Fuzzy Utility Value Analysis and Fuzzy Analytic Hierarchy Process
Susanne Eickemeier, Heinrich Rommelfanger
Faculty of Economics and Business Administration, Institute of Statistics und Mathematics,
Johann Wolfgang Goethe-University, Mertonstr. 17-23, D-60054 Frankfurt, Germany
(e-mail: eickemeier@wiwi.uni-frankfurt.de, rommelfanger@wiwi.uni-frankfurt.de)
1. Introduction
Utility value analysis and Analytic Hierarchical Process (AHP) are well-known, practically
applied procedures for solving multi criteria decision problems. Both consider the restricted
information processing capacity resp. the limited rationality of a decision maker and
consequently follow the experts' demand for more realistic and applicable methods for decision
making. According to empirical researches it is difficult for people to arrange alternatives
consistently, if more than two attributes have to be taken into account, see also May [1954, p. 913]. Consequently in case of utility value analysis and AHP various attributes are ordered by an
hierarchical constructed decision system, in which only few, in most cases two or three
attributes are gradually aggregated to a higher level.
In case of both procedures the aggregation of values is made by weighted addition of the partial
utility values under the precondition of a strongly preferential independence of the objectives
and cardinally scaled numbers. Nevertheless it is generally difficult to achieve cardinally scaled
utility values; especially in case of ordinally scaled or even linguistically made valuations of the
decision criteria it is difficult to determine the utility value.
In order to obtain utility values for the single attributes on the basis of the decision hierarchy
Zangemeister [1972] proposes to introduce a valuation matrix, in which all possible objective
grades are mapped onto the interval [0 , 10] . This precise valuation should be supported by
related interval classes, which are described by verbal valuations from "very bad" to "very
good". It remains however the question whether a decision maker is really able to relate to each
partial objective a true and well-defined utility value. As described in section 5 it seems more
realistic to assume that the decision maker can only map fuzzy utility values on the interval
[0 , 10]. According to Zangemeister the decision maker can chose the objective weights without
restrictions. In order to determine the weights it is however quite often recommended to apply
the pairwise comparison, which means that one has to determine how much the utility value has
to be increased with respect to an objective k, if the utility value of the objective r will be
g
reduced about the absolute value . Thus the resulting substitution rates a kr  r are wellgk
defined, if the weights' sum is normalized to In case of a linear additive decision rule "consistent
preferences" exist, if the respective substitution rates satisfy the consistency rule a kr  a rs  a ks .
2
If all substitution rates ars are positive, the consistency rule and a kk  1 lead to the formula
a rk  1 .
a kr
In order to set up reciprocal pairwise comparison matrices the substitution rates should be
measured on a proportional scale, which is however hardly possible in real life, since singlevalued functions at best exist on an interval scale level. It has therefore so far not been
determined under which conditions reciprocal matrices and the deducted weights vectors can be
regarded as efficiently constructed.
In accordance with the definition of substitution rates a consistent pairwise comparison matrix A
has a special form: all column vectors are multiples of one another and therefore each column
forms an equivalent weights vector, which by normalizing to the weight sum 1 leads to the
normalized weights vector.
In Analytic Hierarchy Process Thomas L. Saaty [1978, 1996] shows an arithmetical alternative
for the determination of the above mentioned weights vector. Saaty takes advantage of the
recognition that in case of a consistent pairwise comparison matrix A the weights vector g
corresponds to the eigenvector of A of the biggest eigenvalue of A. This biggest eigenvalue is
always equal to the rank of the consistent pairwise comparison matrix and all other eigenvalues
are then equal to 0.
In practice it is quite often not possible to construct a consistent pairwise comparison matrix.
The reason for "inconsistent preferences" may be the fact, that the additive decision rule is
inadequate. It is indeed more probable that the linear decision rule is useful and the reasons for
inconsistency can be found in the restricted capacity for information processing or the otherwise
restricted rationality of the decision maker. Accordingly Saaty recommends to apply the
weighted addition even in case of smaller offenses against consistency. As long as Saaty's
constistency index is smaller than 0.1 the normalized eigenvector of the biggest eigenvalue of
the pairwise comparison matrix A should be applied as weights vector.
