„DECISION MAKING IN FUZZY ENVIRONMENT
- WAYS FOR GETTING PRACTICAL DECISON MODELS“
Heinrich J. Rommelfanger*
Abstract. In real decision situations we are often confronted with the problem that the very demanding
conditions of classical decision models are 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 offers 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 practical 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 information 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 disadvantageous 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 additional 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
interactive 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 costbenefit-relations.
Keywords: decision 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 sense of
NEUMANN AND MORGENSTERN [1953]. However, empirical surveys, see e.g. [LILIEN 1987],
[TINGLEY 1987], [MEYER ZU SELHAUSEN 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  {a1, a 2 ,  , a m } ,
2. A set S of possible events associated with each course of action, S  {s1, s 2 ,  , s n } ,
3. A value (result, gain) to be associated with each act-event combination,
g ij  g( a i , s j ) , i  1,2,, m; j  1,2,, n . 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 BERNOULLI-criterion is recommended for rational behavior, i.e. the expected
utility should be maximized:
n
E( a*)  Max E( a i )  Max  u ( g( a i , s j ))  p(s j ) .
a i A
a i A j 1
6. A posteriori probability distribution:
The only chance for improving the solution of a classical decision model is to use additional
information of a test market X  {x1, x 2 , , x K } . Knowing the Likelihoods p( x k | s j ) , the a
priori probability distribution p(s j ) can be substituted by the a posteriori probability distribution
p( x k | s j )  p( s j )
p(s j| x k ) = n
BAYES’s formula .
 p( x k s j )  p( s j )
j1
Since the paper "Fuzzy Sets" of Lofti A. ZADEH was published in 1965, the fuzzy sets theory
has been considered as a new way for modeling more realistic decision models. Especially between
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 h  {(a i ,  D h (a i ))| a i  A}, h  1,, H , TANAKA, OUKUDA; ASAI 1976;
~
2. Fuzzy events Zr  {(s j ,  Z r (s j ))| s j S}, r  1,, R , TANAKA, OUKUDA; ASAI 1976.
~ ~
3. Fuzzy probabilities Pj  P(s j )  {( p,  Pj ( p )| p [0,1]} , WATSON; WEISS; DONELL 1979;
DUBOIS; PRADE 1982; WHALEN 1984.
~
~
4. Fuzzy utility values U ij  U( a i , s j )  {( u,  U ij ( u )| u  U} , where U is the set of given crisp
utilities associated with each act-event combination, JAIN 1976; WATSON; WEISS; DONELL 1979;
YAGER 1979; ROMMELFANGER 1984; WHALEN 1984.
~
5. Fuzzy information Yt  {( x k , Yt ( x k )| x k  X} , TANAKA, OUKUDA; ASAI 1976; SOMMER 1980.
6. Moreover, some authors propose to substitute the probability distribution p(s j ) by a possibility
distribution (s j ) , see e.g. YAGER 1979 and WHALEN 1984. They assume that utilities can be
measured on an ordinal scale only and 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 correct 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 elements; that means that we have
~
~ ~
probability distributions p( Z r ) and p(Yt | Z r ) and values directly associated with the combinations
~
( a i , Z r ), i  1, 2, , m; r  1, 2, , R . Therefore, we can use the classical procedure as we replace
~
~
s j by Z r and x k by Yt . But for simplifying the presentation, we will use crisp notations in this
paper.
 In my opinion persons have 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.
Therefore 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 BERNOULLI-principle can be extended to the fuzzy model. Moreover, if
it is possible to get additional 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 decision 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 probabilities 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 theory is the transformation of the
u ij  u( g( a i , s j )) . Working with fuzzy values
values g ij  g( a i , s j ) in utility values
~
G ij  {( g,  G ij ( g )| g  G )} 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(gij); then the fuzzy results are mapped in the fuzzy utilities
~
U ij  {( u( g ),  G ij ( g )| g  G )} . Alternatively we can suppose that the DM is able to specify directly
~
utility values U ij  {( u,  U ij ( u ))| u  U} , 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 the 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 program FULPAL for solving (multiobjective)
fuzzy linear programming problems, see [ROMMELFANGER 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 appear to be sufficient for practical
applications.
=1
:  U ij ( u ) = 1
=
:  U ij ( u )  
=
:  U ij ( u ) < 
means that u has the highest chance of realization,
means that the decision maker is willing to accept u as an available
value for the time being. A value y with  U ij ( u )   has a good
chance of belonging to the set of utility values associated with the actevent combination (a i , s j ) . Corresponding values of u are relevant for
the decision.
means that u has only a very little chance of belonging to the set of
utility values associated with the act-event combination (a i , s j ) . The
decision maker is willing to neglect the values u with  U ij ( u ) < .
Accordingly the DM should specify numbers u1ij , u1ij , u ij , u ij , u ij , u ij R , so that
   if y [a  , a ij ]
ij
 A ij ( y) 
   else
  1 if y [a1 , a1ij ]
ij
 A ij ( y) 
,
  1 else
and
   if y [a  , a ij ]
ij
 A ij ( y) 


