„DECISION MAKING IN
FUZZY ENVIRONMENT

WAYS FOR GETTING PRA
CTICAL DECISON MODEL
S“
Heinrich J. Rommelfanger
*
Abstract.
In real decision situations we are often confronted with the problem that the very demanding
conditions of classical decision models ar
e often not fulfilled or the costs for getting this information seem too
high. Subsequently, the decision maker usually abstains from constructing a decision model; he fears that this
model is not a real image of his real problem.
The fuzzy set theory offe
rs the possibility to construct decision models with vague data. Consequently, a lot of
decision models with fuzzy components are proposed in literature. In my opinion, only fuzzy utilities (fuzzy
results) and fuzzy probabilities are important for practica
l applications. Therefore, the focus of this paper is
concentrated on these subjects.
At first, fuzzy intervals of the


type are introduced. These special fuzzy sets offer a practical way for
modeling vague data. Moreover, the arithmetic operations can
be calculated with little effort. Afterwards,
different preference orderings on fuzzy intervals are discussed.
Based on these definitions the principle of Bernoulli can easily be extended to decision models with fuzzy
consequences. The use of additional in
formation for improving the a priori probabilities is also possible.
Moreover, fuzzy probabilities can be used combined with crisp or with fuzzy utilities. Here, we introduce
several new algorithms for calculating the fuzzy expected values.
A disadvantageo
us consequence of the use of vague data is the fact that an absolutely best alternative is not
identified in all applications. But normally it is possible to reject the majority of the alternatives as inferior
ones. For getting the optimal alternative addi
tional information on the results of the remaining alternatives can
be used; but this should be done under consideration of cost

benefit

relations.
Apart from the fact that fuzzy models offer a more realistic modeling of decision situations the proposed
in
teractive solution process leads to a reduction of information costs. That circumstance is caused by the fact
that additional information is gathered in correspondence to the requirements and under consideration of cost

benefit

relations.
Keywords:
decisi
on theory, fuzzy utilities, fuzzy probabilities, information costs,
*
Institute of Statistics and Mathematics, J.W. Goethe

University Frankfurt am Main
D

60054 Frankfurt am Main, Mertonstr.17

25
1 Introduction
Looking at modern theories in management sciences and business administration one
recognizes that the majority of these concepts are based on decision theory in the sens
e of
N
EUMANN AND
M
ORGENSTERN
[1953].
However, empirical surveys, see e.g. [
L
ILIEN
1987],
[
T
INGLEY
1987], [
M
EYER ZU
S
ELHAUSEN
1989], reveal that statistical decision models are hardly
used in practice to solve real

life problems. This neglect of recognized
theoretical concepts may be
caused by the fact that the very demanding conditions of classical decision models are often not
fulfilled in real decision situations or the costs for getting this information seem too high.
For modeling a decision problem by a
classical decision system, the decision maker (DM) must
be able to specify the following elements:
1.
A set A of alternative courses of action (acts),
A
a
a
a
m
{
,
,
,
}
1
2
,
2.
A set S of possible events associated with each course of action,
S
s
s
s
n
{
,
,
,
}
1
2
,
3.
A value (result, gain) to be associated with each act

event combination,
g
g
a
s
i
m
j
n
ij
i
j
(
,
)
,
,
,
,
;
,
,
,
1
2
1
2
. G is the set of possible values
g
ij
.
4.
The degree of knowledge with regard
to the chance of each of events occurring. Usually only
partial knowledge is assumed in form of a probability distribution
p
s
j
(
)
.
5.
A criterion by which a course of action is selected:
In literature, the
B
ERNOULLI

criterion is rec
ommended for rational behavior, i.e. the expected
utility should be maximized:
E
a
Max
E
a
Max
u
g
a
s
p
s
a
A
i
a
A
i
j
j
n
j
i
i
(
*)
(
)
(
(
,
))
(
)
1
.
6.
A posteriori probability distribution:
The only chance for improving the solution of a classical decision model is to use additional
informati
on of a test market
X
x
x
x
K
{
,
,
,
}
1
2
. Knowing the Likelihoods
p
x
s
k
j
(

)
, the a
priori probability distribution
p
s
j
(
)
can be substituted by the a posteriori probability distribution
p
s
x
j
k
(

)
=
p
x
s
p
s
p
x
s
p
s
k
j
j
k
j
j
j
n
(

)
(
)
(
)
(
)
1
B
AYES
’
s
formula .
Since the paper "Fuzzy Sets" of Lofti
A.
Z
ADEH
was published in 1965, the fuzzy sets theory
has been considered as a new way for modeling more realistic decision models. Especially betwee
n
1975 and 1985 several decision models with various fuzzy components were introduced. Without
any claim on completeness, the following fuzzy elements have been proposed for use in decision
models:
1.
Fuzzy acts
~
{(
,
(
))

},
,
,
D
a
a
a
A
h
H
h
i
D
i
i
h
1
,
T
ANAKA
,
O
UKUDA
;
A
SAI
1976;
2.
Fuzzy events
~
{(
,
(
))

},
,
,
Z
s
s
s
S
r
R
r
j
Z
j
j
r
1
,
T
ANAKA
,
O
UKUDA
;
A
SAI
1976.
3.
Fuzzy probabilities
~
~
(
)
{(
,
(
)

