Addition of fuzzy numbers based on generalized t-norms

FUZ2Y

sets and systems ELSEVIER

Fuzzy Sets and Systems 98 (1998) 95-99

Addition of fuzzy numbers based on generalized t-norms 1 Xiu-Gang Shang*, Wei-Sun Jiang East China University of Science and Technology, Shanghai 200237, People's Republic of China Received June 1996; revised October 1996

Abstract The addition rules of fuzzy numbers based on the generalized t-norms are given, which generalized the "min"-normbased addition rule of fuzzy numbers. The results are also extended to fuzzy intervals, Its possible applications are pointed. (~) 1998 Elsevier Science B.V. All rights reserved. Keywords." Fuzzy number; Fuzzy addition; Weak noninteraction; Generalized t-norm

I. Introduction

2. Preliminaries

In many scientific areas, such as decision making, system analysis, automatic control and operations research etc., a model cannot be established exactly sometimes, because the data are only approximately known. They should be processed with fuzzy sets, and arithmetic operations may be performed in such cases. Therefore, good arithmetic operations on fuzzy sets are very meaningful, not only for the practical use but also for the theory o f fuzzy sets. More detailed descriptions of applicability and importance o f arithmetic operations on fuzzy sets can be found in [ 1]. The present paper is devoted to the derivation o f formulas o f addition o f LR-fuzzy numbers based on the generalized t-norms proposed by Gu et al. [4], which is the generalization o f the addition rule o f LRfuzzy numbers in the case o f "min"-norm [2, 3]. The rules are also extended to fuzzy intervals.

A fuzzy interval n is a convex fuzzy subset on a real line R with a normalized membership function. Convexity means: Vu, v E R , with u < v, Vw E [u,v],n(w)>~min(n(u),n(v)). When there is only one real number m such that n(m) = 1, n is called a fuzzy number. In this paper, we will consider a special kind o f fuzzy interval defined as [ 1,2]

* Corresponding author. I Supported by the National Natural Science Foundation of China.

~(x) =

1

0

if

m-7<~x<<.m c~>0,

if

m<.x<~n,

if

n.~x<~n+fl f l > 0 ,

otherwise.

(1) where L and R are reference functions: [0, 1] ~ [0, 1], with L(0) = R ( 0 ) = 1 and L(1) = R ( 1 ) = 0 and which are non-increasing, piecewise continuous mappings. [m, n] is the peak of n and m and n are the lower and upper modal values, c~ and fl are the left and right spreads o f n. We shall call such a fuzzy interval of

0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PH S0165-0114(96)00376-4

96

X.-G. Shan9, 14(-S. Jiano/Fuzzy Sets and Systems 98 (1998) 95-99

LR-type and denote it by it = (m, n, c¢,[3)LR. The support of ~ is exactly [m - ~, n+fl]. When m = n, zt will be called fuzzy number and denoted as (m, ~, fl)LR, and m is called the mean value. Consider two fuzzy numbers 7rl and 1r2. The sum of two weakly non-interactive fuzzy numbers II~(z) (r~l + rc2)*(z), in the sense of a triangular norm *, is defined as

Lemma 2 (Gu et al. [4]). Let (L, <<.)be a dense complete lattice, f :L ~ L an order isomorphism satisfying f ( O) = O, f (1 ) = 1, and g be the inverse mapping o f f . Then

/-/i(Z) = sup ~I(X) * 792(y )

Lemma 3 (Gu et al. [4]). Let (L, ~< ) be a dense complete lattice, F :L ~ L an anti-order isomorphism satisfying F ( 0 ) = 1, F(1)----0, and G be the inverse mapping ofF. Then

z=x +y

= sup zq (x) * rr2(z - x)

Vz e R.

(2)

xER

For two fuzzy numbers nl and re2, we have following result.

Lemma 1 (Dubois and Prade [ 1]). Let m and n (>1m) be mean values of nl and 7t2, respectively; then for all w<~m + n the supremum in (2) is reached for a value of x inside the interval [w - n, m]. Let 7zl = (m, c~,fl)LR and ~2 = (n, y, •)LR be two fuzzy numbers. Denoting ~ ( X ) = 7 ~ 2 ( Z - X), then for * = m i n , the supremum in (2) is reached for the intersection of 7r~ and z~l inside [z - n,m] and [1] //~n'n(z) = sup [r~ Fl rq],

(3)

(6)

Tl(x, y) = 9 ( f ( x ) A f ( y ) ) is a generalized t-norm.

