Two-dimensional uncertain linguistic generalized normalized weighted geometric Bonferroni mean and its application to multiple-attribute decision making

2-dimension uncertain linguistic variables (2DULVs) are a powerful tool to express the fuzzy or uncertain information, and the weighted Bonferroni mean can not only take the attribute importance into account but also capture the interrelationship between the attributes. However, the traditional Bonferroni mean can only deal with the crisp numbers. In this paper, Bonferroni mean was extended to process the 2DULVs. Firstly, we proposed the normalized weighted geometric Bonferroni mean (NWGBM) operator and the generalized normalized weighted geometric Bonferroni mean (GNWGBM) operator, which have the characteristics of reducibility and also consider the interrelationships between two attributes. Then we introduced the computation rules, characteristics, the expected value and comparison method of the 2DULVs. Further, we developed the 2-dimension uncertain linguistic normalized weighted geometric Bonferroni mean (2DULNWGBM) and the 2-dimension uncertain linguistic generalized normalized weighted geometric Bonferroni mean (2DULGNWGBM), and explored some properties and discussed some special cases of them. Finally, we developed a new decision making method based on these operators, and an example is given to compare with the existing methods.