Incomplete pairwise comparison matrix was introduced by Harker in 1987 for the case in which the decision maker does not fill in the whole matrix completely due to, e.g., time limitations. However, incomplete matrices occur in a natural way even if the decision maker provides a completely filled in matrix in the end. In each step of the total n(n–1)/2, an incomplete pairwise comparison is given, except for the last one where the matrix turns into complete. Recent results on incomplete matrices make it possible to estimate inconsistency indices CR and CM by the computation of tight lower bounds in each step of the filling in process. Additional information on ordinal inconsistency is also provided. Results can be applied in any decision support system based on pairwise comparison matrices. The decision maker gets an immediate feedback in case of mistypes, possibly causing a high level of inconsistency

Bozóki, Sándor

English

Corvinus Research Archive

AN INCONSISTENCY CONTROL SYSTEM BASED ON INCOMPLETE PAIRWISE COMPARISON MATRICES   Sándor Bozóki Computer and Automation Research Institute  Hungarian Academy of Sciences (MTA SZTAKI); Corvinus University of Budapest (BCE) Budapest, Hungary E-mail: bozoki@sztaki.hu; sandor.bozoki@uni-corvinus.hu    ABSTRACT  Incomplete pairwise comparison matrix was introduced by Harker in 1987 for the case in which the decision maker does not fill in the whole matrix completely due to, e.g., time limitations. However, incomplete matrices occur in a natural way even if the decision maker provides a completely filled in matrix in the end. In each step of the total n(n–1)/2, an incomplete pairwise comparison is given, except for the last one where the matrix turns into complete. Recent results on incomplete matrices make it possible to estimate inconsistency indices CR and CM by the computation of tight lower bounds in each step of the filling in process. Additional information on ordinal inconsistency is also provided. Results can be applied in any decision support system based on pairwise comparison matrices. The decision maker gets an immediate feedback in case of mistypes, possibly causing a high level of inconsistency.  Keywords: Inconsistency, incomplete pairwise comparison matrix, multi-attribute decision models.   1. Introduction   In multi-attribute decision models, weighting the criteria and evaluating the alternatives with respect to the criteria are of the most important steps. In the paper, the method of pairwise comparison matrices [Saaty, 1980] and its generalization for the incomplete case is discussed. For simplicity, assume that the objects to compare are the importances of n criteria (the method is the same when comparing alternatives to each other, voting powers of the individuals in group decision making, or estimated probabilities of future events). A (complete) pairwise comparison matrix A = [aij]i,j=1,…,n is a matrix of size n×n with the properties as follows:  aij  >  0;   (1) aii  =  1;   (2) aij = 1/aji   (3)  for i,j=1,…,n, where aij is the numerical answer given by the decision maker for the question  – “How many times Criterion i is more important than Criterion j?” or  – “How many times Alternative i is better than Alternative j with respect to a given criterion?” or  – “How many times Decision Maker i has larger voting power than Decision Maker j?” in a group decision making problem. – “How many times Event i is more likely than Event j?” in a probability estimation problem.  Proceedings of the International Symposium on the Analytic Hierarchy Process 2011   2 The weighting problem is to find the n-dimensional positive weight vector w = (w1,w2,…,wn)T such that the appropriate ratios of the components of w reflect, or, at least, approximate all the aij values (i,j=1,…,n), given by the decision maker. In the Analytic Hierarchy Process [Saaty, 1980] Eigenvector Method is applied and the approximation wEM of w is defined by  A wEM  = λmax wEM,  where λmax denotes the maximal eigenvalue, also known as Perron eigenvalue, of A and  wEM = (w1EM, w2EM,…,wnEM)T denotes the right-hand side eigenvector of A corresponding to λmax. By Perron's theorem, wEM is positive and unique up to a scalar multiplication. The most often used normalization is ΣwiEM = 1.    