[
,
]}
P
P
s
p
p
p
j
j
P
j
0
1
,
W
ATSON
;
W
EISS
;
D
ONELL
1979;
D
UBOIS
;
P
RADE
1982;
W
HALEN
1984.
4.
Fuzzy utility values
~
~
(
,
)
{(
,
(
)

}
U
U
a
s
u
u
u
U
ij
i
j
U
ij
, where U is th
e set of given crisp
utilities associated with each act

event combination,
J
AIN
1976;
W
ATSON
;
W
EISS
;
D
ONELL
1979;
Y
AGER
1979;
R
OMMELFANGER
1984;
W
HALEN
1984.
5.
Fuzzy information
~
{(
,
(
)

}
Y
x
x
x
X
t
k
Y
k
k
t
,
T
ANAKA
,
O
UKUDA
;
A
SAI
1976;
S
OMMER
1980.
6.
Moreover, s
ome authors propose to substitute the probability distribution
p
s
j
(
)
by a possibility
distribution
(
)
s
j
, see e.g.
Y
AGER
1979 and
W
HALEN
1984. They assume that utilities can be
measured on an ordinal scale only a
nd therefore expected values do not exist.
These new ideas, however were not applied to practice, either because they did not become
known to the public or because they are of little use for real decision problems.
In my opinion the latter statement is cor
rect as far as the points 1, 2, 5 and 6 are concerned:
DMs need workable but not fuzzy acts.
In real problems, the events and the information are usually described in a fuzzy way. In these
cases one is able to assign directly probabilities to those eleme
nts; that means that we have
probability distributions
p
Z
r
(
~
)
and
p
Y
Z
t
r
(
~

~
)
and values directly associated with the combinations
(
,
~
)
,
,
,
,
;
,
,
,
a
Z
i
m
r
R
i
r
1
2
1
2
. Therefore, we can use the classical procedure as we replace
s
j
by
~
Z
r
and
x
k
by
~
Y
t
. But for simplifying the presentation, we will use crisp notations in this
paper.
In my opinion persons h
ave no idea how to interpret possibility degrees in contrast to the
interpretation of probability degrees. Moreover, possibility measures allow no addition or
multiplication but only the comparison of possibility values by using the min

or max

operator.
T
herefore I prefer to use probabilities, even though we have only subjective ones.
I think the best chance for increasing the acceptance of decision models in practice is to use
fuzzy utilities and fuzzy probabilities. Therefore, I will concentrate on these
two extensions. At first,
we will discuss the use of fuzzy utilities or fuzzy values associated with each act

event combination.
In this case the well known
B
ERNOULLI

principle can be extended to the fuzzy model. Moreover, if
it is possible to get additio
nal information of a test market, we can improve the solution by using a
posteriori probability distributions. The concept of „value of additional information“ can also be
extended to fuzzy models by using fuzzy values of information.
Crucial topics of de
cision models with fuzzy utilities are:
a)
The modeling of fuzzy utilities associated with each act

event combination,
b)
the definition of expected utility values,
c)
the preference orderings of expected utility values.
In addition to the fuzzy utilities fuzzy pro
babilities will be used in the second part of the paper.
There the main problem is the calculation of expected utility values.
2 Modeling fuzzy values associated with each act

event combination
One of the most difficult problems in classical decision the
ory is the transformation of the
values
g
g
a
s
ij
i
j
(
,
)
in utility values
u
u
g
a
s
ij
i
j
(
(
,
))
. Working with fuzzy values
~
{(
,
(
)

)}
G
g
g
g
G
ij
G
ij
we have the same difficulties. In this contribution I do not
want to
discuss the question, how to get (fuzzy) utility functions. Therefore, we assume that the DM knows
his utility function u = u(g
ij
); then the fuzzy results are mapped in the fuzzy utilities
~
{(
(
),
(
)

)}
U
u
g
g
g
G
ij
G
ij
. Alternatively we can suppose
that the DM is able to specify directly
utility values
~
{(
,
(
))

}
U
u
u
u
U
ij
U
ij
, where U is the possible set of crisp utility values.
Obviously in the case of risk neutrality, we can use
~
G
ij
instead of
~
U
ij
.
In literature values
~
G
ij
or
~
U
ij
are usually modeled in form of triangular fuzzy numbers. In my
opinion this shape with a mean value is too special, the application of fuzzy intervals is more
realistic. On t
he other side a DM has often more information, so that he can characterize the fuzzy
interval in even more detail.
As an efficient way of getting suitable membership functions we propose the following
procedure, which is in a similar form used in the prog
ram FULPAL for solving (multiobjective)
fuzzy linear programming problems, see [
R
OMMELFANGER
1994].
At first the DM specifies some prominent membership values and relates them to special
meanings. This step can be clarified by using three levels which appe
ar to be sufficient for practical
applications.
㴠=
㨠
U
ij
u
(
)
㴠=
m敡湳⁴桡h⁵慳⁴桥hg桥h桡湣n敡liz慴i潮o
㴠
㨠
U
ij
u
(
)
m敡湳n
桡h桥h摥i獩潮om慫敲i猠willi湧漠慣捥灴甠慳慮a慶慩l慢a攠
v慬略u f潲 桥h im攠 扥b湧⸠ A v慬略u y wi栠
U
ij
u
(
)
桡h 愠 g潯搠
捨c湣n潦扥b潮oi湧漠桥h獥潦畴iliyv慬略u慳獯si慴敤ewi栠桥h慣

