On the orness of Bonferroni mean and its variants

This article addresses orness measures to reflect the or‐like degree of the Bonferroni mean (BM) and its variants. Some properties of these operators associated with their orness measures are portrayed analytically. However, the general orness measure involves the multiple integrals with the integral fold number being the number of the aggregated elements and as a result, the computation becomes complicated when the number of the aggregated elements is large. Furthermore, the analytical formula of the orness measure often cannot be obtained. For this reason, this study concentrates on Monte Carlo simulation to validate the result. We estimate the two parameters of the BM for a predefined orness value and a fixed length of the input vector. Besides the theoretical study of orness measure related to BM and its variants, the article also explores the simulation‐based results. To support this, we provide four numerical examples.

[1]  Humberto Bustince,et al.  Quantitative orness for lattice OWA operators , 2016, Inf. Fusion.

[2]  Jean-Luc Marichal,et al.  Aggregation operators for multicriteria decision aid , 1998 .

[3]  Carlo Bonferroni.  Sulle medie multiple di potenze , 1950 .

[4]  Oihana Aristondo,et al.  The orness value for rank-dependent welfare functions and rank-dependent poverty measures , 2017, Fuzzy Sets Syst..

[5]  Wenyi Zeng,et al.  Monotonic argument‐dependent OWA operators , 2018, Int. J. Intell. Syst..

[6]  Radko Mesiar,et al.  On Some Properties and Comparative Analysis for Different OWA Monoids , 2017, Int. J. Intell. Syst..

[7]  Ramiz M. Aliguliyev,et al.  Aggregating Edge Weights in Social Networks on the Web Extracted from Multiple Sources with Different Importance Degrees , 2012 .

[8]  Huayou Chen,et al.  Hesitant Fuzzy Power Bonferroni Means and Their Application to Multiple Attribute Decision Making , 2015, IEEE Transactions on Fuzzy Systems.

[9]  Dimitar Filev,et al.  On the issue of obtaining OWA operator weights , 1998, Fuzzy Sets Syst..

[10]  Dirk P. Kroese,et al.  Handbook of Monte Carlo Methods , 2011 .

[11]  José M. Merigó,et al.  Weighted‐selective aggregated majority‐OWA operator and its application in linguistic group decision making , 2018, Int. J. Intell. Syst..

[12]  R. Rubinstein,et al.  Variance reduction by the use of common and antithetic random variables , 1985 .

[13]  Christer Carlsson,et al.  Decision making with a fuzzy ontology , 2012, Soft Comput..

[14]  Ronald R. Yager,et al.  Quantifier guided aggregation using OWA operators , 1996, Int. J. Intell. Syst..

[15]  H. Chernoff,et al.  Elementary Decision Theory , 1959 .

[16]  Debashree Guha,et al.  Article in Press G Model Applied Soft Computing Partitioned Bonferroni Mean Based on Linguistic 2-tuple for Dealing with Multi-attribute Group Decision Making , 2022 .

[17]  János C. Fodor,et al.  Characterization of the ordered weighted averaging operators , 1995, IEEE Trans. Fuzzy Syst..

[18]  Radko Mesiar,et al.  Aggregation functions: Means , 2011, Inf. Sci..

[19]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decision-making , 1988 .

[20]  Francisco Chiclana,et al.  Social Network Decision Making with Linguistic Trustworthiness–Based Induced OWA Operators , 2014, Int. J. Intell. Syst..

[21]  H. Kahn,et al.  Methods of Reducing Sample Size in Monte Carlo Computations , 1953, Oper. Res..

[22]  M. O'Hagan,et al.  Aggregating Template Or Rule Antecedents In Real-time Expert Systems With Fuzzy Set Logic , 1988, Twenty-Second Asilomar Conference on Signals, Systems and Computers.

[23]  Tim Wilkin,et al.  Weakly Monotonic Averaging Functions , 2015, Int. J. Intell. Syst..

[24]  Bernadette Bouchon-Meunier,et al.  IPMU '92 : advanced methods in artificial intelligence : 4th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Palma de Mallorca, Spain, July 6-10, 1992 : proceedings , 1993 .

[25]  Holloway Ave,et al.  Properties and Modeling of Partial Conjunction / Disjunction , 2004 .