A pairwise comparison matrix in is called consistent if the transitivity   aij ajk = aik   (4)  holds for all indices i,j,k=1,…,n. Otherwise, the matrix is inconsistent. However, there are different levels of inconsistency, some of which are acceptable in solving real decision problems, some are not. Saaty [Saaty, 1980] defined the inconsistency ratio CR as a positive linear transformation of λmax. It is well known that CR = 0  if and only if the matrix is consistent. According to Saaty, larger value of CR indicates higher level of inconsistency and the 10 percent rule (CR ≤ 10%) separates acceptable matrices from unacceptable ones.   Incomplete pairwise comparison matrix was defined by Harker [Harker, 1987]. It is of the same form as (1)–(3) but one ore more elements, denoted here by *, are not given (or not given yet if we consider the filling in procedure):   1 a12 * . . a1n  1/a12 1 a23 . . * A = * 1/ a23 1 . . a3n  : : :  :  1/ a1n * 1/ a3n . .  1  Variables x1,x2,…,xd (xi > 0, i=1,…,d) are introduced for the missing elements in the upper triangular part of A. Their reciprocals, 1/x1, 1/x2, …, 1/xd are written in the lower triangular part of A. Let x denote the vector (x1, x2 ,…, xd). A(x) is a complete pairwise comparison matrix for any values of x. The total number of missing elements in matrix A is 2d.   1 a12 x1 . . a1n  1/a12 1 a23 . . xd A(x) = A(x1, x2 ,…, xd) =  1/x1 1/a23 1 . . a3n  : : :  : S. Bozóki   3  1/a1n 1/xd 1/a3n . .  1  Based on the correspondence between the CR inconsistency and λmax of a pairwise comparison matrix, the generalization of the Eigenvector Method for the incomplete case is originated from the optimal solution of the eigenvalue minimization problem as follows [Shiraishi, Obata, Daigo, 1998; Shiraishi, Obata, 2002]:  min { λmax (A(x)) | x > 0 }.   (5)  Note again that problem (5) is equivalent to the problem   min { CR(A(x)) | x > 0 }.   (6)  Inconsistency index CM [Koczkodaj, 1993] of a 3×3 pairwise comparison matrix [ [1, a, b] , [1/a, 1, c] , [1/b, 1/c, 1]] is defined as CM = min{ |a-b/c|/a , |b-ac|/b , |c-b/a|/c } = min{ |1-b/(ac)| , |1-(ac)/b|}. In the case of matrices of higher orders, CM(A) is defined as max{ CM(A3×3) |  A3×3 is a 3×3 submatrix of A and A3×3 itself is a pairwise comparison matrix }.   CM of an incomplete pairwise comparison matrix [Bozóki, Fülöp, Koczkodaj, 2011] is defined from the optimization problem as follows:   min { CM(A(x)) | x > 0 }.   (7)  Besides numerical inconsistency indices like CR and CM, ordinal inconsistency has been also analyzed.   Gass [Gass, 1998] investigated the property of ordinal transitivity (aij > 1 and ajk > 1 imply aik > 1), which follows from cardinal transitivity (4) but ordinal transitivity can also hold without cardinal transitivity. By Kéri’s definition [Kéri, 2010], a pairwise comparison matrix A is qualitatively consistent if a consistent matrix A* of the same size can be found so that aij > 1 if and only if aij* > 1 for any pairs of indices. A graph theoretic characterization of qualitatively consistent is also given in [Kéri, 2010].    2. Methodology – incomplete matrices and graphs  Assume that the decision maker compares the relative importance of n criteria and we are able to observe the whole procedure of filling in the pairwise comparison matrix. In each comparison, a direct relation is defined between two criteria, namely, the estimated ratio of their weights. However, two criteria, not compared yet, consequently, having no direct relation, can be in indirect relation, through further criteria and direct relations. It is a natural idea to associate graph structures to (in)complete pairwise comparison matrices.  Given an (in)complete pairwise comparison matrix A of size n×n, graph G is defined as follows:  G := (V,E),  where V ={1, 2, …, n}, the vertices correspond to the objects to compare and E = { e(i,j) | aij and aji is given and i ≠ j }, the undirected edges correspond to the matrix elements. There are no edges corresponding to the missing elements in the matrix. G is an undirected graph.  Proceedings of the International Symposium on the Analytic Hierarchy Process 2011   4 Example. Let a 6×6 pairwise comparison matrix be filled in by the decision maker, B5, B10 and B15 denotes the matrices having 5, 10 and 15 elements, respectively. Matrices B5 and B10 are incomplete and B15 is complete. Graphs associated are also drawn.   