ABSTRACT

Starting with [Goguen, J.A. 1967. Journal of Mathematical Analysis and Applications], several generalizations of the original definition of a fuzzy set have been proposed. In one popular case, one considers as truth values the points in the lower left triangle of the unit square, where their first coordinate is interpreted as “degree of membership”, and their second coordinate as “degree of non-membership”. Generalizing ideas in [Zadeh, L.A. 1968. Journal of Mathematical Analysis and Applications], [Grzegorzewski, P. and E. Mrówka. 2002. In: Soft Methods in Probability, Statistics and Data Analysis, Heidelberg: Physica], [Grzegorzewski, P. 2013. Information Sciences] and [Klement, E.P. and R. Mesiar. 2015. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems], the concept of expected values (based on capacities) of fuzzy events in this general sense is introduced and investigated. Expected values satisfying additional properties such as positive-linearity, comonotone additivity and comonotone maxitivity are studied, as well as an extension to real-valued expected values.

摘要

从[Goguen，JA 1967.数学分析和应用学报]开始，已经提出了模糊集原始定义的几种概括。在一种流行的情况下，人们将单位正方形左下三角形中的点视为真值，其中将其第一坐标解释为“隶属度”，将其第二坐标解释为“非隶属度”。在[Zadeh，LA 1968.数学分析和应用学报]，[Grzegorzewski，P.和E.Mrówka。]中概论化思想。2002。在：《概率论，统计和数据分析中的软方法》，海德堡：物理学，[Grzegorzewski，第2013年。信息科学]和[Klement，EP和R.Mesiar。2015年。《不确定性，模糊性和基于知识的系统国际期刊》，引入并研究了模糊事件期望值（基于容量）的概念。研究了满足其他属性（例如正线性，共调单调加性和共调单调最大值）的期望值，以及对实际值期望值的扩展。