Weak arithmetics of fuzzy numbers

FUIIY

sets and systems ELSEVIER

Fuzzy Sets and Systems 91 (1997) 143-153

Weak arithmetics of fuzzy numbers Milan Mare~ Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P.O. Box 18, 182 08 Praha 8, Czech Republic Received April 1997

Abstract Computation with fuzzy numbers appears to be a perspective branch of fuzzy set theory namely regarding the data processing applications. A very compact survey of the actual state of art is given in Dubois and Prade (1988). This paper aims to offer, in a brief and concentrated form, a survey of some results completing the classical ones of Dubois and Prade. Most of its content summarizes the approach submitted and discussed by Mares (1994). © 1997 Elsevier Science B.V.

Keywords: Fuzzy quantity; Linear space; Group operations with fuzzy numbers; Ordering of fuzzy numbers

1. Basic notions and notations In the whole paper we denote by R the set of real numbers, by R0 = R - {0} the set of non-zero real numbers and

R+={xER: x > 0 } ,

R_

= {xER: x < 0 } ,

Any fuzzy subset a of R is called a

R* = R +

(1)

UR_.

fuzzy quantity with membership function #a: R--~ [0, l] iff

(2)

3xo ER: lta(Xo)= 1, 3xl, x=ER,

XI
(3)

VX~_[XI,X2]; ~a(X)=O.

We denote by ~ the set of all fuzzy quantities and, analogously to (1), ~+ = { a E R:

,Ua(x)>O=e~x>O},

~ _ = { a E R:

,Ua(X)>O=>x
~*

=

~+ U ~_.

(4)

The equality a=b for a, b E R means I~,,(x)=Itb(X) for all xER. If r E R then we denote by (r) the degenerated fuzzy quantity p(r)(r)=l,

p(r)(x)=0

forxcr,

xER.

(5)

If a E ~ then - a E ~ is the fuzzy quantity defined by /~-a(x) = p a ( - x )

for all x E R.

0165-0114/97/$17.00 (E) 1997 Elsevier Science B.V. All rights reserved PH S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 1 3 6 - X

(6)

M. MareSlFuzzy

144

Sets and Systems 91 (1997)

143-153

If a E Iw‘ then l/a E R* is the fuzzy quantity defined by PI/~(X)= P~(l/x)

for all x E Ro.

(7)

The elementary arithmetic operations over fuzzy quantities are derived from so called extension principle. It can be formulated on various levels of generality. For our purposes, the following formulation is quite adequate. Statement 1 (Extension Principle). Let f: R x R -+ R be a binary operation over real numbers. Then it can be extended to the operation over fuzzy quantities, f: IRx R -+R.If we denote for a, b E [wthe quantity c = f(a, b) then the membership function pC is derived from the membership functions pa and pb by

CL&) = sup[min(P&),

pb(y)):

x, Y E R

z = .I+,

Y

)I.

(8)

2. Arithmetic operations

The elementary binary arithmetic operations with fuzzy quantities or with crisp and fuzzy numbers are mostly based on the extension principle (cf. [ 1,2]). 2.1. Crisp sum

Let rER,

ac[W then r+aElW

p,+,(x) = pU(x - r)

is defined by

for any x E R.

2.2. Fuzzy sum

If a,belW then a@bE[W is defined by kBb(x)

= ~-$mWh(y),

pb(x

x E R.

- y))),

It is easy to see that Y+ a = (r) @a. 2.3.

Crisp product

Let r-CR, ac[W then r.aE[W

Pm(X)=

1

/4dr)

is defined for anyxER

for r#O

P(O)(X) for r = 0.

2.4. Fuzzy product

If a, b E [wthen a @ b E [w is defined by k@b(x)

= ;~~(min(h(y)T

k@b@)

= max(k,(0),

itb(x/Y)))?

x E ROY

pb(o)).

It is easy to see that r . a = (r) o a for r # 0.

by

M. MareSlFuzzy Sets and Systems 91 (1997) 143-153

145

2.5. Fuzzy-crisp power

Let a E R+, r E Ro then a’ E [wis defined by x > 0, X
If a E R,, b E R* then ab E lR+ is defined by p&c)

sup(min(k&l’Y)P pb(Y))),

=

Wh 0,

x>@ x
It is easy to see that a’ = a(‘) for r E Ro. 2.7. Crispfuzzy If rER,

power

r>O,

/&‘(x)=

UEIW* then PER+

lnx Inr

V) Pa 0,

’

is defined by

xER+y x
It can be verified that P = (r)“. Generally, all operations over fuzzy quantities given above generalize the operations over crisp operands. For degenerated fuzzy quantities, a = (x), b = (y), x, y E R, r E R, r+a=(r+x),

a@b=(x+y),

r.a=(r.x), u’ = (x’),

a@b=(x.y), lzb= (XY),

(9) ra = (rX).