敶敮e捯c扩湡ni潮o
(
,
)
a
s
i
j
⸠C潲r敳灯湤p湧v慬略u潦甠慲攠r敬敶慮af潲
桥h捩獩潮o
㴠
㨠
U
ij
u
(
)
㰠
m敡湳n桡h甠桡h潮oy愠v敲ylil攠捨c湣n潦扥b潮oi湧漠桥h獥潦
畴ili
yv慬略u慳獯si慴敤ewi栠桥h慣

敶敮e捯c扩湡ni潮o
(
,
)
a
s
i
j
⸠T桥h
摥i獩潮慫敲猠睩sli湧⁴漠湥ol散⁴桥⁶慬略u⁵⁷i栠h
U
ij
u
(
)
㰠
.
A捣潲摩湧ly⁴桥⁄䴠獨潵h搠獰捩fy畭扥b猠
u
u
ij
ij
1
1
,
,
u
u
u
u
ij
ij
ij
ij
,
,
,
R
Ⱐ獯⁴桡h
A
ij
ij
ij
y
if
y
a
a
else
(
)
[
,
]
1
1
1
1
,
A
ij
ij
ij
y
if
y
a
a
else
(
)
[
,
]
慮a
A
ij
ij
ij
y
if
y
a
a
else
(
)
[
,
]
†
.
T桥hl潷敲桥hi湦潲m慴i潮o潦桥h䴬M桥hl慲g敲慲攠桥hi湴敲v慬猠
[
,
]
u
u
ij
ij
Ⱐ
1
,
,
⸠.桥灥pi慬
捡獥
a
ij
1
㴠
a
ij
1
捡渠慬獯s扥bim慧i湥搬n扵bi渠my潰o湩潮oii猠r慲敬yr敡li獴i挠漠慳獵s攠桡h慬l
捯cffi捩敮e猠
~
U
ij
慲攠e畺zy畭扥b献
C潮獥煵敮qly桥h灯pyg潮oli湥nfr潭(
u
ij
Ⱐ
)敲
u
ij
Ⱐ
)Ⱐ,
u
ij
1
ⰠㄩⰠ,
u
ij
1
ⰠㄩⰠ,
u
ij
Ⱐ
)†漠
u
ij
,
⤠
i猠s畩慢a攠e灰p潡o栠h漠o桥敭扥b獨s瀠p畮捴i潮o
~
U
ij
潮o
[
,
]
u
u
ij
ij
.
u
ij
u
ij
u
ij
u
ij
u
ij
1
u
ij
1
Figure 1:
~
U
ij
=
(
;
;
;
;
;
)
,
u
u
u
u
u
u
ij
ij
ij
ij
ij
ij
1
1
We characterize a fuzz
y interval
~
U
ij
with this kind of membership function by
(
;
;
;
;
;
)
,
u
u
u
u
u
u
ij
ij
ij
ij
ij
ij
1
1
. For simplification we call this special fuzzy set a fuzzy interval of


type. If required the DM can specify additional membership levels and additional
points
(
,
(
))
u
u
U
ij
of the polygon line.
An advantage of fuzzy intervals of


type that the arithmetic operations based on
Z
ADEH
's
extension principle can be calculated extremly simple. Moreover the approximation of the product
~
~
A
B
is very much better compared with the terms for fuzzy intervals of L

R

type and it can be
improved by using additional levels.
(
,
,
,
,
,
)
,
a
a
a
a
a
a
1
1
(
,
,
,
,
,
)
,
b
b
b
b
b
b
1
1
=
(
,
,
,
,
,
)
,
a
b
a
b
a
b
a
b
a
b
a
b
1
1
1
1
(
,
,
,
,
,
)
,
a
a
a
a
a
a
1
1
(
,
,
,
,
,
)
,
b
b
b
b
b
b
1
1
=
(
,
,
,
,
,
)
,
a
b
a
b
a
b
a
b
a
b
a
b
1
1
1
1
(
,
,
,
,
,
)
,
a
a
a
a
a
a
1
1
(
,
,
,
,
,
)
,
b
b
b
b
b
b
1
1
=
(
,
,
,
,
,
)
,
a
b
a
b
a
b
a
b
a
b
a
b
1
1
1
1
3 Fuzzy expected values and preference r
elations
As each real number a can be modeled as a fuzzy number
~
{(
,
(
)
}
A
x
x
x
A
R
with
A
(x) =
1 if
x
=
a
0 els
e
,
we assume the general case that each act

event combination
(
,
)
a
s
i
j
is valued by a fuzzy interval
~
U
ij
=
(
;
;
;
;
;
)
,
u
u
u
u
u
u
ij
ij
ij
ij
ij
ij
1
1
.
If the DM is able to specify a priori probabilities
p
s
j
n
j
(
)
,
,
,
,
1
2
, we can calculate the
fuzzy expected value of each act a
i
:
~
U
ij
=
(
;
;
;
;
;
)
,
u
u
u
u
u
u
ij
ij
ij
ij
ij
ij
1
1
=
~
(
)
~
(
)
U
p
s
U
p
s
i
in
n
1
1
,
where
u
u
p
s
i
ij
j
J
n
(
)
,
,
,
1
1
u
u
p
s
i
ij
j
J
n
(
)
,
,
,
1
1
Example
A manufacturer is confronted with the problem of determining the output of a product. Based
on his pattern of production he has the choice between five alternatives which are order
ed according
their size:
a
1
< a
2
< a
3
< a
4
< a
5
.
The profit earned with a specific output depends on the demand, which is not known with
absolute certainty. Due to his amount of information the manufacturer considers either a „high“
(state of nature s
1
) or
an „average“ (state of nature s
2
) or a „low“ (state of nature s
3
) demand. He
assigns the following a

