In this article, we applied the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to multi-criteria group decision making (MCGDM) with bipolar fuzzy rough numbers (BFRN). We present the dominant concept to develop the model of bipolar fuzzy rough sets (BFRS) with a new score, accuracy functions, and essential operations. Based on the concept of BFRS, we proposed the notion of BFR averaging aggregation operators, such as bipolar fuzzy rough weighted averaging (BFRWA), bipolar fuzzy rough ordered weighted averaging (BFROWA), and bipolar fuzzy rough hybrid averaging (BFRHA) aggregation operators. Thereafter, the rudimentary properties of the mentioned aggregation operators (AOs) were given in detail. Further, based on the concept of BFRS, we developed the concept of bipolar fuzzy rough geometric (BFRG) aggregation operators, such as bipolar fuzzy rough weighted geometric (BFRWG), bipolar fuzzy rough ordered weighted geometric (BFROWG), and bipolar fuzzy rough hybrid geometric (BFRHG) AO. Thereafter, prominent properties of the geometric AO were given in detail. Moreover, based on the developed model, we present a stepwise algorithm for applying the TOPSIS approach. The proposed AOs were combined using the concept accumulated geometric operator (AGO) to transform the experts' assessments from the BFR decision matrix into a decision matrix in the form of BFNs to approximate the concept of lower and upper approximations to get a single aggregated BFN. Then, we illustrated a numeric example of the presented concept and discussed the applicability of the proposed approach with the literature to show the significance and consequences of the suggested model. Based on the overall comparative study, we concluded that the proposed approach is superior and more effective than existing methods.