CHAPTER 3
EQUIVALENCE CLASS REPRESENTATION
OF FUZZY NUMBERS *
3.0 Introduction
Decisions are generally made on the basis of ranking-which alternative is best,
which is second best and so on. If the alternatives are fbzzily assessed, there is no linear
ranking among them. In this case, finding the best alternative leads to the problem
of comparing fuzzy numbers. There are different approaches in the literature for
ranking fuzzy numbers [20, 35, 57, 741. In this chapter, we propose a ranking
method by classifyingthe fuzzy numbers into different equivalence classes.
The set of closed bounded intervals of the real line IR is partitioned into
equivalence classes and is used to partition the set of fuzzy numbers in IR. The
arithmetic operations as well as maximum-minimum operations are studied on these
equivalence classes. The partial order relation defined on the equivalence classes
of fuzzy numbers make more fuzzy numbers comparable. Finally, the equivalence
class concept is used to fuzzify certain basic concepts in Evidence Theory and
Possibility Theory in a more meaningful way.
*
Some of the results of this Chapter are to appear in the paper 'Equivalence class
representation of fuzzy numbers-Application to Evidence Theory and Decision
Analysis', The Journal of Fuzzy Mathematics, IFMI, U.S.A.
Notations
Throughout this chapter we use the following notations:
R
-
the set of real numbers.
R+
-
the set of non-negative real numbers.
I(R>
-
the set of all closed bounded intervals in R
7 N,(R)
- the set of all fuzzy numbers in R (see Def.(l.2.13)) whose one
.
cuts are singleton sets.
The usual operations and order relations on I(R) and 7 N,(R) are as defined
in Chapter 1 .
3.1 Equivalence classes of closed intervals of R
In this section we define an equivalence relation on I(R) such that each
equivalence class corresponds to a unique real number in R.
3.1.1 Definition
Let a, b
a
-b
E
I(R) . Then the relation '-' on I(R) given by
iff a - b = [-C,C]
;c
E
R+
is an equivalence relation and it decomposes I(R) into disjoint equivalence
classes.
The set of all equivalence classes of closed intervals in I(R) is denoted by
3.1.2 Note
If [a]
E
[a]={b
[ I(R)], then
E
I(R):a-b)
In particular, the equivalence classes containing
0 = [O, 01 and 1 = [I, 11 are respectively given by
[O]
=
{[-c, c] : c
E
R+)
3.1.3 Definition (mid value function)
Let a = [a,, a,]
E
I(R).
Then the mid value of 'a' denoted by ' Sa ' is defined by
3.1.4 Proposition
Let a = [a,, a,], b = [b,, b,]
a - b i f f Sa
=
E
I(R) . Then
Sb.
Proof:
a - b a a - b =[-c,c]
a a, - b,
= -c
;c
ER+
and a, - b,
=
c
e
Sa=&
3.1.5 Definition
As intervals having same mid value lie in exactly one equivalence class, we
shall define the 'midvalue' of [a]
E
[ r(R)] as S[a] = Sa
3.1.6 Remark
The above definition establishes a one-to-one correspondence between
equivalence classes in [ I(R)] and elements in R.
3.1.7 Proposition
Let [a], [b]
[a]
=
E
[ I(R)]. Then
[b] iff S[a]
=
S[b]
Proof: Obvious.
3.1.8 Proposition
The relation ' ' defined by [a] 5 [b] iff S[a] 5 S[b]
is a total order relation on [ r(R)].
Proof: Obvious.
3.1.9 Remark
The usual order relation defined in I(R) is only a partial order. But the above
relation on [ I(R) ] is total.
3.1.10 Proposition
Let [a], [b]
[ I@)]. Then
E
[a] 5 [b] iff there exist two intervals c, d
E
I(R) such that c
and c 5 d.
Proof:
Let [a], [b]
Let [a,, a,]
E
E
[ I@)] such that [a] 5 [b] .
[a1 and [b,, b,I
E
[bl
Then a, + a, 5 b, + b,and there arise 3 cases.
Case 1: Let a I < b, and a, 5 b,,Then, [a,, a,] 5 [b,, b,]
Case 2:
Let a 1 <
- b, and a, > b,
[a,, a,]
=
- [a, + a, - b,,
[a,+a,-b,, b,]
[b, , b,I
E
a, - (a, - b,)l
[a]
(a:
and
a, + a,- b, 5 b,)
E
[a], d
E
[b]
Let a, > b, and a, I b,
Case 3:
[a,, a,]
=
-
[a, - (a, - b,), a,+ (a, - b,)l
[b,, a, + a, - b,]
E
[a]
and
I [b, b,I
Conversely, if there exists c
S[a]
=
Sc I ~d
=
E
[a] and d
E
[b] such that
c I d, then
S[b] and hence [a] I [b]
This completes the proof.