1 2 5 7 9 9  1/2 1 * * * * B5 = 1/5 * 1 * * *   1/7 * * 1 * *  1/9 * * * 1 *  1/9 * * * * 1     1 2 5 7 9 9  1/2 1 3 6 8 9 B10 = 1/5 1/3 1 4 * *   1/7 1/6 1/4 1 * *  1/9 1/8 * * 1 *  1/9 1/9 * * * 1    1 2 5 7 9 9  1/2 1 3 6 8 9 B15 = 1/5 1/3 1 4 6 7   1/7 1/6 1/4 1 3 5  1/9 1/8 1/6 1/3 1 2  1/9 1/9 1/7 1/5 1/2 1     S. Bozóki   5 3. Summary of the results  Theoretical results are related to the connectedness of the graph associated to the incomplete pairwise comparison matrix. Connectedness is a natural necessary condition, otherwise the problems breaks into several parts the weights of which can change independently from each other.   Theorem [Bozóki, Fülöp, Rónyai, 2010]: The optimal solution of the problem (6) is unique if and only if the graph corresponding to the incomplete pairwise comparison matrix is connected.  It is also shown [Bozóki, Fülöp, Koczkodaj, 2011] that problem (7) can be reduced to an LP problem the solution of which is found fast.    By solving problems (6) and/or (7) real time, the decision maker may get automatic lower tight bounds of CR/CM inconsistency levels in each step of the filling in process. These kinds of information are useful for an immediate recognition of possible misprints. Warning rules and thresholds are under construction as they can be determined based on empirical pairwise comparison matrices.    Pairwise comparison matrices that can be made consistent by the modification of a few (1,2,3) elements are characterized in [Bozóki, Fülöp, Poesz, 2010]. Empirical examples collected from case studies show that these kinds of matrices are not only of theoretical interest but they are present in practice as well [Poesz, 2009].  Algorithms for giving feedback on ordinal (in)consistency are of the same importance as on CR/CM inconsistencies. The extensions of the theorems and propositions cited from Gass [Gass, 1998] and Kéri [Kéri, 2010] to the incomplete case are of the present and future research.   4. Conclusion  There are several types of inconsistency, a number of indices have been defined for both ordinal and cardinal violations of the well known equation aij ajk = aik  (for all indices i,j,k=1,…,n). We focused on two of the cardinal indices, CR and CM, both are extended to the incomplete case. The decision maker gets a lower bounds for the CR/CM-inconsistency level in each step when filling in the pairwise comparison matrix, that can be used for detecting misprints in real time.      REFERENCES  Bozóki, S., Dezső, L., Poesz, A., & Temesi, J. (2011). Pairwise comparison matrices: an empirical research, Proceedings of The International Symposium on the Analytic Hierarchy Process (ISAHP), 15-18 June, 2011, Sorrento (Naples), Italy.  Bozóki, S., Fülöp, J., & Koczkodaj, W.W. (2011). LP-based inconsistency analysis of incomplete pairwise comparison matrices (submitted).  Bozóki, S., Fülöp, J., & Poesz, A. (2010): On pairwise comparison matrices that can be made consistent by the modification of a few elements, Central European Journal of Operations Research , in press, published online: 24 February 2010, DOI 10.1007/s10100-010-0136-9. Proceedings of the International Symposium on the Analytic Hierarchy Process 2011   6  Bozóki, S., Fülöp, J., & Rónyai, L. (2010). On optimal completions of incomplete pairwise comparison matrices. Mathematical and Computer Modelling, 52, 318–333.  Gass, S.I. (1998). Tournaments, transitivity and pairwise comparison. Journal of the Operational Research Society 49, 616–624.  Harker, P.T. (1987). Incomplete pairwise comparisons in the analytic hierarchy process. Mathematical Modelling, 9(11), 837–848.  Kéri, G. (2010). On qualitatively consistent, transitive and contradictory judgment matrices emerging from multiattribute decision procedures, Central European Journal of Operations Research , in press, published online: 24 February 2010, DOI 10.1007/s10100-010-0138-7.  Poesz, A. (2009). Empirical pairwise comparison matrices (EPCM) – an on-line collection from real decisions, http://www.uni-corvinus.hu/index.php?id=29191; http://www.sztaki.hu/~bozoki/epcm  Saaty, T.L. (1980). The analytic hierarchy process, McGraw Hill, New York.  Shiraishi, S., Obata, T., & Daigo, M. (1998). Properties of a positive reciprocal matrix and their application to AHP. Journal of the Operations Research Society of Japan, 41(3), 404–414.  Shiraishi, S., & Obata, T. (2002). On a maximization problem arising from a positive reciprocal matrix in AHP. Bulletin of Informatics and Cybernetics, 34(2), 91–96. 