priori

probabilities to the states of nature:
p(s
1
) = 0,5, p(s
2
) = 0,3, p(s
3
) = 0,2 .
The succeeding matrix of profits
~
U
ij
d
isplays which profits measured in 1.000 $ correspond to
the alternative constellations of output and demand. In order to elude the problem of obtaining
utility values we assume risk neutrality. Then we can simply use the maximization of expected profit
cri
terion for the decision and are sure that the selected acts are those that are consistent with the true
preferences based on expected utilities. As in the case of risk neutrality it does not influence the
decision whether we employ expected profits or expe
cted utilities we are going to apply either of
them in accordance with the content.
s
1
s
2
s
3
a
1
(170; 180; 200; 220; 225; 230)
(70; 83; 90; 100; 110; 120)
(

110;

97;

90;

77;

60,

50)
a
2
(140; 155; 165; 175; 180; 190)
(85; 93; 100; 110; 115; 125)
(

85;

80;

70;

58;

50;

40)
a
3
(120; 135; 145; 150; 160; 170)
(115; 130; 135; 140; 145; 150)
(

30;

20;

10; 0; 5; 10)
a
4
(85; 90; 100; 110; 115; 125)
(85; 93; 100; 105; 108; 115)
(

15;

10;

5; 5; 10; 15)
a
5
(45; 48; 50; 53; 58; 60)
(40; 45; 50;50; 53
; 55)
(35; 40; 45; 50; 55; 60)
Table 1: A priori profit matrix
~
~
(
,
)
U
U
a
s
ij
i
j
expected profit
a
1
a
2
a
3
a
4
a
5
(84; 95.5; 109; 124.6; 133.5; 141)
(78.5; 89.4; 98.5; 108.9; 114.5; 124.5)
(87; 102.5; 111; 117; 124.5; 132)
(65; 70.9; 79; 87.5; 91
.5; 100)
(41.5; 45.5; 49; 51.5; 55.9; 58.5)
Table 2: Expected profit matrix
~
(
)
E
a
i
1
40 50 90 100 120 140
A
0
5
,
0
5
,
a
5
a
4
a
2
a
3
a
1
Figure 2: Membership functions of the expected profits
Comparing the membership functions of the expected profits in Fi
gure 2, it becomes evident
that the alternatives a
4
and a
5
, eventually even alternative a
2
come off a lot worse than the
alternatives a
1
and a
3
. Yet the decision whether a
1
or a
3
should be selected is not as trivial as in the
classical model because of the
fact that fuzzy sets are not well ordered.
In the literature various concepts are proposed for comparing fuzzy sets and for constructing
preference orderings, see e.g. [
D
UBOIS
,
P
RADE
1983], [
B
ORTOLAN
,
D
EGANI
1985],
[
R
OMMELFANGER
1986], [
C
HEN
,
H
WANG
1992]
. Essential preference criteria are the

偲f敲敮e攠
慮搠a桥h

Preference.
Definition:

偲敦敲敮捥:
A set
~
B
is
preferred
to a set
~
C
on the

l敶敬
,
[0, 1]
,
written as
~
~
B
C
, if
i猠桥hl敡獴r敡l
湵n扥bⰠ獯⁴桡h
Inf B
Sup C
††††††††††††
f潲l
Ⱐ,
(ㄩ
慮搠a敡獴湥
Ⱐ
1
桯h摳⁴桥湥煵nliy
ㄩri捴ly⸠
B
x
X
x
B

(
)
††
慮搠†
C
x
X
x
C

(
)
††
慲攠e桥h

l敶敬

獥猠潦
~
B
慮搠
~
C
⸠.
Figure 3: Membership f
unctions of the sets
B
and
C
As long as we only consider fuzzy intervals and fuzzy numbers it is easy to understand that the
statement "
~
~
B
C
" is equivalent to "
~
B
~
C
i猠慬m潳o灯獩iv攠潮o桥hl敶敬h
=
1

"i渠桥h獥湳n潦
桥潬l潷i湧敦i湩i潮Ⱐoiv敮e
T
ANAKA
;
A
SAI
[1984].
As fuzzy intervals of the


type are precisely only
on the levels
,
and 1 it is wise to restrict
the preference observations on these three membership degrees.
Concerning to the expected Fuzzy utilities in Figure 2 we observe the following

preference
relations, where only the most strict relations are
presented:
if
only the following relations are valid:
a
a
a
a
a
a
a
a
4
5
3
5
2
5
1
5
,
,
,
,
i.e. the alternative a
5
is dominated by all other alternatives.
Similarly to this example, in many applications the

preference relations d
oes not lead to a
preference ordering of the given alternatives. The cause of this disadvantage is the pessimistic
attitude of the

preference. Only negative aspects are taken into consideration whereas positive
points are overlooked. Therefore, we consid
er the following preference relation which in its extreme
form goes back to
R
AMIK
;
R
IMANEK
[1985] more appropriate and suitable for application.
Definition:

