On the Properties of the Priority Deriving Procedure in the Pairwise Comparisons Method

The pairwise comparisons method can be used when the relative order of preferences among different concepts (alternatives) needs to be determined. There are several popular imple- mentations of this method, including the Eigenvector Method, the Least Squares Method, the Chi Squares Method and others. Each of the above methods comes with one or more inconsistency in- dices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the optimal decision. This article explores the relationship between inconsistency of input and error of output. An error describes to what extent the obtained results correspond to the single expert's assessments. On the basis of the inconsistency and the error, two properties of the weight deriving procedure are formulated. These properties are proven for eigenvector method and Koczkodaj's inconsistency index. Several estimates using Koczkodaj's inconsistency index for a principal eigenvalue, Saaty's inconsistency index and the Condition of Order Preservation are also provided.

[1]  Alessio Ishizaka,et al.  Review of the main developments in the analytic hierarchy process , 2011, Expert Syst. Appl..

[2]  Carlos A. Bana e Costa,et al.  A critical analysis of the eigenvalue method used to derive priorities in AHP , 2008, Eur. J. Oper. Res..

[3]  Waldemar W. Koczkodaj,et al.  On distance-based inconsistency reduction algorithms for pairwise comparisons , 2010, Log. J. IGPL.

[4]  Gustav Theodor Fechner,et al.  Elements of psychophysics , 1966 .

[5]  O. Perron.  Zur Theorie der Matrices , 1907 .

[6]  J. Fichtner.  On deriving priority vectors from matrices of pairwise comparisons , 1986 .

[7]  Waldemar W. Koczkodaj,et al.  On some convexity properties of the Least Squares Method for pairwise comparisons matrices without the reciprocity condition , 2012, J. Glob. Optim..

[8]  Waldemar W. Koczkodaj,et al.  On Axiomatization of Inconsistency Indicators for Pairwise Comparisons , 2013, Fundam. Informaticae.

[9]  Matthew J. Liberatore,et al.  The analytic hierarchy process in medical and health care decision making: A literature review , 2008, Eur. J. Oper. Res..

[10]  Sushil Kumar,et al.  Analytic hierarchy process: An overview of applications , 2006, Eur. J. Oper. Res..

[11]  Alessio Ishizaka,et al.  Analytic Hierarchy Process and Expert Choice: Benefits and limitations , 2009, OR Insight.

[12]  Kevin Kam Fung Yuen,et al.  Fuzzy Cognitive Network Process: Comparisons With Fuzzy Analytic Hierarchy Process in New Product Development Strategy , 2014, IEEE Transactions on Fuzzy Systems.

[13]  William Ho,et al.  Integrated analytic hierarchy process and its applications - A literature review , 2008, Eur. J. Oper. Res..

[14]  Salvatore Greco,et al.  Dominance-Based Rough Set Approach on Pairwise Comparison Tables to Decision Involving Multiple Decision Makers , 2011, RSKT.

[15]  Nachiappan Subramanian,et al.  A review of applications of Analytic Hierarchy Process in operations management , 2012 .

[16]  János Fülöp,et al.  A method for approximating pairwise comparison matrices by consistent matrices , 2008, J. Glob. Optim..

[17]  Thomas L. Saaty,et al.  The Analytic Hierarchy and Analytic Network Processes for the Measurement of Intangible Criteria and for Decision-Making , 2016 .

[18]  Konrad Kulakowski,et al.  A heuristic rating estimation algorithm for the pairwise comparisons method , 2015, Central Eur. J. Oper. Res..

[19]  W. W. Koczkodaj.  A new definition of consistency of pairwise comparisons , 1993 .

[20]  Michele Fedrizzi,et al.  Incomplete pairwise comparison and consistency optimization , 2007, Eur. J. Oper. Res..

[21]  Ludmil Mikhailov,et al.  Deriving priorities from fuzzy pairwise comparison judgements , 2003, Fuzzy Sets Syst..

[22]  Detlof von Winterfeldt,et al.  Anniversary Article: Decision Analysis in Management Science , 2004, Manag. Sci..

[23]  T. L. Saaty.  A Scaling Method for Priorities in Hierarchical Structures , 1977 .

[24]  Konrad Kulakowski.  Notes on order preservation and consistency in AHP , 2015, Eur. J. Oper. Res..

[25]  Konrad Kulakowski,et al.  Heuristic Rating Estimation Approach to The Pairwise Comparisons Method , 2013, Fundam. Informaticae.

[26]  F. Roberts.  Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences: Measurement Theory , 1984 .

[27]  Konrad Kulakowski,et al.  Notes on discrepancy in the pairwise comparisons method , 2013, ArXiv.

[28]  Ryszard Janicki,et al.  On a pairwise comparison-based consistent non-numerical ranking , 2012, Log. J. IGPL.

[29]  Thomas L. Saaty,et al.  On the measurement of intengibles. a principal Eigenvector approach to relative measurement derived from paired comparisons , 2013 .

[30]  Piotr Faliszewski,et al.  Llull and Copeland Voting Computationally Resist Bribery and Constructive Control , 2009, J. Artif. Intell. Res..

[31]  Konrad Kulakowski,et al.  Heuristic rating estimation: geometric approach , 2014, J. Glob. Optim..

[32]  Tamás Rapcsák,et al.  On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices , 2008, J. Glob. Optim..

[33]  L. Thurstone.  A law of comparative judgment. , 1994 .

[34]  Lajos Rónyai,et al.  On optimal completion of incomplete pairwise comparison matrices , 2010, Math. Comput. Model..

[35]  Josep M. Colomer,et al.  Ramon Llull: from ‘Ars electionis’ to social choice theory , 2011, Social Choice and Welfare.