2,541 research outputs found
Pairwise comparison matrices: an empirical research
Our research focused on testing various characteristics of pairwise comparison (PC) matrices in controlled experiments. About 270 students have been involved in the test exercises and the final pool contained 450 matrices. Our team conducted experiments with matrices of different size obtained from different types of MADM problems. The matrix elements have been generated by different questioning orders, too. The cases have been divided into 18 subgroups according to the key factors to be analyzed. The testing environment made it possible to analyze the dynamics of inconsistency as the number of elements increased in a given case. Various types of inconsistency indices have been applied. The consequent behavior of the decision maker has also been analyzed in case of incomplete matrices using indicators to measure the deviation from the final ranking of alternatives and from the final score vector
An LP-based inconsistency monitoring of pairwise comparison matrices
A distance-based inconsistency indicator, defined by the third author for the consistency-driven pairwise comparisons method, is extended to the incomplete case. The corresponding optimization problem is transformed into an equivalent linear programming problem. The results can be applied in the process of filling in the matrix as the decision maker gets automatic feedback. As soon as a serious error occurs among the matrix elements, even due to a misprint, a significant increase in the inconsistency index is reported. The high inconsistency may be alarmed not only at the end of the process of filling in the matrix but also during the completion process. Numerical examples are also provided
On the geometric mean method for incomplete pairwise comparisons
When creating the ranking based on the pairwise comparisons very often, we
face difficulties in completing all the results of direct comparisons. In this
case, the solution is to use the ranking method based on the incomplete PC
matrix. The article presents the extension of the well known geometric mean
method for incomplete PC matrices. The description of the methods is
accompanied by theoretical considerations showing the existence of the solution
and the optimality of the proposed approach.Comment: 15 page
Notes on the existence of solutions in the pairwise comparisons method using the Heuristic Rating Estimation approach
Pairwise comparisons are a well-known method for modelling of the subjective
preferences of a decision maker. A popular implementation of the method is
based on solving an eigenvalue problem for M - the matrix of pairwise
comparisons. This does not take into account the actual values of preference.
The Heuristic Rating Estimation (HRE) approach is a modification of this method
in which allows modelling of the reference values. To determine the relative
order of preferences is to solve a certain linear equation system defined by
the matrix A and the constant term vector b (both derived from M). The article
explores the properties of these equation systems. In particular, it is proven
that for some small data inconsistency the A matrix is an M-matrix, hence the
equation proposed by the HRE approach has a unique strictly positive solution.Comment: 8 page
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