Preference
A fuzzy set
~
B
is preferred to a fuz
zy set
~
C
on the level
[0,1] , written as
~
~
B
C
, if
i猠桥h
l敡獴敡l畭扥bⰠ獯⁴桡h
卵瀠
卵瀠
†††
慮搠††a湦⁂
I湦⁃
††
f潲l
,
1
(㈩
慮搠a潲敡獴湥
,
1
†
潮攠潦⁴桥h攠e湥煵nlii敳猠s慴i
獦i敤渠n桥ri捴湳n.
Figure 4:

灲pf敲敮捥
For fuzzy intervals
~
(
,
,
,
,
,
)
,
X
x
x
x
x
x
x
i
i
i
i
i
i
i
1
1
of the


type the terms (2) can be
simplified to
~
~
X
X
i
j
x
x
and
x
x
für
i
j
i
j
,
,
1
.
As the

灲敦敲敮e攠r敬慴i潮oi猠w敡k敲桡渠桥h

preference, the alternative a
5
in Figure 3 is
dominated by all other alternatives by using the

灲敦敲敮e攮e
A摤i潮慬ly⁷攠桡v攠湯眠瑨e
灲敦敲敮e攠潲摥i湧猺
a
a
a
a
a
a
a
a
a
a
2
4
3
4
1
4
3
2
1
2
,
,
,
,
.
T桡h猠潮sy⁴桥h捩獩潮整w敥渠n
1
and a
3
is not done.
All the other preference methods are based on defuzzification that means the fuzzy sets are
compressed to a single crisp real number and the prefer
ence ordering is based on this crisp number.
In an empirical study of
R
OMMELFANGER
[1986]
the
criterion of
C
HEN
[1985] and the
level

criterion
of
R
OMMELFANGER
revealed the best accordance with the preference orderings of the
persons who took part in the em
pirical study.
Since we are using utilities or profits of


type it seems rationally to work with the level

criterion for getting a temporary ranking of the acts. For convex fuzzy sets this criterion can be
handled very easily and we get the following
ranking parameter:
R(
~
U
i
) =
u
u
w
u
u
w
u
u
w
i
i
i
i
i
i
1
1
1
2
2
2
with
w
w
w
1
1
.
Normally we would set
w
w
w
1
1
3
, but the DM can also use individually specified
weights.
For our example we define
w
w
w
1
1
3
and get
R(
~
E
3
) =
87
102
5
111
117
124
5
132
6
112
33
,
,
,
< R(
~
E
1
) =
84
95
5
109
124
5
133
5
141
6
114
58
,
,
,
,
I.e. the alternative a
1
has a slightly higher "mean value" than a
3
. On the other hand the fuzz
y
expected values of a
1
has greater spreads than the ones of a
3
. Nevertheless a risk avers DM will
decide in favor of alternative a
1
.
By the application of the level

criterion we receive a guideline for orientation which will
support the DM, the introduce
d ordering, however does not represent a mandatory ranking.
4 The use of additional information
In order to select the best alternative the DM could look for additional information. In classical
decision theory the only chance for improving the solution
is the use of additional information
gathered from a test market for improving the given a priori distribution of the states of nature. The
calculation of expected utilities based on a posteriori distribution values and different definitions of
the term "
value of additional information" are described in detail and discussed in [Rommelfanger
1994, p. 105

108].
Concerning the importance of additional information for improving the probability distribution,
it can be said that using a posteriori probabilities
in decision processes is a complicated procedure
which needs a lot of information from the test market and implies intensive calculations. In practice
the DM has to devote money and time to these activities, before he is actually able to calculate the
valu
e of the information Y. Therefore we are convinced that a posteriori probabilities will hardly
ever be applied to real decision problems.
Apart from the use of a posteriori probabilities there exists another way for improving the
solution in fuzzy decisio
n theory. When gaining additional information the DM can also try to
specify the values associated with act

event combinations more precisely.
In Table 2 and Figure 2 it becomes evident, that as long as the postulated a priori

distribution is
accepted the
alternatives a
5
and a
4
and eventually even alternative a
2
will not be taken into
consideration furthermore. Therefore, assuming the a priori

distribution is correct, additional
information about the preferential alternatives a
1
, a
2
and a
3
should be gathere
d for getting profit
values which are less fuzzy.
We now presume that the additional information results in the following more precise
evaluation of the alternatives a
1
, a
2
and a
3
, see table 5 and figure 4.
s
1
s
2
s
3
a
1
a
2
a
3
(195; 202; 209; 215; 221; 2
25)
(150; 160; 168; 172; 178; 183)
(135; 140; 146; 148; 153; 160)
(88; 93; 98; 100; 105; 110)
(93; 98; 103; 105; 109; 115)
(128; 132; 137; 139; 142; 145)
(

90;

85;

83;

79;

73,

68)
(

70;

66;

62;

60;

55;

50)
(

10;

5;

