Enhanced MCDM Based on the TOPSIS Technique and Aggregation Operators Under the Bipolar pqr-Spherical Fuzzy Environment: An Application in Firm Supplier Selection

Abstract

Multiple criteria decision-making (MCDM) is a significant area of decision-making theory that certainly warrants attention. It might be difficult to accurately convey the necessary decision facts when navigating decision-making problems since we frequently run into complicated issues and unpredictable situations. To address this, introducing the novel idea of the bipolar pqr-spherical fuzzy set (${\mathrm{B}}_{\mathrm{pqr}}$SFS), a hybrid structure of the bipolar fuzzy set (BFS) and the pqr-spherical fuzzy set (pqr-SFS), is the main goal of this work. The fundamental (set-theoretic and algebraic) operations on ${\mathrm{B}}_{\mathrm{pqr}}$SFSs are explained as well as their relations to several known models. A distance measure, such as Euclidean distance, among ${\mathrm{B}}_{\mathrm{pqr}}$SFNs, is provided. Afterward, we expand the fundamental aggregation operators to the pqr-spherical fuzzy (${\mathrm{B}}_{\mathrm{pqr}}$SF) environment by developing bipolar pqr-spherical fuzzy-weighted averaging and bipolar pqr-spherical fuzzy-weighted geometric operators for aggregating ${\mathrm{B}}_{\mathrm{pqr}}$SFNs. According to the aforementioned distance measure and operators, an MCDM approach is established consisting of two algorithms, namely, the TOPSIS method and the method using the proposed operators in the ${\mathrm{B}}_{\mathrm{pqr}}$SF context. Moreover, a numerical example is provided in order to ensure that the presented model is applicable. By using the two algorithms, a comparative analysis of the proposed method with other existing ones is given in order to verify the feasibility of the suggested decision-making procedure.

1. Introduction

This part provides a literature overview of relevant concepts that will be employed in this study.

Extensions of Fuzzy Set Theory: Review

Numerous real-world problems involve fuzziness, such as MCDM, pattern recognition, data mining, etc. The knowledge in these areas is often subject to intrinsic ambiguity or uncertainty and is incomplete. Dealing with this massive information is essential for making informed decisions, gaining meaningful insights, and revealing valuable insights.

To resolve such issues, Zadeh [1] introduced the idea of fuzzy sets (FSs), where each element receives a membership degree (MD) that indicates how closely the element is related to the other elements. These fractions are generally encapsulated as real numbers in the closed $[0,1]$ interval, where 0 means not a part of this group, 1 means completely a member of the group, and everything in between represents some level of participation.

After the concept of FSs, Atanassov [2] proposed the concept of intuitionistic fuzzy sets (IFSs). IFSs provided a more descriptive presentation by accounting for membership degrees (MDs) along with non-membership degrees (¬MDs), representing the intrinsic uncertainty and confusion presented in practical situations. IFS is extensively used for MCDM processing, particularly when MD and ¬MD sums range from zero to one. But in real situations, the sum of MD and ¬MD is sometimes $>1$.

Yager [3] established an innovative notion known as the Pythagorean fuzzy set (PyFS) that generalized IFS. In this regard, the MCDM domain has been extended due to the excellent features of PyFS to describe and control undecided. As we know, PyFS is limited to having the sum of the squares of MD and ¬MD be ≤1.

Yager [4] introduced the q-rung orthopair FS (q-ROFS) as a subsequent evolution of both PyFS and IFS, removing the restrictions mentioned above. Dounis and Stefopoulos [5] used q-ROFS for intelligent medical diagnosis. By extending the latter models, decision-makers have more freedom in selecting values as they wish; on the other hand, it forces them to pick one parameter for all selected values.

Seikh and Mandal [6] introduced p,q-quasirung orthopair fuzzy sets (p,q-QOFSs), which give decision-makers the flexibility to choose different powers for MD and ¬MD, i.e., (${\left(\mathrm{MD}\right)}^p+{(\neg \mathrm{MD})}^q\le 1$, such that p and q could be the same or different). A similar approach appeared in [7] under the name of n,m-rung orthopair fuzzy sets (nm-ROFSs), where $m,n$ are in ${\mathbb{Z}}^{+}$. This change allows for a more flexible encoding of information that accommodates diverse preferences and needs in decision-making contexts. All the aforementioned studies focus on the application of MD and ¬MD while overlooking the impact of the neutral degree, which is a significant factor during the DM process as well.

Cuong and Kreinovich [8] solved the above challenge by introducing the notion of picture fuzzy sets (PFSs). PFSs define the degree of membership as positive (+ve), negative (−ve), and neutral degrees, which correspond to MD, a neutral membership degree (NMD), and ¬MD, respectively, under three distinct functions, inferring the following constraint: $\left(\mathrm{MD}\right)+\left(\mathrm{NMD}\right)+\left(\neg \mathrm{MD}\right)\le 1$. But in some real-life scenarios, the requirement $\left(\mathrm{MD}\right)+\left(\mathrm{NMD}\right)+\left(\neg \mathrm{MD}\right)\le 1$ might not be met.

Mahmood et al. [9] proposed a spherical fuzzy set (SFS), which imposes the new restriction ${\left(\mathrm{MD}\right)}^2+{\left(\mathrm{NMD}\right)}^2+{\left(\neg \mathrm{MD}\right)}^2\le 1$, making uncertainty management more flexible. It was also introduced by Gündoğdu and Kahraman [10]. So, let us try specific values for MD = $0.7$, NMD $=0.4$, and ¬MD $=0.6$; squaring these numbers does not work, as ${\left(0.7\right)}^2+{\left(0.4\right)}^2+{\left(0.6\right)}^2>1$.

Mahmood et al. [9] extended SFSs to T-spherical FSs (T-SFSs), where ${\left(\mathrm{MD}\right)}^n+{\left(\mathrm{NMD}\right)}^n+{\left(\neg \mathrm{MD}\right)}^n\le 1$, for some integer $n\ge 2$. Such an arrangement affords the decision-makers flexibility in assigning any MD, NMD, or ¬MD value inside $[0,1]$. For instance, if we take $n=3$, and MD $=0.7$, NMD $=0.4$, and ¬MD $=0.6$, we obtain ${\left(0.7\right)}^3+{\left(0.4\right)}^3+{\left(0.6\right)}^3\le 1$. Ullah et al. [11] introduced and defined the term ’correlation coefficients of T-SFS’, where levels of membership degrees of the terms should meet the constraints, i.e., ${\left(\mathrm{MD}\right)}^n+{\left(\mathrm{NMD}\right)}^n+{\left(\neg \mathrm{MD}\right)}^n\le 1$. This assumes homogeneity in the levels of MD, NMD, and ¬MD. In actual scenarios, these levels might differ. So, if a decision-maker sets MD $=0.9$, NMD $=0.5$, and ¬MD $=0.7$, for instance, we see that ${\left(0.9\right)}^3+{\left(0.5\right)}^3+{\left(0.7\right)}^3=1.12$ implies this violate linearly aggregation rule. So, a more nuanced approach is needed. Enabling different term levels for MD, NMD, and ¬MD makes it possible to reflect the multi-factorial reality.

Ali and Naeem [12] proposed a new extension of T-SFS, called r,s,t-spherical fuzzy sets (srt-SFSs), which lift all restrictions inflicted by classical sets. Equivalently, Rahim et al. [13] introduced the concept of pqr-spherical fuzzy sets (pqr-SFSs) with little restriction. We follow the latter notation in this work. In pqr-SFSs, the triplet (MD, NMD, and ¬MD) is constrained according to ${\left(\mathrm{MD}\right)}^p+{\left(\mathrm{NMD}\right)}^q+{\left(\neg \mathrm{MD}\right)}^r\le 1$, where $p,r$ are +ve integers with $q=LCM(p,r)$. The pqr-SFSs have been studied in several manners by several researchers; see [14,15]. Karaaslan and Karamaz [16] also extended this concept to interval-valued pqr-spherical fuzzy sets.

Bipolarity is the existence of two opposing or contrasting components, powers, or circumstances. In many areas, it refers to the state of balance between two opposing or complementary poles, commonly considered as good/bad or beneficial/non-beneficial angles. Thus, bipolarity has practical and theoretical applications in everything from psychology and political science to physics and decision-making. Bipolar fuzzy sets are a generalization of conventional fuzzy sets, which are very useful in complex real-life situations where both, +ve, and negative, −ve, feelings need to be considered. These methods can accommodate dual uncertainties, making them applicable in a variety of fields like decision-making, pattern identification, medical diagnosis, sentiment analysis, risk assessment, and AI. As the demand for higher-level models in ambiguous and uncertain situations develops, extended bipolar fuzzy sets are anticipated to play an increasingly important role in enhancing computer intelligence.

The first extension in this form was the bipolar intuitionistic fuzzy set (BIFS) [17] introduced by Ezhilmaran and Sankar. By the same scenario stated in the first paragraph, the bipolar Pythagorean fuzzy set (BPyFS) [18], Fermatean bipolar fuzzy set (FBFS) [19], bipolar q-rung orthopair fuzzy set (Bq-ROFS) [20], bipolar m,n-rung orthopair fuzzy set (Bmn-ROFS) [21], and spherical bipolar fuzzy set (SBFS) [22] are introduced.

Lastly, the bipolar T-spherical fuzzy set (BT-FS) [23], is a model that includes both +ve and −ve values of MD, NMD, and ¬MD. Specifically, ${\mathrm{MD}}^{+}$, NMD⁺, ¬MD⁺ $\in [0,1]$, and MD⁻, ${\mathrm{NMD}}^{-}$, ${\neg \mathrm{MD}}^{-}\in [-1,0]$, satisfy the conditions $0\le {\left({\mathrm{MD}}^{+}\right)}^n+{\left({\mathrm{NMD}}^{+}\right)}^n+{\left({\neg \mathrm{MD}}^{+}\right)}^n\le 1$ and $-1\le {\left({\mathrm{MD}}^{-}\right)}^n+{\left({\mathrm{NMD}}^{-}\right)}^n+{\left({\neg \mathrm{MD}}^{-}\right)}^n\le 0$, for some +ve integer, $n\ge 2$. This requirement forces decision-makers to give an equal n to MD, NMD, and ¬MD. In some cases, decision-makers may have to assign different levels of authority to the MD, NMD, or ¬MD. Such requirements cannot be catered to by BT-SFSs. Therefore, in order to overcome these limitations, we introduce the bipolar pqr-spherical fuzzy set (${\mathrm{B}}_{\mathrm{pqr}}$SFS), which allows decision-makers to choose any +ve integers $p,q,r$, without violating condition $0\le {\left({\mathrm{MD}}^{+}\right)}^p+{\left({\mathrm{NMD}}^{+}\right)}^q+{\left({\neg \mathrm{MD}}^{+}\right)}^r\le 1$, and $-1\le {\left({\mathrm{MD}}^{-}\right)}^p+{\left({\mathrm{NMD}}^{-}\right)}^q+{\left({\neg \mathrm{MD}}^{-}\right)}^r\le 0$. By permitting decision-makers to allocate different parameters, $p,q$, and r to ${\mathrm{MD}}^{+}$, ${\mathrm{NMD}}^{+}$, ${\neg \mathrm{MD}}^{+}\in [0,1]$ and ${\mathrm{MD}}^{-}$, ${\mathrm{NMD}}^{-}$, ${\neg \mathrm{MD}}^{-}\in [-1,0]$, respectively, ${\mathrm{B}}_{\mathrm{pqr}}$SFSs create a relatively open stage for DM processes. More specifically, if a decision-maker sets ${\mathrm{MD}}^{+}=0.95$, ${\mathrm{NMD}}^{+}=0.5$, and ${\neg \mathrm{MD}}^{+}=0.6$; ${\mathrm{MD}}^{-}=-0.03$, ${\mathrm{NMD}}^{-}=-0.4$, and ${\neg \mathrm{MD}}^{-}=-0.2$ for example. In that case, for $p=q=r=4$, one can see that ${\left(0.95\right)}^4+{\left(0.5\right)}^4+{\left(0.6\right)}^4>1$ 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 ${\left(\mathrm{MD}\right)}^n+{\left(\mathrm{NMD}\right)}^n+{\left(\neg \mathrm{MD}\right)}^n\le 1$ for some integer $n\ge 2$, 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 ¬MD) is constrained to ${\left(\mathrm{MD}\right)}^p+{\left(\mathrm{NMD}\right)}^q+{\left(\neg \mathrm{MD}\right)}^r\le 1$, where $p,r\in {\mathbb{Z}}^{+}$ such that $q=LCM(p,r)$. This approach does not allow DM to evaluate MD, NMD, and ¬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 ¬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 ¬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.

2. Background Concepts

The development of the intended model will be aided by keeping in mind the following basic definitions and properties:

Definition 1 

([22]). Let U denote a non-empty finite set. A T-spherical fuzzy set (T-SFS) is defined as follows:

$$\mathcal{T}=\left\{\right(\rho ,\theta \left(\rho \right),\eta \left(\rho \right),\lambda \left(\rho \right))\mid \rho \in U\},$$

where $\theta \left(\rho \right)$ represents a +ve membership, $\eta \left(\rho \right)$ represents neutral membership, and $\lambda \left(\rho \right)$ represents a −ve membership with

$$0\le \theta \left(\rho \right),\eta \left(\rho \right),\lambda \left(\rho \right)\le 1,$$

and

$${\left(\theta \left(\rho \right)\right)}^q+{\left(\eta \left(\rho \right)\right)}^q+{\left(\lambda \left(\rho \right)\right)}^q\le 1,$$

where $q\in {\mathbb{Z}}^{+}$ can be adjusted to a range of values based on the situation or requirements, the triplet $T=(\theta ,\eta ,\lambda )$ is named a T-spherical fuzzy number (abbreviated as T-SFN), such that ${\left(\theta \right)}^q+{\left(\eta \right)}^q+{\left(\lambda \right)}^q\le 1$.

