Hesitant fuzzy geometric Bonferroni means

The Bonferroni mean (BM) can capture the interrelationships among arguments, which plays a crucial role in multi-criteria decision making problems. In this paper, we explore the geometric Bonferroni mean (GBM) considering both the BM and the geometric mean (GM) under hesitant fuzzy environment. We further define the hesitant fuzzy geometric Bonferroni mean (HFGBM) and the hesitant fuzzy Choquet geometric Bonferroni mean (HFCGBM). Then we give the definition of hesitant fuzzy geometric Bonferroni element (HFGBE), which is considered as the basic calculational unit in the HFGBM and reflects the conjunction between two aggregated arguments. The properties and special cases of the HFGBM are studied in detail based on the discussion of the HFGBE. In addition, the weighted hesitant fuzzy geometric Bonferroni mean (WHFGBM) and the weighted hesitant fuzzy Choquet geometric Bonferroni mean (WHFCGBM) are proposed considering the importance of each argument and the correlations among them. In the end, we apply the proposed aggregation operators to multi-criteria decision making, and give some examples to illustrate our results.