proved easily by using Equations (3.1-l) to (3.1-4) at LY= 1 level. The normality of C can be also proved by using interval analysis. Because At) - Af) > 0, Bg) - BP) > 0, any arithmetic operation on two intervals at (Y= 1 level can only result in another interval at that level. Hence C$) cf) > 0. 3.2.2.

Positivity and Negativity of C

Based on the analysis of operations on confidence intervals, the positivity or negativity of the results can be easily derived. One important property is that if A and B are both positive, then C = A - B may be neither positive nor negative. 3.2.3.

Convexity of C

PROPOSITION 3.2.3-l. Under addition or subtraction of two extended fuzzy numbers, convexity is preserued.

EXTENDED

FUZZY

NUMBERS

117

For C = A + B, Cp) and Cg) can be decomposed into one or more subfunctions of (Y. Each of these subfunctions is continuously differentiable with respect to Q in the interior of its domain. It can be easily proved that At’ and BP’ are continuously differentiable with respect to LYin the interior of the domain of each subfunction of Cp) and At) and Bg) are continuously differentiable with respect to (Y in the interior of the domain of each subfunction of Cg). In the interior of the domain of each subfunction of Cp), dcp’ -= dcu

d[ At”’ + BP)] _ tit”’ da

da

I dBt”’ da

’

but

dAp

-a0 da

and

:. In the interior

or the domain dck”’ -= da

dB(La) =aO;

dCfp’ > o -/ da .

of each subfunction

of Cg’,

d[ Ak”)+ Bk”‘] _ dAk”’ I dBk*’ da da da ’

but

dAk*’

---GO da

...

and

dBk”’ ~90;

dck”) <0. da

-

According to Equations (2.3.3-l) and (2.3.3-2), C is a convex fuzzy set. The proof for C = A - B is similar to that for C = A + B. PROPOSITION 3.2.3-2. Under multiplication or divkion of two nonzero [7] extended fuzzy numbers, convexity is preserved. For C = A x B, Cp) and Cg) can be decomposed into one or more subfunctions of CL Each of these subfunctions is continuously differentiable with respect to (r in the interior of its domain. It can be easily proved that At’ and BP) are continuously differentiable with respect to a in the interior of

RENHONG ZHAO AND RAKESH GOVIND

118

the domain of each subfunction of Cp), and A($) and BP) are continuously differentiable with respect to (Y in the interior of the domain of each subfunction of Ck). If A and B are positive, At) > 0 and BP) > 0. In the interior of the domain of each subfunction of Cp), dct”’

d[ Ac,“‘x

da=

Bt”)]

da

=A~)x~+Bp)X~,andO
but

awf’

Bt”‘>Bf)>o,

Ata) > AT’ > 0,

dBp’

~20, dCt? da’

..

daao;

> o *

Likewise dCk”’ -= da

d[ At’ x BP’] da

=Ag’x=

dBk”’

+B~)x~,

d4k”’

but AkQ’> AtO’> 0,

Bg)>Bf)>O,

dAk*) =
dBg’ =
dCk”’ ~ o da -

..

The proof for C = A + B is similar to that for C = A X B. We can also prove that whether A and B are positive or negative, after multiplication or division, the resulting fuzzy number C is always a convex set. 3.2.4.

Regularity of C

PROPOSITION3.2.4. On addition or subtraction of two regular extended j&y numbers, regularity will be preserved. But on multiplication or division, regularity may not be preserved. Proof. If A and B are both regular extended fuzzy numbers, from Equation (2.3.4.1-1) we have A, = ;kl(Ap’+

Ag))da

and

B, = $&‘(Bp)+

Bk)) da.

(3.2.4-l)

EXTENDED

FUZZY

From Equation

NUMBERS

119

(2.3.4.2-1) we can have

2A~)~11(Al”)+Ak”))da~2A~) cl

and

2BE)~jo1(B~)+Bk”))du~2BC). (3.2.4-2)

For

c =

A

+

B +

[A?’ + BP’, Ag’ + B~'I = [Cp’, ck”‘l,

jol(cp+Ck”‘)dn=Q( AT’ + BP’)

+ (Ak”’ + BK))] da

=~‘(aL”‘+ak”‘)dn+lol(B~)+

From Equation

(3.2.4-2) and Equation

Bg))da.

(3.2.4-3)

(3.2.4-3),

but Ct” = A’:’ + B”’

L,

:.

Ck” = A$’ + B(l). R,

2Cf’cjo1(C~)+Ck”‘)da~2C~).