Furthermore Saaty proposes to apply pairwise comparison matrices as well to determine the
utility values. Thus the hierarchy of goals gets another level, since each subgoal on the former
basis level will branch out in alternatives. Anyhow this procedure is only efficient in case of few
alternatives.
Central element of such multi criteria decisions is therefore the reciprocal pairwise comparison
matrix. In case of real life decisions it is however very difficult for the decision maker to define
the substitution rates resp. the partial utility values for alternatives. Saaty therefore proposes to
apply a scale, which ranges from 1 to 9 to represent judgement entries. Saatys 9-point scale has
not been rationally proved and is therefore vulnerable to criticism. On the one hand different
scales can lead to different ranking orders. On the other hand the definition of preference is
rather spongy in case of pairwise comparison and the term „fuzziness“ used by Saaty has little in
common with the fuzzy set theory. For example he does not exactly define the meaning of
"weakly more important", "strongly more important" and "absolutely more important". It is
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however even more decisive that Saaty's 9-point scale does not at all provide cardinally scaled
numbers and therefore a weighted addition of these values or the definition of reciprocal values
can not be accepted from the theoretical point of view.
Since experience has proved that it is more likely for users to think of substition rates in the way
that one goal is x-times more important than the other one, in real life it seems more useful to
follow this evident opinion and leave Saaty's interpretation. Furthermore only roughly defined
pairwise comparisons like "goal 1 is about three times more important than goal 2" should not
be artificially condensed, but modeled mathematically including the given vague information.
An adequate and more realistic modeling will be achieved by using fuzzy values, as proposed by
Buckley [1985]. In the following section 2 a new version of the Fuzzy- approach will be
described.
2. Definition of the weights vector from a pairwise comparison matrix with fuzzy
substitution rates
In real life the decision maker is quite often not in the position to define exactly all substitution
rates of a pairwise comparison and deduct from this a consistent pairwise comparison matrix. In
case of some pairwise comparisons he has only a vague idea, how much more important one
goal seems to be compared with the other one. Such roughly defined data can be mathematically
described by fuzzy sets. In most cases it is sufficient to apply the practically proved (form of)
fuzzy intervals of the --type, which furthermore has the advantage that arithmetical
calculations can be made more easily. Fuzzy intervals ~aij of the --type have piecewise linear
membership functions and can be described sufficiently by 6 values:
~a = (a; a; a ; a ; a ; a ) ,  , see Fig. 3 and Rommelfanger [1986, 1994]. As special cases
ij ij ij ij ij ij
ij
they also include trapezoid fuzzy intervals, fuzzy numbers and also real numbers. If there is only
few information given the -level might be neglected.
1

a ij
a ij
a ij
a
a ij
ij
a ij
Fig. 1: ~aij = (aij; aij ; aij; aij; aij; aij) , 
If the pairwise comparison matrix with fuzzy intervals would be interpreted as a collection of six
pairwise comparison matrices, for which only the values of a ij , a ij , a ij, aij, a ij or a ij would be
applied, then for each of these pairwise comparison matrices the eigenvector for the largest
4
weight vector could be calculated. It is indeed quite unlikely that the thus calculated and
normalized eigenvectors would come out well-ordered and could be condensed to one "fuzzy
eigenvector". We would rather have to decide, which of these eigenvectors should be used as
weights vector.
Therefore we want to take another, in our opinion more reasonable way to determine the
weights vector, which rather meets the concept of the fuzzy set theory and uses fuzzy intervals
of the --type as components. The chosen procedure is based on the fact that in a consistent
pairwise comparison matrix all column vectors are multiples of one another and in normalized
form result in the weights vector resp. the eigenvector of the largest eigenvalue. If the
consistency condition is not fulfilled, the weights vector logically will be determined by
averaging the normalized column vectors.