Empirical pairwise comparison matrices (EPCM) – an on-line collection from real decisions, http://www.uni-corvinus.hu/index.php?id=29191; http://www.sztaki.hu/~bozoki/epcm Saaty,

Incomplete pairwise comparisons in the analytic hierarchy process.

LP-based inconsistency analysis of incomplete pairwise comparison matrices (submitted).

On a maximization problem arising from a positive reciprocal matrix in AHP.

On pairwise comparison matrices that can be made consistent by the modification of a few elements,

On qualitatively consistent, transitive and contradictory judgment matrices emerging from multiattribute decision procedures,

Pairwise comparison matrices: an empirical research,

Properties of a positive reciprocal matrix and their application to AHP.

Tournaments, transitivity and pairwise comparison.

An inconsistency control system based on incomplete pairwise comparison matrices

AN INCONSISTENCY CONTROL SYSTEM BASED ON INCOMPLETE 
PAIRWISE COMPARISON MATRICES  
 
Sándor Bozóki 
Computer and Automation Research Institute  
Hungarian Academy of Sciences (MTA SZTAKI); 
Corvinus University of Budapest (BCE) 
Budapest, Hungary 
E-mail: bozoki@sztaki.hu; sandor.bozoki@uni-corvinus.hu 
 
 
 
ABSTRACT 
 
Incomplete pairwise comparison matrix was introduced by Harker in 1987 for the case in which the 
decision maker does not fill in the whole matrix completely due to, e.g., time limitations. However, 
incomplete matrices occur in a natural way even if the decision maker provides a completely filled in 
matrix in the end. In each step of the total n(n–1)/2, an incomplete pairwise comparison is given, except 
for the last one where the matrix turns into complete. Recent results on incomplete matrices make it 
possible to estimate inconsistency indices CR and CM by the computation of tight lower bounds in each 
step of the filling in process. Additional information on ordinal inconsistency is also provided. Results 
can be applied in any decision support system based on pairwise comparison matrices. The decision 
maker gets an immediate feedback in case of mistypes, possibly causing a high level of inconsistency. 
 
Keywords: Inconsistency, incomplete pairwise comparison matrix, multi-attribute decision 
models.  
 
