基於P測度之改進模糊測度及其模糊積分

An Improved Fuzzy Measure Based on P Measure and Its Fuzzy Integrals

劉湘川(Hsiang-Chuan Liu)

《測驗統計年刊》 14期_上 (2006/06) Pp. 1-14

https://doi.org/10.6773/JRMS.200606.0001

摘要

在模糊測度中，常用之Sugeno之λ測度雖靈敏，但不恆存在非可加性測度，Zadeh之P測度，恆存在非可加性測度，且計算簡易，但測度較不靈敏，且只能處理特定之次可加性測度。本文提出基於P測度之改進模糊測度，則能兼顧前二者之優點，藉以求取Choquet積分值或Segeno積分值，可改進與整合計分有關之決策方法之分析功效。

關鍵字

λ測度 ； P測度 ； m測度 ； Choquet積分 ； Sugeno積分

並列摘要

Sugeno's λ-measure is the most often used fuzzy measure to aggregate criteria in decision making problems with the assumption that there are interactions among criteria. It is a sensitive fuzzy measure. But its solution of nonadditive measure does not always exist. Zadeh's P-measure always has the easy computing solution of nonadditive measure. But its solution of non-additive measure is not sensitive enough as the Sugeno's λ-measure.
In this study, we propose an improved fuzzy measure based on P-measure and its solution of nonadditive measure is not only always existed but also easy computed and sensitive.
Three fuzzy measures, including λ-measure, P-measure, and our proposed m-measure are used to calculate two different kinds of fuzzy integral, Choquet integral and Sugeno integral for student’s performance based on a Basic Competence Test by 10 simple examples. The results show that our proposed m-measure is the best among the three fuzzy measures to aggregate criteria in decision making problems when the interactions among criteria exist.

並列關鍵字

λ-measure ； P-measure ； m-measure ； Choquet integral ； Sugeno integral

參考文獻

Choquet, G.(1953).Theory of capacities.Annales de l’Institut Fourier.5,131-295.

Dempster,A.P(1967).Upper and lower probabilities induced by multi-valued mapping.Annals of Mathematical Statistics.38,325-339.

Shafer,G.(1976).A Mathematical Theory of Evidence.Princeton, New Jersey:Princeton University Press.

Sugeno,M.(1974).Theory of fuzzy integrals and its applications.Tokyo Institute of Thchnology.

Wang, Z.,Klir, G. J.(1992).Fuzzy measure theory.New York:Plenum Press.