Abstract
Motivated by bipolar fuzzy ( $$B_{p}$$ F) sets ( $$B_{p}$$ FSs) and picture fuzzy sets ( $$P_{c}$$ FSs), we introduced the aggregation operators for the novel extension of $$P_{c}$$ FSs named as bipolar picture fuzzy sets (B $$P_{c}$$ FSs). Firstly, various arithmetic rules, scores and accuracy functions of the B $$P_{c}$$ FSs are presented. Secondly, some aggregation operators of B $$P_{c}$$ FSs are constructed to accumulate the bipolar picture fuzzy (B $$P_{c}$$ F) data. Thirdly, the features of these aggregation operators are penned and then based on these aggregation operators a multiple criteria decision-making (MCDM) approach is developed to resolve the vague data. In the end, an illustrated example of the investment of money is put forward to show the authenticity and efficacy of the suggested approach. Moreover, B $$P_{c}$$ F-TOPSIS, B $$P_{c}$$ F-VIKOR, and sensitivity analysis (SA) have been used to provide the strength and practicality of the proposed MCDM model.
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