1. Introduction 
 
 In multi-attribute decision models, weighting the criteria and evaluating the alternatives with respect to 
the criteria are of the most important steps. In the paper, the method of pairwise comparison matrices 
[Saaty, 1980] and its generalization for the incomplete case is discussed. For simplicity, assume that the 
objects to compare are the importances of n criteria (the method is the same when comparing alternatives 
to each other, voting powers of the individuals in group decision making, or estimated probabilities of 
future events). A (complete) pairwise comparison matrix A = [aij]i,j=1,…,n is a matrix of size n×n with the 
properties as follows: 
 
aij  >  0;   (1) 
aii  =  1;   (2) 
aij = 1/aji   (3) 
 
for i,j=1,…,n, where aij is the numerical answer given by the decision maker for the question  
– “How many times Criterion i is more important than Criterion j?” or  
– “How many times Alternative i is better than Alternative j with respect to a given criterion?” or  
– “How many times Decision Maker i has larger voting power than Decision Maker j?” in a group 
decision making problem. 
– “How many times Event i is more likely than Event j?” in a probability estimation problem. 
 
Proceedings of the International Symposium on the Analytic Hierarchy Process 2011 
 
 2 
The weighting problem is to find the n-dimensional positive weight vector w = (w1,w2,…,wn)
T
 such that 
the appropriate ratios of the components of w reflect, or, at least, approximate all the aij values 
(i,j=1,…,n), given by the decision maker. In the Analytic Hierarchy Process [Saaty, 1980] Eigenvector 
Method is applied and the approximation w
EM
 of w is defined by 
 
A w
EM
  = λmax w
EM
, 
 
where λmax denotes the maximal eigenvalue, also known as Perron eigenvalue, of A and  
w
EM
 = (w1
EM
, w2
EM
,…,wn
EM
)
T
 denotes the right-hand side eigenvector of A corresponding to λmax. By 
Perron's theorem, w
EM
 is positive and unique up to a scalar multiplication. The most often used 
normalization is Σwi
EM
 = 1.   
 
A pairwise comparison matrix in is called consistent if the transitivity  
 
aij ajk = aik   (4) 
 
holds for all indices i,j,k=1,…,n. Otherwise, the matrix is inconsistent. However, there are different levels 
of inconsistency, some of which are acceptable in solving real decision problems, some are not. Saaty 
[Saaty, 1980] defined the inconsistency ratio CR as a positive linear transformation of λmax. It is well 
known that CR = 0  if and only if the matrix is consistent. According to Saaty, larger value of CR 
indicates higher level of inconsistency and the 10 percent rule (CR ≤ 10%) separates acceptable matrices 
from unacceptable ones.  
 
Incomplete pairwise comparison matrix was defined by Harker [Harker, 1987]. It is of the same form as 
(1)–(3) but one ore more elements, denoted here by *, are not given (or not given yet if we consider the 
filling in procedure): 
 
 1 a12 * . . a1n 
 1/a12 1 a23 . . * 
A = * 1/ a23 1 . . a3n 
 : : :  : 
 1/ a1n * 1/ a3n . .  1 
 
Variables x1,x2,…,xd (xi > 0, i=1,…,d) are introduced for the missing elements in the upper triangular part 
of A. Their reciprocals, 1/x1, 1/x2, …, 1/xd are written in the lower triangular part of A. Let x denote the 
vector (x1, x2 ,…, xd). A(x) is a complete pairwise comparison matrix for any values of x. The total 
number of missing elements in matrix A is 2d. 
 
 1 a12 x1 . . a1n 
 1/a12 1 a23 . . xd 
A(x) = A(x1, x2 ,…, xd) =  1/x1 1/a23 1 . . a3n 
 : : :  : 
S. Bozóki 
 
 3 
 1/a1n 1/xd 1/a3n . .  1 
 
Based on the correspondence between the CR inconsistency and λmax of a pairwise comparison matrix, the 
generalization of the Eigenvector Method for the incomplete case is originated from the optimal solution 
of the eigenvalue minimization problem as follows [Shiraishi, Obata, Daigo, 1998; Shiraishi, Obata, 
2002]: 
 
min { λmax (A(x)) | x > 0 }.   (5) 
 
Note again that problem (5) is equivalent to the problem  
 
min { CR(A(x)) | x > 0 }.   (6) 
 
Inconsistency index CM [Koczkodaj, 1993] of a 3×3 pairwise comparison matrix [ [1, a, b] , [1/a, 1, c] , 
[1/b, 1/c, 1]] is defined as CM = min{ |a-b/c|/a , |b-ac|/b , |c-b/a|/c } = min{ |1-b/(ac)| , |1-(ac)/b|}. In the 
case of matrices of higher orders, CM(A) is defined as max{ CM(A3×3) |  A3×3 is a 3×3 submatrix of A and 
A3×3 itself is a pairwise comparison matrix }.  
 