As operator we recommend the arithmetic mean, since the thus received weights will be used for
the weighted addition of the partial utilities. Buckley [1985] uses the geometric mean to
calculate the weights without further discussion .He only points out that in case of a consistent
pairwise comparison matrix (with real numbers) the geometric mean will lead to the same
weight vector as Saaty's eigenvector method. This statement as well applies to the arithmetic
mean. The method of taking the arithmetic mean of the normalized column vectors offers the
advantage, that the difference between the normalized column vectors can be revealed by simple
comparison.
If we transfer this considerations on fuzzy intervals of the --type, one first has to calculate the
~,
~
~
column sums 
1 2,  , n in order to normalize the column vectors of a pairwise comparison
matrix:
~  ( ;  ;  ;  ;  ;  ) ,  = ~

a1j  ~
a2 j    ~
anj
j
j
j
j
j
j
j
n
n
n
n
n
n
i 1
i 1
i 1
i 1
i 1
i 1
= (  aij ;  aij ;  aij ;  aij ;  aij ;  aij ) ,  .
(1)
By normalizing
aij aij aij aij aij aij , 
~
~
~
norm
aij
 aij   j = (  ;  ;
; ;
; )
 j  j  j  j j j
the weights
~
gi  1  (~
a norm  ~
ainorm
 ~
ainnorm) ,
2
n i1
g '  (~
g1, ~
g2,  , ~
gn ) will be calculated.
which form the weight vector ~
(2)
(3)
Unfortunately this normalization procedure, based on Zadeh's extension principle, has the
disadvantage that the normalized pairwise comparisons become even more fuzzy. Therefore
weight constellations, for which the sum of single weights is larger than 1 might occur. This is
however not very useful. Furthermore the "traditional" way of normalizing will change the
character of the substitution rates: Exact numbers as well as fuzzy numbers will lead to fuzzy
intervals of the --type. Therefore, we want to introduce a new way of normalizing the column
5
vectors, which is, in our opinion, more convincing, since it keeps the given fuzziness and adopts
the given numbers by their type.
3. A new procedure for normalizing the columns of a pairwise comparison matrix
In order to come up the term "normalizing to 1" it is in our opinion more suitable to divide all
parameters of a fuzzy interval of the --type by the same real number. For normalizing the jth
column we especially refer to the arithmetic mean *j  12 ( j   j) . Though it is more difficult
to calculate, we can, however, as well use the arithmetic mean 16 (j  j   j   j  j  j ) .
Normalization is done by formula
~
aij
aij aij aij aij aij aij , 
~
*
.
aij  * = ( * ; * ; * ; * ; * ; * )
j j j j j j
j
(4)
With the thus normalized substitution rates the weights
~
*),
gi*  1  (~
a*  ~
ai*2    ~
ain
n i1
g '*  (~
g*, ~
g*,  , ~
g* ) will be calculated.
which form the weights vector ~
1
2
(5)
n
One advantage of the new normalization method is that the given fuzziness of the substitution
rates remains unchanged and is not increased. Furthermore the shape of the fuzzy sets is kept.
Another consequence of this kind of normalization is the fact that the total utility values, which
are now calculated according to formula
~
uk  ~
g*  u1k  ~
g*  u2k    ~
g*  unk
1
2
n
(6)
are also less fuzzy and therefore the preference order is more clearly.
4. Application of fuzzy weights
In this section we want to discuss the question, how to choose the best alternative, if the
decision maker can define the weights of the single objectives only vaguely, in form of fuzzy
intervals of the --type.
We will analyze three different applications and illustrate the procedures by numerical
examples:
-
The decision maker is able to define crisp partial utility values
-
The decision maker is only able to define fuzzy partial utility values
-
WE use the AHP-approach for the valuation of attributes and determine the utility values of
the attributes by pairwise comparison matrices.
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5. Conclusions
The above explanations clearly show that it is not necessary artificially to condense fuzzy values
to crisp data. Even in case of fuzzy substitution rates and/or fuzzy partial utility values fuzzy
utility value analysis or AHP may lead to convincing rankings/orderings of the alternatives.
They do not always deliver a clear ranking of all alternatives, since an artificially produced
sharp distinction is no longer possible on the level of real numbers. Usually it is however
possible to reduce the number of possible alternatives. Of course fuzzy extensions may be
applied to large hierarchical goal systems as well.