T2(x, y) = G(F(x ) V F ( y ) )

(7)

is a generalized t-norm. Subsequently, we will replace * in (2) with T~ and 7'2 to get the exact calculation formulas of addition of fuzzy numbers based on these two generalized t-norms. (A) * = T~:

Theorem 1. Assume that 1~1 and re2 are defined as above; then when the triangular norm is 1"1,

x where A denotes the usual fuzzy set intersection. Solving the equation rrl(X)=~z~(x) on the interval [z - n, m] yields the result. For LR-fuzzy numbers, it is

L ( ~ - ~ - f ) = L ( X - z + 'n ) 7

(4)

Proof. For triangular norm/'1 (x, y ) = g ( f ( x ) A f ( y ) ) , the function f is increasing and so is 9. Therefore, f(gl(X))

which yields the m a x i m u m ~T= m7 + (z - n)~

_--

and H~nin = ( m + n , Ot+ V, fl+~)LR.

(5)

3. Addition of fuzzy numbers Gu et al. [4] proposed the generalized t-norms as follows:

(8)

IIf ~= ( m + n , ~ + 7,fl+f)LR.

f(L(~-))

if

1

if x = m ,

fIRC~-~)I

if

0

otherwise.

I

m-~<~x<~m c¢>0,

m<~x<~m+fl f l > 0 ,

(9)

Since L and R are piecewise continuous and nonincreasing, the functions f ( L ( . ) ) and f ( R ( . ) ) are also piecewise continuous and non-increasing. Let f ( L ( . ) ) = L ' ( . ) and f ( R ( . ) ) = R ' ( . ) , so (9)

X.-G. Shang, W.-S. Jiang/Fuzzy Sets and Systems 98 (1998) 95-99

becomes

97

Remark. From Theorem 1, we can see that the addi-

f(TCl(X))

t/L,(_~_f )

if m-~<~x<<.m ~ > 0 , if x = m ,

=~1 /

/

(10)

\

IR'(~--~)

if m<~x<~m+]3 ]3>0,

(O----

otherwise.

tion rule of fuzzy numbers based on the generalized t-norm T1, which is in fact a class of t-norms by the variance of the function f , is just the same as the "min"-norm. It is the generalization of "min"-normbased addition of fuzzy numbers. Theorem 1 can be easily extended to n fuzzy numbers, i.e.,

Therefore, f ( r q ) is UR'-type fuzzy number: f ( r q ) = (m, ~,/~)uR,; f(n2) has a complete similar form, i.e., f(It2) = (n, 7, 3)L,R,. From (2),

Corollary 1. Let ~i = (mi, o~i,]3i)LR for i = 1..... n be LR-fuzzy numbers; then the sum of these fuzzy numbers based upon T1 is

[If' = sup Tl(Tq(x), 7~2(2 X)), -

-

xER

= sup o(f(Tzl (x)) A f(rc2(z - x))). xCR

(11 )

Since 9 is increasing, we have

[If'(z)=g(SUPxcR f(lt,(x)) A f ( ~ 2 ( z - - x ) ) ) .

(12)

First, we calculate Z ( z ) = SUPxcRf(nl(X)) A f(rc2 (z - x ) ) , the argument of (12). We can see that Z has the similar form as • = rain (i.e. (3)), then

Z(z) = sup [f(rtl(X)) A f(rc2(z - x))].

(13)

x

Since f ( r q (x)) and f(n2 (z - x)) are L'R'-fuzzy numbers, Z(z) can be derived by solving the equation f ( r q ( x ) ) = f(Tt2(z - x ) ) on [z - n, m], that is

~gi

:

i=1

~-~mi, i~=lOti, ~ f l i i=1

'

(18)

•

i=1

R

The above result can be readily extended to fuzzy intervals, provided that the LR representation be slightly adapted, namely by splitting the mean value mi into two numbers m~ and m~', then we have:

Corollary 2. Let 7"(,i' -- -

( m i ,' m " i , ~i, ]3i)LRfor i= l .... ,n be LR-fuzzy intervals; then the addition of these fuzzy intervals based on the generalized t-norm T1 is

~zi =

~ mi, i=1

L'(~U~f-~ x ) = L ' ( x-z+n)7

'=

mi , i=1

~i,

t~i

i=1

(19) R "

(14) (B) • =/'2:

which yields the maximum _ m7 + (z - n)c¢ c~+ 7 and

(15)

The calculation of the addition of fuzzy numbers based on /'2, i.e., [i~2 is more complicated. Before calculating H~"2, we will give the following lemma first:

Z = ( m + n,~ + 7,]3 + 6)L,R,.

(16)

Lemma 4. For any u and v,

(17)

sup Tz(u, v) = G

Substituting (16) into (12), we have

[It, (z) = g(Z(z)).

U~/)

Since g(f(L(.)))=L(.) and 9(f(R(.)))=R(.), 9(Z(z)) is again a LR-type fuzzy number, i.e.,

[I~' =(m + n,C~ + 7,]3 + 6)LR.

[]

sup FC(u) A FC(v) U~/)

, /'

(20)

//

where A c stands for the common complement of A, i.e., AC = l - A.