1; 0; 5; 8)
Table 6: A prior
i

profit matrix with additional information
~
(
,
)
U
a
s
I
i
j
expected profit
a
1
a
2
a
3
(105.9; 111.9; 117.3; 121.7; 127.4; 131.9)
(88.9; 96.2; 102.5; 105.5; 110.7; 116)
(96.1; 108.6; 113.9; 115.7; 120.1; 125.1)
Table 7: Expected profit matrix
with additional information
~
(
)
E
a
U
I
i
90 110 130 140
a
2
a
3
a
1
Figure 5: Membership functions of the expected profits with additional information
Figure 5 indicates that act a
1
is the best alternative. This identification is confi
rmed by the
level

criterion which leads to the preference parameter
R (
~
(
)
E
a
I
2
) = 103.3 < R(
~
(
)
E
a
I
3
) = 113.25 < R(
~
(
)
E
a
I
1
) = 119.35.
Having used additional information the DM can be more confiden
t that the chosen act is really
the best in accordance to the preference criterion „maximization of the expected utility“. Even so it
is almost impossible to define the value of this information and in real applications the DM must
still decide to collect
and to process additional information without knowing anything about the
result of his efforts. Only in the unlikely case that the profits are described by fuzzy numbers and the
additional information has no influence on their mean values it is possible to
specify the value of
this additional information, see [
T
ANAKA
,
I
SHIHASHI
,
A
SAI
1986].
A possible advice for additional gathering of information could be that the information costs
should not be greater than R(
~
(
)
E
a
I
1
)

R(
~
(
)
E
a
I
3
) = 114.6

112.3 = 2.3 [1000 $].
This course of calculations in which the values associated with act

event combinations are
modeled by fuzzy intervals of


type should be repeated and by doing so the evaluations can be
improved step
by step through additional information. In my opinion the essential advantage of this
interactive procedure is that it presents an adequate answer to the information dilemma of real
problems.
One way to limit the extensive information process could be tha
t one starts designing a model of
the real problem with only the information which can be obtained with little effort and at reasonable
costs. When modeling by fuzzy intervals we then accept the disadvantage that some of the
parameters show great spreads.
Using the preference criterion „maximization of the expected
utility“, we will in general get no clear ranking of the acts, but usually we can observe that only few
alternatives are taken into consideration. Only the evaluations of these decisive acts shou
ld be
improved by collecting additional information. By doing so, the costs for additional information can
be reduced. Opposed to the extensive gathering of information ex ante

which is inevitable in
classical models

, the acquisition of additional info
rmation will then be designed in accordance to
the set aims and carried out under consideration of cost

benefit

relations.
5 Fuzzy probabilities
The case that extensive information about the entry of the states of nature may not be available
has also to
be considered. As a consequence it could occur that the a priori

probabilities are not
described precisely, but only vaguely by means of fuzzy intervals of


type
~
(
)
(
;
;
;
;
;
)
,
P
s
p
p
p
p
p
p
j
j
j
j
j
j
1
, j = 1, 2,..., n.
For our example we assume that the fuzzy p
robabilities in Table 8 are given:
s
j
~
(
)
(
;
;
;
;
;
)
,
P
s
p
p
p
p
p
p
j
j
j
j
j
j
1
s
1
~
(
)
(
,
;
,
;
,
;
,
;
,
;
,
)
,
P
s
1
0
45
0
48
0
49
0
51
0
53
0
55
s
2
~
(
)
(
,
;
,
;
,
;
,
;
,
;
,
)
,
P
s
2
0
26
0
28
0
29
0
3
0
31
0
33
s
3
~
(
)
(
,
;
,
;
,
;
,
;
,
;
,
)
,
P
s
3
0
17
0
18
0
2
0
2
0
21
0
23
Table 8: Fuzzy Probabilities
~
(
)
P
s
j
The following formulas offer a
simple approximation method for calculating the fuzzy expected
utilities:
~
(
;
;
;
;
;
)
~
~
(
)
~
~
(
)
,
E
E
E
E
E
E
E
U
P
s
U
P
s
i
A
i
i
i
i
i
i
i
in
n
1
1
1
1
,
where
E
u
p
i
ij
j
J
n
1
1
,
,
,
E
u
p
i
ij
j
J
n
1
1
,
,
,
At first we discuss the special case that the utilities are crisp. Moreover we assume that th
e DM
is risk neutral and the profits are given by Table 9.
s
1
s
2
s
3
a
1
210
100

80
a
2
170
105

60
a
3
150
140

10
a
4
105
102
0
a
5
50
50
50
Table 9: Profits, measured in [1.000 $]
Then we get the fuzzy expected profits:
~
(
;
;
;
;
;
)
,
E
E
E
E
E
E
E
i
A
i
i
i
i
i
i
1
1
a
1
(1
02,1 ; 112 ; 115,9 ; 121,1 ; 127,9 ; 134,9)
a
2
(90 ; 98,4 ; 101,75 ; 106,2 ; 111,85 ; 117,95)
a
3
(101,6 ; 109,1 ; 112,1 ; 116,5 ; 121,1 ; 127)
a
4
(73,77 ; 78,96 ; 81,03 ; 84,15 ; 87,27 ; 91,41)
a
5
(44 ; 47 ; 49 ; 50,5 ; 52,5 ; 55,5)
Tabelle 10: Fuzzy
Expected Profits
~
E
i
A
40 50
75
100
125 135
1
=0,5
=0,05
a
5
a
4
a
2
a
3
a
1
Figure 6: Membership fuctions of fuzzy expected profits
~
E
i
A
Figure 6 reveals that according the