3. Further concepts As shown in [ 1,2], e.g., the operations shown above do not fulfill some of the properties being selfevidently fulfilled in the crisp case. Handling this fact needs the introduction of several auxiliary concepts. This approach is discussed in [2] and, especially, in [3]. More detailed properties of the notions given below are summarized, e.g., in [2]. 3.1. Symmetry If y E R, a E Iwthen we say that a is y-symmetric iff p&y+x)=p&y-X)

for all XER.

(10)

M. MareMFuzzy Sets and Systems 91 (1997) 143-153

146

The set of all y-symmetric fuzzy quantities is denoted by ~y, and by ~ we denote the union ~=

U ~Y

(11)

yER

Evidently, if a E ~y, y E R, then there exist s E ~o such that a E (y) ® s.

3.2. Transversibility If y E Ro, a E ~* then we say that a is y-transversible iff

g a ( y / x ) = { ~ a ( y . x ) for x > 0all , f ox~<0. r

(12)

The set of all y-symmetric fuzzy quantities is denoted by Yy, and by ~- we denote the union ~-= U ~-s"

(13)

yERo

Also in this case, if a E Ny, y E Ro, then there exists t E ~-1 such that a = (y) @ t.

3.3. Equivalences Let a, b E R. Then we say that a is additively equivalent to b, and write a ,-~ b iff there exist s b s2 E No such that

a G s l =b®s2.

(14)

It can be easily seen that for a, b, c E

a~®a, a,.~®bce~b~ea, a ~ b A b ~ ® c ~ a~®c,

(15)

and that for a E 5y, a ~ (Y/. Let a, b E ~*. Then we say that a is multiplicatively equivalent to b, and write a ~ o b iff there exist q, t2 E ~-1 such that

a G t j = b @ t2.

(16)

It can be easily seen that for a, b, c E N* a~Ga,

a~ebc~,bN oa,

a~GbAb~Gc

~ a~oc,

(17)

and for a E ~-y, a N o (y)It is easy to verify that for any r E R, a, b, c E R,

a~ e b~r .a@c~

r. b®c,

(18)

and for any r E R , a, bcE[~*

a~®bc~r.a@[email protected]@c.

(19)

M. MareJ/Fuzzy Sets and Systems 91 (1997) 143-153

147

3.4. Trapezoidality and triangularity

Let a E ~ and let there exist real numbers al <~ao<~aPo<~a2 E R such that pa(x)=O = 1

forxa2, for ao ~
----(x-- al)/(ao -- al)

for x E ( a l , a o ] ,

= (X -- a 2 ) / ( d 0 -- a2)

for x E [a~,a2).

(20)

Then a is called trapezoidal. Equality a 1 = ao naturally means #a(x) = 0 for x < a0, pa(ao)= 1 and, analogously, if a2 =a~ then p a ( x ) = 0 for x>a'o, ]Aa(ato)=1. If ao =a'o then the fuzzy quantity a is called triangular. Fuzzy quantity b E ~ is called almost trapezoidal iff there exist trapezoidal fuzzy quantity a such that a,~ e b. The auxiliary concepts formulated above enable us to formulate weaker algebraic properties valid for larger classes of fuzzy quantities. Namely, the 0-symmetric ones are able to play the role of fuzzy zero, and the 1-transversible ones play the role of fuzzy units meanwhile the equivalences can bridge the gap between the fuzzy quantitative results of algebraic operations and deterministic character of some of the demanded properties (see [3]).

4. Survey of properties Particular algebraic properties of the operations summarized in Section 2 are valid in various strength - as strict equalities or as equivalences (additive or multiplicative). Their survey based on results referred in [1,2], is briefly shown here. The main problems with processing fuzzy quantities are connected with the validity of group properties and distributivity. Namely, if we take (0) for the zero-element of ~ and (1) for the unit of ~* then it can be easily seen that a®(O)=a

and

a®(l>=a

for any a E ~. But these zero and unit do not satisfy their expected properties analogous to the deterministic case if the opposite and reciprocal elements are to be introduced. It is natural to consider - a for the opposite to a and 1/a for its reciprocal but (except the degenerated cases) a @ ( - a ) is not equal (0) and a Q ( l / a ) is not equal (1). This is natural as crisp number cannot be a result of operations defined in Sections 2.2 and 2.4 for fuzzy inputs. The solution can be easily found in generalization of the concepts of fuzzy zero and fuzzy unit to the elements from $0 and ~-1, respectively. Such generalization unavoidably implies also certain transformation of the strict equality relation into more adequate weaker equivalence - the additive, respective multiplicative one. More principal problem is connected with one of the distributivity conditions. Namely, if rl, r2 E R and a E ~ then (rl + r2). a is not generally equal to ( r l " a)E~)(r2- a). It means, e.g., that a • a need not be the same like 2. a. This inconsistency cannot be explained by too strong definition of some objects, and it seems