Definition 2 

([22]). Let $T_1=({\theta }_1,{\eta }_1,{\lambda }_1)$, $T_2=({\theta }_2,{\eta }_2,{\lambda }_2)$, and $T=(\theta ,\eta ,\lambda )$ be three T-SFNs. Then, we have the following:

1. 

$$T_1\oplus T_2=\left(\sqrt[2q]{{\left({\theta }_1\right)}^q+{\left({\theta }_2\right)}^q-{\left({\theta }_1\right)}^q{\left({\theta }_2\right)}^q},{\eta }_1{\eta }_2,{\lambda }_1{\lambda }_2\right).$$

2. 

$$T_1\otimes T_2=\left({\theta }_1{\theta }_2,\sqrt[2q]{{\left({\eta }_1\right)}^q+{\left({\eta }_2\right)}^q-{\left({\eta }_1\right)}^q{\left({\eta }_2\right)}^q},\sqrt[2q]{{\left({\lambda }_1\right)}^q+{\left({\lambda }_2\right)}^q-{\left({\lambda }_1\right)}^q{\left({\lambda }_1\right)}^q}\right).$$

3. 

$$ϵ·T=(\sqrt[2q]{1-{(1-{\left(\theta \right)}^q)}^{ϵ}},{\left(\eta \right)}^{ϵ},{\left(\lambda \right)}^{ϵ}).$$

4. 

$$T^{ϵ}=({\left(\theta \right)}^{ϵ},\sqrt[2q]{1-{(1-{\left(\eta \right)}^q)}^{ϵ}},\sqrt[2q]{1-(1-{\left(\lambda \right)}^q){)}^{ϵ}}).$$

Definition 3 

([23]). Let U be an initial universe. A bipolar T-spherical fuzzy set (BT-SFS) over U is defined as follows:

$$\ddot{\mathcal{B}}=\{(\rho ,{\theta }^{+}\left(\rho \right),{\eta }^{+}\left(\rho \right),{\lambda }^{+}\left(\rho \right),{\theta }^{-}\left(\rho \right),{\eta }^{-}\left(\rho \right),{\lambda }^{-}\left(\rho \right)):\rho \in U\},$$

where ${\theta }^{+},{\eta }^{+},{\lambda }^{+}:U\to [0,1]$ refer to the +ve degree membership, neutral, and non-membership functions, respectively, and ${\theta }^{-},{\eta }^{-},{\lambda }^{-}:U\to [-1,0]$ refer to the −ve degree membership, neutral, and non-membership functions, respectively, such that

$$0\le {\left({\theta }^{+}\left(\rho \right)\right)}^q+{\left({\eta }^{+}\left(\rho \right)\right)}^q+{\left({\lambda }^{+}\left(\rho \right)\right)}^q\le 1,$$

and

$$-1\le {\left({\theta }^{-}\left(\rho \right)\right)}^q+{\left({\eta }^{-}\left(\rho \right)\right)}^q+{\left({\lambda }^{-}\left(\rho \right)\right)}^q\le 0,$$

for each $\rho \in U$ and some $q\in {\mathbb{Z}}^{+}$. The 6-tuple $B=({\theta }^{+},{\eta }^{+},{\lambda }^{+},{\theta }^{-},{\eta }^{-},{\lambda }^{-})$ is conveniently called a bipolar T-spherical fuzzy number (BT-SFN).

Definition 4 

([23]). Let $B_1=({\theta }_1^{+},{\eta }_1^{+},{\lambda }_1^{+},{\theta }_1^{-},{\eta }_1^{-},{\lambda }_1^{-})$ and $B_2=({\theta }_2^{+},{\eta }_2^{+},{\lambda }_2^{+},{\theta }_2^{-},{\eta }_2^{-},{\lambda }_2^{-})$ denote two BT-SFNs. Then, we have the following:

1. 

$$\begin{array}{cc}\hfill B_1\oplus B_2=& (\sqrt{{\left({\left({\theta }_1^{+}\right)}^q\right)}^2+{\left({\left({\theta }_2^{+}\right)}^q\right)}^2-{\left({\left({\theta }_1^{+}\right)}^q\right)}^2{\left({\left({\theta }_2^{+}\right)}^q\right)}^2},{\left({\eta }_1^{+}\right)}^q·{\left({\eta }_2^{+}\right)}^q,{\left({\lambda }_1^{+}\right)}^q·{\left({\lambda }_2^{+}\right)}^q,\hfill \\ \hfill & -{\left({\theta }_1^{+}\right)}^q·{\left({\theta }_2^{+}\right)}^q,-\sqrt{{\left({\left({\eta }_1^{-}\right)}^q\right)}^2+{\left({\left({\eta }_2^{-}\right)}^q\right)}^2-{\left({\left({\eta }_1^{-}\right)}^q\right)}^2{\left({\left({\eta }_2^{-}\right)}^q\right)}^2},\hfill \\ \hfill & -\sqrt{{\left({\left({\lambda }_1^{-}\right)}^q\right)}^2+{\left({\left({\lambda }_2^{-}\right)}^q\right)}^2-{\left({\left({\lambda }_1^{-}\right)}^q\right)}^2{\left({\left({\lambda }_2^{-}\right)}^q\right)}^2}).\hfill \end{array}$$

2. 

$$\begin{array}{cc}\hfill B_1\otimes B_2=& ({\left({\theta }_1^{+}\right)}^q·{\left({\theta }_2^{+}\right)}^q,\sqrt{{\left({\left({\eta }_1^{+}\right)}^q\right)}^2+{\left({\left({\eta }_2^{+}\right)}^q\right)}^2-{\left({\left({\eta }_1^{+}\right)}^q\right)}^2{\left({\left({\eta }_2^{+}\right)}^q\right)}^2},\hfill \\ \hfill & \sqrt{{\left({\left({\lambda }_1^{+}\right)}^q\right)}^2+{\left({\left({\lambda }_2^{+}\right)}^q\right)}^2-{\left({\left({\lambda }_1^{+}\right)}^q\right)}^2{\left({\left({\lambda }_2^{+}\right)}^q\right)}^2},\hfill \\ \hfill & -\sqrt{{\left({\left({\theta }_1^{-}\right)}^q\right)}^2+{\left({\left({\theta }_2^{-}\right)}^q\right)}^2-{\left({\left({\theta }_1^{-}\right)}^q\right)}^2{\left({\left({\theta }_2^{-}\right)}^q\right)}^2},\hfill \\ \hfill & -{\left({\eta }_1^{-}\right)}^q·{\left({\eta }_2^{-}\right)}^q,-{\left({\lambda }_1^{-}\right)}^q·{\left({\lambda }_2^{-}\right)}^q).\hfill \end{array}$$

3. 

$$\begin{array}{cc}\hfill ϵ·B=& (\sqrt{1-{(1-{\left({\left({\theta }_B^{+}\right)}^q\right)}^2)}^{ϵ}},{\left({\left({\eta }_B^{+}\right)}^q\right)}^{ϵ},{\left({\left({\lambda }_B^{+}\right)}^q\right)}^{ϵ},-{(-{\left({\theta }_B^{+}\right)}^q)}^{ϵ},\hfill \\ \hfill & -\sqrt{1-{(1-(-{\left({\left({\eta }_B^{+}\right)}^q\right)}^2))}^{ϵ}},-\sqrt{1-{(1-(-{\left({\left({\lambda }_B^{+}\right)}^q\right)}^2))}^{ϵ}}).\hfill \end{array}$$

4. 

$$\begin{array}{cc}\hfill B^{ϵ}=& ({({\theta }_B^{+})}^q{)}^{ϵ},\sqrt{1-{(1-{\left({\left({\eta }_B^{+}\right)}^q\right)}^2)}^{ϵ}},\sqrt{1-{(1-{\left({\left({\lambda }_B^{+}\right)}^q\right)}^2)}^{ϵ}},\hfill \\ & -\sqrt{1-{(1-(-{\left({\left({\theta }_B^{+}\right)}^q\right)}^2))}^{ϵ}},-{(-{\left({\eta }_B^{-}\right)}^q)}^{ϵ},-{(-{\left({\lambda }_B^{-}\right)}^q)}^{ϵ}).\hfill \end{array}$$

Definition 5 

([12,13]). Let U denote a non-empty finite set. A pqr-spherical fuzzy set (pqr-SFS) is defined as follows:

$$\mathcal{S}=\left\{\right(\rho ,\theta \left(\rho \right),\eta \left(\rho \right),\lambda \left(\rho \right))\mid \rho \in U\},$$

such that $\theta \left(\rho \right)$ represents a +ve membership, $\eta \left(\rho \right)$ represents neutral membership, and $\lambda \left(\rho \right)$ represents a −ve membership, where

$$0\le \theta \left(\rho \right),\eta \left(\rho \right),\lambda \left(\rho \right)\le 1,$$

and

$${\left(\theta \left(\rho \right)\right)}^p+{\left(\eta \left(\rho \right)\right)}^q+{\left(\lambda \left(\rho \right)\right)}^r\le 1,$$

for any +ve integers $p,q,r$. The triplet $S=(\theta ,\eta ,\lambda )$ such that ${\left(\theta \right)}^p+{\left(\eta \right)}^q+{\left(\lambda \right)}^r\le 1$ is called a pqr-spherical fuzzy number (pqr-SFN).

Definition 6 

([13]). Let $S_1=({\theta }_1,{\eta }_1,{\lambda }_1)$, $S_2=({\theta }_2,{\eta }_2,{\lambda }_2)$, and $S=(\theta ,\eta ,\lambda )$ denote three pqr-SFNs. Then, we have the following:

1. 

$$S_1\oplus S_2=\left(\sqrt[2p]{{\left({\theta }_1\right)}^p+{\left({\theta }_2\right)}^p-{\left({\theta }_1\right)}^p{\left({\theta }_2\right)}^p},{\eta }_1{\eta }_2,{\lambda }_1{\lambda }_2\right).$$

2. 

$$S_1\otimes S_2=({\theta }_1{\theta }_2,\sqrt[2q]{{\left({\eta }_1\right)}^q+{\left({\eta }_2\right)}^q-{\left({\eta }_1\right)}^q{\left({\eta }_2\right)}^q},\sqrt[2r]{{\left({\lambda }_1\right)}^r+{\left({\lambda }_2\right)}^r-{\left({\lambda }_1\right)}^r{\left({\lambda }_1\right)}^r}).$$

3. 

$$ϵ·S=(\sqrt[2p]{1-{(1-{\left(\theta \right)}^p)}^{ϵ}},{\left(\eta \right)}^{ϵ},{\left(\lambda \right)}^{ϵ}).$$

4. 

$$S^{ϵ}=({\left(\theta \right)}^{ϵ},\sqrt[2q]{1-{(1-{\left(\eta \right)}^q)}^{ϵ}},\sqrt[2r]{1-(1-{\left(\lambda \right)}^r){)}^{ϵ}}).$$

3. Bipolar pqr-Spherical Fuzzy Sets

In this section, a bipolar pqr-spherical fuzzy set, which is a hybrid model that incorporates both BFS and pqr-SFS, is defined, and we provide its fundamental operations:

Definition 7. 

Let U be an initial universe. A bipolar pqr-spherical fuzzy set (${\mathrm{B}}_{\mathrm{pqr}}$SFS) over U is defined as follows:

$$\ddot{\mathcal{T}}=\{(\rho ,{\theta }^{+}\left(\rho \right),{\eta }^{+}\left(\rho \right),{\lambda }^{+}\left(\rho \right),{\theta }^{-}\left(\rho \right),{\eta }^{-}\left(\rho \right),{\lambda }^{-}\left(\rho \right)):\rho \in U\},$$

where ${\theta }^{+},{\eta }^{+},{\lambda }^{+}:U\to [0,1]$ refer to the +ve degree membership, neutral, and non-membership functions, respectively, and ${\theta }^{-},{\eta }^{-},{\lambda }^{-}:U\to [-1,0]$ refer to the −ve degree membership, neutral, and non-membership functions, respectively, such that

$$0\le {\left({\theta }^{+}\left(\rho \right)\right)}^p+{\left({\eta }^{+}\left(\rho \right)\right)}^q+{\left({\lambda }^{+}\left(\rho \right)\right)}^r\le 1,$$

and

$$-1\le {\left({\theta }^{-}\left(\rho \right)\right)}^p+{\left({\eta }^{-}\left(\rho \right)\right)}^q+{\left({\lambda }^{-}\left(\rho \right)\right)}^r\le 0,$$

for each $\rho \in U$ and some $p,q,r\in {\mathbb{Z}}^{+}$. For convenience, we call $\ddot{\tau }=({\theta }^{+},{\eta }^{+},{\lambda }^{+},{\theta }^{-},{\eta }^{-},{\lambda }^{-})$ a bipolar pqr-spherical fuzzy number (${\mathit{B}}_{pqr}$SFN).

This model can be effectively applied to various real-world situations, including electoral voting systems, business strategies, finance, agricultural optimization, and medical treatments. More precisely, profit and loss statements are commonly used in the business and finance fields. BpqrSFS allows us to link business decision-making activities. When people invest their money in certain firms, their primary goal is to maximize profits within a set time frame. BpqrSFS has the ability to deal with such situations and issues by BpqrSFN, which is represented by $({\theta }^{+},{\eta }^{+},{\lambda }^{+},{\theta }^{-},{\eta }^{-},{\lambda }^{-})$, where ${\theta }^{+}$ is the degree of satisfaction that produces profit, ${\eta }^{+}$ is the degree of abstinence that produces profit, and ${\lambda }^{+}$ is the degree of dissatisfaction that produces profit, where ${\theta }^{-}$ is the degree of satisfaction that causes a loss, ${\eta }^{-}$ is the degree of abstinence that causes a loss, and ${\lambda }^{-}$ is the degree of dissatisfaction that causes a loss.

In accordance with existing literature, we shall refer to the proposed set as a bipolar (p,q,r)-spherical (or bipolar (r,s,t)-spherical) fuzzy set. However, we prefer to stick with the way mentioned above.

Remark 1. 

Definition 7 yields the following definitions in certain cases:

1. 

bipolar T-spherical fuzzy set (BT-SFS) [23] if $p=q=r$.

2. 

spherical bipolar fuzzy set (SBFS) [22] if $p=q=r=2$.

3. 

bipolar picture fuzzy set (BPFS) [24] if $p=q=r=1$.