else

.

The lower the information of the DM, the larger are the intervals [u 
ij , u ij ] ,   1,  ,  . The special
case a1ij = a 1ij can also be imagined, but in my opinion it is rarely realistic to assume that all
~
coefficients U ij are fuzzy numbers.
Consequently the polygon line from ( u ij , ) over ( u ij , ), ( u1ij , 1), ( u1ij , 1), ( u ij , ) to ( u ij ,)
~
is a suitable approach to the membership function of U ij on [u ij , u ij ] .

u ij
u1ij
u
u1ij
ij
u ij
u ij
~
Figure 1: U ij = ( u ij ; u ij ; u1ij; u1ij; u ij ; u ij ) ,
We characterize a fuzzy interval
~
U ij with this kind of membership function by
( u ij ; u ij ; u1ij; u1ij; u ij ; u ij ) , . 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 ij ( u )) of the polygon line.
An advantage of fuzzy intervals of --type that the arithmetic operations based on ZADEH'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  , a1 , a1 , a  , a  ),  ( b  , b  , b1 , b1 , b  , b  ), =
( a   b  , a   b  , a1  b1 , a1  b1 , a   b  , a   b  ),
( a  , a  , a1 , a1 , a  , a  ),  ( b  , b  , b1 , b1 , b  , b  ), =
( a   b  , a   b  , a1  b1 , a1  b1 , a   b  , a   b  ),
( a  , a  , a1 , a1 , a  , a  ),  ( b  , b  , b1 , b1 , b  , b  ), =
( a   b  , a   b  , a1  b1 , a1  b1 , a   b  , a   b  ),
3 Fuzzy expected values and preference relations
As each real number a can be modeled as a fuzzy number
~
if x = a
1
A  {( x,  A ( x ) x  R} with A(x) = 
,
else
0
we assume the general case that each act-event combination ( a i , s j ) is valued by a fuzzy interval
~
U ij = ( u ij ; u ij ; u1ij; u1ij; u ij ; u ij ) , .
If the DM is able to specify a priori probabilities p(s j ), j  1, 2, , n , we can calculate the
fuzzy expected value of each act ai:
~
~
~
U ij = ( u ij ; u ij ; u1ij; u1ij; u ij ; u ij ) , = U i1  p(s1 )  U in  p(s n ) ,
where
n

u
i   u ij  p(s j ) ,   , , 1
n

u
i   u ij  p(s j ) ,   1, , 
J 1
J 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 ordered according
their size: a1< a2 < a3 < a4 < a5.
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 s1) or an „average“ (state of nature s2) or a „low“ (state of nature s3) demand. He
assigns the following a-priori-probabilities to the states of nature:
p(s1) = 0,5, p(s2) = 0,3, p(s3) = 0,2 .
~
The succeeding matrix of profits U ij displays 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
criterion 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 expected utilities we are going to apply either of
them in accordance with the content.
s1
s2
s3
a1 (170; 180; 200; 220; 225; 230) (70; 83; 90; 100; 110; 120) (-110; -97; -90; -77; -60, -50)
a2 (140; 155; 165; 175; 180; 190) (85; 93; 100; 110; 115; 125) (-85; -80; -70; -58; -50; -40)
a3 (120; 135; 145; 150; 160; 170) (115; 130; 135; 140; 145; 150)
(-30; -20; -10; 0; 5; 10)
a4 (85; 90; 100; 110; 115; 125)
(85; 93; 100; 105; 108; 115)
(-15; -10; -5; 5; 10; 15)
a5
(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 ij  U(a i , s j )
expected profit
a1
a2
a3
a4
a5
(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 )
a5
1
a4
a2
a3
a1
 A  0,5
  0,5
40
50
90
100
120
140
Figure 2: Membership functions of the expected profits
Comparing the membership functions of the expected profits in Figure 2, it becomes evident
that the alternatives a4 and a5, eventually even alternative a2 come off a lot worse than the
alternatives a1 and a3. Yet the decision whether a1 or a3 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. [DUBOIS, PRADE 1983], [BORTOLAN, DEGANI 1985],
[ROMMELFANGER 1986], [CHEN, HWANG 1992]. Essential preference criteria are the -Preference
and the -Preference.
Definition: -Preference:
~
~
~
~
A set B is preferred to a set C on the -level,   [0, 1], written as B  C , if  is the least real
number, so that
Inf B  Sup C
for all   , 1
and at least one   , 1 holds the inequality (1) strictly.
B  x  X |  B ( x )   and C  x  X |  C ( x )  
(1)
~
~
are the -level-sets of B and C .
 and C