148

M. Mare~/Fuzzy Sets and Systems 91 (1997) 143-153

to be rather an inherent property o f vagueness. Numbers should be strict (or, at least, specifically uncertain) if this type of distributivity is to be guaranteed. In the following tables we briefly summarize the degree of validity o f the group, linear space and other properties. Before doing so it is useful to remember one statement presented in [1]. Statement 2 (Distributivity). Let a, b, e E ~. Then a ® (b @ c) = (a ® b) ® (a ® c) if and only if at least one of the following conditions is fulfilled: • a is a crisp number, i.e., a = (y) for some y E R, • #a, #b, /~c are upper semicontinuous and either b E ~+, c E R+ or b E ~ _ , c E ~ _ , • /~, #b, #c are upper semicontinuous and b E 50, c E 50. Then the following groups of properties can be determined.

4.1. Relations valid as stron9 equalities

Relation

In formulas

Valid for

Commutativity

a@b=b@a a@b=b@a

a, b E ~ a, b E ~

Associativity

rl + ( r z + a ) = ( r l + r 2 ) + a , (a®b)@c=a@(boc)

rl, r 2 E R , a E ~ a, b, c E

rl

rl, r2 ER, a E R

Properties of zero

Properties o f unit

• (r2

•

a) = ( r t • 1"2)" a

aQ(bGc)=(aQb)@c

a, b, c E R

0+a=a a®(0)=a 0" a = (0)

aE~ aE~ a E[~

1 ,a=a

a

a@(1)=a Distributivity

Properties o f powers

r. (a@b)=(r.a)G(r. b) (rl + rl )- a = (rl • a) ® (r2 • a) a Q ( b ® c ) = ( a 6) b ) ® ( a ( ~ e) a -r = 1/a r = ( l / a ) r (ar) s = a r's a r 6) b r = (a 6) b ) r a-b = 1/a b = (1/a) b (a b)c = ab®c

ER a E~

a, b E ~, r E R r~, r2 E R, rl • r2 > 0, a E ~, a is trapezoidal Due to Statement 2 aE~+,

rERo a E ~+, r, s E R o a, b E ~ + , r E R o

aEi+, aER+,

bER* b, e E i *

4.2. Relations valid as additive equivalences

It is easy to see that for any a, b E ~, a -- b implies a ~ e b and, consequently, all relations introduced in Section 4.1 are valid also as additive equivalences. Moreover

M. Mare.f/Fuzzy Sets and Systems 91 (1997) 143-153

Relation

In formulas

Valid for

Opposite element

a ® (--a) ~ e (0) ~ e s

aE~,

sES0

Properties of zero

a@s~¢a a 6) s ~q~ (0) a@s~es

aE~, aE~, aE~,

sE~o sES0 sE~o

Distributivity

(rl + r2)" a ~ e (rl • a) @ (r2 • a)

rl, r z E R , a E ~ , a is almost trapezoidal or a E 5

149

4.3. Relations valid as multipficative equivalences It is easy to see that for any a, b E R, a = b implies a ~® b and, consequently, all relations mentioned in 4.1 are valid also as multiplicative equivalences. Moreover, Relation

In formulas

Valid for

Inverse element

a ® (l/a)"~o (1) a®(1/a)~®t

a E ~* aE~*, tE~l

Properties of unit

a @ t ~® a

a E 0~, t E 7]-1

Distributivity

t@(a@b)~G((y)®a)@((y)@b

Properties of power

t ~ ~@ t tb~®t ar @aS ~oar+s a r @ a -r ~® (1) ar@a-~ot a b @ a-b ~® (1) ab@a-b"~®t

)

a, b E ~ ,

tE-~y

t E -[]-1, r E Ro t E 11-1, b E ~ 0 aEqFy, y > 0 , r, s E R o , r + s ¢ O a E ff~+, r ER0 a E ~ + , r E R o , tE-~l a E ~+, b E ~* a E ~ + , b E ~ * , tE~-~

