Waldemar W. Koczkodaj

A new definition of consistency of pairwise comparisons

A new definition of consistency is introduced. It allows us to locate the roots of inconsistency and is easy to interpret. It also forms a better basis than the old eigenvalue consistency for selecting a threshold based on common sense. The new definition of consistency is applicable to expert systems and to knowledge acquisition. It is instrumental in applications of fuzzy sets and the theory of evidence where the definitions of the membership and belief functions are fundamental issues.

A Monte Carlo study of pairwise comparison

The pairwise comparison methodology introduced by Thurston in 1927 (see [12]) can be used as a powerful inference tool and knowledge acquisition technique in knowledge-based systems. The goal of pairwise comparisons is to establish the relative preferences of n stimuli in situations in which it is impractical (or meaningless) to provide estimates for the stimuli. To this end, an expert provides pairwise comparison coefficients a ij >0 which are meant to be a substitute for the quotients s i /s j of the unknown (or undefined) values s i ,s j >0. There has been an ongoing discussion about which method fdor finding solutions to a pairwise comparison matrix is better [11,2]. It is not clear whether or not an analytical proof can be devised. This seems to be a good reason for formulating the problem as an empirical experiment using a Monte Carlo approach.

On Axiomatization of Inconsistency Indicators for Pairwise Comparisons

This study examines the notion of inconsistency in pairwise comparisons for providing an axiomatization for it. It also proposes two inconsistency indicators for pairwise comparisons. The primary motivation for the inconsistency reduction is expressed by a computer industry concept “garbage in, garbage out”. The quality of the output depends on the quality of the input.

On distance-based inconsistency reduction algorithms for pairwise comparisons

A complete proof of convergence of a certain class of reduction algorithms for distance-based inconsistency (defined in 1993) for pairwise comparisons is presented in this paper. Using pairwise comparisons is a powerful method for synthesizing measurements and subjective assessments. From the mathematical point of view, the pairwise comparisons method generates a matrix (say A) of ratio values (aij )o f theith entity compared with the jth entity according to a given criterion. Entities/criteria can be both quantitative or qualitative allowing this method to deal with complex decisions. However, subjective assessments often involve inconsistency, which is usually undesirable. The assessment can be refined via analysis of inconsistency, leading to reduction of the latter. The proposed method of localizing the inconsistency may conceivably be of relevance for nonclassical logics (e.g., paraconsistent logic) and for uncertainty reasoning since it accommodates inconsistency by treating inconsistent data as still useful information.

Computing a consistent approximation to a generalized pairwise comparisons matrix

Abstract This paper presents an algorithm for computing a consistent approximation to a generalized pairwise comparisons matrix (that is, without the reciprocity property or even 1s on the main diagonal). The algorithm is based on a logarithmic transformation of the generalized pairwise comparisons matrix into a linear space with the Euclidean metric. It uses both the row and (reciprocals of) column geometric means and is thus a generalization of the ordinary geometric means method. The resulting approximation is not only consistent, but also closest to the original matrix, i.e., deviates least from an experts original judgments. The computational complexity of the algorithm is O ( n 2 ).

On the quality evaluation of scientific entities in Poland supported by consistency-driven pairwise comparisons method

Comparison, rating, and ranking of alternative solutions, in case of multicriteria evaluations, have been an eternal focus of operations research and optimization theory. There exist numerous approaches at practical solving the multicriteria ranking problem. The recent focus of interest in this domain was the event of parametric evaluation of research entities in Poland. The principal methodology was based on pairwise comparisons. For each single comparison, four criteria have been used. One of the controversial points of the assumed approach was that the weights of these criteria were arbitrary. The main focus of this study is to put forward a theoretically justified way of extracting weights from the opinions of domain experts. Theoretical bases for the whole procedure are based on a survey and its experimental results. Discussion and comparison of the two resulting sets of weights and the computed inconsistency indicator are discussed.

Statistically Accurate Evidence of Improved Error Rate by Pairwise Comparisons

A statistical experiment was designed to check whether the pairwise comparisons method, introduced by Thurstone in 1927, can really improve the accuracy of estimation of stimuli. This method was compared with direct rating. The experiment was designed and implemented to minimize statistical bias. Randomly generated bars were used since everyone is an expert on estimating lengths. The statistical analysis favoured the pairwise comparisons method The obtained results are decisive; more than a 300% improvement in accuracy was gained with at least 10 level of confidence.

A different perspective on a scale for pairwise comparisons

One of the major challenges for collective intelligence is inconsistency, which is unavoidable whenever subjective assessments are involved. Pairwise comparisons allow one to represent such subjective assessments and to process them by analyzing, quantifying and identifying the inconsistencies. We propose using smaller scales for pairwise comparisons and provide mathematical and practical justifications for this change. Our postulates aim is to initiate a paradigm shift in the search for a better scale construction for pairwise comparisons. Beyond pairwise comparisons, the results presented may be relevant to other methods using subjective scales.

Testing the accuracy enhancement of pairwise comparisons by a Monte Carlo experiment

A statistical experiment was designed to check if the pairwise comparisons method, which was introduced by Fechner in 1860 and developed by Thurstone in 1927, really improves the accuracy of estimation of stimuli. The experiment has been designed and implemented to minimize statistical bias. The accuracy improvement by the pairwise comparisons method (when compared with the direct rating method) is decisive: the mean value of the improvement exceeds 500% and a 95% confidence interval is (4.657, 5.389).

A weak order approach to group ranking

A new approach to the problem of a group ranking is presented. The solution satisfies four of Arrows six axioms [1,2] and is sufficient for a wide range of applications. A constructive algorithm is proposed for finding a group ranking on the basis of individual rankings of which some may collide with others