[26]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .

[27]  LeSheng Jin,et al.  Fuzzy orness measure and new orness axioms , 2015, Kybernetika.

[28]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decisionmaking , 1988, IEEE Trans. Syst. Man Cybern..

[29]  Xinwang Liu,et al.  An Orness Measure for Quasi-Arithmetic Means , 2006, IEEE Transactions on Fuzzy Systems.

[30]  Zhou Rongxi,et al.  A method for obtaining the maximum entropy OWA operator weights with uncertain orness measure , 2008, 2008 Chinese Control and Decision Conference.

[31]  Jozo J. Dujmovic.  Andness and orness as a mean of overall importance , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[32]  Ronald R. Yager,et al.  An extension of the naive Bayesian classifier , 2006, Inf. Sci..

[33]  Ben Niu,et al.  Aggregation of Heterogeneously Related Information with Extended Geometric Bonferroni Mean and Its Application in Group Decision Making , 2018, Int. J. Intell. Syst..

[34]  Ronald R. Yager.  New modes of OWA information fusion , 1998, Int. J. Intell. Syst..

[35]  Henrik Legind Larsen,et al.  Generalized conjunction/disjunction , 2007, Int. J. Approx. Reason..

[36]  Henrik Legind Larsen,et al.  Efficient Andness-Directed Importance Weighted Averaging Operators , 2003, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[37]  Radko Mesiar,et al.  A Model Based on Linguistic 2-Tuples for Dealing With Heterogeneous Relationship Among Attributes in Multi-expert Decision Making , 2015, IEEE Transactions on Fuzzy Systems.

[38]  Ronald R. Yager,et al.  Fuzzy modeling for intelligent decision making under uncertainty , 2000, IEEE Trans. Syst. Man Cybern. Part B.

[39]  Ronald R. Yager,et al.  OWA aggregation over a continuous interval argument with applications to decision making , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[40]  Stephan Reiff-Marganiec,et al.  A Method for Automated Web Service Selection , 2008, 2008 IEEE Congress on Services - Part I.

[41]  Vicenç Torra,et al.  Empirical analysis to determine Weighted OWA orness , 2001 .

[42]  Jesús Manuel Fernández Salido,et al.  Extending Yager's orness concept for the OWA aggregators to other mean operators , 2003, Fuzzy Sets Syst..

[43]  Xinwang Liu,et al.  The orness measures for two compound quasi-arithmetic mean aggregation operators , 2010, Int. J. Approx. Reason..

[44]  Ronald R. Yager,et al.  Generalized Bonferroni mean operators in multi-criteria aggregation , 2010, Fuzzy Sets Syst..

[45]  Zeshui Xu,et al.  Geometric Bonferroni means with their application in multi-criteria decision making , 2013, Knowl. Based Syst..

[46]  Amar Oukil,et al.  A systematic approach for ranking football players within an integrated DEA‐OWA framework , 2017 .

[47]  Robert Fullér,et al.  On Obtaining Minimal Variability Owa Operator Weights , 2002, Fuzzy Sets Syst..

[48]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[49]  Gleb Beliakov,et al.  Aggregation Functions: A Guide for Practitioners , 2007, Studies in Fuzziness and Soft Computing.

[50]  Xinwang Liu,et al.  A general model of parameterized OWA aggregation with given orness level , 2008, Int. J. Approx. Reason..

[51]  R. Yager.  Families of OWA operators , 1993 .

[52]  Inmaculada Lizasoain,et al.  Orness For Idempotent Aggregation Functions , 2017, Axioms.

[53]  Yi Yang,et al.  On Generalized Extended Bonferroni Means for Decision Making , 2016, IEEE Transactions on Fuzzy Systems.

[54]  Thomas Gerstner,et al.  Dimension–Adaptive Tensor–Product Quadrature , 2003, Computing.

[55]  Nikhil R. Pal,et al.  Orness Measure of OWA Operators: A New Approach , 2014, IEEE Transactions on Fuzzy Systems.

[56]  Ronald R. Yager,et al.  Extending multicriteria decision making by mixing t‐norms and OWA operators , 2005, Int. J. Intell. Syst..

[57]  Thierry Arnould,et al.  From "And" to "Or" , 1992, IPMU.