preference the alternative a
1
is the be
st one.
These results are also valid if the approximately calculated values
~
E
i
A
are replaced by the
exactly calculated values
~
E
i
P
. Because an additional restriction,
p
j
j
n
1
1
, is observed, we have
~
E
i
P
~
E
i
A
E
E
i
P
i
A
u
u
(
)
(
)
.
Therefore, the calculation of
~
E
i
P
is only necessary for the alternatives which are not excluded
on basis of
~
E
i
A
's.
For calcul
ating the fuzzy expected utilities
~
(
;
;
;
;
;
)
,
E
E
E
E
E
E
E
i
P
i
i
i
i
i
i
1
1
we can use the
following terms:
E
Min
u
p
p
p
p
and
p
i
ij
j
j
n
j
j
j
j
j
n
{
[
,
]
}
,
,
,
1
1
1
1
E
Max
u
p
p
p
p
and
p
i
ij
j
j
n
j
j
j
j
j
n
{
[
,
]
}
,
,
1
1
1
1
.
For simplifying the calculation, we can use the following algorithms:
Algorithm
for calculate the probabili
ties
p
i
j
(
)
belonging to
E
i
,
,
,
1
.
1.
Specify for all probabilities the smallest value:
p
i
p
j
j
(
)
.
2.
Increase the probabilities of the state of nature with the highest utility value. If this is given
for
s
1
, we have
p
i
Max
p
p
p
p
p
j
j
n
1
1
1
2
1
(
)
{
[
,
]

}
3.
If the inequality is fulfilled in the strong sense, than we increase the probability of the state of
nature with the second highest utility value. If this is given for s
2
, we make the calculation
p
i
Max
p
p
p
p
p
p
j
j
n
2
2
2
1
3
1
(
)
{
[
,
]

}
4.
This procedure is to continue as long as the inequality is not fulfilled as equation.
Algorithm
for calculate the probabilities
p
i
j
(
)
belonging to
E
i
,
,
,
1
.
1.
Specify for all probabilities the
smallest value:
p
i
p
j
j
(
)
.
2.
Increase the probabilities of the state of nature with the smallest utility value. If this is given for
s
n
, we have
p
i
Max
p
p
p
p
p
n
n
n
j
j
n
(
)
{
[
,
]

}
1
1
1
3.
If the inequality is fulfilled in the strong sense, than
we increase the probability of the state of
nature with the second smallest utility value. If this is given for s
n

1
, we make the calculation
p
i
Max
p
p
p
p
p
p
n
n
n
j
j
n
n
1
1
1
1
2
1
(
)
{
[
,
]

}
4.
This procedure is to continue as long as the inequality is not fulfilled as equat
ion.
Using these algorithms we get for the example with crisp profit values the probabilities
p
i
und
p
i
j
j
(
)
(
)
, which are in this special example independent of i, because the profits in Table 10
comply with the ordering relation
x
x
x
i
i
i
1
2
3
for i = 1, 2,...,5.
1
1
p
1
0,45
0,48
0,50
p
1
0,51
0,53
0,55
p
2
0,32
0,31
0,30
p
2
0,29
0,29
0,28
p
3
0,
23
0,21
0,20
p
3
0,20
0,18
0,17
Table 11: Probabilities
p
i
p
i
j
j
(
)
(
)
and
With these probabilities
p
and
p
j
j
we calculate the fuzzy expected utilities
~
E
i
P
:
~
(
;
;
;
;
;
)
,
E
E
E
E
E
E
E
i
P
i
i
i
i
i
i
1
1
a
1
(108,1 ; 115 ; 119 ; 120,1 ; 125,2 ; 129,9)
a
2
(96,3 ; 101,6 ; 104,5 ; 105,2 ; 109,8 ; 112,7)
a
3
(110 ; 113,3 ; 115 ; 115,1 ; 118,3 ; 120)
a
4
(79,9 ; 82,0 ; 83,1 ; 83,1 ; 85,2 ; 86,3)
a
5
(50 ; 50 ; 50 ; 50 ; 50 ; 50)
Table 12: Fuzzy Expected Utilit
ies
~
E
i
P
=0,5
=0,05
50 75 100 125
1
a
5
a
4
a
2
a
3
a
1
Figure 7: Membership Functions of the Fuzzy Expected Profits
~
E
i
P
Comparing the membership functions of the
~
E
i
P
in Figure 12 with the functions of
~
E
i
A
in
Figure 7 we can clearly recognize, that the fuzzy values
~
E
i
P
are less fuzzier as their approximations
~
E
i
A
.
Remark:
A special case of decision models with crisp results
x
g
a
s
ij
i
j
(
,
)
or
crisp utilities
u
u
x
u
g
a
s
ij
ij
i
j
(
)
(
,
)
and fuzzy probabilities
~
(
)
(
;
;
;
;
;
)
,
P
s
p
p
p
p
p
p
j
j
j
j
j
j
1
is the LPI

