1 Cross entropy measures of bipolar and interval 2 bipolar neutrosophic sets and their application for 3 multi - attribute decision making 4 5

Bipolar neutrosophic set is an important extension of bipolar fuzzy set. This set is a 15 hybridization of bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic 16 set consists of three independent positive membership functions and three independent negative 17 membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic 18 sets and prove its properties. We also define cross entropy measures of interval bipolar 19 neutrosophic sets and prove its properties. Thereafter, we develop two novel multi-attribute 20 decision making methods based on the proposed cross entropy measures. In the decision making 21 framework, we calculate the weighted cross entropy measures between each alternative and the 22 ideal alternative to rank the alternatives and choose the best one. We solve two illustrative 23 examples of multi-attribute decision making problems and compare the obtained result with the 24 results of other existing methods to show the applicability and effectiveness of the developed 25 method. In the end, the main conclusion and future scope of research are summarized. 26