3.1.11 Definition
[a]
E
[ I(R)] is said to be an equivalence class of closed intervals in [0, 11 if
0 I S[a] I 1. In this case, there exists atleast one closed interval [a,, a,]
E
[a]
such that [a,, a,] c [0, 11.
3.1.12 Definition
[a]
E
[ I(R)] is said to be nonnegative iff S[al 2 0.
3.1.13 Proposition
Let a, b, c, d
hold.
E
I(R) be such that a - c and b - d. Then the following results
(i)
a+b-c+d
(ii)
a- b
(iii)
If h
-c - d
E
-
R, then a c
2
ha
- hc
Proof: Obvious.
3.1.14 Proposition
The following operations are well defined on [ r(R)]
(i)
[a ] + [b]
(ii)
[a] - [ b]
(iii)
[a + b]
=
=
[a - b]
h [a] = [ h a] ; h E R
[a] A [b]
=
[a]
[b]
if S[a] I S[b]
if B[a] > Sib]
(v> [a] v [bl
=
[a]
[b]
if S[a] 2 S[b]
if 4 a I <
(iv)
(vii)
(
[a] / [b] = [c] , where
flc]
=
S[a] provided S[b] + 0
S[bl
Proof: Obvious.
3.1.15 Note
The mid value function
(iv)
~ ( [ a/[b])
]
=
satisfies the following properties.
&[a] provided S[b] # 0
4bI
(v)
S([al
A
[bl)
=
S [a1 A Sib1
3.1.16 Note
[I(R)] forms a normed linear space w.r.t. the above defined addition and
scalar multiplication and the norm ' 1 ' defined by II[x]II
(1
=
16[x]I
The metric induced by the norm is given by
and the two spaces R and [I([R)] are topologically equivalent w.r.t. the
metric topologies induced respectively by the usual metric and d.
3.2 Equivalence classes of fuzzy numbers
In this section we define an equivalence relation on 7 N, (R) and verify some
properties of the corresponding equivalence classes.
3.2.1 Definition
The relation '-' on 7 N, (R) defined by
Z - 6 iff ";i-"g
Va
E
(0, 11
is an equivalence relation on 7 N, (IR) .
The set of all such equivalence classes of fuzzy numbers in 7 N, (R) be
denoted by [7N, (R)].
3.2.2 Proposition
Let Z,6
Z
-
E
F N , (R) .Then
iff ( ( r ) = ( Z - ( r )
Vr
Proof:
( ( r )= -
r
)
Vr
E
E
[R
e
"~-"
= i"6-"Z
b'a
["a'-, "Z+] -
=
[%-, "k ]
-
-a,.,+,ag+- S ]
0
[a;-
e
"Z-+"Z+= "6-+"b
e
"z-
e
ii- b
[%-, "b ] - ["E-, "E+]
-
=
b'a
(O,l]
E
[at,--
"E+, $+ -"I]
E
b'a E (0, 11
-
Hence the result.
3.2.3 Note
We embed the set of real numbers into the set of fuzzy numbers.
The equivalence class containing 0
z
E
E
R is denoted by [ fj]
[ t i ] iff
Z (r)
=
Z (-r)
V ~ E I R , andlE =[0,0]
The equivalence class containing 1 E R is denoted by [T 1.
iT
(
.
[Tliff
)
Z
(
+
)
Y r E R , and
';i
=[1,1]
3.2.4 Definition
If [ Z ]
E
[ 7 N1(R)], then the a -cut of [ Z ] denoted "[Z] is defined by
3.2.5 Proposition
Proof:
Hence the proof.
3.2.6 Proposition
Let [El, [ L I
E
[ F N , (WI
Define[ii]< [ g ] i f f a [ Z ]
<
a[g]
Va~(o,'l]
Then ' 5 ' is a partial order relation on [ F N, @)I.
Proof: Obvious.
3.2.7 Proposition
[ g ] 5 [glifthereexists F E [ Z ] and
a
E
such that F 5
[El
a.
Proof:
LetF E [ P ] and
-c 5 -d =
a
[$I
"E 5 "2
suchthat F 5
a.
Va E ( O , ~ ]
=
["El<[";]
=
" [ z ] "[;I
~
Va~(O,11
=
" [ ; i ] ~"[GI
Va~(O,ll
=
[ z l 5 [5]
V~E(O,~I
Hence the result.
3.2.8 Remark
According to the above proposition two equivalence classes of fuzzy numbers
are comparable if any two fuzzy numbers in the respective equivalence classes are
comparable.
3.2.9 Proposition
The following operationsare well defined on [ 7 N, (R)].