According to Equation (2.3.4-31, C is regular. The proof for C = A - B is similar to the above. But for A x B and A + B, the resulting C is not necessarily regular. For example, if AtO)> 0, Bf) > 0, and C = A X B + [A?’ X BP’, &’ X BP’1 = [Cp), Cg)], then

/,‘(Cp' +Cp))

da

@L”‘x

BP’+

Ak”)x Bk”))da

= Ct”

At) x Bt’

120

RENHONG

ZHAO AND RAKESH

GOVIND

Here

At”)XBf)

<1

Ak”‘XBk”) >1

and

At) x Bt’)

Ay) x Bt’)

’

but At”’

x

Bt”)

Ay) x Bt’)

is not necessarily

greater

&’ +

x

BP’

AC’ x Bt’)

than 2. So

jol(ct”‘+Ck”‘)da >2,

Cp

or

/,‘( c(La)+ ck))

da B 2c’,“,

does not necessarily hold. To explain this conclusion, an example is given as follows: If A and B are the two regular extended fuzzy numbers shown in Figures 5 and 6, then At’=1+4a

At’=

7

8-2~~ 7

A(f) = 1 > 0,

A,=~j’(A~‘+Ak”‘)dol=5=Ae’, 0

BP)=

1+2t~,

BN =

C= AX B+

BP)=

8-4a,

c(L)+ BP’) ;kltBL [cp,cp]

Ct’)=(1+4)x(1+2)=15,

BP) = 1 > 0,

da = 4 = Bk’),

= [A~$x

B~),A~)x

Ck”= (8-2)x(8-4)

/1(CI”‘+Ck”‘)dcu=j1[(l+401)x(1+2cu)+(8-2a)x(8-4a)]da=493 0

0

> 2x Ck”= 48.

~g)]

= 24,

EXTENDED

FUZZY

121

NUMBERS

*

10

Fig. 5.

According

to Equation

5.0

6.0

x

8.0

Extended fuzzy number A in Section 3.2.4.

(2.3.4.2-l),

C is not regular.

3.2.4.1. Deriving the Best Nonfuzzy Two Regular Extended Fuzzy Numbers fuzzy numbers, for C = A + B,

Value for Addition and Subtraction of If A and B are two regular extended

(3.2.4.1-1)

P(Y)

4 B

10

~--------------

00

-

10

Fig. 6.

30

40

8.0

Extended fuzzy number B in Section 3.2.4.

Y

RENHONG ZHAO AND RAKESH GOVIND

122

Then substitute Equation (3.2.4.1-1) into Equation (3.2.4.1-1): (3.2.4.1-2)

CN=AN+BN.

A similar result can be obtained for C = A - B: (3.2.4.1-3)

&=A,-B,.

Equation (3.2.4.1-2) and Equation (3.2.4.1-3) are useful formulas for deriving the best nonfuzzy value of the sum and difference of two regular extended fuzzy numbers. These two formulas are extensions of Equation (1.5-l) from the genera1 fuzzy number field into the extended fuzzy number field. But for multipli~tion or division, no extension of Equation (1.5-l) can hold in the extended fuzzy number field.

3.2.5.

Symmetry of C

PROPOSITION 3.2.5. infer addition or s~traction tended fuzzy numbers, symmetry will be preserved. C = A t B, symmetry will not be preserved.

of two symmet~ca~ exBut for C = A x B or

The proof can be derived by using similar methods to the above.

3.2.6. Absolute Spread and Relative Spread of C

3.2.6.1. Absolute Spread of C, PROPOSITION 3.2.6.1.For addit~n or s~tr~ction of two extended fuzzy numbers A and B, the absolute spread of the resulting fuzzy number C is greater than that of both A and B for all 0 6 (Y=G1 levels.

If A and B are two extended fuzzy numbers, Proof of Proposition 3.2.6.1. from the discussion in Section 2.3.6, we have ASDY)

=

At) - At”) > 0,

O
(3.2.6.1-1)

ASDg)

=

Bg) - Rt”’ > 0,

OCcX61.