CM of an incomplete pairwise comparison matrix [Bozóki, Fülöp, Koczkodaj, 2011] is defined from the 
optimization problem as follows:  
 
min { CM(A(x)) | x > 0 }.   (7) 
 
Besides numerical inconsistency indices like CR and CM, ordinal inconsistency has been also analyzed.   
Gass [Gass, 1998] investigated the property of ordinal transitivity (aij > 1 and ajk > 1 imply aik > 1), which 
follows from cardinal transitivity (4) but ordinal transitivity can also hold without cardinal transitivity. By 
Kéri’s definition [Kéri, 2010], a pairwise comparison matrix A is qualitatively consistent if a consistent 
matrix A* of the same size can be found so that aij > 1 if and only if aij* > 1 for any pairs of indices. A 
graph theoretic characterization of qualitatively consistent is also given in [Kéri, 2010].  
 
 
2. Methodology – incomplete matrices and graphs 
 
Assume that the decision maker compares the relative importance of n criteria and we are able to observe 
the whole procedure of filling in the pairwise comparison matrix. In each comparison, a direct relation is 
defined between two criteria, namely, the estimated ratio of their weights. However, two criteria, not 
compared yet, consequently, having no direct relation, can be in indirect relation, through further criteria 
and direct relations. It is a natural idea to associate graph structures to (in)complete pairwise comparison 
matrices. 
 
Given an (in)complete pairwise comparison matrix A of size n×n, graph G is defined as follows:  
G := (V,E),  where V ={1, 2, …, n}, the vertices correspond to the objects to compare and E = { e(i,j) | aij 
and aji is given and i ≠ j }, the undirected edges correspond to the matrix elements. There are no edges 
corresponding to the missing elements in the matrix. G is an undirected graph. 
 
Proceedings of the International Symposium on the Analytic Hierarchy Process 2011 
 
 4 
Example. Let a 6×6 pairwise comparison matrix be filled in by the decision maker, B5, B10 and B15 
denotes the matrices having 5, 10 and 15 elements, respectively. Matrices B5 and B10 are incomplete and 
B15 is complete. Graphs associated are also drawn. 
 
 1 2 5 7 9 9 
 1/2 1 * * * * 
B5 = 1/5 * 1 * * * 
  1/7 * * 1 * * 
 1/9 * * * 1 * 
 1/9 * * * * 1 
 
 
 
 1 2 5 7 9 9 
 1/2 1 3 6 8 9 
B10 = 1/5 1/3 1 4 * * 
  1/7 1/6 1/4 1 * * 
 1/9 1/8 * * 1 * 
 1/9 1/9 * * * 1 
 
 
 1 2 5 7 9 9 
 1/2 1 3 6 8 9 
B15 = 1/5 1/3 1 4 6 7 
  1/7 1/6 1/4 1 3 5 
 1/9 1/8 1/6 1/3 1 2 
 1/9 1/9 1/7 1/5 1/2 1 
 
 
 
 
S. Bozóki 
 
 5 
3. Summary of the results 
 
Theoretical results are related to the connectedness of the graph associated to the incomplete pairwise 
comparison matrix. Connectedness is a natural necessary condition, otherwise the problems breaks into 
several parts the weights of which can change independently from each other.  
 
Theorem [Bozóki, Fülöp, Rónyai, 2010]: The optimal solution of the problem (6) is unique if and only if 
the graph corresponding to the incomplete pairwise comparison matrix is connected. 
 
It is also shown [Bozóki, Fülöp, Koczkodaj, 2011] that problem (7) can be reduced to an LP problem the 
solution of which is found fast.   
 
