On pairwise comparison matrices that can be made consistent by the modification of a few elements. (English) Zbl 1213.90132

Summary: Pairwise comparison matrices are often used in Multi-attribute Decision Making for weighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.


90B50 Management decision making, including multiple objectives
Full Text: DOI Link


[1] Bozóki S, Lewis R (2005) Solving the least squares method problem in the AHP for 3 {\(\times\)} 3 and 4 {\(\times\)} 4 matrices. Central Eur J Oper Res 13: 255–270 · Zbl 1136.90396
[2] Bozóki S, Rapcsák T (2008) On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. J Glob Optim 42(2): 139–148 · Zbl 1177.90205
[3] Bozóki S (2008) Solution of the least squares method problem of pairwise comparisons matrices. Central Eur J Oper Res 16: 345–358 · Zbl 1173.90588
[4] Chu ATW, Kalaba RE, Spingarn K (1979) A comparison of two methods for determining the weight belonging to fuzzy sets. J Optim Theory Appl 4: 531–538 · Zbl 0377.94002
[5] Gass SI (1998) Tournaments, transitivity and pairwise comparison matrices. J Oper Res Soc 49: 616–624 · Zbl 1131.90381
[6] Gass SI, Standard SM (2002) Characteristics of positive reciprocal matrices in the analytic hierarchy process. J Oper Res Soc 53: 1385–1389 · Zbl 1138.90405
[7] Kéri G (2005) Criteria for pairwise comparison matrices. Szigma 36: 139–148 (in Hungarian)
[8] Koczkodaj WW (1993) A new definition of consistency of pairwise comparisons. Math Comput Model 18: 79–84 · Zbl 0804.92029
[9] Poesz A (2008) Analysis of the inconsistency of empirical pairwise comparison matrices. Master’s Thesis, Department of Decisions in Economics, Corvinus University of Budapest
[10] Poesz A (2009) Empirical pairwise comparison matrices (EPCM)–an on-line collection from real decisions, version EPCM-October-2009. http://www.uni-corvinus.hu/index.php?id=29191 ; http://www.sztaki.hu/\(\sim\)bozoki/epcm
[11] Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New-York
[12] Standard SM (2000) Analysis of positive reciprocal matrices. Master’s Thesis, Graduate School of the University of Maryland
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.