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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
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
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