(3.2.6.1-2)

EXTENDED

FUZZY

NUMBERS

123

For C=A+B, ASDg)=Cg)--Cp)=(

Ak")+Bk"))-(At)+BP)) =(Ak")-Ap))+(Bg)- Bt"') =

From Equations ASD$+

(3.2.6.1-l),

> 07

ASDP)

ASDY)

+ ASD(,n),

O
(3.2.6.1-3)

(3.2.6.1-2), and (3.2.6.1-3), we have

> ASDY),

ASD(C’ > ASD(B’,

0 <(Y Q 1. (3.2.6.1-4)

For C=A-B,

ASD(C)=C~)-Ctn)=(Aku)-B~))-(Ata)-B~)) =(Ak"'- At))+(Bg)- BfP)) = From Equations (3.2.6.1-l), also valid for C = A - B.

ASDY)

+ ASD(B),

O
(3.2.6.1-2), and (3.2.6.1-9,

Equation

(3.2.6.1-5) (3.2.6.1-4) is

3.2.6.2. Relative Spread of C PROPOSITION 3.2.6.2. For multiplication or divkion of two nonzero extended fLz.zy numbers A and B, the relative spread of the resulting fuzzy number C is greater than that of both A and B for all 0 G LYG 1 levels. Proof of Proposition 3.2.6.2. both positive, from (3.1-31,

If two extended

fuzzy numbers

A and B are

Since C is positive, then from (2.3.6.2-l),

RSD(a)

_

C

a?’ _ 4’ x BP = Ct"'

At"'xBt")

RSDY)

x RSD$+,

0 g a < 1. (3.2.6.2-l)

RENHONG

124 From Equation

ZHAO AND RAKESH

(2.3.6.2-3), RSD$) > 1 and RSD(B)>

GOVIND

1 for 0 d (YQ 1. So

RSD(ca)>1, RSD(CU'>RSDy), RSD(Cn'>RSD(B',

0
Using the same method, we can prove that (3.2.6.2-2) holds for both C = A x B and C = A + B whether A and B are positive or negative. 4. DISCUSSION OF THE FUZZY INVOLVING ADDITION

EQUATIONS

The discussion of additive and multiplicative inverses has practical significance in solving algebraic equations which contain fuzzy parameters. Misumoto and Tanaka’s conclusion [7, 81 that there are no additive and multiplicative inverses concerns general fuzzy numbers. In this paper, it has been shown that general fuzzy numbers are special cases of extended fuzzy numbers. For generalization, in this section we will study in particular the addition operation on two extended fuzzy numbers. 4.1.

THE SUM OF Two EMENDER

FUZZY

NUMBERS

PROPOSITION 4.1. After any arithmetic operation on two extended fuzzy numbers, the result is also an extendedfuzzy number.

Proposition 4.1 is a consequence of Proposition 3.2.6.1 and Proposition 3.2.6.2. It can be expressed mathematically as follows: For any two extended fuzzy numbers A and B,

A*B#n.

(4.1-1)

Here * stands for + , - , x , + , and n is any crisp number. A special case of Equation (4.1-l) is that it is impossible to find two extended fuzzy numbers A and B which satisfy A + B = 0. Here 0 is a crisp number. 4.2.

FVZZIFICATION

OF A CRISP NUMBER

For a crisp number a, an extended fuzzy number

a +( - a>= 0. A natural extension of this result into A is

= [At”) _ Aa”‘, At$) _ At”)]

EXTENDED

FUZZY

NUMBERS

125

Based on the discussion in Section 2.3.5, C = A +( - A) is symmetrical and C, = f(Cp) + Cg’) = 0. So, when the extension principle is used for addition of two extended fuzzy numbers A and B, (B = - A) only A + B = 0 (extended fuzzy number) rather than A + B = 0 (crisp number) is reasonable and possible. Here the extended fuzzy number B is a fuzzification of the crisp number 0. 4.2.1.

Definition of Fuzzification for a Crisp Number

Given the definition of “best nonfuzzy value” for an extended fuzzy number, fuzzification of a crisp number can be defined as follows: for an extended fuzzy number A, if its best nonfuzzy value is A,, then A is said to be a fuzzification of A,. Our definition of fuzzification is as the inverse operation of defuzzification. It is different from the definition given in literature [14]. INFERENCE.

Any crisp number has infinitely many fuzzifications.

The above study can be extended further to a more general case, that is, to find B from A + B = 9 when A and 8 are known. 4.3.

