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Saaty‘s matrix revisited: Securing the consistency of
pairwise comparisons
Robert Hlavatý
1
Abstract. The paper is intended to outline the ways of proper pairwise comparison
matrix data filling. The decision makers are sometimes undisciplined while filling in
the pairwise comparison matrix making their own pairwise comparison of criteria
contradictory itself. The consistency of the matrix then reaches over a feasible limit
and the information value of the data is ruined. The inconsistency originates in in-
discipline or unawareness of the decision maker. The goal is to help or force the de-
cision maker to fill the Saaty’s matrix in a proper manner and make the data useful
and consistent. For most common pairwise comparison matrices of the size 3x3 and
4x4, the intervals of stability are defined. If a Saaty’s matrix appears to be incon-
sistent, the following text describes how the certain values in the matrix should be
modified to receive a feasible result in terms of consistency.
Keywords: Saaty’s matrix, decision making, inconsistency index, characteristic
equation.
JEL Classification: C44
AMS Classification: 90B50
1 Introduction
The tools for Multiple Criteria Decision Making (MCDM) have its use in various fields of human acting. Ac-
cording to Hwang and Yoon [3] The MCDM problem can be classified into two categories in this respect: Mul-
tiple Attribute Decision Making (MADM), and Multiple Objective Decision Making (MODM). In other words
the set of decision alternatives can be either discrete finite or continuous infinite. In either of them, the criteria
act as an important factor with essential impact on comparison of variants. Obviously, if all the criteria in a mod-
el had equal importance, the MCDM problem would be reduced to simpler problem. Usually, the importance of
a criterion is expressed by its weight. The means of weights assessment were discussed many times. Entropy
method proposed by Shannon [8] as well as the LINMAP method developed by Srinivasan and Shocker [10] can
be applied if the decision matrix is known. These concepts are generally difficult to understand for a decision
maker. When the pairwise comparison matrix is given, the Least square method and the Eigenvector method can
be applied for weights assessment.
The easiest and most intuitive way of weights assessment in pairwise comparison matrix is use of various
scoring scales. If one seeks to get “expert’s opinion” in the form of weights then this is the most comfortable
way of criteria evaluation for unqualified evaluators. It can be undertaken in a comparison matrix telling which
one of two criteria is more important than the other and also to what extent is it preferred before the other. Saaty
[5] ,[7] defined a pairwise comparison matrix in which, using the scale of
to , the comparison of criteria
can be done. The matrix itself must be reciprocal and the preference consistency must hold. Filling the prefer-
ences into the matrix is generally difficult to understand for common people whose opinions and preferences we
are trying to obtain. It is even more difficult to force the decision makers to keep their opinions consistent, even
for the low number (i.e. 3 or 4) of criteria compared.
After the preference information is gathered from the experts, the data might be damaged due to inconsisten-
cy or even missing because some of them had to be excluded due to inconsistency. Shirashi, Obata and Daigo [9]
show the way of estimation of one missing datum in incomplete matrices using their own heuristic method. A
different heuristic method for estimation of more missing data is described by Harker [2]. Also the fuzzy ap-
proach to preference in pairwise comparison matrix can be applied, as shown by Ramík [4]. It doesn’t have to be
necessary to exclude the damaged inconsistent data from the matrix. Inconsistency issues are widely discussed
by Bozóki and Rapcsák [1] showing some interesting general properties of inconsistent matrices and their eigen-
values. In the following section, the analytical description of inconsistency is shown for 3x3 and 4x4 pairwise
comparison matrices, using the brute-force approach to inference.
1
University of Life Sciences, Department of Systems Engineering, Kamýcká 129, 165 21, Prague.
2 Material and methods
The idea is to find out how the certain values in pairwise comparison matrix can be changed while the consisten-
cy still holds. Saaty (1980) proposed a method for calculating inconsistency. The largest eigenvalue
of
pairwise comparison matrix is computed. It was shown [5] that
and is equal to if and only if
the matrix is fully consistent. The inconsistency index was defined by
(1)
The index alone has no true meaning until any critical values are introduced. At first, Saaty proposed that the
inconsistency ratio lesser than is generally acceptable. Later, Saaty [6] modified the critical value of incon-
sistency ratio to for matrices and for matrices. The values of and will be taken
into account during the further calculations for corresponding matrices.
2.1 Pairwise comparison matrix
Let us have a general pairwise comparison matrix of 3 criteria given by elements :
(2)
The characteristic polynomial of is expressed as !"#$=0. For the particular pairwise comparison
matrix we receive
%
%
(3)
and thus the characteristic equation of the matrix :
&
'
(
'
'
)
(4)
It can be proved that the characteristic equation has the roots
*
+,-
*
./
(
&
+0 or
*
(
+,-
*
(
/
&
+0 if the matrix is fully consistent (i.e.
