A theoretical approach to Liu's generalized Lambda-fuzzy measure

摘要

λ-measure, P-measure, additive measure and B-measure are four well known fuzzy measure, all of them are univalent fuzzy measure with only one formulaic fuzzy measure solution. In this paper, first of all, it is proved that both λ-measure and additive measure are larger than P-measure and less than B-measure, in other words, both λ-measure and additive measure are never equal to the smallest fuzzy measure, P-measure and the largest fuzzy measure, B-measure. Furthermore, a novel multivalent fuzzy measure with infinitely many fuzzy measure solutions based on λ-measure, called a generalized λ-measure, is proposed. It is proved that the measure function of this new measure is continuous and increasing on λ, each different λ. value corresponding different fuzzy measure, from the smallest fuzzy measure, P-measure to the largest fuzzy measure, B-measure, and containing λ-measure itself, in other words, all of above four univalent fuzzy measures are special cases of this new fuzzy measure. Moreover, this new generalized λ-measure is also satisfying the monotone condition rather than the old generalized λ-measure proposed by L.-Z. Yang, M.-H. Ha, X.-J. Wang and Z.-R. Zhang. However, the old one is just the λ-measure without the monotone condition. It is a generalized measure but no more a fuzzy measure. Hence, this new and true generalized fuzzy measure is called Liu's generalized λ-measure. Some properties about this new measure are also discussed.