THE SOLUTION

TO A + B = 0 WHEN A AND 0 ARE KNOWN

Finding the solution to A + B = 8 when A and 8 are known is similar to solving a fuzzy equation A + X = C. E. Sanchez [12] and some others have published many papers in this field. Some specially defined set operations (a, y, etc.) are used. Usually some superset of the exact solution, but not the exact solution itself, is derived [3]. For application, an exact solution and a more direct, simple, and convenient method are required. In the following sections the exact solution of A + X = C is derived, based on the previous discussion in this paper. 4.3.1. The Absolute tended Fuzzy Numbers

Spread

A general form of algebraic

Difference

equation

Requirement

for

Two Known Ex-

which has one unknown

fuzzy term X

is A+X=C. Here A and C are two known extended

(4.3.1-1) fuzzy numbers.

126

RENHONG

ZHAO AND RAKESH

GOVIND

PROPOSITION 4.3.1. A necessary condition for the existence of a solution to A+X=Ck O
ASD(C’ > ASDY),

Proof of Proposition 4.3.1. at cY-level,

Using the expressions

A+X=C-+~),Ck”‘] Solve Equation

functions

=[A~)+X~),A~‘+Xk”‘].

condition

= Cp,

_ At”‘,

Xk”’

for the existence

xp < xp, Substitute

of membership

(4.3.1-3)

(4.3.1-3) for X: Xp,

One necessary

(4.3.1-2)

Equation

= Cg’

_ At)_

of an extended

(4.3.1-4) fuzzy number

O
(4.3.1-4) into Equation

X is

(4.3.1-5)

(4.3.1-5):

Cp) - Ap) < Ck”’ - Ati). After rearrangement, Ck”’ - Ct”’ > Ak”) - At”‘,

In view of the definition (4.3.1-6) is

of the absolute

spread of an extended

ASD(C’ > ASDY),

If the extended

fuzzy number

O,
fuzzy number,

O
X degenerates

x+ [xp’,x#p] =[x,x], Then Equation

(4.3.1-6)

into a crisp number O,
(4.3.1-7) x, (4.3.1-8)

(4.3.1-4) becomes x = Ct”’ _ At) = Ck’ _ A@

(4.3.1-9)

EXTENDED Rearrange

FUZZY Equation

NUMBERS (4.3.1-9): ck”’

. .

Combining proved.

127

Equation

- ct”’

= &’

_ AL”‘;

ASD(C’ = ASDP),

(4.3.1-7)

(4.3.1-10)

0 < (Y < 1.

and Equation

(4.3.1-10,

(4.3.1-11) Proposition

4.3.1 is

4.3.2. The-Differential Difference Requirement for Two Known Extended Fuzzy Numbers PROPOSITION 4.3.2. If A(f), A($, Cp), and Cg) are differentiable with respect to (Y for 0 < (Y< 1, it can be easily proved that Xp) and Xf) are also differentiable with respect to (Y at 0 < (Y< 1. Another necessary condition for the exktence of a solution to A + X = C is dCp’ z aLAp’ and dCp’ Q dAk”’ da da do do ’

O
Proof of Proposition 4.3.2. From Section 2.3.3, the requirement ity of the extended fuzzy number X is

dxp

-/

Substitute a?’ ‘da!

Equation dct”’ =-da

da

>O and

dx”
(4.3.1-4) into Equation

d-it”’ > o and dXk”’ da’

-=da

’

O<(Y
(4.3.2-l)

for convex-

(4.3.2-2)

(4.3.2-2): dCk”’ ---GO, da

dAk”’ do

O
From the equations

(4.3.2-3),

dCt”’ ~ tit”’ do da

>, o and

-dCk*’ d -dk*’ do da

~ o ’

0 <(Y < 1. (4.3.2-4)

128

RENHONG

If X is a crisp number

dxp -=da

GOVIND

X, then (4.4.2-3) becomes

dCt”) --COd&j da

ZHAO AND RAKESH

!k!-&%&@&O,

and

dn

O
’

(4.3.2-5)

Then dCp’ -=da

Combining proved.

tit’

Equation

(4.3.2-4)

Based on Proposition derived. INFERENCE.

Zf A?),

and

da

dCk”’ -=da

and

d#’ da

Equation

4.3.1 and Proposition

A($,

Cp),

O<(Y
’

(4.3.2-6),

Proposition

(4.3.2-6)

4.3.2 is

4.3.2, the following inference

are continuously differentiable

and Cg)

is

with

respect to (Y for 0 < CY< 1, a sufficient and necessary condition for the existence of a solution to A + X = C is

ASD(c+

dCt”’ ~ a24t”’ and da da

4.3.3.

ASDjqa)

dCg’ da

for

$?!F

da

O
for

(4.3.2-7)

0 < LY< 1.