model
proposed by
K
OFLER
and
M
ENGES
[1976]. This model with linear partial information (LPI) can be
interpreted as the special case where all
~
(
)
P
s
j
have constant membership functions, i.e.
p
p
p
and
p
p
p
j
j
j
j
j
1
. In my opinion this assumption is not very realistic; in practical
problems a DM has in general more information.
Finally we will calculate fuzzy expected utilities
~
E
i
P
where the fuzzy profits of Table 1 and the
fuzzy probabilities of Table 8 are given. In doing so, we have to observe that in 3 cases the ordering
x
x
x
i
i
i
1
2
3
is not given. Divergent of Table 11 we get in these cases
the probabilities:
p
1
(4)
0,51
p
1
(5)
0,53
0,55
p
2
(4)
0,28
p
2
(5)
0,28
0,26
p
3
(4)
0,21
p
3
(5)
0,19
0,19
Table 13: Probabilities
p
p
p
j
j
j
(
),
(
)
(
)
4
5
5
and
~
(
;
;
;
;
;
)
,
E
E
E
E
E
E
E
i
P
i
i
i
i
i
i
1
1
a
1
(73,6 ; 93,6 ; 109 ; 125,8 ; 140,4 ; 151,6)
a
2
(70,7 ; 86,4 ; 98,5 ; 109,6 ; 119,8 ; 132,7)
a
3
(83,9 ; 100,9 ; 111 ; 117,1 ; 127,8 ; 137,2)
a
4
(62 ; 69,8 ; 79 ; 87,
6 ; 94 ; 103,5)
a
5
(41,1 ; 44,3 ; 49 ; 51,5 ; 56 ; 58,7)
Table 14: Fuzzy Expected Profits
~
E
i
P
=0,5
=0,05
1
40
50
75
100 125 150
a
5
a
4
a
2
a
3
a
1
Figur 8: Membership Functions of the
Fuzzy Expected Profits
~
E
i
P
Figure 8 reveals that ev
en this imprecise information is sufficient to exclude act a
5
from further
consideration, whereas it now becomes more difficult to establish a preference order between a
1
and
a
3
.
In analogy to the procedure in section 5, additional information should be c
ollected to get more
precise descriptions of the values of the remaining acts and the probabilities
~
(
)
P
s
j
.
7 Conclusions
In this contribution we demonstrated that the modeling of real decision problems by means of
fuzzy models leads
to a reduction of information costs; that circumstance is caused by the fact that
within the interactive solution process additional information is gathered in correspondence to the
requirements and under consideration of cost

benefit

relations. Therefore
we recommend to start
with transferring the real problem into a fuzzy model instead of trying to select the best alternative
right away. Even though no dominant alternative can be selected at the beginning, inferior ones can
certainly be eliminated. In or
der to come to a decision between the remaining courses of action
additional information should be gathered to clarify the situation.
7 References
[1]
Chen, S.H.:
Ranking of Fuzzy Numbers with Maximizing and Minimizing set.
Fuzzy
Sets and Systems 17 (198
5), pp 113

129
[2]
Dubois, D. and Prade, H.:
The Use of Fuzzy Numbers in Decision Analysis
. in: Gupta,
M.M. and Sanchez, E.:
Fuzzy Information and Decision Processes
. Amsterdam New
York Oxford 1982, pp 309

321
[3]
Dubois, D. and Prade, H.:
Ranking of Fuzz
y Numbers in the Setting of Possibility
Theory
. Information Sciences, 30 (1989), pp 183

224
[4]
Lilien, G.:
MS/OR: A mid

life crises
. Interfaces 17 (1987), pp 53

59
[5]
Menges, G. and Kofler, E.:
Entscheidungen bei unvollständiger Information.
Springer
Ve
rlag Berlin Heidelberg 1976
[6]
Meyer zu Selhausen, H.:
Repositioning OR's Products in the Market.
Interfaces 19
(1989), pp 79

87
[7]
Neumann, J.v. and Morgenstern, O.:
Theory of games and economic behavior.
Princeton 1953
[8]
Ramik, J. and Rimanek, J.:
I
nequality between Fuzzy Numbers and its Use in Fuzzy
Optimization
. Fuzzy Sets and Systems 16 (1985), pp 123

138.
[9]
Rommelfanger H.:
Entscheidungsmodelle mit Fuzzy

Nutzen
. Operations Research
Proceedings 1983 (1984), pp 559

567
[10]
Rommelfanger H.:
Rango
rdnungsverfahren für unscharfe Mengen
. OR

Spektrum 8
(1986), pp 219

228
[11]
Rommelfanger, H.:
Entscheiden bei Unschärfe

Fuzzy Decision Support

Systeme
.
Springer Verlag Berlin Heidelberg, Second edition 1994
[12]
Sommer, G.:
Bayes

Entscheidungen mit uns
charfer Problembeschreibung.
Peter Lang

Verlag, Frankfurt am Main 1980
[13]
Tanaka, H.; Okuda, T. and Asai, K.:
A Formulation of Fuzzy Decision Problems and its
Application to an Investment Problem
. Kybernetes 5 (1976), pp 25

30
[14]
Tanaka, H. and Asai,
K.:
Fuzzy Linear Programming with Fuzzy Numbers
. Fuzzy Sets
and Systems 13 (1984), pp 1

10
[15]
Tanaka, H.; Ichihashi, H. and Asai, K.:
A Formulation of Linear Programming
Problems
based on Comparison of Fuzzy Numbers
. Control and Cybernetics 13
(1984), p
p 185

194
[16]
Tingley, G.A.:
Can MS/OR sell itself well enough?
Interfaces 17 (1987), pp 41

52
[17]
Watson, S.R.; Weiss, J.J. and Donell, M.L.:
Fuzzy Decision Analysis
. IEEE,
Transactions on Systems, Man and Cybernetics 9 (1979), 1

9
[18]
Whalen, T.:
De
cision Making under Uncertainty with various Assumptions about
available Information
. IEEE, Transactions on Systems, Man and Cybernetics 14
(1984), 888

900
[19]
Yager, R.R.:
Possibilistic Decision Making
. IEEE, Transactions on Systems, Man and
Cybernetics
9 (1979),
[20]
Zadeh, L.A.:
Fuzzy Sets
. Information and Control 8 (1965), pp 338

353
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