4. 

bipolar mn-rung orthopair fuzzy set (Bmn-ROFS) [21] if ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)=0$.

5. 

Fermatean bipolar fuzzy set (FBFS) [19] if $p=q=r=3$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)=0$.

6. 

bipolar Pythagorean fuzzy set (BPyFS) [18] if $p=q=r=2$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)=0$.

7. 

bipolar intuitionistic fuzzy set (BIFS) [17] if $p=q=r=1$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)=0$.

8. 

pqr-spherical fuzzy set (pqr-SFS) [12,13] if ${\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

9. 

T-spherical fuzzy set (T-SFS) [9] if $p=q=r$ and ${\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

10. 

spherical fuzzy set (SFS) [9,25] if $p=q=r=2$ and ${\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

11. 

picture fuzzy set (PFS) [8] if $p=q=r=1$ and ${\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

12. 

p,q-quasirung orthopair fuzzy set (pq-QROFS) [6] if ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

13. 

q-rung orthopair fuzzy set (q-ROFS) [4] if $p=q$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

14. 

Fermatean fuzzy set (FFS) [26] if $p=q=3$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

15. 

Pythagorean fuzzy set (PyFS) [3] if $p=q=2$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

16. 

intuitionistic fuzzy set (IFS) [2] if $p=q=1$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

17. 

fuzzy set (FS) [1] if $p=1$ and ${\eta }_{\ddot{\tau }}^{+}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{+}\left(\rho \right))={\theta }_{\ddot{\tau }}^{-}\left(\rho \right)={\eta }_{\ddot{\tau }}^{-}\left(\rho \right)={\lambda }_{\ddot{\tau }}^{-}\left(\rho \right))=0$.

Next,

Table 1. Superiority of the proposed model compared to existing ones.

Model	MD	NMD	¬MD	FSDP *	Bipolarity
FS [1]	Yes	No	No	No	No
IFS [2]	Yes	No	Yes	No	No
PyFS [3]	Yes	No	Yes	No	No
FFS [26]	Yes	No	Yes	No	No
q-ROFS [4]	Yes	No	Yes	No	No
mn-ROFS [7]	Yes	No	Yes	Yes	No
PFS [8]	Yes	Yes	Yes	No	No
SFS [9,25]	Yes	Yes	Yes	No	No
T-SFS [9]	Yes	Yes	Yes	No	No
pqr-SFS [12,13]	Yes	Yes	Yes	Yes	No
BIFS [17]	Yes	No	Yes	No	Yes
BPyFS [18]	Yes	No	Yes	No	Yes
FBFS [19]	Yes	No	Yes	No	Yes
Bmn-ROFS [21]	Yes	No	Yes	Yes	Yes
BPFS [24]	Yes	Yes	Yes	No	Yes
SBFS [22]	Yes	Yes	Yes	No	Yes
BT-SFS [23]	Yes	Yes	Yes	No	Yes
Proposed	Yes	Yes	Yes	Yes	Yes

* FSDP means Flexibility in Selecting Different Parameters.

highlights the exceptional performance of our novel model compared to existing models, excelling in membership degree (MD), neutral degree (NMD), non-membership degree (¬MD), offering greater flexibility in parameter selection, and effectively capturing bipolarity.

Now, we present the fundamental operations on ${\mathrm{B}}_{\mathrm{pqr}}$SFNs.

Definition 8. 

Let $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ and $\ddot{\ell }=({\theta }_{\ddot{\ell }}^{+},{\eta }_{\ddot{\ell }}^{+},{\lambda }_{\ddot{\ell }}^{+},{\theta }_{\ddot{\ell }}^{-},{\eta }_{\ddot{\ell }}^{-},{\lambda }_{\ddot{\ell }}^{-})$ be two ${\mathit{B}}_{pqr}$SFNs. Then, we have the following:

1. 

$\ddot{\tau }\tilde{\subseteq }\ddot{\ell }$, if ${\theta }_{\ddot{\tau }}^{+}\le {\theta }_{\ddot{\ell }}^{+}$, ${\eta }_{\ddot{\tau }}^{+}\le {\eta }_{\ddot{\ell }}^{+}$, ${\lambda }_{\ddot{\tau }}^{+}\ge {\lambda }_{\ddot{\ell }}^{+}$, ${\theta }_{\ddot{\tau }}^{-}\ge {\theta }_{\ddot{\ell }}^{-}$, ${\eta }_{\ddot{\tau }}^{-}\ge {\eta }_{\ddot{\ell }}^{-}$, and ${\lambda }_{\ddot{\tau }}^{-}\le {\lambda }_{\ddot{\ell }}^{-}$.

2. 

$\ddot{\tau }=\ddot{\ell }$ if $\ddot{\tau }\tilde{\subseteq }\ddot{\ell }$ and $\ddot{\ell }\tilde{\subseteq }\ddot{\tau }$.

3. 

${\ddot{\tau }}^c=({\lambda }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\theta }_{\ddot{\tau }}^{-})$, where ${\ddot{\tau }}^c$ means the complement of $\ddot{\tau }$.

4. 

$\ddot{\tau }\tilde{\cap }\ddot{\ell }=\left(\begin{array}{c}min({\theta }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\ell }}^{+}),min({\eta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\ell }}^{+}),max({\lambda }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\ell }}^{+}),\\ max({\theta }_{\ddot{\tau }}^{-},{\theta }_{\ddot{\ell }}^{-}),max({\eta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\ell }}^{-}),min({\lambda }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\ell }}^{-})\end{array}\right)$.

5. 

$\ddot{\tau }\tilde{\cup }\ddot{\ell }=\left(\begin{array}{c}max({\theta }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\ell }}^{+}),min({\eta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\ell }}^{+}),min({\lambda }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\ell }}^{+}),\\ min({\theta }_{\ddot{\tau }}^{-},{\theta }_{\ddot{\ell }}^{-}),max({\eta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\ell }}^{-}),max({\lambda }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\ell }}^{-})\end{array}\right)$.

Proposition 1. 

Let $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$, $\ddot{\ell }=({\theta }_{\ddot{\ell }}^{+},{\eta }_{\ddot{\ell }}^{+},{\lambda }_{\ddot{\ell }}^{+},{\theta }_{\ddot{\ell }}^{-},{\eta }_{\ddot{\ell }}^{-},{\lambda }_{\ddot{\ell }}^{-})$, and $\ddot{\wp }=({\theta }_{\ddot{\wp }}^{+},{\eta }_{\ddot{\wp }}^{+},{\lambda }_{\ddot{\wp }}^{+},{\theta }_{\ddot{\wp }}^{-},{\eta }_{\ddot{\wp }}^{-},{\lambda }_{\ddot{\wp }}^{-})$ denote ${\mathit{B}}_{pqr}$SFNs. Then, we have the following:

1. 

$\ddot{\tau }\tilde{\subseteq }\ddot{\ell }$ and $\ddot{\ell }\tilde{\subseteq }\ddot{\wp }\Rightarrow \ddot{\tau }\tilde{\subseteq }\ddot{\wp }$.

2. 

$\ddot{\tau }\tilde{\cap }\ddot{\ell }=\ddot{\ell }\tilde{\cap }\ddot{\tau }$.

3. 

$\ddot{\tau }\tilde{\cup }\ddot{\ell }=\ddot{\ell }\tilde{\cup }\ddot{\tau }$.

4. 

$\ddot{\tau }\tilde{\cap }\left(\ddot{\ell }\tilde{\cap }\ddot{\wp }\right)=\left(\ddot{\tau }\tilde{\cap }\ddot{\ell }\right)\tilde{\cap }\ddot{\wp }$.

5. 

$\ddot{\tau }\tilde{\cup }\left(\ddot{\ell }\tilde{\cup }\ddot{\wp }\right)=\left(\ddot{\tau }\tilde{\cup }\ddot{\ell }\right)\tilde{\cup }\ddot{\wp }$.

6. 

$\left(\ddot{\tau }\tilde{\cap }\ddot{\ell }\right)\tilde{\cup }\ddot{\ell }=\ddot{\ell }$.

7. 

$\left(\ddot{\tau }\tilde{\cup }\ddot{\ell }\right)\tilde{\cap }\ddot{\ell }=\ddot{\ell }$.

8. 

$\ddot{\tau }\tilde{\cap }\left(\ddot{\ell }\tilde{\cup }\ddot{\wp }\right)=\left(\ddot{\tau }\tilde{\cap }\ddot{\ell }\right)\tilde{\cup }\left(\ddot{\tau }\tilde{\cap }\ddot{\wp }\right)$.

9. 

$\ddot{\tau }\tilde{\cup }\left(\ddot{\ell }\tilde{\cap }\ddot{\wp }\right)=\left(\ddot{\tau }\tilde{\cup }\ddot{\ell }\right)\tilde{\cap }\left(\ddot{\tau }\tilde{\cup }\ddot{\wp }\right)$.

Proof. 

See Appendix A (1). □

Proposition 2. 

Let $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ and $\ddot{\ell }=({\theta }_{\ddot{\ell }}^{+},{\eta }_{\ddot{\ell }}^{+},{\lambda }_{\ddot{\ell }}^{+},{\theta }_{\ddot{\ell }}^{-},{\eta }_{\ddot{\ell }}^{-},{\lambda }_{\ddot{\ell }}^{-})$ denote ${\mathit{B}}_{pqr}$SFNs. Then, we have the following:

1. 

${\left(\ddot{\tau }\tilde{\cap }\ddot{\ell }\right)}^c={\ddot{\ell }}^c\tilde{\cup }{\ddot{\tau }}^c$.

2. 

${\left(\ddot{\tau }\tilde{\cup }\ddot{\ell }\right)}^c={\ddot{\ell }}^c\tilde{\cap }{\ddot{\tau }}^c$.

3. 

${\left({\ddot{\tau }}^c\right)}^c=\ddot{\tau }$.

Proof. 

See Appendix A (2). □

Remark 2. 

Given any ${\mathit{B}}_{pqr}$SFNs $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ and $\ddot{\ell }=({\theta }_{\ddot{\ell }}^{+},{\eta }_{\ddot{\ell }}^{+},{\lambda }_{\ddot{\ell }}^{+},{\theta }_{\ddot{\ell }}^{-},{\eta }_{\ddot{\ell }}^{-},{\lambda }_{\ddot{\ell }}^{-})$, then ${\ddot{\tau }}^c$, $\ddot{\tau }\tilde{\cap }\ddot{\ell }$, $\ddot{\tau }\tilde{\cup }\ddot{\ell }$ are also ${\mathit{B}}_{pqr}$SFNs.

Example 1. 

Consider the initial universe $U=\left\{{\rho }_1\right\}$, and let $\ddot{\tau }=\{0.4,0.3,0.8,-0.5,-0.3,-0.6\}$, $\ddot{\ell }=\{0.5,0.2,0.7,-0.4,-0.6,-0.8\}$ be ${\mathit{B}}_{pqr}$SFNs. Then, we have the following:

1. 

Neither $\ddot{\tau }\tilde{\subseteq }\ddot{\ell }$ nor $\ddot{\ell }\tilde{\subseteq }\ddot{\tau }$, since ${\eta }_{\ddot{\tau }}^{+}\nleqq {\eta }_{\ddot{\ell }}^{+}$ and ${\theta }_{\ddot{\ell }}^{+}\nleqq {\theta }_{\ddot{\tau }}^{+}$.

2. 

${\ddot{\tau }}^c=\left\{0.8,0.3,0.4,-0.6,-0.3,-0.5\right\}$.

3. 

$\ddot{\tau }\tilde{\cap }\ddot{\ell }=\left\{0.4,0.2,0.8,-0.4,-0.3,-0.8\right\}$.

4. 

$\ddot{\tau }\tilde{\cup }\ddot{\ell }=\left\{0.5,0.2,0.7,-0.5,-0.3,-0.6\right\}$.

Next, we present the sum, product, scalar product, and exponent operations on the ${\mathrm{B}}_{\mathrm{pqr}}$SFSs.

Definition 9. 

Let $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ and $\ddot{\ell }=({\theta }_{\ddot{\ell }}^{+},{\eta }_{\ddot{\ell }}^{+},{\lambda }_{\ddot{\ell }}^{+},{\theta }_{\ddot{\ell }}^{-},{\eta }_{\ddot{\ell }}^{-},{\lambda }_{\ddot{\ell }}^{-})$ denote ${\mathit{B}}_{pqr}$SFNs (${\mathit{B}}_{pqr}$SFNs) and let $ϵ\in {\mathbb{R}}^{+}$. Then, we have the following:

1. 

$$\begin{array}{cc}\hfill \ddot{\tau }\oplus \ddot{\ell }=& (\sqrt[2p]{{\left({\theta }_{\ddot{\tau }}^{+}\right)}^{2p}+{\left({\theta }_{\ddot{\ell }}^{+}\right)}^{2p}-{\left({\theta }_{\ddot{\tau }}^{+}\right)}^{2p}{\left({\theta }_{\ddot{\ell }}^{+}\right)}^{2p}},{\eta }_{\ddot{\tau }}^{+}·{\eta }_{\ddot{\ell }}^{+},{\lambda }_{\ddot{\tau }}^{+}·{\lambda }_{\ddot{\ell }}^{+},-{\theta }_{\ddot{\tau }}^{-}·{\theta }_{\ddot{\ell }}^{-},\hfill \\ & -\sqrt[2q]{{\left({\eta }_{\ddot{\tau }}^{-}\right)}^{2q}+{\left({\eta }_{\ddot{\ell }}^{-}\right)}^{2q}-{\left({\eta }_{\ddot{\tau }}^{-}\right)}^{2q}{\left({\eta }_{\ddot{\ell }}^{-}\right)}^{2q}},-\sqrt[2r]{{\left({\lambda }_{\ddot{\tau }}^{-}\right)}^{2p}+{\left({\lambda }_{\ddot{\ell }}^{-}\right)}^{2p}-{\left({\lambda }_{\ddot{\tau }}^{-}\right)}^{2p}{\left({\lambda }_{\ddot{\ell }}^{-}\right)}^{2p}}).\hfill \end{array}$$

2. 

