NASA-CR-I90709
Fuzzy Sets, Rough Sets, and Modeling Evidence: Theory and Application A Dempster-Shafer Based Approach to Compromise D e cisio n Ma kin g wLth Multiattrib utes Applied to Product Selection
-
-
Korvin
An|l---tie
O, ,.4-
--
University
p.. t_ 0w _0
to
.=======
of Houston-Downtown
_
,,,,..,_,..,,...,.,
!
U
-
_y
1992
0
-.....
,.o
.........
Cooperative Agreement NCC 9-16 ResearchAct_v_ty No. SR.01: Using Roug[!_Sets
for Knowledge
Acquisition
L_
=_0 OUJU_
u
LU
_
Q C) _
L
NASA ,Johnson Space Center Information Systems Directorate Informatzo_cihnology Division
_. +J 0 u_
i, I W I--.LL! _ (/)
-=4
0
co Z I-- :E _J_O _" 0 OL s_
, 0 _'.-
0
0
:_wOO_.jr 0
set
denotes
on
re(A)
re(A)
of
elements
the
B
AAB,
expression
of
Subsets
of
fuzzy
and
their
last
a mass
subsets
and
_u
functions
By
If
is
the
function.
m
U
is
masses by
a
zero
focal
on
X,
(m I ® mz)
element
then (A)
are
the
=
called
of
m
direct
Z_. A
if
focal m(U)
sum
>0.
of
m1(B ) mz(C ) /
mI
Z_..
V
m1(B ) m z (C) focal
elements
(fuzzy)
focal
m I • m z are and
m 2. We
Shafer
m
of
m I and
element
obtained set
A
m 2.
of by
_
o
. Here
Of
Thus
intersecting =
0.
and
course,
m I • m z.
(m I • mz)(e)
B
A the
the (For
C
denote
denotes focal
focal
(fuzzy) a
typical
elements
elements
additional
of of
m_
details,
see
m z come
from
[7].) This
m w
if
rule
of
composition
applies
5
when
m I and
independent
sources
generated
by
of
these
information
two
sources.
and
The
represent
direct
sum
the
is
mass
a construct z
that
sometimes
independent
models
sources
well
of
the
information
information,
but
gathered
this
is not
from
always
w
the l
case.
For
article
a discussion by
L.A.
of
Zadeh
this,
the
[12].
In
reader
this
is
referred
context,
the
to
the
set
X
is W
often
called A
the
mass
universe
function,
generates
two
X.
are
of m,
important
discourse.
on
the
universe
functions
set
of
discourse,
defined
on
the
X,
sets
of m m
These
Bel
(B)
the
-ZA_
belief
re(A)
and
plausibility
functions
an
probability
(unknown)
some
area
where
oll
and (S)
Pls denote
plausibility -
a
Z_,.
may
be
m(A).
lower
function.
functions
and
For
The
an
belief
and
bound
for
upper
example,
:
let
S
denote
J
I
present. w
x
m
w
In
Figure
i:
Figure
i,
could
be
Figure we
have
found.
five
Three
of
experts the
2:
locating
five
points
where
experts
have
located
have
located
oil
oil
I
oil m
V
inside
the
outside
area
of
S.
S,
We
and
could
two say
experts that
the
probability
of
to
oil
be
inside J
S
is
have sure
3/5,
since
seven of
we
experts
have
three
locating
themselves
so
the
hits oil.
out These
i th expert
of
five.
experts locates
In are the
Figure not oil
2 we
totally
v
to
m
be
w
anywhere
in
A t rather
than
at
one
specific
point.
Under
these
6
m mmm
L_ w
circumstances, not
the
defined,
that
the
lowest
since
the
might
be
oil
possible
hits
out
that
we
of
belief
and If
...AT}
If
the
with -
It
is
clear
points,
then
Be1
a
formal w
Bel(e)
For
every
to
a
{l,2,...,n)
and can
= eZ(-l)IA'el
a
=
be Bel
set
A n
-B.
