The ordinal consistency of an incomplete reciprocal preference relation

The ordinal consistency is the usual weak transitivity condition that a logical and consistent person should use if he/she does not want to express inconsistent opinions, and therefore becomes the minimum requirement condition that a consistent reciprocal preference relation should verify. In this paper, we define and study the ordinal consistency of an incomplete reciprocal preference relation. We then develop an algorithm to judge whether an incomplete reciprocal preference relation is ordinally consistent. This proposed algorithm can also find all cycles of length 3 to n in the incomplete digraph of the incomplete reciprocal preference relation. Based on this proposed algorithm and two rules, we develop another algorithm to repair an inconsistent incomplete reciprocal preference relation and to convert it to one with ordinal consistency. Our algorithm eliminates the cycles of length 3 to n in the digraph of an incomplete reciprocal preference relation most effectively. Our proposed method can preserve the initial preference information as much as possible. Furthermore, the proposed method can be used for an incomplete reciprocal preference relation with strict comparison and non-strict comparison information. Finally, the effectiveness and validity of the proposed method are illustrated with examples.

[1]  Zbigniew Switalski,et al.  General transitivity conditions for fuzzy reciprocal preference matrices , 2003, Fuzzy Sets Syst..

[2]  J. Kacprzyk.  Group decision making with a fuzzy linguistic majority , 1986 .

[3]  Zeshui Xu,et al.  Incomplete linguistic preference relations and their fusion , 2006, Inf. Fusion.

[4]  Soung Hie Kim,et al.  Group decision making procedure considering preference strength under incomplete information , 1997, Comput. Oper. Res..

[5]  S. Orlovsky.  Decision-making with a fuzzy preference relation , 1978 .

[6]  Francisco Herrera,et al.  A Consensus Model for Group Decision Making With Incomplete Fuzzy Preference Relations , 2007, IEEE Transactions on Fuzzy Systems.

[7]  T. Tanino.  Fuzzy preference orderings in group decision making , 1984 .

[8]  Efe A. Ok.  Utility Representation of an Incomplete Preference Relation , 2002, J. Econ. Theory.

[9]  Bernard De Baets,et al.  On the cycle-transitive comparison of artificially coupled random variables , 2008, Int. J. Approx. Reason..

[10]  J. Kacprzyk.  Group decision-making with a fuzzy majority via linguistic quantifiers. part II: , 1985 .

[11]  Peter P. Wakker,et al.  On solving intransitivities in repeated pairwise choices , 1995 .

[12]  Francisco Herrera,et al.  A Learning Procedure to Estimate Missing Values in Fuzzy Preference Relations Based on Additive Consistency , 2004, MDAI.

[13]  B. De Baets,et al.  Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity , 2005, Fuzzy Sets Syst..

[14]  Miroslaw Kwiesielewicz,et al.  Inconsistent and contradictory judgements in pairwise comparison method in the AHP , 2004, Comput. Oper. Res..

[15]  S. Orlovsky.  Decision-making with a fuzzy preference relation , 1978 .

[16]  Francisco Herrera,et al.  Some issues on consistency of fuzzy preference relations , 2004, Eur. J. Oper. Res..

[17]  Zeshui Xu,et al.  Algorithms for improving consistency or consensus of reciprocal [0, 1]-valued preference relations , 2013, Fuzzy Sets Syst..

[18]  Francisco Herrera,et al.  Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[19]  Yejun Xu,et al.  Eigenvector method, consistency test and inconsistency repairing for an incomplete fuzzy preference relation , 2013 .

[20]  Soung Hie Kim,et al.  An interactive procedure for multiple attribute group decision making with incomplete information: Range-based approach , 1999, Eur. J. Oper. Res..

[21]  T. Saaty,et al.  The Analytic Hierarchy Process , 1985 .

[22]  Z. S. Xu,et al.  Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation , 2004, Int. J. Approx. Reason..

[23]  Zbigniew Switalski,et al.  Transitivity of fuzzy preference relations - an empirical study , 2001, Fuzzy Sets Syst..

[24]  Francisco Herrera,et al.  Individual and Social Strategies to Deal with Ignorance Situations in Multi-Person Decision Making , 2009, Int. J. Inf. Technol. Decis. Mak..

[25]  Francisco Herrera,et al.  Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity , 2009, IEEE Transactions on Fuzzy Systems.

[26]  Bernard De Baets,et al.  Cyclic Evaluation of Transitivity of Reciprocal Relations , 2006, Soc. Choice Welf..

[27]  Kazutomo Nishizawa,et al.  A CONSISTENCY IMPROVING METHOD IN BINARY AHP , 1995 .

[28]  Soung Hie Kim,et al.  Interactive group decision making procedure under incomplete information , 1999, Eur. J. Oper. Res..

[29]  Yejun Xu,et al.  Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations , 2013 .

[30]  F. Herrera,et al.  A consistency-based procedure to estimate missing pairwise preference values , 2008 .

[31]  Jian Ma,et al.  A method for repairing the inconsistency of fuzzy preference relations , 2006, Fuzzy Sets Syst..

[32]  Zeshui Xu,et al.  Group decision-making procedure based on incomplete reciprocal relations , 2008, Soft Comput..

[33]  Francisco Herrera,et al.  A Note on the Estimation of Missing Pairwise Preference Values: a Uninorm Consistency Based Method , 2008, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[34]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[35]  Zeshui Xu,et al.  A Procedure for Decision Making Based on Incomplete Fuzzy Preference Relation , 2005, Fuzzy Optim. Decis. Mak..

[36]  José Luis García-Lapresta,et al.  Consistent models of transitivity for reciprocal preferences on a finite ordinal scale , 2008, Inf. Sci..

[37]  John A. Keane,et al.  A heuristic method to rectify intransitive judgments in pairwise comparison matrices , 2012, Eur. J. Oper. Res..

[38]  Yejun Xu,et al.  The ordinal consistency of a fuzzy preference relation , 2013, Inf. Sci..

[39]  Alexis Tsoukiàs,et al.  Preference Modelling , 2004, Preferences.

[40]  Hsuan-Shih Lee,et al.  Estimating Missing Values in Incomplete Additive Fuzzy Preference Relations , 2007, KES.

[41]  Francisco Herrera,et al.  Group decision making with incomplete fuzzy linguistic preference relations , 2009, Int. J. Intell. Syst..

[42]  Yejun Xu,et al.  Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations , 2012, Fuzzy Sets Syst..

[43]  Francisco Herrera,et al.  A web based consensus support system for group decision making problems and incomplete preferences , 2010, Inf. Sci..

[44]  Zaiwu Gong,et al.  Least-square method to priority of the fuzzy preference relations with incomplete information , 2008, Int. J. Approx. Reason..

[45]  Zeshui Xu,et al.  Algorithms for estimating missing elements of incomplete intuitionistic preference relations , 2011, Int. J. Intell. Syst..

[46]  Enrique Herrera-Viedma,et al.  A Selection Process Based on Additive Consistency to Deal with Incomplete Fuzzy Linguistic Information , 2010, J. Univers. Comput. Sci..

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

[48]  Bernard De Baets,et al.  Closing reciprocal relations w.r.t. stochastic transitivity , 2014, Fuzzy Sets Syst..