). Due to the nature of the cubic polyno-
mial and the values of +"1it is possible to treat the as
. There is an obvious dependency be-
tween the matrix elements and the maximum eigenvalue . The matrix inconsistency depends on the en-
tire triad of values. Thus we can assume that for each combination ")$ of values there will be
such third value that
. Assuming the inconsistency index must not reach over the threshold of
, the
formula (1) for this particular case must hold:
2
(5)
2
)
'
(6)
Substituting the
(6) for into the characteristic equation (4), it is possible to obtain graphical represen-
tation of dependency between either and
. Let us show at least one case e.g.
3"$ with chosen
4 and 4. The function shows how the value of affects the consistency index:
Figure 1
3"$+,
5
. (in Desmos graphing calculator)
The dash line expresses the threshold value of for inconsistency index. If the function reaches under the
threshold then
2. The intercepts of the function with the threshold show the range of as the con-
sistency holds. The range for this particular case is 67)89. The lower and upper bound of express
the critical value of where the
. The local minimum of 3"$ lies in :; expressing the ideal func-
tion of the value where the
. Practically, going back to the pairwise comparison matrix, the value of
should lie in the range 7)89. Using the Saaty’s scale, the should be then chosen from the following set
of numbers: +<
*
(
)= so the inconsistency of the matrix would be acceptable. The same principle works for
expressing
3"$ and
3"$. Clearly, it is possible to define such feasible interval for either or
if the two others from the triad are known and the acceptable inconsistency index
is set. Also the ideal value
for zero inconsistency can be calculated.
For practical application it is desirable to calculate lower and upper bounds for either and . It is possible
to express from the characteristic equation (4). Since the parameters in the equation are found in the
fractions’ numerators and denominators at the same time, the expression of these parameters will lead to quadrat-
ic equations. The solutions of these equations are expressed as follows:
"
&
(
'
)
>
?
@
4
A
'
B
'
&
)
(
)
"
&
(
')>?
@
4
A
'
B
'
&
)
(
)
"
&
(
')>?
@
4
A
'
B
'
&
)
(
)
(7)
The polynomial in discriminant returns positive values for all C. This must hold true since it was proved
that
.Considering the polynomial in discriminant it can be said that either of these equations will al-
ways have 2 different solutions in , or one solution in , of multiplicity 2 if , i.e. the inconsistency index is
0. The range of feasible values of will always lie within the range of two roots of each quadratic equation.
2.2 DD pairwise comparison matrix
Let us describe the situation for matrices of the size . A general pairwise comparison matrix E given by
elements 3 is constructed:
E
F
G
H
3
3
I
J
K
(8)
The characteristic polynomial of E is expressed as !"E#$=0. For the particular pairwise comparison
matrix E we receive
E
F
G
H
3
3
I
J
K
(9)
The expression of the characteristic equation becomes somehow difficult and time consuming, although
achievable. The Laplace expansion "$
L5M
N
LM
for OP-th minor of E is used and extensive characteristic equation
is found:
B
&
'
"
Q
$
'
QQ
(10)
where
Q
3
'
3
'
'
'
'
'
3
'
3
(11)
and
QQ
Q
3
3
3
3
)
(12)
It can be proved that the characteristic equation (10) has the roots
*
(
+,-
*
./
&
B
+0 or
*
(
+,-
*
(
/
&
B
+0 if the matrix E is fully consistent (i.e.
). Again, the from the
characteristic equation can be treated as
. This time, however, the assumption is not true that for every five
values of 3 there would be the sixth value such that
. As the polynomial became more com-
plex than in the case of matrix, it may happen that for five known values of 3 there is no such
remaining value that would make the whole matrix fully consistent. The dependency between a random value of
3 and the consistency index
can be shown as graphical representation. Using the inconsistency
formula (1) we can express
as
2
'
(13)
and then substitute
(13) for into the characteristic equation (10). Let us show two examples, the first
e.g.
3"$ with chosen 43 . Then the relation shows how the value of
affects consistency index:
Figure 2
3"$+,
5
. (in Desmos graphing calculator)
The dash line again expresses the threshold value of . The intercepts of the dash line with the curve ex-
press the range of as the consistency holds. The actual range is 67))89. Compared to the case,
the curve will not reach the value of zero and so there is no such value of making the matrix fully consistent.
Although the local minimum of the curve represents the value of at which the
reaches its minimum for
positive values of . We will now change the parameter to R. The situation changes as depicted on
the following picture:
Figure 3
3"$+,
5
. (in Desmos graphing calculator)
Clearly the curve does not cross the threshold of , at least not in terms of positive numbers. Theoretical
values of for this case could be found in the range 6748))9 where the
2 is satisfied. This
is not applicable since we operate only with positive values of preference. Practically, there is no such value of
that would satisfy the inconsistency index and more values of 3 had to be changed at once.
Once again, it is desirable to determine the feasible range for the values of 3 so the inconsistency
index would be acceptable. Expressing individual parameters 3 from the characteristic equation (10)
results in complicated expressions that complete a quadratic equation. For instance the expression of parameter
is shown:
S
>
?
S
(
TU
)
T
(14)
where
T
(
3
'
(
3
(
3
(
(
S"
B
&
')$ 3'
(
'
(
'
(
3
(
'
(
3
(
(
(
3
(
(
3
U
(
3'
(
3
(
3
(
(
(15)
The expressions of 3 will be not shown because they lead to very similar complicated results, only
with differing parameters of 3.