Figure 3: Membership functions of the sets B
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 is almost positive on the level h = 1-" in the sense of
the following definition, given by TANAKA; ASAI [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 4  a 5 , a 3  a 5 , a 2  a5 , a1  a 5 ,
i.e. the alternative a5 is dominated by all other alternatives.
Similarly to this example, in many applications the -preference relations does 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 consider the following preference relation which in its extreme
form goes back to RAMIK; RIMANEK [1985] more appropriate and suitable for application.
Definition: -Preference
~
~
~
~
A fuzzy set B is preferred to a fuzzy set C on the level   [0,1] , written as B  C , if  is the
least real number, so that
Sup B  Sup C
and Inf B  Inf C
for all    , 1
(2)




and for at least one   , 1 one of these inequalities is satisfied in the strict sense.
Figure 4: -preference
~
For fuzzy intervals Xi  ( xi , xi , x1i , x1i , xi , xi ), of the --type the terms (2) can be
simplified to
~
~



für   , ,1 .
Xi  X j  x 
i  x j and x i  x j
As the -preference relation is weaker than the - preference, the alternative a5 in Figure 3 is
dominated by all other alternatives by using the - preference.
Additionally we have now the preference orderings:
a 2  a 4 , a 3  a 4 , a1  a 4 , a 3  a 2 , a1  a 2 .
That is only the decision between a1 and a3 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 preference ordering is based on this crisp number.
In an empirical study of ROMMELFANGER [1986] the criterion of CHEN [1985] and the levelcriterion of ROMMELFANGER revealed the best accordance with the preference orderings of the
persons who took part in the empirical study.
Since we are using utilities or profits of --type it seems rationally to work with the levelcriterion 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:
 u1  u1i

u i  u i
u i  u i
~
i
 w1 
 w 
 w   with w1  w   w   1 .
R( U i ) = 
2
2
 2

Normally we would set w1  w   w   13 , but the DM can also use individually specified
weights.
For our example we define w1  w   w   13 and get
~
~
,5133,5141
R( E 3 ) = 87102,51116117124,5132  112,33 < R( E1 ) = 8495,5109124
 114,58
6
I.e. the alternative a1 has a slightly higher "mean value" than a3. On the other hand the fuzzy
expected values of a1 has greater spreads than the ones of a3 . Nevertheless a risk avers DM will
decide in favor of alternative a1.
By the application of the level-criterion we receive a guideline for orientation which will
support the DM, the introduced 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
value 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 decision 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 a5 and a4 and eventually even alternative a2 will not be taken into
consideration furthermore. Therefore, assuming the a priori-distribution is correct, additional
information about the preferential alternatives a1, a2 and a3 should be gathered 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 a1, a2 and a3 , see table 5 and figure 4.
s1
s2
s3
a1 (195; 202; 209; 215; 221; 225) (88; 93; 98; 100; 105; 110)
(-90; -85; -83; -79; -73, -68)
a2 (150; 160; 168; 172; 178; 183) (93; 98; 103; 105; 109; 115) (-70; -66; -62; -60; -55; -50)
a3 (135; 140; 146; 148; 153; 160) (128; 132; 137; 139; 142; 145)
(-10; -5; -1; 0; 5; 8)
~
Table 6: A priori-profit matrix with additional information U I ( a i , s j )
expected profit
a1
a2
a3
(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 U I ( a i )
a2