$$\begin{array}{cc}\hfill \ddot{\tau }\otimes \ddot{\ell }=& ({\theta }_{\ddot{\tau }}^{+}·{\theta }_{\ddot{\ell }}^{+},\sqrt[2q]{{\left({\eta }_{\ddot{\tau }}^{+}\right)}^{2q}+{\left({\eta }_{\ddot{\ell }}^{+}\right)}^{2q}-{\left({\eta }_{\ddot{\tau }}^{+}\right)}^{2q}{\left({\eta }_{\ddot{\ell }}^{+}\right)}^{2q}},\sqrt[2r]{{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^{2r}+{\left({\lambda }_{\ddot{\ell }}^{+}\right)}^{2r}-{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^{2r}{\left({\lambda }_{\ddot{\ell }}^{+}\right)}^{2r}},\hfill \\ & -\sqrt[2p]{{\left({\theta }_{\ddot{\tau }}^{-}\right)}^{2p}+{\left({\theta }_{\ddot{\ell }}^{-}\right)}^{2p}-{\left({\theta }_{\ddot{\tau }}^{-}\right)}^{2p}{\left({\theta }_{\ddot{\ell }}^{-}\right)}^{2p}},-{\eta }_{\ddot{\tau }}^{-}·{\eta }_{\ddot{\ell }}^{-},-{\lambda }_{\ddot{\tau }}^{-}·{\lambda }_{\ddot{\ell }}^{-}).\hfill \end{array}$$

3. 

$$\begin{array}{cc}\hfill ϵ.\ddot{\tau }=& (\sqrt[2p]{1-{(1-{\left({\theta }_{\ddot{\tau }}^{+}\right)}^{2p})}^{ϵ}},{\left({\eta }_{\ddot{\tau }}^{+}\right)}^{ϵ},{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^{ϵ},-{(-{\theta }_{\ddot{\tau }}^{-})}^{ϵ},-\sqrt[2q]{1-{(1-{\left({\eta }_{\ddot{\tau }}^{-}\right)}^{2q})}^{ϵ}},\hfill \\ \hfill & -\sqrt[2r]{1-{(1-{\left({\lambda }_{\ddot{\tau }}^{-}\right)}^{2r})}^{ϵ}}).\hfill \end{array}$$

4. 

$$\begin{array}{cc}\hfill {\ddot{\tau }}^{ϵ}=& ({\left({\theta }_{\ddot{\tau }}^{+}\right)}^{ϵ},\sqrt[2q]{1-{(1-{\left({\eta }_{\ddot{\tau }}^{+}\right)}^{2q})}^{ϵ}},\sqrt[2r]{1-{(1-{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^{2r})}^{ϵ}},-\sqrt[2p]{1-{(1-{\left({\theta }_{\ddot{\tau }}^{-}\right)}^{2p})}^{ϵ}},\hfill \\ \hfill -{(-{\eta }_{\ddot{\tau }}^{-})}^{ϵ},& -{(-{\lambda }_{\ddot{\tau }}^{-})}^{ϵ}).\hfill \end{array}$$

Example 2. 

Recall the ${\mathit{B}}_{pqr}$SFNs $\ddot{\tau }=\{0.4,0.3,0.8,-0.5,-0.3,-0.6\}$ and $\ddot{\ell }=\{0.5,0.2,0.7,-0.4,-0.6,-0.8\}$ considered in Example 1 such that $p=4,q=3,r=6$ and let $ϵ=0.3$. Then, we have the following:

1. 

$\ddot{\tau }\oplus \ddot{\ell }=\{0.2598,0.06,0.56,-0.2,-0.3618,-0.6431\}$.

2. 

$\ddot{\tau }\otimes \ddot{\ell }=\{0.2,0.0926,0.6586,-0.2598,-0.18,-0.48\}$.

3. 

$ϵ.\ddot{\tau }=\{0.1184,0.6968,0.9352,-0.8123,-0.0603,-0.2946\}$.

4. 

${\ddot{\tau }}^{ϵ}=\{0.7597,0.0603,0.5258,-0.1851,-0.6968,-0.8579\}$.

Theorem 1. 

Let ${\ddot{\tau }}_1=({\theta }_{{\ddot{\tau }}_1}^{+},{\eta }_{{\ddot{\tau }}_1}^{+},{\lambda }_{{\ddot{\tau }}_1}^{+},{\theta }_{{\ddot{\tau }}_1}^{-},{\eta }_{{\ddot{\tau }}_1}^{-},{\lambda }_{{\ddot{\tau }}_1}^{-})$, ${\ddot{\tau }}_2=({\theta }_{{\ddot{\tau }}_2}^{+},{\eta }_{{\ddot{\tau }}_2}^{+},{\lambda }_{{\ddot{\tau }}_2}^{+},{\theta }_{{\ddot{\tau }}_2}^{-},{\eta }_{{\ddot{\tau }}_2}^{-},{\lambda }_{{\ddot{\tau }}_2}^{-})$, and ${\ddot{\tau }}_3=({\theta }_{{\ddot{\tau }}_3}^{+},{\eta }_{{\ddot{\tau }}_3}^{+},{\lambda }_{{\ddot{\tau }}_3}^{+},{\theta }_{{\ddot{\tau }}_3}^{-},{\eta }_{{\ddot{\tau }}_3}^{-},{\lambda }_{{\ddot{\tau }}_3}^{-})$ denote ${\mathit{B}}_{pqr}$SFSs. Then, we have the following:

1. 

${\ddot{\tau }}_1\oplus {\ddot{\tau }}_2={\ddot{\tau }}_2\oplus {\ddot{\tau }}_1$.

2. 

${\ddot{\tau }}_1\otimes {\ddot{\tau }}_2={\ddot{\tau }}_2\otimes {\ddot{\tau }}_1$.

3. 

$({\ddot{\tau }}_1\oplus {\ddot{\tau }}_2)\oplus {\ddot{\tau }}_3={\ddot{\tau }}_1\oplus ({\ddot{\tau }}_2\oplus {\ddot{\tau }}_3)$.

4. 

$({\ddot{\tau }}_1\otimes {\ddot{\tau }}_2)\otimes {\ddot{\tau }}_3={\ddot{\tau }}_1\otimes ({\ddot{\tau }}_2\otimes {\ddot{\tau }}_3)$.

Proof. 

Straightforward. □

Theorem 2. 

Let $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$, ${\ddot{\tau }}_1=({\theta }_{{\ddot{\tau }}_1}^{+},{\eta }_{{\ddot{\tau }}_1}^{+},{\lambda }_{{\ddot{\tau }}_1}^{+},{\theta }_{{\ddot{\tau }}_1}^{-},{\eta }_{{\ddot{\tau }}_1}^{-},{\lambda }_{{\ddot{\tau }}_1}^{-})$, and ${\ddot{\tau }}_2=({\theta }_{{\ddot{\tau }}_2}^{+},{\eta }_{{\ddot{\tau }}_2}^{+},{\lambda }_{{\ddot{\tau }}_2}^{+},{\theta }_{{\ddot{\tau }}_2}^{-},{\eta }_{{\ddot{\tau }}_2}^{-},{\lambda }_{{\ddot{\tau }}_2}^{-})$ denote three ${\mathit{B}}_{pqr}$SFSs and $ϵ,{ϵ}_1,{ϵ}_2>0$. Then, we have the following:

1. 

$ϵ·\left({\ddot{\tau }}_1\oplus {\ddot{\tau }}_2\right)=\left(ϵ·{\ddot{\tau }}_1\right)\oplus \left(ϵ·{\ddot{\tau }}_2\right)$.

2. 

$({ϵ}_1\oplus {ϵ}_2)·\ddot{\tau }=\left({ϵ}_1·\ddot{\tau }\right)\oplus \left({ϵ}_2·\ddot{\tau }\right)$.

3. 

${\left(\ddot{\tau }\right)}^{{ϵ}_1+{ϵ}_2}={\left(\ddot{\tau }\right)}^{{ϵ}_1}\otimes {\left(\ddot{\tau }\right)}^{{ϵ}_2}$.

4. 

${\left({\ddot{\tau }}_1\otimes {\ddot{\tau }}_2\right)}^{ϵ}={\left({\ddot{\tau }}_1\right)}^{ϵ}\otimes {\left({\ddot{\tau }}_2\right)}^{ϵ}$.

Proof. 

See Appendix A (3). □

One of the most important tools in the fuzzy set analysis is the distance measure. It is commonly used when making decisions. The most commonly used distance measurement is the Euclidean distance. This distance measure between ${\mathit{B}}_{\mathrm{pqr}}$SFNs, which will be utilized later in the TOPSIS method, is provided in this section.

Definition 10. 

Let $\ddot{\tau }$ and $\ddot{\ell }$ be two ${\mathit{B}}_{pqr}$SFNs on a set $U=\{{\rho }_1,{\rho }_2,\cdots ,{\rho }_n\}$. A distance measure between $\ddot{\tau }$ and $\ddot{\ell }$ is $\ddot{D}:{\left(B_{pqr}SFN\right)}^2\to [0,1]$, if the following conditions are satisfied:

1. 

$0\le \ddot{D}(\ddot{\tau },\ddot{\ell })\le 1$;

2. 

$\ddot{D}(\ddot{\tau },\ddot{\ell })=\ddot{D}(\ddot{\ell },\ddot{\tau })$;

3. 

$\ddot{D}(\ddot{\tau },\ddot{\ell })=0$ if and only if $\ddot{\tau }=\ddot{\ell }$.

Definition 11. 

Let $\ddot{\tau }$ and $\ddot{\ell }$ be two ${\mathit{B}}_{pqr}$SFNs on a set $U=\{{\rho }_1,{\rho }_2,\cdots ,{\rho }_n\}$. We define the distance measure between $\ddot{\tau }$ and $\ddot{\ell }$ as follows:

$$\ddot{D}(\ddot{\tau },\ddot{\ell })=\sqrt{\begin{array}{cc}& {\displaystyle \frac{1}{6}}[{\left(({\theta }_{\ddot{\tau }}^{+}{)}^p-{\left({\theta }_{\ddot{\ell }}^{+}\right)}^p\right)}^2+{\left({\left({\eta }_{\ddot{\tau }}^{+}\right)}^q-{\left({\eta }_{\ddot{\ell }}^{+}\right)}^q\right)}^2+{\left({\left({\lambda }_{\ddot{\tau }}^{+}\right)}^r-{\left({\lambda }_{\ddot{\ell }}^{+}\right)}^r\right)}^2\hfill \\ \hfill & +{\left({\left({\theta }_{\ddot{\tau }}^{-}\right)}^p-{\left({\theta }_{\ddot{\ell }}^{-}\right)}^p\right)}^2+{\left({\left({\eta }_{\ddot{\tau }}^{-}\right)}^q-{\left({\eta }_{\ddot{\ell }}^{-}\right)}^q\right)}^2+{\left({\left({\lambda }_{\ddot{\tau }}^{-}\right)}^r-{\left({\lambda }_{\ddot{\ell }}^{-}\right)}^r\right)}^2].\hfill \end{array}}$$

(1)

Notice that characteristics of Definition 10 are clearly met by the distance measure mentioned in Equation (1).

Example 3. 

Recalling again the ${\mathit{B}}_{pqr}$SFNs $\ddot{\tau }=\{0.4,0.3,0.8,-0.5,-0.3,-0.6\}$ and $\ddot{\ell }=\{0.5,0.2,0.7,-0.4,-0.6,-0.8\}$ given in Example 1 such that $p=2,q=3,r=4$. Then, we have the following: $\ddot{D}(\ddot{\tau },\ddot{\ell })=0.1923.$

4. Bipolar pqr-Spherical Fuzzy Aggregation Operators

This section extends the basic aggregation operators into the ${\mathrm{B}}_{\mathrm{pqr}}$SFS environment by creating bipolar pqr-spherical fuzzy-weighted averaging and bipolar pqr-spherical fuzzy-weighted geometric operators for aggregating the ${\mathrm{B}}_{\mathrm{pqr}}$SFNs.

4.1. Bipolar pqr-Spherical Fuzzy-Weighted Average Operators

In this subsection, the weighted average operator for aggregating ${\mathrm{B}}_{\mathrm{pqr}}$SFNs is introduced, and its key characteristics are examined.

Definition 12. 

Let ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})$, for $j=1,2,\cdots ,l$, be a collection of ${\mathit{B}}_{pqr}$SFNs, then the formulation of the bipolar pqr-spherical fuzzy-weighted average (${\mathit{B}}_{pqr}$SFWA) operator is expressed as follows:

$$B_{pqr}SFWA({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)=\underset{j=1}{\overset{l}{⨁}}\left({ϵ}_j·{\ddot{\tau }}_j\right),$$

whereas $ϵ=({ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_l)$ is meant to the weight of ${\ddot{\tau }}_j$ for $j=1,2,\cdots ,l$ such that ${\sum }_{j=1}^l{ϵ}_j=1$ and each ${ϵ}_j>0$.

The following theorems are derived from the ${\mathrm{B}}_{\mathrm{pqr}}$SFNs operating rules.

Theorem 3. 

The ${\mathit{B}}_{pqr}$SFWA operator is defined as follows:

$$\begin{array}{cc}\hfill B_{pqr}SFWA({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)=& (\sqrt[2p]{1-\prod _{j=1}^l{(1-{\left({\theta }_{\ddot{\tau }}^{+}\right)}^{2p})}^{{ϵ}_j}},\prod _{j=1}^l{\left({\eta }_{\ddot{\tau }}^{+}\right)}^{{ϵ}_j},\prod _{j=1}^l{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^{{ϵ}_j},-\prod _{j=1}^l{(-{\theta }_{\ddot{\tau }}^{-})}^{{ϵ}_j},\hfill \\ \hfill & -\sqrt[2q]{1-\prod _{j=1}^l{(1-{\left({\eta }_{\ddot{\tau }}^{-}\right)}^{2q})}^{{ϵ}_j}},-\sqrt[2r]{1-\prod _{j=1}^l{(1-{\left({\lambda }_{\ddot{\tau }}^{-}\right)}^{2r})}^{{ϵ}_j}}).\hfill \end{array}$$

(2)

Proof. 