(4.3.2-8)

Exact Solution for A + X = C

When Equations (4.3.2-7) and (4.3.2-8) are satisfied, the exact solution can be found by using the reverse operation separately on bound points of all a-level set intervals. No inverse operations on the known extended fuzzy numbers are required. Thus we have given a unique method for solving A + X = C in this paper. It is different from the usual way of solving a general algebraic equation. To explain the above comment in more detail, assume A and C are two extended fuzzy numbers which satisfy Equations (4.3.2-7) and (4.3.2-8). Since [A?), Ag)] +[X$_@, Xg)] = [Cp),Cg)], if the inverse operation is used for a known extended fuzzy number, A, A + X - A = C - A then interval arithmetic

EXTENDED

FUZZY

129

NUMBERS

can be applied:

After rearrangement,

X can not be isolated in this way. But if the two end points Xp) and Xg) are solved for separately, then from

At) + Xp) = Cp),

XP) = CP) - At”‘;

from

At) + Xk*’ = Cg’,

Xk*’ = Ck”’ - Ak*‘.

Then X + [ Xp), Xg)]. Since Al”), Ag), Cp), and Cg) are known functions of (Y,the solution so derived is the required exact solution. It is the satisfaction of Equations (4.3.2-7) and (4.3.2-g) that guarantees the existence of this exact solution. When C = 8 (a fuzzification of the crisp number 0) and Equations (4.3.2-7) and (4.3.2-8) are satisfied, the solution of A + X = C is xz”’

= (jp

- At”‘,

xk”’

= ($3

- Ak”‘.

(4.3.3-l)

Based on the inference in Section 4.2.1 and the discussion in Sections 4.3.1 and 4.3.2, there are infinitely many fuzzifications of 0 which satisfy Equations (4.4.2-7) and (4.3.2-8). So the exact solution of A + X = B is restricted by the specific membership of 8. Thus 8 + [At”) - Ag), At) - &)I is a special solution when X = A. And we have A - A = 0, which is the natural extension from a - a = 0 in the nonfuzzy field. 5.

CONCLUSION

The goal of this paper was to present a systematic discussion of extended fuzzy numbers, which have been called fuzzy intervals in the literature. It is proved in this paper that using arithmetic operations on a-level set intervals is much more simple and convenient than using general set operations to apply the extension principle. By using the derived sufficient and necessary condition for solving the fuzzy equation A + X = C, the exact solution rather than some superset of the exact solution can be easily obtained.

130

RENHONG

ZHAO AND RAKEBH GOVIND

REFERENCES 1. Didier Dubois and Henri Prade, Operations on fuzzy numbers, Internat. J. System Sci. 9(6):613-626 (1978). 2. Didier Dubois and Henri Prade, Addition of interactive fuzzy numbers, IEEE Trans. Automat. Control AC-26(4):926-936 (1981). 3. Siegfried Gottwald, On the existence of solution of systems of fuzzy equations, Fuzzy Sefs and Systems, 1984, pp. 301-302. 4. Abraham Kandel, Fuzzy Mathematical Techniques with Applications, Addison-Wesley, Reading, Mass., 1986, pp. 38-44. 5. Arnold Kaufmann and Madan M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold, New York, 1985, pp. 9-14. 6. P. J. King and E. H. Mamdani, The application of fuzzy control systems to industrial processes, Automaticu 13:235-242 (1977). 7. M. Mizumoto and K. Tanaka, Algebraic properties of fuzzy numbers, in Proceedings: IEEE International Conference on Cybernetics and Society, 1976, pp. 559-563. 8. M. Mizumoto and K. Tanaka, Some properties of fuzzy numbers, in Aduances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam, 1979, pp. 153-164. 9. R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1966, pp. 8-9. 10. S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems, 1978, pp. 97-110. recent results, 11. Richard M. Tong, Synthesis of fuzzy models for industrial progress-some Internat. J. Gen. Systems 4:143-162 (1978). 12. E. Sanchez, Solution of fuzzy equations with extended operations, Fuzzy Sets and Systems, 1984, pp. 237-248. 13. Felix S. Wong and Weimin Dong, The vertex method and its use in earthquake engineering, in First International Symposium on Fuzzy Mathematics in Earthquake Research, Seismology Press, China, 1985. 14. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. 8:199-251 (1975). 15. Hans J. Zimmermann, Fuzzy Set Theory--and Its Applications, Kluwer-Nijhoff, Hingham, Mass., 1985, pp. 47-59. Received 4 November 1987; revised 4 December 1987, 31 May 1988