a 3 a1


90
110
130
140
Figure 5: Membership functions of the expected profits with additional information
Figure 5 indicates that act a1 is the best alternative. This identification is confirmed by the
level-criterion which leads to the preference parameter
~
~
~
R ( E I ( a 2 ) ) = 103.3 < R( E I ( a 3 ) ) = 113.25 < R( E I ( a1 ) ) = 119.35.
Having used additional information the DM can be more confident 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 [TANAKA, ISHIHASHI, ASAI 1986].
A possible advice for additional gathering of information could be that the information costs
should not be greater than
~
~
R( EI ( a1 ) ) - R( EI (a3 ) ) = 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 that 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 should 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 information 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 j )  ( p  ; p  ; p; p1j ; p j ; p j ), , j = 1, 2,..., n.
j
j
For our example we assume that the fuzzy probabilities in Table 8 are given:
sj
s1
s2
s3
~
P(s j )  ( p  ; p  ; p; p1j ; p j ; p j ),
j
j
~
P(s1 )  ( 0,45 ; 0,48 ; 0,49 ; 0,51; 0,53; 0,55), 
~
P(s 2 )  ( 0,26 ; 0,28 ; 0,29 ; 0,3; 0,31; 0,33), 
~
P(s3 )  ( 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:
~
~ ~
~
~
EiA  ( Ei ; Ei ; E1i ; E1i ; Ei ; Ei ),   U i1  P(s1 )  U in  P(s n ) ,
n
n


E
i   u ij  p j ,   , , 1
where

Ei   u 
ij  p j ,   1, , 
J 1
J 1
At first we discuss the special case that the utilities are crisp. Moreover we assume that the DM
is risk neutral and the profits are given by Table 9.
a1
a2
a3
a4
a5
s1
s2
s3
210
170
150
105
50
100
105
140
102
50
-80
-60
-10
0
50
Table 9: Profits, measured in [1.000 $]
Then we get the fuzzy expected profits:
~
EiA  ( Ei ; Ei ; E1i ; E1i ; Ei ; Ei ), 
a1
(102,1 ; 112 ; 115,9 ; 121,1 ; 127,9 ; 134,9)
a2
(90 ; 98,4 ; 101,75 ; 106,2 ; 111,85 ; 117,95)
a3
(101,6 ; 109,1 ; 112,1 ; 116,5 ; 121,1 ; 127)
a4
(73,77 ; 78,96 ; 81,03 ; 84,15 ; 87,27 ; 91,41)
a5
(44 ; 47 ; 49 ; 50,5 ; 52,5 ; 55,5)
~
Tabelle 10: Fuzzy Expected Profits E iA
a5
1
a4
a2
a3
a1
=0,5
=0,05
40
50
75
100
~
Figure 6: Membership fuctions of fuzzy expected profits E iA
Figure 6 reveals that according the  -preference the alternative a1 is the best one.
125
135
~
These results are also valid if the approximately calculated values E iA are replaced by the
n
~
exactly calculated values E iP . Because an additional restriction,  p j  1 , is observed, we have
j1
~
~
E iP  E iA   P ( u )   A ( u ) .
Ei
Ei
~P
Therefore, the calculation of E i is only necessary for the alternatives which are not excluded
~
on basis of E iA 's.
~
For calculating the fuzzy expected utilities EiP  ( Ei ; Ei ; E1i ; E1i ; Ei ; Ei ),  we can use the
following terms:
n
n



E
i  Min{  u ij  p j p j [p j , p j ] and  p j  1} ,   ,  , 1
j1
j1
n
n


E i  Max{  u 
ij  p j p j [p j , p j ] and  p j  1}   1,  ,  .
j1
j1
For simplifying the calculation, we can use the following algorithms:

Algorithm for calculate the probabilities p j ( i ) belonging to E i ,   1, ,  .
1.
2.