See Appendix A (4). □

Theorem 4 

(Idempotency). Let ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})$ denote a family of ${\mathit{B}}_{pqr}$SFNs, for $j\in \{1,2,\cdots ,l\}$. If ${\ddot{\tau }}_j=\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ for all j, then

$$B_{pqr}SFWA({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)=\ddot{\tau }.$$

Proof. 

See Appendix A (5). □

Theorem 5 

(Monotonicity). If ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})$ and ${\ddot{\ell }}_j=({\theta }_{{\ddot{\ell }}_j}^{+},{\eta }_{{\ddot{\ell }}_j}^{+},{\lambda }_{{\ddot{\ell }}_j}^{+},{\theta }_{{\ddot{\ell }}_j}^{-},{\eta }_{{\ddot{\ell }}_j}^{-},{\lambda }_{{\ddot{\ell }}_j}^{-})$ are two families of ${\mathit{B}}_{pqr}$SFNs indexed by $j=1,2,\cdots ,l$, and if ${\ddot{\tau }}_j\le {\ddot{\ell }}_j$ for all j, then

$$B_{pqr}SFWA({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)\le B_{pqr}SFWA({\ddot{\ell }}_1,{\ddot{\ell }}_2,\cdots ,{\ddot{\ell }}_l).$$

Proof. 

See Appendix A (6). □

Theorem 6 

(Boundedness). Let ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})(j=1,2,\cdots ,l)$ denote a collection of ${\mathit{B}}_{pqr}$SFNs. Then, we have the following:

$$\underset{j=1,2,\cdots ,l}{min}\left\{{\ddot{\tau }}_j\right\}\le B_{pqr}SFWA({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)\le \underset{j=1,2,\cdots ,l}{max}\left\{{\ddot{\tau }}_j\right\}.$$

(3)

Proof. 

See Appendix A (7). □

4.2. Bipolar pqr-Spherical Fuzzy-Weighted Geometric Operators

The novel concept of the weighted geometric operator for aggregating ${\mathrm{B}}_{\mathrm{pqr}}$SFNs is presented in this subsection, along with an examination of its key characteristics.

Definition 13. 

Let ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})$, for $j=1,2,\cdots ,l$, be a collection of ${\mathit{B}}_{pqr}$SFNs. Then the formulation of the bipolar pqr-spherical fuzzy-weighted geometric (${\mathit{B}}_{pqr}$SFWG) operator is expressed as follows:

$$B_{pqr}SFWG({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)=\underset{j=1}{\overset{l}{⨂}}\left({ϵ}_j·{\ddot{\tau }}_j\right),$$

where $ϵ=({ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_l)$ denotes the weight of ${\ddot{\tau }}_j$ for $j=1,2,\cdots ,l$ such that ${\sum }_{j=1}^l{ϵ}_j=1$ and ${ϵ}_j>0$.

Theorem 7. 

The ${\mathit{B}}_{pqr}$SFWG operator is defined as follows:

$$\begin{array}{cc}\hfill B_{pqr}SFWG({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)=& (\prod _{j=1}^l{\left({\theta }_{\ddot{\tau }}^{+}\right)}^{{ϵ}_j},\sqrt[2q]{1-\prod _{j=1}^l{(1-{\left({\eta }_{\ddot{\tau }}^{+}\right)}^{2q})}^{{ϵ}_j}},\sqrt[2r]{1-\prod _{j=1}^l{(1-{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^{2r})}^{{ϵ}_j}},\hfill \\ \hfill & -\sqrt[2p]{1-\prod _{j=1}^l{(1-{\left({\theta }_{\ddot{\tau }}^{-}\right)}^{2p})}^{{ϵ}_j}},-\prod _{j=1}^l{(-{\eta }_{\ddot{\tau }}^{-})}^{{ϵ}_j},-\prod _{j=1}^l{(-{\lambda }_{\ddot{\tau }}^{-})}^{{ϵ}_j}).\hfill \end{array}$$

(4)

Proof. 

The same steps as those outlined in the proof of Theorem 3. □

Theorem 8 

(Idempotency). Let ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})$ denote a family of ${\mathit{B}}_{pqr}$SFNs indexed by $j=1,2,\cdots ,l$. If ${\ddot{\tau }}_j=\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ for all j, then

$$B_{pqr}SFWG({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)=\ddot{\tau }.$$

Proof. 

The proof is similar to the proof of Theorem 4. □

Theorem 9 

(Monotonicity). If ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})$ and ${\ddot{\ell }}_j=({\theta }_{{\ddot{\ell }}_j}^{+},{\eta }_{{\ddot{\ell }}_j}^{+},{\lambda }_{{\ddot{\ell }}_j}^{+},{\theta }_{{\ddot{\ell }}_j}^{-},{\eta }_{{\ddot{\ell }}_j}^{-},{\lambda }_{{\ddot{\ell }}_j}^{-})$ are two families of ${\mathrm{B}}_{\mathrm{pqr}}$SFNs indexed by $j=1,2,\cdots ,l$, and if ${\ddot{\tau }}_j\le {\ddot{\ell }}_j$ for all j, then

$$B_{pqr}SFWG({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)\le B_{pqr}SFWG({\ddot{\ell }}_1,{\ddot{\ell }}_2,\cdots ,{\ddot{\ell }}_l).$$

Proof. 

The proof is similar to the proof of Theorem 5. □

Theorem 10 

(Boundedness). Let ${\ddot{\tau }}_j=({\theta }_{{\ddot{\tau }}_j}^{+},{\eta }_{{\ddot{\tau }}_j}^{+},{\lambda }_{{\ddot{\tau }}_j}^{+},{\theta }_{{\ddot{\tau }}_j}^{-},{\eta }_{{\ddot{\tau }}_j}^{-},{\lambda }_{{\ddot{\tau }}_j}^{-})(j=1,2,\cdots ,l)$ denote a collection of ${\mathit{B}}_{pqr}$SFNs. Then,

$$\underset{j=1,2,\cdots ,l}{min}\left\{{\ddot{\tau }}_j\right\}\le B_{nqr}SFWG({\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_l)\le \underset{j=1,2,\cdots ,l}{max}\left\{{\ddot{\tau }}_j\right\}.$$

Proof. 

The proof is similar to the proof of Theorem 6. □

Definition 14. 

Let $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ denote a ${\mathit{B}}_{pqr}$SFN. Then, the score function of ${\mathit{B}}_{pqr}$SFN $\ddot{\tau }$ is defined as follows:

$$scor\left(\ddot{\tau }\right)={\displaystyle \frac{1}{2}}(1+{\left({\theta }_{\ddot{\tau }}^{+}\right)}^p(1-{\left({\eta }_{\ddot{\tau }}^{+}\right)}^q-{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^r)-{\left|{\theta }_{\ddot{\tau }}^{-}\right|}^p(1-{\left|{\eta }_{\ddot{\tau }}^{-}\right|}^q-{\left|{\lambda }_{\ddot{\tau }}^{-}\right|}^r\left)\right).$$

Definition 15. 

Let $\ddot{\tau }=({\theta }_{\ddot{\tau }}^{+},{\eta }_{\ddot{\tau }}^{+},{\lambda }_{\ddot{\tau }}^{+},{\theta }_{\ddot{\tau }}^{-},{\eta }_{\ddot{\tau }}^{-},{\lambda }_{\ddot{\tau }}^{-})$ denote a ${\mathit{B}}_{pqr}$SFN. Then, the accuracy function of ${\mathit{B}}_{pqr}$SFN $\ddot{\tau }$ is defined as follows:

$$accu\left(\ddot{\tau }\right)={\displaystyle \frac{1}{2}}({\left({\theta }_{\ddot{\tau }}^{+}\right)}^p+{\left({\eta }_{\ddot{\tau }}^{+}\right)}^q+{\left({\lambda }_{\ddot{\tau }}^{+}\right)}^r+{\left|{\theta }_{\ddot{\tau }}^{-}\right|}^p+{\left|{\eta }_{\ddot{\tau }}^{-}\right|}^q+{\left|{\lambda }_{\ddot{\tau }}^{-}\right|}^r).$$

Definition 16. 

Let ${\ddot{\tau }}_1=({\theta }_{{\ddot{\tau }}_1}^{+},{\eta }_{{\ddot{\tau }}_1}^{+},{\lambda }_{{\ddot{\tau }}_1}^{+},{\theta }_{{\ddot{\tau }}_1}^{-},{\eta }_{{\ddot{\tau }}_1}^{-},{\lambda }_{{\ddot{\tau }}_1}^{-})$ and ${\ddot{\tau }}_2=({\theta }_{{\ddot{\tau }}_2}^{+},{\eta }_{{\ddot{\tau }}_2}^{+},{\lambda }_{{\ddot{\tau }}_2}^{+},{\theta }_{{\ddot{\tau }}_2}^{-},{\eta }_{{\ddot{\tau }}_2}^{-},{\lambda }_{{\ddot{\tau }}_2}^{-})$ be ${\mathit{B}}_{pqr}$SFNs. Then, from Definitions 14 and 15, we have the following:

1. 

If $scor\left({\ddot{\tau }}_1\right)<scor\left({\ddot{\tau }}_2\right)$, then ${\ddot{\tau }}_1\prec {\ddot{\tau }}_2$ (${\ddot{\tau }}_1$ is inferior to ${\ddot{\tau }}_2$);

2. 

If $scor\left({\ddot{\tau }}_1\right)>scor\left({\ddot{\tau }}_2\right)$, then ${\ddot{\tau }}_1\succ {\ddot{\tau }}_2$ (${\ddot{\tau }}_1$ is superior to ${\ddot{\tau }}_2$);

3. 

If $scor\left({\ddot{\tau }}_1\right)=scor\left({\ddot{\tau }}_2\right)$, then

                        if $accu\left({\ddot{\tau }}_1\right)<accu\left({\ddot{\tau }}_2\right)$, then ${\ddot{\tau }}_1\prec {\ddot{\tau }}_2$ (${\ddot{\tau }}_1$ is "inferior” to ${\ddot{\tau }}_2$);

                      if $accu\left({\ddot{\tau }}_1\right)>accu\left({\ddot{\tau }}_2\right)$, then ${\ddot{\tau }}_1\succ {\ddot{\tau }}_2$ (${\ddot{\tau }}_1$ is “superior” to ${\ddot{\tau }}_2$);

                     if $accu\left({\ddot{\tau }}_1\right)=accu\left({\ddot{\tau }}_2\right)$, then ${\ddot{\tau }}_1={\ddot{\tau }}_2$ (${\ddot{\tau }}_1$ is “equivalent” to ${\ddot{\tau }}_2$).

5. MCDM Algorithms

In this part, two fundamental methods (algorithms) are applied to MCDM. The first one is the TOPSIS technique, and the second one involves using the proposed operators in the ${\mathrm{B}}_{\mathrm{pqr}}$SF context, which introduces an approach for handling MCDM problems in a bipolar pqr-spherical environment using the two aforementioned aggregation operators. Let $\ddot{\tau }=\{{\ddot{\tau }}_1,{\ddot{\tau }}_2,\cdots ,{\ddot{\tau }}_n\}$ denote a family of alternatives and $\ddot{C}=\{{\ddot{C}}_1,{\ddot{C}}_2,\cdots ,{\ddot{C}}_l\}$ denote a set of criteria. Using the ${\mathrm{B}}_{\mathrm{pqr}}$SFNs, the decision-maker assesses the n options in light of the l criteria.

5.1. Extension of the TOPSIS Method with Bipolar pqr-Spherical Fuzzy Sets

TOPSIS is a multi-criteria evaluation approach, established in 1981 by Hwang and Yoon [27], with subsequent improvements by Yoon in [28] and by Hwang et al. in [29]. TOPSIS was founded on the premise that the preferred option should have the lowest geometrical distance from the +ve ideal solution (PIS) and the greatest geometrical distance from the −ve ideal solution (NIS). In 2021, a booklet on TOPSIS in the fuzzy setting was released [30]. The TOPSIS approach comprises several essential metrics. A few researchers introduced the TOPSIS approach based on similarity measures (Hamming, Euclidean, and Dice similarity measures); others discovered the presence of various clustering operators and addressed several real-world issues. As a result, effective and suitable actions will be implemented. However, many strategic decision-making problems inside businesses will be mutually exclusive. Maldonado-Macías et al. [31] analyzed intuitionistic fuzzy TOPSIS. Junior et al. [32] compared the AHP technique with the TOPSIS technique in their application to specific supplier selections. Yilmaz et al. [33] developed the interval-valued type-2 fuzzy TOPSIS method to help international law practitioners and engineers make better decisions while dealing with unpredictable and dynamic systems. Jin et al. [34] introduced the picture fuzzy TOPSIS method to apply it to risk management problems. In 2019, Gündoğdu [10] formulated the fuzzy TOPSIS technique based on spherical fuzzy sets. Akram and Arshad [35] identified bipolar fuzzy TOPSIS methodologies and applications. Recently, Karaaslan and Karamaz [16] proposed the interval-valued pqr-spherical fuzzy TOPSIS method with some applications. This work developed the fuzzy TOPSIS method with ${\mathrm{B}}_{\mathrm{pqr}}$SFs, named the ${\mathrm{B}}_{\mathrm{pqr}}$SF TOPSIS method. The method’s stepwise procedure is explained as follows:

1.