By solving problems (6) and/or (7) real time, the decision maker may get automatic lower tight bounds of 
CR/CM inconsistency levels in each step of the filling in process. These kinds of information are useful 
for an immediate recognition of possible misprints. Warning rules and thresholds are under construction 
as they can be determined based on empirical pairwise comparison matrices.   
 
Pairwise comparison matrices that can be made consistent by the modification of a few (1,2,3) elements 
are characterized in [Bozóki, Fülöp, Poesz, 2010]. Empirical examples collected from case studies show 
that these kinds of matrices are not only of theoretical interest but they are present in practice as well 
[Poesz, 2009]. 
 
Algorithms for giving feedback on ordinal (in)consistency are of the same importance as on CR/CM 
inconsistencies. The extensions of the theorems and propositions cited from Gass [Gass, 1998] and Kéri 
[Kéri, 2010] to the incomplete case are of the present and future research. 
 
 
4. Conclusion 
 
There are several types of inconsistency, a number of indices have been defined for both ordinal and 
cardinal violations of the well known equation aij ajk = aik  (for all indices i,j,k=1,…,n). We focused on two 
of the cardinal indices, CR and CM, both are extended to the incomplete case. The decision maker gets a 
lower bounds for the CR/CM-inconsistency level in each step when filling in the pairwise comparison 
matrix, that can be used for detecting misprints in real time.  
 
 
 
 
REFERENCES 
 
Bozóki, S., Dezső, L., Poesz, A., & Temesi, J. (2011). Pairwise comparison matrices: an empirical 
research, Proceedings of The International Symposium on the Analytic Hierarchy Process (ISAHP), 15-18 
June, 2011, Sorrento (Naples), Italy. 
 
Bozóki, S., Fülöp, J., & Koczkodaj, W.W. (2011). LP-based inconsistency analysis of incomplete 
pairwise comparison matrices (submitted). 
 
Bozóki, S., Fülöp, J., & Poesz, A. (2010): On pairwise comparison matrices that can be made consistent 
by the modification of a few elements, Central European Journal of Operations Research , in press, 
published online: 24 February 2010, DOI 10.1007/s10100-010-0136-9. 
Proceedings of the International Symposium on the Analytic Hierarchy Process 2011 
 
 6 
 
Bozóki, S., Fülöp, J., & Rónyai, L. (2010). On optimal completions of incomplete pairwise comparison 
matrices. Mathematical and Computer Modelling, 52, 318–333. 
 
Gass, S.I. (1998). Tournaments, transitivity and pairwise comparison. Journal of the Operational 
Research Society 49, 616–624. 
 
Harker, P.T. (1987). Incomplete pairwise comparisons in the analytic hierarchy process. Mathematical 
Modelling, 9(11), 837–848. 
 
Kéri, G. (2010). On qualitatively consistent, transitive and contradictory judgment matrices emerging 
from multiattribute decision procedures, Central European Journal of Operations Research , in press, 
published online: 24 February 2010, DOI 10.1007/s10100-010-0138-7. 
 
Poesz, A. (2009). Empirical pairwise comparison matrices (EPCM) – an on-line collection from real 
decisions, http://www.uni-corvinus.hu/index.php?id=29191; http://www.sztaki.hu/~bozoki/epcm 
 
Saaty, T.L. (1980). The analytic hierarchy process, McGraw Hill, New York. 
 
Shiraishi, S., Obata, T., & Daigo, M. (1998). Properties of a positive reciprocal matrix and their 
application to AHP. Journal of the Operations Research Society of Japan, 41(3), 404–414. 
 
Shiraishi, S., & Obata, T. (2002). On a maximization problem arising from a positive reciprocal matrix in 
AHP. Bulletin of Informatics and Cybernetics, 34(2), 91–96. 


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AN INCONSISTENCY CONTROL SYSTEM BASED ON INCOMPLETE PAIRWISE COMPARISON MATRICES

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An inconsistency control system based on incomplete pairwise comparison matrices

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