Then
the
have can
is
to
be
3 say
called
{A1,
reduce
to
to as
=
5/7.
specific
a probability a belief
are
(or
points.
may for
A2,
function;
m(Ai)
reduced
not a
be
a
The viewed
probability
In
fact,
the
subsets
of
are:
A2,...,A
the
where
iA-Bi Z m(A). A_ 7
finite
cardinality
terms
=
n
A,)
non-empty
in
BeI(B)
we we
function.
At,
w
the
seek
1
denotes
(B)
S,
Z A_s,.
axioms
belief
all
defined
we
a mass
function
k Z(-l)i'l ÷' Bel(_
]IJ
m i is
elements
subsets,
over
m
viewed
function
BeI(X) of
be
is
plausibility.
of
=
and
usual
bellef
collection
ranges
re(A)
apply
in
called
are
a belief
the
0 and
I
function
PIs(S)
can
If
probability
Then
equal
S.
S
indicating
probability
is
Ai
inside
are
exists
lowest
focal
e.g.
of
elements
sets
the
BeI(AIuA2u...A,)
where
The
being
expert
highest
and
are
and
for
=
oil
1 _ i S 7.
Pls
true;
not
axioms
(1)
(ii)
not
do
that
focal
the
where
probability
function
outside
7.
3/7
if and
or
a probability
plauslbillty)
as
=
oil
fifth
probability
- I/6;
that
is
of
the
and
th@
the
m(Ai)
Thus
converse
out
define
Z_cs
function.
want
highest
m(Al)
BeI(S)
fourth inside
we
5 hits
we
of
probability
7.
have
probability
of is
a
of mass
the
I. m
Any
defined
cardinality
such by of
A belief (i)
Bel(o)
function
is
= 0, Bei(x)
called
Baysian
if
= 1 m
(ii)Bel(AuB)
=
whenever
A
BeI(A)
and
B
+ BeI(B)
are
disjoint.
It
may
be
shown
that
the mm_
following (i)
conditions
Bel
(ll)
is
are
equivalent
Baysian
Focal
w
elements
are
points m
(ill)
BeI(A)
+
In
present
the
BeI(-A)
=
1
work,
it
will
be
very
natural
to
extend J
this
setting
subsets
of
to the
considered
the
case
universe
by
Zadeh
where
of
focal
discourse.
[ii],
but
elements
This
Yager
are
setting
[8],
among
fuzzy
was
first
others,
m
has m
done on
similar the
work.
theory
of
We
would
masses
also
over
refer
fuzzy
the
sets
reader
by
Yen
to
[9],
a
paper
since
we i
will
define
the
different
attributes
and
the
ideal
in
those
terms. To of
begin
alternatives
these
will
selection
hl,
by
the
of
of
example,
Documentation,
F2 the if
software Let to
the
could
=
FI,
be
i;
F2,
Fn
Adequate
..., to
the
For
Help,
fl2,
For under
1 S
a
denote
a
for
list
of
maker
in
F i could
be
decision
fiKi
i S n and
set
example,
assessment
Let
fl3 could
Documentation,
our
example,
etc.
Fi, where flI,
process,
packages
important
be
making
defined.
alternatives.
attribute, i
decision
...h t is
user.
considered
Documentation,
For
h2,
different
evaluation
elements
compromise
be
attributes the
the
t
denote
1 S k i _ n i.
denote
Inadequate
and
Extensive
m
8 W
o
Documentation, hi,
we
respectively.
have
n
fuzzy
Associated
sets
with
corresponding
each
alternative,
the
n different
to "f
attributes.
Thus,
hj
is
associated
with
Z
a_j_i /
fi_i where
Kt-1
aljK! is
the
present
amount
in
example,
to
which
alternative
hi;
the
first
Documentation
.1/Extenslve
Documentation
with
attributes,
We
recognize
particular more is
Help
the
another
such
Using
an
.! Z d_!
/
+
and
+
Help
from
the
attributes
i _< n and
n
For
our with
Documentation
if
+
Help
we
are
+
concerned
may
under
proposed
for
objects given
fuzzy
desire
a
consideration
[4]
alternative as
t.
Help.