The alternatives are evaluated using l criteria. The final values of the alternatives in relation to each criterion create a ${\mathrm{B}}_{\mathrm{pqr}}$SF decision matrix, which is formed as follows:

$$\begin{array}{c}\hfill \ddot{D}={\left[{\ddot{s}}_{ij}\right]}_{n\times l}=\left\{{\theta }_{ij}^{+},{\eta }_{ij}^{+},{\lambda }_{ij}^{+},{\theta }_{ij}^{-},{\eta }_{ij}^{-},{\lambda }_{ij}^{-}\right\}=\left[\begin{array}{ccc}{\ddot{s}}_{11}& {\ddot{s}}_{12}& \cdots {\ddot{s}}_{1l}\\ {\ddot{s}}_{21}& {\ddot{s}}_{22}& \cdots {\ddot{s}}_{2l}\\ ⋮\\ {\ddot{s}}_{n1}& {\ddot{s}}_{n2}& \cdots {\ddot{s}}_{nl}\end{array}\right]\end{array}$$

$$\begin{array}{c}\hfill =\left[\begin{array}{ccc}\left\{{\theta }_{11}^{+},{\eta }_{11}^{+},{\lambda }_{11}^{+},{\theta }_{11}^{-},{\eta }_{11}^{-},{\lambda }_{11}^{-}\right\}& \left\{{\theta }_{12}^{+},{\eta }_{12}^{+},{\lambda }_{12}^{+},{\theta }_{12}^{-},{\eta }_{12}^{-},{\lambda }_{12}^{-}\right\}& \cdots \left\{{\theta }_{1l}^{+},{\eta }_{1l}^{+},{\lambda }_{1l}^{+},{\theta }_{1l}^{-},{\eta }_{1l}^{-},{\lambda }_{1l}^{-}\right\}\\ \left\{{\theta }_{21}^{+},{\eta }_{21}^{+},{\lambda }_{21}^{+},{\theta }_{21}^{-},{\eta }_{21}^{-},{\lambda }_{21}^{-}\right\}& \left\{{\theta }_{22}^{+},{\eta }_{22}^{+},{\lambda }_{22}^{+},{\theta }_{22}^{-},{\eta }_{22}^{-},{\lambda }_{22}^{-}\right\}& \cdots \left\{{\theta }_{2l}^{+},{\eta }_{2l}^{+},{\lambda }_{2l}^{+},{\theta }_{2l}^{-},{\eta }_{2l}^{-},{\lambda }_{2l}^{-}\right\}\\ ⋮\\ \left\{{\theta }_{n1}^{+},{\eta }_{n1}^{+},{\lambda }_{n1}^{+},{\theta }_{n1}^{-},{\eta }_{n1}^{-},{\lambda }_{n1}^{-}\right\}& \left\{{\theta }_{n2}^{+},{\eta }_{n2}^{+},{\lambda }_{n2}^{+},{\theta }_{n2}^{-},{\eta }_{n2}^{-},{\lambda }_{n2}^{-}\right\}& \cdots \left\{{\theta }_{nl}^{+},{\eta }_{nl}^{+},{\lambda }_{nl}^{+},{\theta }_{nl}^{-},{\eta }_{nl}^{-},{\lambda }_{nl}^{-}\right\}\end{array}\right]\end{array}$$

(5)

where ${\theta }_{ij}^{+}\in [0,1]$ and ${\theta }_{ij}^{-}\in [-1,0]$ represent the +ve and −ve satisfaction degrees of alternative ${\ddot{\tau }}_i$ under criterion ${\ddot{C}}_j$; ${\eta }_{ij}^{+}\in [0,1]$ and ${\eta }_{ij}^{-}\in [-1,0]$ represent the +ve and −ve neutrality degrees; and ${\lambda }_{ij}^{+}\in [0,1]$ and ${\lambda }_{ij}^{-}\in [-1,0]$ represent the +ve and −ve dissatisfaction degrees.

2.

If the criteria are not on the same scale, normalizing is needed, which is the normalizing of the decision matrix according to each criterion’s nature as follows:

$$\begin{array}{cc}\hfill {\left[{\ddot{s}}_{ij}\right]}_{n\times l}& =\{{\theta }_{ij}^{+},{\eta }_{ij}^{+},{\lambda }_{ij}^{+},{\theta }_{ij}^{-},{\eta }_{ij}^{-},{\lambda }_{ij}^{-}\}\hfill \\ \hfill & =\left\{\begin{array}{c}\{{\theta }_{ij}^{+},{\eta }_{ij}^{+},{\lambda }_{ij}^{+},{\theta }_{ij}^{-},{\eta }_{ij}^{-},{\lambda }_{ij}^{-}\}\mathrm{for}\mathrm{benefit}\mathrm{criteria}\hfill \\ \{{\lambda }_{ij}^{+},{\eta }_{ij}^{+},{\theta }_{ij}^{+},{\lambda }_{ij}^{-},{\eta }_{ij}^{-},{\theta }_{ij}^{-}\}\mathrm{for}\mathrm{non}-\mathrm{benefit}\left(\cos \mathrm{t}\right)\mathrm{criteria}\hfill \end{array}\right.\hfill \end{array}$$

(6)

3.

The decision-makers give each criterion a weight value in order to accomplish the normalcy condition, $i.e.,$ regarding the weight vector $ϵ=\{{ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_l\}$ with

$$\sum _{j=1}^l{ϵ}_j=1$$

4.

The ${\mathrm{B}}_{\mathrm{pqr}}$SF-weighted normalized decision matrix is calculated using the attribute weight vector in the manner described below:

$$\begin{array}{cc}\hfill & \ddot{D}\otimes ϵ={\left[{\ddot{\alpha }}_{ij}\right]}_{n\times l}=\left[\begin{array}{ccc}{\ddot{\alpha }}_{11}& {\ddot{\alpha }}_{12}& \cdots {\ddot{\alpha }}_{1l}\\ {\ddot{\alpha }}_{21}& {\ddot{\alpha }}_{22}& \cdots {\ddot{\alpha }}_{2l}\\ ⋮\\ {\ddot{\alpha }}_{n1}& {\ddot{\alpha }}_{n2}& \cdots {\ddot{\alpha }}_{nl}\end{array}\right],\hfill \end{array}$$

(7)

where each $\ddot{\alpha }={\left[(a_{ij},c_{ij},d_{ij},e_{ij},m_{ij},n_{ij})\right]}_{n\times l}=\left({ϵ}_j·{\theta }_{ij}^{+},{ϵ}_j·{\eta }_{ij}^{+},{ϵ}_j·{\lambda }_{ij}^{+},{ϵ}_j·{\theta }_{ij}^{-},{ϵ}_j·{\eta }_{ij}^{-},{ϵ}_j·{\lambda }_{ij}^{-}\right)$ is calculated via Equation (3), for each $i=1,2,\cdots ,n$ and $j=1,2,\cdots ,l$.

5.

The bipolar pqr-spherical fuzzy +ve ideal solution (${\mathrm{B}}_{\mathrm{pqr}}$SFPIS) and the bipolar pqr-spherical fuzzy −ve ideal solution (${\mathrm{B}}_{\mathrm{pqr}}$SFNIS) are calculated as follows:

$$\begin{array}{cc}\hfill B_{pqr}SFPIS=\{& \left\{{\left(M_1^{+}\right)}_P,{\left(N_1^{+}\right)}_P,{\left(V_1^{+}\right)}_P,{\left(M_1^{-}\right)}_P,{\left(N_1^{-}\right)}_P,{\left(V_1^{-}\right)}_P\right\},\hfill \\ & \left\{{\left(M_2^{+}\right)}_P,{\left(N_2^{+}\right)}_P,{\left(V_2^{+}\right)}_P,{\left(M_2^{-}\right)}_P,{\left(N_2^{-}\right)}_P,{\left(V_2^{-}\right)}_P\right\},\hfill \\ \hfill & \cdots ,\left\{{\left(M_l^{+}\right)}_P,{\left(N_l^{+}\right)}_P,{\left(V_l^{+}\right)}_P,{\left(M_l^{-}\right)}_P,{\left(N_l^{-}\right)}_P,{\left(V_l^{-}\right)}_P\right\}\},\hfill \end{array}$$

where for $j=1,2,\cdots ,l$;

$$\begin{array}{cc}\hfill & {\left(M_j^{+}\right)}_P=\underset{i}{max}\left\{{\theta }_{ij}^{+}\right\},{\left(N_j^{+}\right)}_P=\underset{i}{min}\left\{{\eta }_{ij}^{+}\right\},{\left(V_j^{+}\right)}_P=\underset{i}{min}\left\{{\lambda }_{ij}^{+}\right\},\hfill \\ \hfill & {\left(M_j^{-}\right)}_P=\underset{i}{min}\left\{{\theta }_{ij}^{-}\right\},{\left(N_j^{-}\right)}_P=\underset{i}{max}\left\{{\eta }_{ij}^{-}\right\},{\left(V_j^{-}\right)}_P=\underset{i}{max}\left\{{\lambda }_{ij}^{-}\right\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill B_{pqr}SFNIS=\{& \left\{{\left(M_1^{+}\right)}_N,{\left(N_1^{+}\right)}_N,{\left(V_1^{+}\right)}_N,{\left(M_1^{-}\right)}_N,{\left(N_1^{-}\right)}_N,{\left(V_1^{-}\right)}_N\right\},\hfill \\ & \left\{{\left(M_2^{+}\right)}_N,{\left(N_2^{+}\right)}_N,{\left(V_2^{+}\right)}_N,{\left(M_2^{-}\right)}_N,{\left(N_2^{-}\right)}_N,{\left(V_2^{-}\right)}_N\right\},\hfill \\ \hfill & \cdots ,\left\{{\left(M_l^{+}\right)}_N,{\left(N_l^{+}\right)}_N,{\left(V_l^{+}\right)}_N,{\left(M_l^{-}\right)}_N,{\left(N_l^{-}\right)}_N,{\left(V_l^{-}\right)}_N\right\}\},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & {\left(M_j^{+}\right)}_N=\underset{i}{min}\left\{{\theta }_{ij}^{+}\right\},{\left(N_j^{+}\right)}_N=\underset{i}{min}\left\{{\eta }_{ij}^{+}\right\},{\left(V_j^{+}\right)}_N=\underset{i}{max}\left\{{\lambda }_{ij}^{+}\right\},\hfill \\ \hfill & {\left(M_j^{-}\right)}_N=\underset{i}{max}\left\{{\theta }_{ij}^{-}\right\},{\left(N_j^{-}\right)}_N=\underset{i}{max}\left\{{\eta }_{ij}^{-}\right\},{\left(V_j^{-}\right)}_N=\underset{i}{min}\left\{{\lambda }_{ij}^{-}\right\}.\hfill \end{array}$$

6.

The normalized Euclidean distance of alternatives, ${\ddot{\tau }}_i$ for $(i=1,2,\cdots ,n)$, from bipolar pqr-spherical fuzzy +ve and −ve ideal solutions, is calculated as follows:

$${\ddot{D}}_l({\ddot{\tau }}_i,B_{pqr}SFPIS)=\sqrt{\begin{array}{cc}\hfill {\displaystyle \frac{1}{6l}}\sum _{j=1}^l& [{\left(({\theta }_{{\ddot{\tau }}_{ij}}^{+}{)}^p-{\left({\left(M_j^{+}\right)}_P\right)}^p\right)}^2+{\left({\left({\eta }_{{\ddot{\tau }}_{ij}}^{+}\right)}^q-{\left({\left(N_j^{+}\right)}_P\right)}^q\right)}^2\hfill \\ \hfill & +{\left({\left({\lambda }_{{\ddot{\tau }}_{ij}}^{+}\right)}^r-{\left({\left(V_j^{+}\right)}_P\right)}^r\right)}^2+{\left({\left({\theta }_{{\ddot{\tau }}_{ij}}^{-}\right)}^p-{\left({\left(M_j^{-}\right)}_P\right)}^p\right)}^2\hfill \\ \hfill & +{\left({\left({\eta }_{{\ddot{\tau }}_{ij}}^{-}\right)}^q-{\left({\left(N_j^{-}\right)}_P\right)}^q\right)}^2+{\left({\left({\lambda }_{{\ddot{\tau }}_{ij}}^{-}\right)}^r-{\left({\left(V_j^{-}\right)}_P\right)}^r\right)}^2]\hfill \end{array}}$$

(8)

and

$${\ddot{D}}_l({\ddot{\tau }}_i,B_{pqr}SFNIS)=\sqrt{\begin{array}{cc}\hfill {\displaystyle \frac{1}{6l}}\sum _{j=1}^l& [{\left(({\theta }_{{\ddot{\tau }}_{ij}}^{+}{)}^p-{\left({\left(M_j^{+}\right)}_N\right)}^p\right)}^2+{\left({\left({\eta }_{{\ddot{\tau }}_{ij}}^{+}\right)}^q-{\left({\left(N_j^{+}\right)}_N\right)}^q\right)}^2\hfill \\ \hfill & +{\left({\left({\lambda }_{{\ddot{\tau }}_{ij}}^{+}\right)}^r-{\left({\left(V_j^{+}\right)}_N\right)}^r\right)}^2+{\left({\left({\theta }_{{\ddot{\tau }}_{ij}}^{-}\right)}^p-{\left({\left(M_j^{-}\right)}_N\right)}^p\right)}^2\hfill \\ \hfill & +{\left({\left({\eta }_{{\ddot{\tau }}_{ij}}^{-}\right)}^q-{\left({\left(N_j^{-}\right)}_N\right)}^q\right)}^2+{\left({\left({\lambda }_{{\ddot{\tau }}_{ij}}^{-}\right)}^r-{\left({\left(V_j^{-}\right)}_N\right)}^r\right)}^2]\hfill \end{array}}$$

(9)

7.

The relative closeness degree of the corresponding alternative to the bipolar pqr-spherical fuzzy +ve ideal solution, denoted by ${\varpi }_i$ for $i=1,2,\cdots ,n$, is computed by the following formula:

$${\varpi }_i={\displaystyle \frac{{\ddot{D}}_l({\ddot{\tau }}_i,B_{pqr}SFNIS)}{{\ddot{D}}_l({\ddot{\tau }}_i,B_{pqr}SFPIS)+{\ddot{D}}_l({\ddot{\tau }}_i,B_{pqr}SFNIS)}}.$$

(10)

8.

The alternative that has the highest closeness degree value is selected as the best alternative.

5.2. Bipolar pqr-Spherical Fuzzy Aggregation Operator Method

Aggregation tools are essential in MCDM issues that utilize this method, as aggregation involves merging multiple numerical values into a single value; thus, we can refer to this method as an aggregator [36,37,38]. Numerous researchers have focused on the aggregation operators associated with different types of fuzzy numbers. After exploring certain operations concerning intuitionistic fuzzy numbers (IFNs), both geometric and averaging aggregation operators based on IFNs were developed, as described by Xu and Yager [39] and Xu [40]. Subsequently, picture fuzzy geometric and averaging aggregation operators were studied in [41]. Then, spherical aggregation operators were discussed in [25] in the context of MCDM problems. The latter operators were extended to the so-called T-spherical aggregation operators [42]. The concepts of pqr-spherical weighted (geometric and averaging) aggregation operators were proposed in [12,13], generalizing all earlier operators. The T-spherical aggregation operators were further extended to the bipolar T-spherical aggregation operators that appeared in [23], aggregating the given +ve and −ve values. In this study, we build upon the previous ideas by extending bipolar T-spherical aggregation operators to ${\mathrm{B}}_{\mathrm{pqr}}$SF aggregation operators, which are the basis of the new method in MCDM. The stepwise procedure of the proposed method is explained as follows:

1.

