Construction of Choquet integrals through unimodal weighting vectors

Semiuninorm‐based ordered weighted averaging (SUOWA) operators are a specific case of Choquet integrals that allow us to generalize simultaneously weighted means and ordered weighting averaging (OWA) operators. Although SUOWA operators possess some very interesting properties, their main weakness is that, sometimes, the game used in their construction is not monotonic and it is necessary to calculate its monotonic cover. In this paper, we introduce a new family of weighting vectors, called unimodal weighting vectors, which embrace some of the most outstanding weighting vectors used in the framework of OWA operators, and we show that when using these weighting vectors and a specific semiuninorm we directly get normalized capacities. Moreover, we also show that these operators satisfy some properties which are very useful in practice.

[1]  Joan Torrens,et al.  Balanced Discrete Fuzzy Measures , 2000, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[2]  Bonifacio Llamazares,et al.  Constructing Choquet integral-based operators that generalize weighted means and OWA operators , 2015, Inf. Fusion.

[3]  Bonifacio Llamazares,et al.  Closed-form expressions for some indices of SUOWA operators , 2018, Inf. Fusion.

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

[5]  Vicenç Torra,et al.  The weighted OWA operator , 1997, Int. J. Intell. Syst..

[6]  Bonifacio Llamazares,et al.  SUOWA operators: Constructing semi-uninorms and analyzing specific cases , 2016, Fuzzy Sets Syst..

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

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

[9]  R. Stanley.  Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a , 1989 .

[10]  Esther de Ves,et al.  Applying logistic regression to relevance feedback in image retrieval systems , 2007, Pattern Recognit..

[11]  Humberto Bustince,et al.  A Practical Guide to Averaging Functions , 2015, Studies in Fuzziness and Soft Computing.

[12]  G. Choquet.  Theory of capacities , 1954 .

[13]  Huawen Liu,et al.  Semi-uninorms and implications on a complete lattice , 2012, Fuzzy Sets Syst..

[14]  Bonifacio Llamazares,et al.  An Analysis of Some Functions That Generalizes Weighted Means and OWA Operators , 2013, Int. J. Intell. Syst..

[15]  Bonifacio Llamazares,et al.  A Behavioral Analysis of WOWA and SUOWA Operators , 2016, Int. J. Intell. Syst..

[16]  Bonifacio Llamazares,et al.  SUOWA operators: An analysis of their conjunctive/disjunctive character , 2019, Fuzzy Sets Syst..

[17]  R. Yager,et al.  PARAMETERIZED AND-UKE AND OR-LIKE OWA OPERATORS , 1994 .

[18]  Xinwang Liu,et al.  A Review of the OWA Determination Methods: Classification and Some Extensions , 2011, Recent Developments in the Ordered Weighted Averaging Operators.

[19]  M. Maschler,et al.  The Structure of the Kernel of a Cooperative Game , 1967 .

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

[21]  Michel Grabisch,et al.  On equivalence classes of fuzzy connectives-the case of fuzzy integrals , 1995, IEEE Trans. Fuzzy Syst..

[22]  V. Torra.  On some relationships between the WOWA operator and the Choquet integral , 2004 .

[23]  Bonifacio Llamazares Rodríguez.  SUOWA operators: Constructing semi-uninorms And analyzing specific cases , 2016 .

[24]  Ronald R. Yager,et al.  Uninorm aggregation operators , 1996, Fuzzy Sets Syst..

[25]  Ren Zhang,et al.  A New Ordered Weighted Averaging Operator to Obtain the Associated Weights Based on the Principle of Least Mean Square Errors , 2017, Int. J. Intell. Syst..

[26]  E. Lieb.  Concavity properties and a generating function for stirling numbers , 1968 .

[27]  Nasrin Rahmati,et al.  Tokyo Institute of Technology , 2001, PACIS.

[28]  Ronald R. Yager,et al.  Centered OWA Operators , 2007, Soft Comput..

[29]  Zeshui Xu,et al.  An overview of methods for determining OWA weights , 2005, Int. J. Intell. Syst..

[30]  Jian Wang,et al.  S‐H OWA Operators with Moment Measure , 2017, Int. J. Intell. Syst..

[31]  Solomon Tesfamariam,et al.  Probability density functions based weights for ordered weighted averaging (OWA) operators: An example of water quality indices , 2007, Eur. J. Oper. Res..

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

[33]  Gleb Beliakov,et al.  A method of introducing weights into OWA operators and other symmetric functions , 2017 .

[34]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[35]  Janusz Kacprzyk,et al.  Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice , 2011, Studies in Fuzziness and Soft Computing.

[36]  Xinwang Liu,et al.  Models to determine parameterized ordered weighted averaging operators using optimization criteria , 2012, Inf. Sci..

[37]  M. Grabisch.  Fuzzy integral in multicriteria decision making , 1995 .

[38]  B. Llamazares.  A Study of SUOWA Operators in Two Dimensions , 2015 .

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

[40]  Byeong Seok Ahn,et al.  On the properties of OWA operator weights functions with constant level of orness , 2006, IEEE Transactions on Fuzzy Systems.

[41]  Bernard De Baets,et al.  Aggregation Operators Defined by k-Order Additive/Maxitive Fuzzy Measures , 1998, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[42]  L. Shapley,et al.  The kernel and bargaining set for convex games , 1971 .