(i)
[zl+[gl
= [ z + i;1
Ci)
[zl-[El
=[z- G I
h [ ~ =l [ h Z ]
; h E R - (0)
(iv)
[PI [El
where "[El
(9
=
"[Z] a[b]
V a E (0, 11
[ z I / [ b l =[EI
where "[TI
=
a[;i]/a[b~
provided 6 "[El z 0
V a E (0, 11
V a E (0,1]
(
i
[)
z l A [El =[EI
where "[C]
=
A
a[b]
Va
E
((),I]
[ z l v [El =[zl
where "[TI
=
"[Z] v "[El
Va~ ( 0 ~ 1 1
Proof:
Since the 6 values of the a -cuts in the same equivalence class must be equal,
the results follow fkom the Prop. (3.2.5).
3.2.10 Remark
Addition and subtraction of elements in [ 7 N, (R)] are conformal with the
usual addition and subtraction of fuzzy numbers. But the multiplication and hence
division of elements in [ 7 N, @)I are not conformal with the usual multiplication and
division operations on fuzzy numbers.
3.3 Application to Evidence Theory and Possibility Theory
In this section we introduce the concepts of Evidence Theory and Possibility
Theory using equivalence class representation of fuzzy numbers.
Throughout we assume that X is a nonempty finite set, P(X)-the power set of
X and [ F N, ([O, I])] is the set of all equivalence classes of fuzzy numbers whose
1-cuts are singleton sets and whose supports are contained in [0, 11.
i.e., [ g ]
E
[FN1([O,l])]if 0 5 S a [ Z ] II
v
(0, 11
3.3.1 Definition
Afuzzy valued basicprobability assignment is a function
m : PO() + [FN,([O, l])]satisfymgthe conditions:
m,. m ( + ) = [ @ I
Cm(A)
=[TI
m2' A c X
3.3.2 Remark
m ( + ) = [ @ ] iff Sam(+)=O
and C m ( A )
AcX
=[TI
Vcx~(O,l]
iff C 6 a m ( ~ ) =1 V a ~ ( 0 , 1 ]
AcX
3.3.3 Definition
F c X is afocal element of m if 6"m(F) > 0
Va
E
(0,1]
3.3.4 Definition
Fuzzy valued belief measure and fuzzy valued plausibility measure induced
by afuzzy valued basicprobability assignment m can be defined respectively as
PI (A)
-
Cmcn
FnAt4
3.3.5 Proposition
(i)
Be1 (A) 5 PI (A)
(ii)
PI(A)=[T]-B~~(A')
VAEP(X)
Proof:
(i)
Be1 (A) 5 PI (A) iff daBel (A) <- daPl (A) V a
The inequality on the R.H.S. holds V a
(ii)
E
E
(0,1]. Hence the result.
PI(A)=[T]-Bel(Ac)iff
6" PI (A)
=
1- 6" Be1 (Ac)
and this holds V a
E
V a E (0,1]
(0, I]. Hence the result.
3.3.6 Definition
Pos : P(X)
+ [ F N , ([O, 1111
is afuzzy valuedpossibility measure if
(iii) Pos (A uB) = Pos (A) v Pos (B)
(0,1]
;A, B E P(X)
3.3.7 Definition
A function Nec : P(X)
+ [ F N , ([O, I])]
is afuzzy valued necessity measure if
O
Nec($)=[GI
(ii)
Nec(X) = [ ]
(ii
Nec (A nB) = Nec (A)
A
Nec (B)
;A, B
E
P(X)
3.3.8 Note
If
is a fuzzy valued possibility measure, then the function p defined by
p(A)=[TI
-
A(A')
;A
E
P(X)
is a fuzzy valued necessity measure.
3.3.9 Definition
A function r : X
+ [ FN, ([O, I])]
is afuzzy valuedpossibility distribution
The corresponding possibility and necessity measures are respectively given by
PosJA)
=
V r(x)
XEA
3.3.10 Remark
It can be easily verified that the basic properties of possibility and necessity
measures which do hold in the crisp case, also hold true for the corresponding
fuzzy valued measures.
In this Chapter, the set of fuzzy numbers in R whose one-cuts are singleton sets
is partitioned into equivalence classes. This facilitates the fuzzification of
several concepts in Evidence Theory and Possibility Theory, more meaningfully. In
real situations, the representation of degrees of beliefs and possibilities as fuzzy
numbers seems to be more realistic. However, the equivalence class representation
of fuzzy numbers is found more suitable as it provides a typical representation for
fuzzy number valued basic probability assignments and possibilitiy distributions
satisfying the respective normality conditions. Moreover, as the partition of fuzzy
numbers into equivalence classes reduces the incomparability among fuzzy numbers,
it provides a method for ordering fuzzy numbers. In the next chapter, we apply these
techniques for the fuzzy decision analysis of a problem in decision theory.