3 Results and discussion
It was shown how the values in the pairwise comparison matrix influence the inconsistency index. In case of
matrix it is always possible to define the triad of parameters such that the inconsistency index is
zero. Although in some cases the triad of values cannot be practically used for weights estimation. For
example we can consider the following pairwise comparison matrix of three criteria with known parameters
and :
R
(16)
Obviously if the preference of 1
st
criterion to 3
rd
criterion is very strong (i.e. =9) and simultaneously, the
preference of the 3
rd
criterion to the 2
nd
is very strong, then the preference of the 1
st
criterion to the 2
nd
would
have to be “very strong very strong” (i.e. ). We shall use the formula (7) for estimation of the value
so the matrix inconsistency would be acceptable. We receive the range of 67)49 and the ideal
value of as expected. This is practically unacceptable since the scale of
*
V
W is used. This kind of deci-
sion maker’s preference is wrong in its own nature as the 2
nd
criterion here has negligible importance and per-
haps should not be included into decision making process at all.
In case of matrix, the way of preference filling into matrix appears to be more complicated when an
acceptable inconsistency index is desired. It emerged from the properties of the characteristic equation (10) and
formula (14) that 4 situations could possibly occur for estimation of the parameters 3. Let us denote
an arbitrary parameter of 3 as the general parameter X with the ideal value X
Y
for
and ac-
ceptable range X+7X X9 for
2. The situations are then following:
• X
Y
+7X X9Z7
*
V
9-X+7X X9Z7
*
V
9. If the value of X is set within the range, than the incon-
sistency value is acceptable. Simultaneously the ideal value of X
Y
is found where the inconsistency in-
dex
. This situation would appear rarely as the ideal value usually cannot be found due to com-
plexity of the characteristic equation. Full consistency of the matrix can be reached by changing of the
single parameter.
• X
Y
[7X X9Z7
*
V
9-X+7X X9Z7
*
V
9. If the value of X is set within the range, than the incon-
sistency value is acceptable. Ideal value of X
Y
is not found. It is not possible to reach full consistency of
the matrix only by changing a single parameter.
• X
Y
[7X X9Z7
*
V
9-X+7X X9Z"\
*
V
$]"\$. The consistency is acceptable only for such
parameter value X out of the feasible range.
• X
Y
[,-X[,. The range of the parameter X cannot be found because the solution of (10) has no real
roots. There is no such value of X that would make the consistency of the matrix acceptable.
The four situations show expressions of but one parameter of the sextuplet 3. If all the parameters
are expressed at the same time, the six intervals will be shown for each of them. It is than necessary to find such
intervals where the 1
st
or 2
nd
situation of above occurred. Changing the appropriate parameter will make the
inconsistency index acceptable.
4 Conclusion
The way how to treat preference values in the and pairwise comparison matrices was proposed. The
small size matrices were subjected to analysis because the higher the size is the more difficult it gets to hold on
to consistent opinions. The methodology could be as well applied for matrices of larger size with some difficul-
ties. Finding the characteristic equation for larger matrix can be conceivable using the Newton’s identities, alt-
hough this task becomes very time consuming as the matrix size grows larger. Even for the matrix of size
the expression of the individual parameters seems to be complicated but if the formulae are implemented into the
computer environment (e.g. MS Excel will do nicely) than we gain an easy tool for consistency evaluation. If a
decision maker’s preferences are gathered and placed into pairwise comparison matrix, the inconsistency check
should be done afterwards. Should the inconsistency index of the matrix be unacceptable then the feasible inter-
vals for single preference values can be found. These intervals will show the problematic preference values and
in addition, how these values could be possibly changed ex-post – if it is possible for a decision maker to do so.
References
[1] Bozóki, S., and Rapcsák, T.: On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrix.
Journal of Global Optimization 42 (2008), 157–175.
[2] Harker, P. T.: Alternative modes of questioning in analytic hierarchic process. Mathematical modelling 9
(1987), 353–360.
[3] Hwang, C.L. and Yoon, K.: Multiple attribute decision making:methods and applications: a state-of-the-art
survey. Springer-Verlag, 1981.
[4] Ramík, J.: Fuzzy preference matrix with missing elements and its application to ranking of alternatives.
Mathematical Methods in Economics 31(2013), 767-772.
[5] Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, 1980.
[6] Saaty, T.L.: Fundamentals of decision making. RSW Publications, 1994.
[7] Saaty, T.L: Relative measurement and its generalization in decision making. RACSAM, 102(2) (2008), 251–
318.
[8] Shannon, C. E.: Prediction and entropy of printed English. Bell system technical journal 30 (1951), 50–64.
[9] Shiraishi, S., Obata, T.,Daigo, M.: Properties of a positive reciprocal matrix and their application to AHP.
Journal of the operations research 41 (1998),404–414.
[10] Srinivasan, V., and Shocker A.D.: Linear programming techniques for multidimensional analysis of prefer-
ence. Psychometrika 38 (1973), 337–342.