The alternatives are evaluated using the l criterion. The final values of the alternatives with regard to each criterion are used to generate a ${\mathrm{B}}_{\mathrm{pqr}}$SF decision matrix similar to Equation (5).

2.

If the criteria are not on the same scale, normalizing is needed, which is the normalizing of the decision matrix according to each criterion’s nature, as in Equation (6).

3.

The decision-makers give each criterion a weight value in order to accomplish the normalcy condition, $i.e.,$ regarding the weight vector $ϵ=\{{ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_l\}$ with

$$\sum _{j=1}^l{ϵ}_j=1.$$

4.

The ${\mathrm{B}}_{\mathrm{pqr}}$SFWA operator or ${\mathrm{B}}_{\mathrm{pqr}}$SFWG operator is computed via ${\ddot{s}}_i$, where

$${\ddot{s}}_i=\left\{{\ddot{\theta }}_i^{+},{\ddot{\eta }}_i^{+},{\ddot{\lambda }}_i^{+},{\ddot{\theta }}_i^{-},{\ddot{\eta }}_i^{-},{\ddot{\lambda }}_i^{-}\right\}=B_{pqr}SFWA\left({\ddot{s}}_{i1},{\ddot{s}}_{i2},\cdots ,{\ddot{s}}_{il}\right)$$

and is obtained by Equation (2), or

$${\ddot{s}}_i=\left\{{\ddot{\theta }}_i^{+},{\ddot{\eta }}_i^{+},{\ddot{\lambda }}_i^{+},{\ddot{\theta }}_i^{-},{\ddot{\eta }}_i^{-},{\ddot{\lambda }}_i^{-}\right\}=B_{pqr}SFWG\left({\ddot{s}}_{i1},{\ddot{s}}_{i2},\cdots ,{\ddot{s}}_{il}\right)$$

is obtained by Equation (4), for $i=1,2,\cdots ,n$.

5.

Score function of ${\ddot{\tau }}_i$, denoted by $scor\left({\ddot{\tau }}_i\right)$, is calculated using Definition 14.

6.

We sort the alternatives in decreasing order using the score values we obtained in step 5.

6. A Numerical Example Example of the Supplier Assessment Process

The supplier assessment process is critical for any business that wants to ensure its entire supply chain will be both efficient and reliable while also being cost-effective. The supplier assessment process involves evaluating potential and existing suppliers to determine their ability to meet the firm’s requirements in terms of quality, price, delivery time, and other defining criteria.

Supplier assessment data were collected in order to create long-term relationships with suppliers who can always provide products or services that meet company standards. This process also allows businesses to mitigate risks, like supply chain disruptions or compromised quality, and can lead to cost savings by making supplier partnerships more effective.

In this part, three criteria are used by a firm to assess four suppliers (${\ddot{\tau }}_1$, ${\ddot{\tau }}_2$, ${\ddot{\tau }}_3$, ${\ddot{\tau }}_4$) for a long-term contract. The criteria are as follows: $C_1:$ cost (minimization), where lower costs are preferred; $C_2:$ quality (maximization): superior quality is appreciated; $C_3:$ delivery time (minimization): It is preferable to have faster delivery times, where the weight vector for the criteria is defined as $ϵ=\left[0.40.40.2\right].$ To apply the first algorithm:

1.

We obtain the ${\mathrm{B}}_{\mathrm{pqr}}$SF evaluation matrix $\ddot{D}=\left({\ddot{s}}_{ij}\right)$ in

Table 2. ${\mathrm{B}}_{\mathrm{pqr}}$SF evaluation matrix.

${\ddot{\mathit{\tau }}}_{\mathit{j}}$	Cost ${\mathit{C}}_1$	Quality ${\mathit{C}}_2$	Delivery Time ${\mathit{C}}_3$
${\ddot{\tau }}_1$	$\left\{0.85,0.01,0.05,-0.1,-0.3,-0.06\right\}$	$\left\{0.69,0.1,0.02,-0.21,-0.03,-0.33\right\}$	$\left\{0.54,0.4,0.09,-0.42,-0.05,-0.1\right\}$
${\ddot{\tau }}_2$	$\left\{0.79,0.03,0.08,-0.2,-0.4,-0.02\right\}$	$\left\{0.72,0.2,0.03,-0.17,-0.1,-0.065\right\}$	$\left\{0.68,0.09,0.21,-0.32,-0.23,-0.08\right\}$
${\ddot{\tau }}_3$	$\left\{0.83,0.01,0.05,-0.13,-0.03,-0.3\right\}$	$\left\{0.7,0.03,0.14,-0.21,-0.02,-0.11\right\}$	$\left\{0.73,0.1,0.17,-0.22,-0.03,-0.36\right\}$
${\ddot{\tau }}_4$	$\left\{0.8,0.09,0.04,-0.24,-0.04,-0.32\right\}$	$\left\{0.82,0.06,0.01,-0.12,-0.07,-0.5\right\}$	$\left\{0.81,0.09,0.1,-0.12,-0.03,-0.26\right\}$

.

2.

The weight vector for the criteria above is $ϵ=\left[0.40.40.2\right]$.

3.

In this step, normalizing is needed as criteria are not on the same scale; in order to normalize them, we use step (3) in Definition 8. Criteria $C_1$ and $C_3$ need to be normalized by using Equation (6), as shown in

Table 3. Normalized ${\mathrm{B}}_{\mathrm{pqr}}$SF evaluation matrix.

${\ddot{\mathit{\tau }}}_{\mathit{j}}$	Cost ${\mathit{C}}_1$	Quality ${\mathit{C}}_2$	Delivery Time ${\mathit{C}}_3$
${\ddot{\tau }}_1$	$\left\{0.05,0.01,0.85,-0.06,-0.3,-0.1\right\}$	$\left\{0.69,0.1,0.02,-0.21,-0.03,-0.33\right\}$	$\left\{0.09,0.4,0.54,-0.1,-0.05,-0.42\right\}$
${\ddot{\tau }}_2$	$\left\{0.08,0.03,0.79,-0.02,-0.4,-0.2\right\}$	$\left\{0.72,0.2,0.03,-0.17,-0.1,-0.065\right\}$	$\left\{0.21,0.09,0.68,-0.08,-0.23,-0.32\right\}$
${\ddot{\tau }}_3$	$\left\{0.05,0.01,0.83,-0.3,-0.03,-0.13\right\}$	$\left\{0.7,0.03,0.14,-0.21,-0.02,-0.11\right\}$	$\left\{0.17,0.1,0.73,-0.36,-0.03,-0.22\right\}$
${\ddot{\tau }}_4$	$\left\{0.04,0.09,0.8,-0.32,-0.04,-0.24\right\}$	$\left\{0.82,0.06,0.01,-0.12,-0.07,-0.5\right\}$	$\left\{0.1,0.09,0.81,-0.26,-0.03,-0.12\right\}$

.

4.

We calculate the weighted normalized ${\mathrm{B}}_{\mathrm{pqr}}$SF decision matrix values by using Equation (7), where $p=3$, $q=4$, and $r=5$, as obtained in

Table 4. Weighted normalized ${\mathrm{B}}_{\mathrm{pqr}}$SF evaluation matrix.

${\ddot{\mathit{\tau }}}_{\mathit{j}}$	Cost C1	Quality C2	Delivery Time C3
${\ddot{\tau }}_1$	{0.0018, 0.1585, 0.9524, −0.3245,−0.0716, −0.0083}	{0.3548, 0.3981, 0.3092, −0.5357,−0.0007, −0.0907}	{0.0047, 0.8326, 0.8841, −0.631,−0.0017, −0.1279}
${\ddot{\tau }}_2$	{0.0047, 0.246, 0.9317, −0.2091,−0.1272, −0.0333}	{0.3876, 0.5253, 0.3492, −0.4922,−0.008, −0.0035}	{0.0258, 0.6178, 0.9258, −0.6034,−0.0354, −0.0742}
${\ddot{\tau }}_3$	{0.0018, 0.1585, 0.9456, −0.6178,−0.0007,−0.0141}	{0.3655, 0.246, 0.5544, −0.5357,−0.0003,−0.0101}	{0.0169, 0.631, 0.939, −0.8152,−0.0006,−0.0351}
${\ddot{\tau }}_4$	{0.0012, 0.3817, 0.9352, −0.634,−0.0013, −0.048}	{0.5129, 0.3245, 0.2512, −0.4282,−0.0039, −0.2082}	{0.0058, 0.6178, 0.9587, −0.7638,−0.0006,−0.0104}

.

5.

The ${\mathrm{B}}_{\mathrm{pqr}}$SFPIS and ${\mathrm{B}}_{\mathrm{pqr}}$SFNIS are computed in

Table 5. B_pqrSFPIS and B_pqrSFNIS.

Ideal Solutions	Cost C1	Quality C2	Delivery Time C3
BpqrSFPIS	{0.0047, 0.1585, 0.91, −0.634, −0.0007, −0.0083}	{0.5129, 0.246, 0.1585, −0.5357, −0.0003, −0.0035}	{0.0258, 0.6178, 0.8841, −0.8152, −0.0006, −0.0104}
BpqrSFNIS	{0.0012, 0.1585, 0.9371, −0.2091, −0.0007, −0.048}	{0.3548, 0.246, 0.4555, −0.4282, −0.0003, −0.2082}	{0.0047, 0.6178, 0.9587, −0.6034, −0.0006, −0.1279}

:

6.

The Euclidean distances of alternatives ${\ddot{\tau }}_i$ for $(i=1,2,\cdots ,n)$ from the bipolar pqr-spherical fuzzy +ve and −ve ideal solutions are calculated from Equations (8) and (9), respectively, in

Table 6. The Euclidean distances of all alternatives with ${\mathrm{B}}_{\mathrm{pqr}}$SFPIS and ${\mathrm{B}}_{\mathrm{pqr}}$SFNIS, with the weighted normalized ${\mathrm{B}}_{\mathrm{pqr}}$SF evaluation matrix.

Alternatives	$\ddot{\mathit{D}}({\ddot{\mathit{\tau }}}_{\mathit{i}},{\mathit{B}}_{\mathit{pqr}}\mathit{SFPIS})$	$\ddot{\mathit{D}}({\ddot{\mathit{\tau }}}_{\mathit{i}},{\mathit{B}}_{\mathit{pqr}}\mathit{SFNIS})$
${\ddot{\tau }}_1$	$0.121$	$0.1036$
${\ddot{\tau }}_2$	$0.1043$	$0.0435$
${\ddot{\tau }}_3$	$0.052$	$0.0967$
${\ddot{\tau }}_4$	$0.0701$	$0.0841$

.

7.

The relative closeness degree of the corresponding alternative to the bipolar pqr-spherical fuzzy +ve ideal solution, denoted by ${\varpi }_i$ for $i=1,2,\cdots ,n$, is computed by Formula (10), which is shown in

Table 7. The relative closeness degrees.

Alternatives	${\mathit{\varpi }}_{\mathit{i}}$
$i=1$	$0.4612$
$i=2$	$0.2942$
$i=3$	$0.6505$
$i=4$	$0.5452$

and Figure 2.

8.

The alternatives are ranked based on their final scores (relative closeness values) as follows:

$${\ddot{\tau }}_3>{\ddot{\tau }}_4>{\ddot{\tau }}_1>{\ddot{\tau }}_2.$$

Therefore, ${\ddot{\tau }}_3$ is the best supply to be selected.

To apply the second algorithm to the same numerical example used for the first algorithm:

1.

We obtain the ${\mathrm{B}}_{\mathrm{pqr}}$SF evaluation matrix $\ddot{D}=\left({\ddot{\tau }}_{ij}\right)$, as in Table 2.

2.

The weight vector for criteria is $ϵ=\left[0.40.40.2\right]$.

3.

Normalizing, in this step, is needed, as criteria are not on the same scale; and in order to normalize them, we use step 3 in Definition 8. Criteria $C_1$ and $C_3$ need to be normalized by using Equation (6), as shown in Table 3.

4.

In order to aggregate the information presented in Table 3, we calculate the ${\mathrm{B}}_{\mathrm{pqr}}$SFWA operator or the ${\mathrm{B}}_{\mathrm{pqr}}$SFG operator, computed via ${\ddot{s}}_i$ as shown below:

$$\begin{array}{cc}\hfill {\ddot{s}}_1& =B_{pqr}SFWA({\ddot{s}}_{11},{\ddot{s}}_{12},{\ddot{s}}_{13})=\{0.3548,0.0525,0.1732,-0.1097,-0.0716,-0.132\}\hfill \\ \hfill {\ddot{s}}_2& =B_{pqr}SFWA({\ddot{s}}_{21},{\ddot{s}}_{22},{\ddot{s}}_{23})=\{0.3877,0.0798,0.2072,-0.0621,-0.1274,-0.0745\}\hfill \\ \hfill {\ddot{s}}_3& =B_{pqr}SFWA({\ddot{s}}_{31},{\ddot{s}}_{32},{\ddot{s}}_{33})=\{0.3655,0.0246,0.397,-0.2698,-0.0008,-0.0352\}\hfill \\ \hfill {\ddot{s}}_4& =B_{pqr}SFWA({\ddot{s}}_{41},{\ddot{s}}_{42},{\ddot{s}}_{43})=\{0.5129,0.0765,0.139,-0.2074,-0.0039,-0.2082\}\hfill \end{array}$$

5.

Score values; the aggregated data are provided in

Table 8. The score values.

Alternatives	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_{\mathit{i}}\right)$
$i=1$	$0.605$
$i=2$	$0.6205$
$i=3$	$0.5996$
$i=4$	$0.6793$

.