5. Interpretations As follows from the previous section, the fuzzy quantities do not form either an additive or multiplicative group if the strict equality is demanded, and (0) or (1) are considered for the zero or unit element, respectively. This fact can complicate some theoretical considerations or applied procedures. Nevertheless, if we substitute strict equality by additive equivalence and take the class 5o for "fuzzy zeros" then ~ becomes a group, and some of its significant subsets (e.g., the set of almost trapezoidal fuzzy quantities, or S ) become linear spaces. Similarly, if we use multiplicative equivalence instead of the strict equality and take 1-transversible quantities for "fuzzy units" then ~* becomes a multiplicative group.

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M. Mated~Fuzzy Sets and Systems 91 (1997) 143-153

6. O r d e r i n g s - several c o m m e n t s

There exist numerous definitions of the ordering relation over fuzzy quantities. They are modified due to the specific features of particular applications or theoretical models. Anyhow, all these concepts of ordering can be grouped into two clusters, In the first one each fuzzy quantity a E g~ is represented by some real value or, generally, by a deterministic object and the ordering of fuzzy quantities is reduced to the ordering of these representatives. It is, of course, a deterministic relation. In the second case, the ordering is a binary fuzzy relation over the set R. Its membership function maps x ~ on [0, 1], and its values are derived from the membership functions of the entering fuzzy quantities. In the following paragraphs we consider fuzzy quantities a,b,c E ~ which are to be ordered, and we introduce only a few typical examples of relations to illustrate their characteristic properties.

6. I. Ordering via deterministic representation Let us consider, first, the case in which each fuzzy quantity is represented by a deterministic object and these objects can be ordered (may be partially) by a crisp relation. The simplest representatives are real values somehow connected with the considered fuzzy quantity, e.g.

Modal values. We can order a ~ b iff sup{x: #~(x)= 1} ~> sup{x: I~b(X)= 1}.

(21)

The ordering of modal values can be rather generalized.

~-modal values. Relation a ~ b is valid iff for e E R, 0 < e,< 1 sup{x: #,(x) ~> 1 - e} ~> sup{x: #b(X)>~ 1 -- e,}.

(22)

Extremal possible values. Analogously to the previous case, a ~ b iff sup{x: ]Aa(X)>0} ~ sup{x: #b(X)>0}.

(23)

This approach can be generalized in the following way:

e-extremal possible values. In this case a ~ b iff for some e > 0, e E R, sup{x: I~a(X)> ~} ~> sup{x: l~b(X)> ~}.

(24)

In all these cases the ordering over ~ is transformed into complete ordering of real numbers with its well known properties. Namely a ~ a,

(25)

a ~ b A b ~ a implies a specific kind of equivalence a ~ b,

(26)

a ~ b, but not b ~ a then we may define a strong ordering relation a >-b,

(27)

a ~ b A b>~ c~a>~ c,

(28)

M. MareglFuzzy Sets and Systems 91 (1997) 143-153

always

a~bVb~-a, or a ~ b V b ~ a ,

151

(29)

a ~ b is equivalent to a ,,~ b V a ~- b,

(30)

relation a ~ b excludes b ~ a,

(31)

and some others. The orderings based on more global representatives of fuzzy quantities, usually derived from the formal shape of membership functions are almost as transparent as the previous (and other similar) concepts. The first one of them offers an ordering relation over fuzzy quantities but it respects quite different purpose of the ordering.

Implicative ordering. Due to this approach a ~ b iff p~(x)>>-pb(X) for all x.

(32)

This relation does not order the achievable values of fuzzy quantities but their "dispersion" or extent of fuzziness and in this sense it does not fit to the other orderings presented here. We can only note that it fulfills (25) and (28), equivalence suggested in (26) is the strict equality, (27),(30) and (31) are evidently fulfilled and, as this ordering is not complete, property (29) is not generally valid. Other type of the ordering, in this case comparing the achievable values, indeed, rather extends the idea being present in the ordering of extremal or modal values to the whole areas of the support sets.

Disjoint ordering. The relation a ~ b is valid iff sup(min(#a(X),

x
Pb(Y))) ----0

(33)

which is equivalent with the condition that if for x E R, [2b(X) > 0 then p ~ ( y ) = 0 for any y ~
e-disjoint ordering. The relation a >- b is valid iff for some e > 0 sup(min(/~(x),

x
Itb(y))) < e.

