## 1 Introduction
Therefore, in order to overcome these limitations, we introduce the bipolar pqr-spherical fuzzy set (BpqrSFS), which allows decision-makers to choose any \(+\)ve integers \(p,q,r\), without violating condition \(0\leq(\text{MD}^{+})^{p}+(\text{NMD}^{+})^{q}+(\
eg\text{MD}^{+})^{r}\leq 1\), and \(-1\leq(\text{MD}^{-})^{p}+(\text{NMD}^{-})^{q}+(\
eg\text{MD}^{-})^{r}\leq 0\). By permitting decision-makers to allocate different parameters, \(p,q\), and \(r\) to \(\text{MD}^{+}\), NMD\({}^{+}\), \(\
eg\text{MD}^{+}\in[0,1]\) and MD\({}^{-}\), NMD\({}^{-}\in[-1,0]\), respectively, BpqrSFSs create a relatively open stage for DM processes. More specifically, if a decision-maker sets \(\text{MD}^{+}=0.95\), NMD\({}^{+}=0.5\), and \(\
eg\text{MD}^{+}=0.6\); MD\({}^{-}=-0.03\), NMD\({}^{-}=-0.4\), and \(\
eg\text{MD}^{-}=-0.2\) for example. In that case, for \(p=q=r=4\), one can see that \((0.95)^{4}+(0.5)^{4}+(0.6)^{4}>1\) and \((-0.03)^{4}+(-0.4)^{4}+(-0.2)^{4}>0\) violate the restricted conditions for the BT-F model. However, for \(p=4\), \(q=5\), and \(r=6\), both restrictions hold true, showing the power of choosing different parameters and, consequently, the robustness of the proposed model. Figure 1 illustrates its position and demonstrates how the most well-known extended fuzzy sets are derived from the conventional fuzzy set. A T-spherical fuzzy set, which satisfies the restriction \((\text{MD})^{n}+(\text{NMD})^{n}+(\
eg\text{MD})^{n}\leq 1\) for some integer \(n\geq 2\), is a verbal generalization of various types of extended fuzzy sets (discussed above). However, it is not free from limitations. Then, the concept of pqr-spherical fuzzy sets lifts approximately all restrictions imposed by earlier sets. In a pqr-spherical fuzzy set, the triplet (MD, NMD, and \(\
eg\text{MD}\)) is constrained to \((\text{MD})^{p}+(\text{NMD})^{q}+(\
eg\text{MD})^{r}\leq 1\), where \(p,r\in\mathbb{Z}^{+}\) such that \(q=LCM(p,r)\). This approach does not allow DM to evaluate MD, NMD, and \(\
eg\text{MD}\) in a \(+\)ve as well as \(-\)ve manner. The bipolar T-spherical fuzzy set [23] appeared in the literature, which contains both \(+\)ve and \(-\)ve values of MD, NMD, and \(\
eg\text{MD}\). However, this approach fails to provide an additional flexible atmosphere for DM procedures by permitting decision-makers to set multiple values for \(p,q,r\) for \(+\)ve and \(-\)ve values of MD, NMD, and \(\
eg\text{MD}\), respectively. This encourages us to put forward the bipolar pqr-spherical fuzzy set, a model that overcomes the preceding restrictions. Furthermore, bipolar pqr-spherical fuzzy sets generalize various fuzzy models in the literature and many MCDM techniques can be extended based on them. This article offers a new notion of bipolar pqr-spherical fuzzy sets along with its principal operations, which expand on the existing bipolar T-spherical fuzzy sets and pqr-spherical fuzzy sets. Both averaging and geometric aggregation operators are put forward to aggregate bipolar pqr-spherical fuzzy information in the MCDM issues based on the prescribed operations. An extension of the TOPSIS technique for bipolar pqr-spherical fuzzy sets is proposed, which generalizes other methods. The structure of this paper is outlined as follows: Section 2 presents some fundamental definitions and associated concepts succinctly. In Section 3, we present the operational laws and aggregation operators pertaining to pqr-spherical fuzzy numbers. Section 4 introduces the MCDM and TOPSIS methods, along with a numerical example to illustrate the suggested approach. Section 5 conducts an intensive comparative evaluation of the suggested methods in relation to previously established models. The conclusion of this paper is found in Section 6.