X.-G. Shano, W.-S. Jiang/Fuzzy Sets and Systems 98 (1998) 95-99

98

Proof. Since G is decreasing, then

SO

sup T2(u, v) = sup G(F(u) V F(v)), /g,/3

FC(rtl(X))

{

U~I3

1-F(L(~-f)) 1

B~, De Morgan law of fuzzy connective,

/ /

if

m-~<<.x<~m ~ > 0 ,

if

x=m,

N\

1 - F ( R ( ~ - - - m - ) ) if

F(u) V F(v) = (FC(u) A FC(v)) c,

X \

F

m<~x<<.m+fl f l > 0 ,

//

0

otherwise.

SO,

(25) SUPu, v T2(u'v)= G \u.v(inf(FC(u) A FC(v))C). Since the complement AC is decreasing with respect to A, we get supT2(u,v)=G

supFC(u)AFC(v)

.

[]

Now, we can state the addition rule of fuzzy numbers based on the generalized t-norm T2 as follows: Theorem 2. When the triangular norm & 1"2,the sum of rq and 7tz is

II~2 = (m + n, c~+ 7, fl + 6)LR.

(21)

Proof. The addition of fuzzy numbers based on the generalized t-norm 7'2 is //2r2(z)= sup G(F(rq(x))VF(Tz2(y))).

(22)

z=x+y

By Lemma 4, we can rewrite (22) as

5)

\ , , xeR

.

(23) First, we will calculate F(~l(x)). Since F is decreasing and F ( 0 ) = 1 and F(1 ) = 0, then F(rq(x))

0

1

if

m-~<~x<~m c~>0,

if

x =m,

if

m<~x<~m+fl f l > 0 ,

otherwise,

(24)

Because F is decreasing and L and R are nonincreasing, F(L(.)) and F(R(.)) are non-decreasing. Let L I ( . ) = I - F ( L ( . ) ) for m - ~<~x<~m and RI(.) = 1 - F ( R ( . ) ) for m<~x<~m +fl; then Ll andR1 are non-increasing. Therefore, FC(~tx(X)) is also LRtype fuzzy number, denoting FC(rq ) = ( m, c~,fl )L,R, . Similarly, FC(rt2) = (n, 7, 6)L,R~. Denoting Z(z) = supx~R FC(rh(x)) AFC(n2(z -x)), then Z can be easily derived just like (5), i.e., (26)

Z = ( m + n,c~ + y,3 + 6k,R,.

Substituting the above expression into (23), we get H~2(z) = G((Z(z))C).

(27)

It is easy to see that ZC(z) has the similar form as (24). Since G ( 0 ) = 1 and G ( 1 ) = 0 , then 7~x+yl"2 =(m + n,~ + ?,fl + 6)LR"

[]

Remark. From Theorem 2, we can see that the addition rule of fuzzy numbers based on the generalized t-norm/'2, which is in fact a class of t-norms according to the variance of the function F, is just the same as the "min"-norm. It is also a generalization of "min"norm-based addition of fuzzy numbers.

Theorem 2 is also applicable to n fuzzy numbers, i.e., Corollary 1. Let ~i=(mi,~i, fli)LRfor i = 1..... n be LR-fuzzy numbers; then the sum of these fuzzy numbers based upon T2 is

(28) i=1

i=1

i=1

/LR

X.-G. Shan o, W.-S. Jiano/Fuzzy Sets and Systems 98 (1998) 95-99

The above result can be readily extended to fuzzy intervals similarly, i.e.,

Corollary 2. Let 7z~= (m~, m~', ~i, f l i ) L R f Or i = 1,..., n be LR f u z z y intervals; then the addition o f these f u z z y intervals based on the generalized t-norm T2 is

rciI

=

~ m'i' S-" ~ mr' t' O~i' fli • i=1 i=1 "= i=1 R

(29)

The above results can be easily extended to subtraction of two fuzzy numbers based on the generalized t-norms [2]. We will not give detailed expressions here.

99

on the generalized t-norms seem independent. Our result greatly extends the addition rules of fuzzy sets. In some real process if some variables are described with fuzzy numbers and they are weakly non-interactive in the sense of a generalized t-norm, their addition operations satisfy the rules in the present paper, so they can be processed as if they are independent. This will simplify such problems greatly. This principle is just like linear transformation principle, which makes two variables apart from each other by coordinate transformation. Our results have potential applications.

References 4. Conclusion We obtain the addition rules of weakly noninteractive fuzzy numbers based on the generalized t-norms proposed by Gu et al. They have the same forms as the addition rule of non-interactive fuzzy numbers, i.e., based on "min"-norm. It shows that the weakly non-interactive fuzzy numbers based

[1] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Systems Sci. 9 (1978) 613-626. [2] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [3] D. Dubois and H. Prade, Addition of interactive fuzzy numbers, IEEE Trans. Automat. Control 26 (1981) 926-936. [4] W.-X. Gu, S.-Y. Li et al., The generalized t-norms and the TLPF-groups, Fuzzy Sets and Systems 72 (1995) 357-364.