(34)

None of these relations is a strict ordering of the type (27), they are incomplete and transitive (28). Their extension to weak ordering a ~ b fulfilling the other properties mentioned above is possible in several ways, e.g., a ~ b if neither b ~ a nor a ~- b are true. Another global approach to the ordering of fuzzy quantities can be formulated in the following way.

Uniform ordering. Let us denote b0 = inf{x:

pb(X)= 1},

a0 ---- sup{x: pa(X)---- 1}.

Then a ~ b iff

pa(x)<~Itb(X) for all x<<.bo, pa(x)>~#b(X) for all x>>.ao.

(35)

This ordering is not complete, which means that (29) looses its sense but (25) and (28) are fulfilled, (26) and (27) can be used and also (30) and (31) have sense.

M. Mare~/Fuzzy Sets and Systems 91 (1997) 143-153

152

6.2. Fuzzy ordering relation It can be considered naturally that the relations between vague objects are also vague. It means that the ordering relation over fuzzy quantities should be a fuzzy relation as well. To illustrate the problems and reasonable advantages of such approach we introduce only one but quite representative example. As it evidently reflects the general paradigm of the extension principle (8), we will refer it as the

Extensional fuzzy ordering. As the complementarity principles (30),(31),(29) and also (26),(27) gain specific form in the fuzzy case, it is reasonable to define three fuzzy relations and to study their mutual position. For a, b E ~ we define fuzzy relations a>~b, a ~ b , a>-b with membership functions v~: R x ~---~[0,1], v~: ~ x ~---~[0,1] and v>_: ~ × ~--~[0,1] where v~(a, b) = sup(min(/~a(X),/zb(y))),

(36)

x>_-y

v~(a, b) : sup(min(#a(X), #b(x))),

(37)

xER

v>_(a,b) = sup(min(#a(X), #b(Y))),

(38)

x>y

respectively. Then for any a E

v~(a,a) = v~(a,a) = 1 = v>_(a,a)

(39)

which corresponds to (25). Moreover, for any a, b E

v>~(a,b) = max(v>_(a, b), v~(a, b))

(40)

which is consistent with (30). It is also easy to see

v~(a,b) = v~(b,a).

(41)

On the other hand, the complementarity of these three relations (analogous to (29), (30), (31)) is valid in weaker form. Namely

max(v>_(a,b), v>_(b,a), v~(a,b))= 1,

(42)

which means that also

max(v~(a,b), v>.(b,a))= max(v~(a,b), v~(b,a))= 1.

(43)

The transitivity relation, analogous to (28) is not generally guaranteed. In its stronger form, it could be represented by an equality

v>~(a,c) : min(v>~(a, b), v>~(b,c))

(44)

or, in a weakened form (if we admit also other possibilities of achieving a ~>c than via b), at least

v~(a, c) >t min(v~>(a, b), v~(b, c)).

(45)

None of these relations is generally fulfilled, and it is not difficult to construct examples (cf. [2]) in which v~(a,c) < min(v~(a,b), v~(b,c))). Of course, examples of the validity of (44) and (45) can be constructed, as well.

M. Mare,~/Fuzzy Sets and Systems 91 (1997) 143-153

153

7. Conclusive remark The sections above offer a brief overview of results regarding the algebraic processing of fuzzy quantities based on the extension principle. It is to be stressed that there exist also other approaches to the arithmetics of fuzziness. Some of them may be based on the convolution (see [2]), where some of the results are similar to that ones given above. Other, in this case significantly generalized, methods are connected with the concept of triangular norms. The results obtained for the t-norm based fuzzy numbers (see [4], for example) do not contradict the special case of extension principle and bring some general views on the algebra of fuzzy numbers.

Acknowledgements This paper summarizes results of research supported by several grants. The author wishes to express thanks namely for the support by the Academy of Sciences of the Czech Republic grant No. A 1075503, by the Academy Key Project No. 1, by the Grant Agency of the Czech Republic grant No. 402/96/0414 and by the European Union ACE Project No. P 95-2014-R.

References [1] D. Dubois, H. Prade, Fuzzy numbers: an overview, in: J.C. Bezdek (Ed.), Analysis of Fuzzy Information, CRC-Press, Boca Raton 1988, vol. 2, 3-39. [2] M. MareS, Computation Over Fuzzy Quantities, CRC Press, Boca Raton, 1994. [3] M. MareS, Fuzzy zero, algebraic equivalence: Yes or No? Kybernetika 32 (4) (1994) 343-351. [4] R. Mesiar, A note to the T-sum of L-R fuzzy numbers, Fuzzy Sets and Systems 79 (1996) 259-261.