6.

Based on the score values in Table 8, the ranking of alternative suppliers, as depicted in Figure 3, is as follows:

$${\ddot{\tau }}_4>{\ddot{\tau }}_2>{\ddot{\tau }}_1>{\ddot{\tau }}_3.$$

7. Sensitivity Analysis

In this section, we performed a sensitivity analysis of the suggested approaches by varying the parameters $p,q$, and r. As a result, the ${\mathrm{B}}_{\mathrm{pqr}}$SF TOPSIS procedure was divided into three situations. First, we set values to the parameters $p=3$, $q=4$, and $r=5$. Second, we swapped the values of p and r while keeping the value of q. Finally, we assigned a similar value to each parameter to analyze the performance of the proposed operators. Data from the preceding DM problem were gathered again using the processes outlined in the pqr-TOPSIS technique. Later, the relative closeness degrees for the alternatives were determined, and in all situations, ${\ddot{\tau }}_3$ was found to be the best option and ${\ddot{\tau }}_2$ to be the worst; see

Table 9. ${\mathrm{B}}_{\mathrm{pqr}}$SF TOPSIS ranking results.

${\mathrm{B}}_{\mathbf{pqr}}$SFS	${\mathit{\varpi }}_1$	${\mathit{\varpi }}_2$	${\mathit{\varpi }}_3$	${\mathit{\varpi }}_4$	Ranking Results
$p=3,q=4,r=5$	0.454	0.2931	0.6215	0.5595	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$
$p=5,q=4,r=3$	0.463	0.3406	0.5909	0.5419	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$
$p=3,q=3,r=3$	0.4389	0.3202	0.6468	0.6125	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$

and Figure 4.

In the second part of the analysis, we altered the parameters $p,q$, and r. As a consequence, the operation started with the parameters $p=3$, $q=4$, and $r=5$, in order to investigate the performance of the proposed operator. Data from the aforementioned DM problem were aggregated once more with the ${\mathrm{B}}_{\mathrm{pqr}}$SFWA operator. The score function was then used to calculate the score values for the alternatives, which were ranked by ${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$. Fixing $q=4$ and switching p and r values yielded alternatives ranked as ${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$ (the rank of the final two options flips). This adjustment occurred when the value of $q\le 5$.

Table 10. ${\mathrm{B}}_{\mathrm{pqr}}$SFWA ranking results.

${\mathrm{B}}_{\mathbf{pqr}}$SFWA	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_1\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_2\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_3\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_4\right)$	Ranking Results
$p=3,q=4,r=5$	0.605	0.6205	0.5996	0.6793	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$p=5,q=4,r=3$	0.5494	0.561	0.5493	0.6193	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$p=4,q=5,r=3$	0.5722	0.5859	0.5732	0.6468	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$p=3,q=3,r=3$	0.6045	0.6194	0.5938	0.679	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$p=2,q=4,r=6$	0.6503	0.6699	0.6244	0.71	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$p=3,q=5,r=7$	0.605	0.6206	0.601	0.6792	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$p=7,q=5,r=3$	0.5234	0.5315	0.5244	0.5794	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$p=5,q=9,r=13$	0.5496	0.5615	0.5527	0.6196	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$p=13,q=9,r=5$	0.5025	0.5044	0.503	0.524	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$

and Figure 5 display the scores for different values of $p,q$, and r and show that ${\ddot{\tau }}_3$ is always the best.

8. Comparative Analysis

This section of the paper provides a comparative analysis to ensure the stability of the suggested work. Before proceeding, it is recommended that you review Remark 1 in Section 3 and Table 1 to comprehend the superiority of the provided work. For example, if we set $p=q=r=2$, we can derive the BT-SF TOPSIS technique. In addition, if we remove the neutrality degree membership and set $p=r$, the q-ROF TOPSIS technique can be achieved. For additional study, we consider the numerical data in Table 9,

Table 11. q-ROF TOPSIS ranking results.

q-ROFS	${\mathit{\varpi }}_1$	${\mathit{\varpi }}_2$	${\mathit{\varpi }}_3$	${\mathit{\varpi }}_4$	Ranking Results
$q=1\left(IFS\right)$	0.1672	0.309	0.7211	0.3239	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$
$q=2\left(PyFS\right)$	0.1749	0.3062	0.6176	0.5051	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$
$q=3\left(FFS\right)$	0.2213	0.3804	0.562	0.6809	${\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$
$q=5$	0.2535	0.4317	0.6063	0.7735	${\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$
$q=15$	0.2636	0.3804	0.6181	0.7653	${\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$

,

Table 12. BT-SF TOPSIS ranking results.

BT-SFS	${\mathit{\varpi }}_1$	${\mathit{\varpi }}_2$	${\mathit{\varpi }}_3$	${\mathit{\varpi }}_4$	Ranking Results
$q=1\left(BPFS\right)$	0.474	0.4438	0.6549	0.5778	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$
$q=2\left(BSFS\right)$	0.3661	0.4233	0.7135	0.6862	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$
$q=3$	0.4406	0.3051	0.6776	0.5063	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$
$q=5$	0.3557	0.50008	0.659	0.6084	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$

. We used our proposed approaches along with existing ones to the data supplied by the decision expert. The relative closeness of the alternatives suggests that ${\ddot{\tau }}_3$ is the optimum alternative for all the previously described techniques, except in q-ROF TOPSIS, where $q=3,5,15$, ${\ddot{\tau }}_4$ is optimal, whereas either ${\ddot{\tau }}_1$ or ${\ddot{\tau }}_2$ is ranked last; see Figure 4, Figure 6, Figure 7.

For the second method, Remark 1 in Section 3 justifies that different existing aggregation operators (in the contexts of IFSs, PyFSs, FFSs, q-ROFSs, mn-ROFSs, PFSs, SFSs, T-SFSs, pqr-FSs, BIFSs, BPyFSs, FBFSs, Bq-ROFSs, Bmn-ROFSs, BPFSs, SBFSs, and BT-SFSs) can be deduced from the Bpqr-SFWA operator. The aggregated results presented in Table 10,

Table 13. pqr-SFWA Ranking Results.

pqr-SFWA	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_1\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_2\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_3\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_4\right)$	Ranking Results
$p=3,q=4,r=5$	0.5737	0.5863	0.5738	0.6372	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$p=5,q=4,r=3$	0.5307	0.5373	0.507	0.5833	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$p=2,q=4,r=6$	0.6149	0.631	0.6187	0.6809	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$p=3,q=5,r=7$	0.5737	0.5865	0.5774	0.6373	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$p=7,q=5,r=3$	0.5131	0.5167	0.4884	0.5529	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$

,

Table 14. Ranking results of q-ROFWA operators for different values of q.

q-ROFWA	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_1\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_2\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_3\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_4\right)$	Ranking Results
$q=1\left(IFWA\right)$	0.225	0.2383	0.0199	0.3759	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=2\left(PyFWA\right)$	0.1999	0.2191	0.0838	0.3424	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=3\left(FFWA\right)$	0.1422	0.1633	0.093	0.2718	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=5$	0.0656	0.0821	0.0611	0.1691	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=15$	0.0015	0.0029	0.0019	0.0207	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$

,

Table 15. Ranking results of ${\mathrm{B}}_{\mathrm{q}}$-ROFWA operators.

${\mathrm{B}}_{\mathbf{q}}$-ROFWA	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_1\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_2\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_3\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_4\right)$	Ranking Results
$q=1\left(BIFWA\right)$	0.1914	0.176	−0.0544	0.2533	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$
$q=2\left(BPyFWA\right)$	0.094	0.0913	−0.0183	0.1585	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$
$q=3\left(BFFWA\right)$	0.0379	0.0424	−0.0035	0.0818	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=5$	0.0056	0.0087	0.0007	0.0201	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=15$	0.000024	0.000003	0.00000052	0.0000005	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$

,

Table 16. Ranking results of ${\mathrm{B}}_{nm}$ROFWA operators for different values of m,n.

${\mathrm{B}}_{\mathbf{nm}}$-ROFWA	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_1\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_2\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_3\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_4\right)$	Ranking Results
$n=1,m=2$	0.1718	0.1896	0.0055	0.2102	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$n=2,m=1$	0.156	0.1134	−0.044	0.2589	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$
$n=3,m=7$	0.0734	0.089	0.0672	0.1343	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$n=7,m=3$	0.0276	0.0212	−0.013	0.0201	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$

,

Table 17. Ranking results of T-SFWA operators for different values of q.

T-SFWA	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_1\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_2\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_3\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_4\right)$	Ranking Results
$q=1\left(PFWA\right)$	−0.3743	−0.254	−0.4908	−0.3733	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=2\left(SFWA\right)$	−0.3678	−0.2659	−0.4548	−0.3925	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=5$	−0.2139	−0.1521	−0.2182	−0.212	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=15$	−0.2359	−0.0124	−0.0267	−0.0227	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$

and

Table 18. Ranking results of BT-SFWA for different values of q.

BT-SFWA	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_1\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_2\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_3\right)$	$\mathit{scor}\left({\ddot{\mathit{\tau }}}_4\right)$	Ranking Results
$q=1\left(BPSFWA\right)$	0.563	0.5854	0.4691	0.5755	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=2\left(BSFWA\right)$	0.5487	0.5589	0.5481	0.606	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$
$q=5$	0.5049	0.5076	0.5057	0.5287	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$
$q=15$	0.500003	0.50001	0.500005	0.50052	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$

suggest that ${\ddot{\tau }}_4$ is the best alternative to be selected in almost all the operators except in the T-SF-weighted average (T-SFWA) and BTSF-weighted average (BTSFWA) operators for $q=1$, where ${\ddot{\tau }}_2$ is the best. The good news is that the score values generated by the proposed operators for various values of $p,q$, and r demonstrate that the results almost match their findings, implying the proposed work’s stability. Figure 5, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show how the prospective operators’ score values compare to those of existing operators. This demonstrates that the proposed operator yields stable mathematical conclusions and can be utilized in place of existing operators in the context of Bpqr-SFSNs, and any extensions that may be derived from them. For some additional details, see the comparison in

Table 19. A comparison between the proposed method and existing methods.

Method	Parameter(s)	Ranking Results	Optimal Alternative
IFS TOPSIS [31]	n/a	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_3$
PyFS TOPSIS [43]	n/a	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_3$
FFS TOPSIS [44]	n/a	${\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
q-ROF TOPSIS [45]	$q=5$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$q=15$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
BT-SFS TOPSIS [23]	$q=1$	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$	${\ddot{\tau }}_3$
	$q=2$	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_3$
	$q=3$	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$	${\ddot{\tau }}_3$
	$q=5$	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_3$
${\mathrm{B}}_{\mathrm{pqr}}$SF TOPSIS (proposed)	$p=3,q=4,r=5$	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$	${\ddot{\tau }}_3$
	$p=5,q=4,r=3$	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$	${\ddot{\tau }}_3$
	$p=3,q=3,r=3$	${\ddot{\tau }}_3\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2$	${\ddot{\tau }}_3$
q-ROFWA [20]	$q=1$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=2$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=5$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=15$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
T-SFWAO [42]	$q=1$	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_2$
	$q=2$	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_2$
	$q=5$	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_2$
	$q=15$	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_2$
pqr-SFWAO [12]	$p=3,q=4,r=5$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$p=5,q=4,r=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$p=2,q=4,r=6$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$p=3,q=5,r=7$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$p=7,q=5,r=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
${\mathrm{B}}_{\mathrm{q}}$-ROFWAO [20]	$q=1$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=2$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=5$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=15$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
${\mathrm{B}}_{nm}$-ROFWAO [21]	$n=1,m=2$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$n=2,m=1$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$n=3,m=7$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$n=7,m=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
BT-SFWA [23]	$q=1$	${\ddot{\tau }}_2\succ {\ddot{\tau }}_4\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_2$
	$q=2$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$q=5$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$q=15$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
${\mathrm{B}}_{\mathrm{pqr}}$SFWAO (proposed)	$p=3,q=4,r=5$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$p=5,q=4,r=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$p=4,q=5,r=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$p=3,q=3,r=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$p=2,q=4,r=6$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$p=3,q=5,r=7$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_1\succ {\ddot{\tau }}_3$	${\ddot{\tau }}_4$
	$p=7,q=5,r=3$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$p=5,q=9,r=13$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$
	$p=13,q=9,r=5$	${\ddot{\tau }}_4\succ {\ddot{\tau }}_2\succ {\ddot{\tau }}_3\succ {\ddot{\tau }}_1$	${\ddot{\tau }}_4$

.

9. Conclusions

In this work, we define a new notion, named bipolar pqr-spherical fuzzy sets (${\mathrm{B}}_{\mathrm{pqr}}$SFS), which is a combination of bipolar fuzzy sets and pqr-spherical fuzzy sets. Then, we investigate some basic properties of ${\mathrm{B}}_{\mathrm{pqr}}$FSs, including the basic operational laws, complement operation, union operation, and intersection operation. This new framework allows for a more nuanced representation of uncertainty by accounting for different degrees of membership, non-membership, and neutrality, in addition to bipolarity. This adaptability of bipolar pqr-SFS has proven to be immensely valuable in intricate decision-making processes that involve evaluating both affirmative and adverse perceptions. Its utility in multi-criteria group decision-making problems has been demonstrated, and it has been shown that this framework is well-suited for capturing the complexities of bipolar judgmental reasoning during the decision-making process. This has been conducted by using two algorithms—the first one is referred to as the `technique for order of preference by similarity to ideal solution’ (TOPSIS) and the second one involves using the proposed operators themselves in the ${\mathrm{B}}_{\mathrm{pqr}}$SF context, as well as providing a numerical example, which demonstrates the usefulness of the proposed set. Targeted applications were developed in the multi-criteria decision-making field to address problems in the ${\mathrm{B}}_{\mathrm{pqr}}$SF environment using the two aforementioned aggregation operators. Additionally, a comparative study was used to show the applicability and viability of the proposed approach.

Future works will be applied with more decision-making methods, like VIKOR. In addition, a few other aggregation operators can be introduced to enhance the existing approaches owing to the generation of a more precise ranking outcome. $℞₱$