Specify for all probabilities the smallest value: p j ( i )  p j .
Increase the probabilities of the state of nature with the highest utility value. If this is given for
s1 , we have
n

p1 ( i )  Max{p [p  , p1 ] | p   p   1}
1
j
j 2
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 s2, we make the calculation
n

p 2 ( i )  Max{p [p  , p 2 ] | p1  p   p   1}
2
j
j 3
4.
This procedure is to continue as long as the inequality is not fulfilled as equation.
Algorithm for calculate the probabilities p ( i ) belonging to E
i ,   , , 1 .
j
1.
2.
Specify for all probabilities the smallest value: p  ( i )  p  .
j
j
Increase the probabilities of the state of nature with the smallest utility value. If this is given for
sn , we have
n 1

p  ( i )  Max{p [p  , p 
n ] |  p j  p  1}
n
n
j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 sn-1, we make the calculation
n 2


p  ( i )  Max{p [p  , p 
n 1 ]|  p j  p  p n  1}
n

1
n 1
j1
4.
This procedure is to continue as long as the inequality is not fulfilled as equation.
Using these algorithms we get for the example with crisp profit values the probabilities

p j ( i ) und p  ( i ) , which are in this special example independent of i, because the profits in Table 10
j
comply with the ordering relation x i1  x i 2  x i3 for i = 1, 2,...,5.

p
1
p
2

p
3

1
0,45
0,48
0,50
0,32
0,31
0,30
0,23
0,21
0,20


0,53
0,55
0,29
0,28
0,18
0,17
1
Table 11: Probabilities

0,51
p1

0,29
p2

0,20
p3

p j ( i ) and p  ( i )
j

~
With these probabilities p j and p  we calculate the fuzzy expected utilities E iP :
j
~
EiP  ( Ei ; Ei ; E1i ; E1i ; Ei ; Ei ), 
a1
(108,1 ; 115 ; 119 ; 120,1 ; 125,2 ; 129,9)
a2
(96,3 ; 101,6 ; 104,5 ; 105,2 ; 109,8 ; 112,7)
a3
(110 ; 113,3 ; 115 ; 115,1 ; 118,3 ; 120)
a4
(79,9 ; 82,0 ; 83,1 ; 83,1 ; 85,2 ; 86,3)
a5
(50 ; 50 ; 50 ; 50 ; 50 ; 50)
~
Table 12: Fuzzy Expected Utilities E iP
1
a5
a4
a2
a3 a1
=0,5
=0,05
50
75
100
125
~
Figure 7: Membership Functions of the Fuzzy Expected Profits E iP
~
~
Comparing the membership functions of the E iP in Figure 12 with the functions of E iA in
~
Figure 7 we can clearly recognize, that the fuzzy values E iP are less fuzzier as their approximations
~
E iA .
Remark:
A special case of decision models with crisp results x ij  g( a i , s j ) or crisp utilities
~
u ij  u( x ij )  u  g( a i , s j ) and fuzzy probabilities P(s j )  ( p  ; p  ; p; p1j ; p j ; p j ), is the LPI-model
j j
proposed by KOFLER and MENGES [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 p1j  p j  p j . In my opinion this assumption is not very realistic; in practical
j
j
problems a DM has in general more information.
~
Finally we will calculate fuzzy expected utilities E iP 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 i1  x i 2  x i3 is not given. Divergent of Table 11 we get in these cases the probabilities:

p  (4)
1
p  (4)
2
p  (4)
3

0,51
p1 (5)
0,28
p 2 (5)
0,21
p3 (5)




0,53
0,55
0,28
0,26
0,19
0,19


Table 13: Probabilities p  ( 4), p j (5) and p j (5)
j
~
EiP  ( Ei ; Ei ; E1i ; E1i ; Ei ; Ei ), 
a1
(73,6 ; 93,6 ; 109 ; 125,8 ; 140,4 ; 151,6)
a2
(70,7 ; 86,4 ; 98,5 ; 109,6 ; 119,8 ; 132,7)
a3
(83,9 ; 100,9 ; 111 ; 117,1 ; 127,8 ; 137,2)
a4
(62 ; 69,8 ; 79 ; 87,6 ; 94 ; 103,5)
a5
(41,1 ; 44,3 ; 49 ; 51,5 ; 56 ; 58,7)
~
Table 14: Fuzzy Expected Profits E iP
a5
1
a4
a2
a3
a1
=0,5
=0,05
40
50
75
100
125
150
~
Figur 8: Membership Functions of the Fuzzy Expected Profits E iP
Figure 8 reveals that even 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
a3 .
In analogy to the procedure in section 5, additional information should be collected 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 order to come to a decision between the remaining courses of action
additional information should be gathered to clarify the situation.
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