FUZZY ENVIRONMENT - WAYS FOR GETTING PRACTICAL DECISON MODEL

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30 Νοε 2013 (πριν από 4 χρόνια και 1 μήνα)

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„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=



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









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iliyv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af潲
桥⁤h捩獩潮o









U
ij
u
(
)




m敡湳n桡h甠桡h潮oy愠v敲ylil攠捨c湣n潦扥b潮oi湧漠桥h獥潦
畴ili
yv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
(
)



.

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







1
,
,

⸠.桥⁳灥pi慬
捡獥
a
ij
1


a
ij
1

捡渠慬獯s扥bim慧i湥搬n扵bi渠my潰o湩潮oii猠r慲敬yr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



)敲

u
ij



)Ⱐ,
u
ij
1
ⰠㄩⰠ,
u
ij
1
ⰠㄩⰠ,
u
ij



)†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







1


桯h摳⁴桥⁩湥煵nliy
ㄩ⁳ri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nlii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.



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