Coombs
alternative
_
associated
maker
attributes
any
j
be
,
and
decision
of
_
F_ is
.3/Undesirable
the
C.H.
attribute 1
could
that
for
I $
n
of
.S/Adequate
.1/Desirable
level
ideal
fl_! ,
_
Documentation
notation
the
i
fl_t
package
element.
ideal
the
express
+
element
than
I _
computer
.4/Inadequate
.6/Acceptable
element
that of
there
choice.
above,
we
may
sets:
I _< k i S n!
where
di_! expresses
Kf-I
to
what
We
may
degree use
[5]
to
his
degree
assist
The pairwise decision the
a
of
decision
process
such
the
DM
of
comparisons maker(s)
value
in
preference
assigning
elements
relative
the
of of
each the
as the
each
of
ffK_
weights
each
of
attribute's a total in
elements
9
the
of
of
F i.
method
to
determine
F t. importance
through
elements
same with
ft_(
constant-sum process
distribute
two
element
evaluation
relative
pair
wants
Guilford's
for
of to
maker
I00
asks
points
between
proportion
respect
to
the
as each
the other
[3].
After
all
subjective
of
values
are
recovered
The
use
the
comparisons
implicit
through
in
use
of
the
have
been
decision
a ratio
made,
maker's
scale
the W
judgment
method
([3],
[5]). m
of
decision It
Is
with
Guilford's
maker's
each
ratio
consistency
necessary respect
[5]
that to
the
of
DM's
consistent
attribute's
scale judgment
ideal
be
weights
elements
method
since
to as
of
also be
allows
monitored
accurate
relative
these
the
as
[I].
possible
importance
values
m
for
form
mass nm
functions
that
plausibility As
ultimately
of
each
Zeleny
[14]
influence
the
belief
and
alternative. suggests,
the
ideal
serves
as
a
minimum
,r
.... re qulrement
for
intelligent
discourse.
This
ideal
of
each
as
generated m
by
the
relative
importance
weights
attribute's i
elements
reflects
psychological, a
decision
societal,
relatively
and
cultural,
these
situation
[13].
weights Thus,
genetic,
background
context-dependent
importance,
decision
maker's
environmental
unstable,
informational given
the
As
concept
are
the
[13].
of
reflective
relatlve
of
k
a
importance D
values sets
determined of
attribute
associated These viewed can of
as
express
with mass fuzzy each
alternatives.
by
the
decision
elements, each
thereby
subsets
contain
vary
altering
the
focal
of
alternatives.
of
an
element
.3/h I +.5/h 2 +...+.8/h
may
for
different
mass
function
W
attribute.
functions
For
maker
our t if
elements In
attribute
example, h1's
Help
we
as can
has
other a
fuzzy
write
been
which
can
words, set,
be we
FjK_,
F,ELpU"_sfr'bLe =
evaluated
m
as
W
.3
i0 i
7
Undesirable,
h2's
etc.
previous
Using
Help
has
been
evaluated
notation,
as
we
.5 Undesirable,
determine,
F_K_
as
t
_.I alibi / with
each
fuzzy A----
hj where; element,
focal
terms
We
of
the
the
nine
of
fuzzy
focal
attributes
determine A,,
of
the focal
over
which
define
of
elements
section
masses,
each
we
can
mi
element,
and
elements,
m.
For
our
A,,
the
be
_< i _< n) each
1 _ kf < n o. combination
m
over
FIK_.
the
We
let
A,
example,
we
have
from
the
two
of
over
can
(1
the
which
definition
defined
define
f_[f for
formed
each
Using
Fl[i of
use
focal
Help,
m,
we
function
sets,
III).
elements,
n
associated
functions
mass
of
function,
F_,
mass
m n where
forming sets
Thus,
- d01t; 1 < i _ n and
...e
element
mass
of
m!(FIll)
Documentation (See
attribute,
weight
finite
1 _< k_ _< n_.
each
can
= mI •
focal
fuzzy
elements.
we
thereby
intersection be
m
and
Fill,
that
m by,
[7 ],
of
ideal
F t so
define
rule
fi[_
Indeed,
attribute,
i < n
elements,
determined. in
1
