606 research outputs found
Axiomatization of Inconsistency Indicators for Pairwise Comparisons
This study proposes revised axioms for defining inconsistency indicators in
pairwise comparisons. It is based on the new findings that "PC submatrix cannot
have a worse inconsistency indicator than the PC matrix containing it" and that
there must be a PC submatrix with the same inconsistency as the given PC
matrix.
This study also provides better reasoning for the need of normalization. It
is a revision of axiomatization by Koczkodaj and Szwarc, 2014 which proposed
axioms expressed informally with some deficiencies addressed in this study.Comment: This paper should have been withdrawn by the first author a long time
ago. The work has been finished with another researcher, I have been pushed
out the projec
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
Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process
We propose to extend the aggregation scheme of Saatyâs AHP, from the stan- dard weighted averaging to the more general Choquet integration. In our model, a measure of inconsistency between criteria is derived from the main pairwise comparison matrix and it is used to construct a non-additive capacity, whose associated Choquet integral reduces to the standard weighted mean in the con- sistency case. In the general inconsistency case, however, the new aggregation scheme based on Choquet integration tends to attenuate (resp. emphasize) the priority values of the criteria with higher (resp. lower) average inconsistency with the remaining criteria.Aggregation Functions, Multiple Criteria Analysis, AHP, Inconsintency, non-additive measures, Choquet integral, and Shapley values.
The state of the art development of AHP (1979-2017): A literature review with a social network analysis
Although many papers describe the evolution of the analytic hierarchy process (AHP), most adopt a subjective approach. This paper examines the pattern of development of the AHP research field using social network analysis and scientometrics, and identifies its intellectual structure. The objectives are: (i) to trace the pattern of development of AHP research; (ii) to identify the patterns of collaboration among authors; (iii) to identify the most important papers underpinning the development of AHP; and (iv) to discover recent areas of interest. We analyse two types of networks: social networks, that is, co-authorship networks, and cognitive mapping or the network of disciplines affected by AHP. Our analyses are based on 8441 papers published between 1979 and 2017, retrieved from the ISI Web of Science database. To provide a longitudinal perspective on the pattern of evolution of AHP, we analyse these two types of networks during the three periods 1979?1990, 1991?2001 and 2002?2017. We provide some basic statistics on AHP journals and researchers, review the main topics and applications of integrated AHPs and provide direction for future research by highlighting some open questions
The state of the art development of AHP (1979-2017): a literature review with a social network analysis
Although many papers describe the evolution of the analytic hierarchy process (AHP), most adopt a subjective approach. This paper examines the pattern of development of the AHP research field using social network analysis and scientometrics, and identifies its intellectual structure. The objectives are: (i) to trace the pattern of development of AHP research; (ii) to identify the patterns of collaboration among authors; (iii) to identify the most important papers underpinning the development of AHP; and (iv) to discover recent areas of interest. We analyse two types of networks: social networks, that is, co-authorship networks, and cognitive mapping or the network of disciplines affected by AHP. Our analyses are based on 8441 papers published between 1979 and 2017, retrieved from the ISI Web of Science database. To provide a longitudinal perspective on the pattern of evolution of AHP, we analyse these two types of networks during the three periods 1979â1990, 1991â2001 and 2002â2017. We provide some basic statistics on AHP journals and researchers, review the main topics and applications of integrated AHPs and provide direction for future research by highlighting some open questions
The Cycle Inconsistency Index in Pairwise Comparisons Matrices
AbstractThe method of pairwise comparisons is widely applied in the decision making process. The inconsistency of data may significantly affect the final result. Since the notion of consistency is based on triads or cycles, there is a great need for defining the measure of a triad or cycle inconsistency.In the paper a set of properties of a good cycle inconsistency index is proposed. Two construction methods of a cycle-based inconsistency index for a pairwise comparisons matrix are introduced. All is supported by the examples
Axiomatizations of inconsistency indices for triads
Pairwise comparison matrices often exhibit inconsistency, therefore many
indices have been suggested to measure their deviation from a consistent
matrix. A set of axioms has been proposed recently that is required to be
satisfied by any reasonable inconsistency index. This set seems to be not
exhaustive as illustrated by an example, hence it is expanded by adding two new
properties. All axioms are considered on the set of triads, pairwise comparison
matrices with three alternatives, which is the simplest case of inconsistency.
We choose the logically independent properties and prove that they
characterize, that is, uniquely determine the inconsistency ranking induced by
most inconsistency indices that coincide on this restricted domain. Since
triads play a prominent role in a number of inconsistency indices, our results
can also contribute to the measurement of inconsistency for pairwise comparison
matrices with more than three alternatives.Comment: 12 page
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