Intuitionistic fuzzy normalized weighted geometric Bonferroni means with reducibility and boundedness and application in decision making1

Abstract

Bonferroni mean (BM) is an important aggregation operator in decision making. The desirable characteristic of the BM is that it can capture the interrelationship between the aggregation arguments or the individual attributes. The optimized weighted geometric Bonferroni mean (OWGBM) and the generalized optimized weighted geometric Bonferroni mean (GOWGBM) proposed by Jin et al in 2016 are the extensions of the BM. However, the OWGBM and the GOWGBM have neither the reducibility nor the boundedness, which will lead to the illogical and unreasonable aggregation results and might make the wrong decision. To overcome these existing drawbacks, based on the normalized weighted Bonferroni mean (NWBM) and the GOWGBM, we propose the normalized weighted geometric Bonferroni mean (NWGBM) and the generalized normalized weighted geometric Bonferroni mean (GNWGBM), which can not only capture the interrelationship between the aggregation arguments, but also have the reducibility and the boundedness. Further, we extend the NWGBM and the GNWGBM to the intuitionistic fuzzy decision environment respectively, and develop the intuitionistic fuzzy normalized weighted geometric Bonferroni mean (IFNWGBM) and the generalized intuitionistic fuzzy normalized weighted geometric Bonferroni mean (GIFNWGBM). Subsequently, we prove some properties of these operators. Moreover, we present a new intuitionistic fuzzy decision method based on the IFNWGBM and the GIFNWGBM. Two application examples and comparisons with other existing methods are used to verify the validity of the proposed method.