An Ordinal Consistency Indicator for Pairwise Comparison Matrix

The pairwise comparison (PC) matrix is often used to manifest human judgments, and it has been successfully applied in the analytic hierarchy process (AHP). As a PC matrix is formed by making paired reciprocal comparisons, symmetry is a striking characteristic of a PC matrix. It is this simple but p...

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Autor principal:	Ting Kuo
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	analytic hierarchy process multiple criteria decision-making ordinal consistency pairwise comparisons rank Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/73b76479386e41b7a6c23d4b9ce31218
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spelling	oai:doaj.org-article:73b76479386e41b7a6c23d4b9ce312182021-11-25T19:07:24ZAn Ordinal Consistency Indicator for Pairwise Comparison Matrix10.3390/sym131121832073-8994https://doaj.org/article/73b76479386e41b7a6c23d4b9ce312182021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2183https://doaj.org/toc/2073-8994The pairwise comparison (PC) matrix is often used to manifest human judgments, and it has been successfully applied in the analytic hierarchy process (AHP). As a PC matrix is formed by making paired reciprocal comparisons, symmetry is a striking characteristic of a PC matrix. It is this simple but powerful means of resolving multicriteria decision-making problems that is the basis of AHP; however, in practical applications, human judgments may be inconsistent. Although Saaty’s rule for the consistency test is commonly accepted, there is evidence that those so-called “acceptable” PC matrices may not be <i>ordinally</i> consistent, which is a necessary condition for a PC matrix to be accepted. We propose an <i>ordinal</i> consistency indicator called SDR (standard deviation of ranks), derive the upper bound of the SDR, suggest a threshold for a decision-maker to assess whether the ordinal consistency of a PC matrix is acceptable, and reveal a surprising fact that the degree of ordinal inconsistency of a small PC matrix may be more serious than a large one. We made a comparative analysis with some other indicators. Experimental results showed that the <i>ordinal</i> inconsistency measured by the SDR is invariant under heterogeneous judgment measurements with a varied spectrum of scales, and that the SDR is superior to the two compared indicators. Note that the SDR not only works for a <i>multiplicative</i> PC matrix but can also be used for <i>additive</i> and <i>fuzzy</i> PC matrices.Ting KuoMDPI AGarticleanalytic hierarchy processmultiple criteria decision-makingordinal consistencypairwise comparisonsrankMathematicsQA1-939ENSymmetry, Vol 13, Iss 2183, p 2183 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	analytic hierarchy process multiple criteria decision-making ordinal consistency pairwise comparisons rank Mathematics QA1-939
spellingShingle	analytic hierarchy process multiple criteria decision-making ordinal consistency pairwise comparisons rank Mathematics QA1-939 Ting Kuo An Ordinal Consistency Indicator for Pairwise Comparison Matrix
description	The pairwise comparison (PC) matrix is often used to manifest human judgments, and it has been successfully applied in the analytic hierarchy process (AHP). As a PC matrix is formed by making paired reciprocal comparisons, symmetry is a striking characteristic of a PC matrix. It is this simple but powerful means of resolving multicriteria decision-making problems that is the basis of AHP; however, in practical applications, human judgments may be inconsistent. Although Saaty’s rule for the consistency test is commonly accepted, there is evidence that those so-called “acceptable” PC matrices may not be <i>ordinally</i> consistent, which is a necessary condition for a PC matrix to be accepted. We propose an <i>ordinal</i> consistency indicator called SDR (standard deviation of ranks), derive the upper bound of the SDR, suggest a threshold for a decision-maker to assess whether the ordinal consistency of a PC matrix is acceptable, and reveal a surprising fact that the degree of ordinal inconsistency of a small PC matrix may be more serious than a large one. We made a comparative analysis with some other indicators. Experimental results showed that the <i>ordinal</i> inconsistency measured by the SDR is invariant under heterogeneous judgment measurements with a varied spectrum of scales, and that the SDR is superior to the two compared indicators. Note that the SDR not only works for a <i>multiplicative</i> PC matrix but can also be used for <i>additive</i> and <i>fuzzy</i> PC matrices.
format	article
author	Ting Kuo
author_facet	Ting Kuo
author_sort	Ting Kuo
title	An Ordinal Consistency Indicator for Pairwise Comparison Matrix
title_short	An Ordinal Consistency Indicator for Pairwise Comparison Matrix
title_full	An Ordinal Consistency Indicator for Pairwise Comparison Matrix
title_fullStr	An Ordinal Consistency Indicator for Pairwise Comparison Matrix
title_full_unstemmed	An Ordinal Consistency Indicator for Pairwise Comparison Matrix
title_sort	ordinal consistency indicator for pairwise comparison matrix
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/73b76479386e41b7a6c23d4b9ce31218
work_keys_str_mv	AT tingkuo anordinalconsistencyindicatorforpairwisecomparisonmatrix AT tingkuo ordinalconsistencyindicatorforpairwisecomparisonmatrix
_version_	1718410284777865216

An Ordinal Consistency Indicator for Pairwise Comparison Matrix

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