An Approach to Select the Investment Based on Bipolar Picture Fuzzy Sets

Abstract

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

1 Introduction

Coung [6] presented a concept of picture fuzzy sets ($P_{c}$FSs), a generalization of fuzzy sets (FSs) [35] in 2013. $P_{c}$FSs consists of three well-known degrees, belonging degree (BD), non-belonging degree (NBD) and neutral degree (ND) so that $0 \le BD + NBD + ND \le 1$. $P_{c}$FSs become a significant tool to handle the situations that have more answers like, yes, no, neutral and refusal. Later on, Cuong and Kreinovich [7] developed some operational rules for $P_{c}$FSs to deal with the picture fuzzy information accurately. Aggregation operators (AOs) are extensively used to accumulate the data under different extensions of FSs, for example, Xia et al. [31] introduced a chain of AOs for hesitant fuzzy (HF) data by using quasi arithmetic means, Wei [27] constructed variously prioritized AOs for aggregating HF data, Wei et al. [28] introduced HF Choquet integral AOs, HF Choquet ordered averaging (HFCOA) operator and HF Choquet ordered geometric (HFCOG) operator. Similarly, numerous experts work to develop AOs for $P_{c}$FSs [9, 13, 18, 29]. Recently, Zhang et al. [36] developed some Heronian mean ($H_{r}$M) AOs to accumulate the picture fuzzy numbers ($P_{c}$FNs) and formulated an MCDM approach for solving the multiple criteria problems. The Hamy mean (HM) operators are extended by Li et al. [15] for Pythagorean fuzzy sets (PFSs) and then they implemented these to get the solution of MCDM problems. Recently, HM operators are also applied by Wei et al. [30] under the framework of dual hesitant PFSs.

Nowadays, DMs are using $B_{p}$FSs [37, 38] as a significant tool to handle the vague and uncertain data in MCDM problems. A couple of elements, named as, the positive degree of membership ($P_{v}$DM) and the negative degree of membership ($N_{v}$DM), are used to represent an entity in $B_{p}$FSs and the range of these degrees always bounded in $[-1, 1]$. Many DMs have used the $B_{p}$FSs in their research articles [11, 14, 39,40,41]. Later on, Gul [10] developed accumulation operators for $B_{p}$F information. An idea of hesitant $B_{p}$FSs and its operational rules are presented by Wei et al. [32] in 2017. Lu et al. [16] introduced the idea of bipolar 2-tuple linguistic fuzzy sets ($B_{p}$2TLFSs). Moreover, Xu and Wei [34] presented the concept of dual $B_{p}$FSs and developed various operational rules to handle it. Based on $B_{p}$FSs, Akram and Arshad [1] proposed the $B_{p}$F linguistic variables and $B_{p}$F numbers. Alghamdi et al. [3] established the MCDM technique with the help of $B_{p}$F framework. Akram et al. developed the $B_{p}$F-TOPSIS and $B_{p}$F-ELECTRE-I techniques for medical diagnosis [2]. Shumaiza et al. [20] proposed the Trapezoidal $B_{p}$F numbers to investigate the group decision-making problems.

Based on the literature review, it can be seen that both $P_{c}$FSs and $B_{p}$FSs have got much attention from the DMs. That is why we have created a novel extension called B$P_{c}$FSs by combining both $P_{c}$FSs and $B_{p}$FSs. B$P_{c}$FSs have additional information in the form of $P_{v}$BD and $N_{v}$BD which are not present in the $P_{c}$FSs due to which it helps the DMs more effectively than $P_{c}$FSs and $B_{p}$FSs in the decision-making process. In this article, we pay heed to three aspects: (1) presented some core operational laws for B$P_{c}$FSs and (2) established two aggregation operators, called bipolar picture fuzzy weighted average (B$P_{c}$FWA)and geometric (B$P_{c}$FWG) operators to accumulate B$P_{c}$F information. (3) developed score and accuracy functions to compare the outcomes of two B$P_{c}$F numbers.

Vanderbei [24] presented the idea of linear programming (LP) model that allows some objective function to be maximized or minimized according to the circumstances. LP model is capable the DMs to tackle the issues which they face in MCDM procedures. Many DMs implemented the LP model to handle the MCDM problems in various area of lives [4, 8, 12, 25]. Sometimes the decisions alter with the change of weights of criteria. It means that weights of criteria have a vital influence on decisions. Assigning the weights to criteria are difficult task for the DMs, however, Sindhu et al. [21, 22] have applied the LP technique to evaluate the weights. To avoid biases, we have used the TOPSIS to find out the objective function, and then LP technique has applied to determine the criteria’s weights in this work.

The Rest of the article is planned as Sect. 2 encloses some preliminaries regarding bipolar picture fuzzy sets, LP model, and score function. The AOs like B$P_{c}$FWA and B$P_{c}$FWG operators are developed, and their properties are discussed in Sect. 3, respectively. Based on these operators an MCDM model is proposed in Sect. 4, and the developed model is then applied on a practical example about the selection of investment company in Sect. 5 to elaborate the validity and effectiveness. A comprehensive comparative analysis and a sensitivity analysis is performed to empower the proposed MCDM model in Sects. 6 and 7, respectively. A brief discussion and conclusions are penned in Sect. 8.

2 Preliminaries

This section comprises some notions like FSs, $P_{c}$FSs, $B_{p}$FSs and LP to support the B$P_{c}$FSs and MCDM model. Also, to compare the B$P_{c}$F numbers, novel score and accuracy functions are presented here.

Definition 1

[35] A fuzzy set (FS) F on a discourse set $X=\{x_{1}, x_{2},\ldots , x_{n}\}$ is presented as:

$$F=\Big \{\Big \langle x_{i},m_{F}(x_{i})\Big \rangle \Big |x_{i}\in X\Big \},$$

where, $m:X\rightarrow [0,1]$.

Definition 2

[6] Let $X=\{x_{1}, x_{2},\ldots , x_{n}\}$ be a fixed set, a picture fuzzy set $P_{c}$ on X is defined as:

$$P_{c}=\Big \{ \Big \langle x_{i},\alpha _{P_{c}(x_{i})}, \gamma _{P_{c}}(x_{i}), \beta _{P_{c}}(x_{i})\Big \rangle \Big | x_{i}\in X, i=1, 2, \ldots , n\Big \},$$

where $\alpha _{P_{c}}(x_{i})$, $\beta _{P_{c}}(x_{i})$, $\gamma _{P_{c}}(x_{i})\in [0,1]$ are called the acceptance membership, neutral and rejection membership degrees of $x_{i}\in X$ to the set $P_{c}$, respectively, and $\alpha _{P_{c}}(x_{i})$, $\gamma _{P_{c}}(x_{i})$ and $\beta _{P_{c}}(x_{i})$ fulfil the condition: $0\le \alpha _{P_{c}}(x_{i})+ \gamma _{P_{c}}(x_{i})+ \beta _{P_{c}}(x_{i}) \le 1$, for all $x_{i}\in X$. Also $\zeta _{P_{c}}(x_{i})=1- \alpha _{P_{c}}(x_{i})- \gamma _{P_{c}}(x_{i})- \beta _{P_{c}}(x_{i})$, then $\zeta _{P_{c}}(x_{i})$ is said to be a degree of refusal membership of $x_{i}\in X$ in $P_{c}$. For our convenience, we can write $P_{k}=(\alpha ^{k}_{P_{c}}(x_{i})$, $\beta ^{k}_{P_{c}}(x_{i}), \gamma ^{k}_{P_{c}}(x_{i}))$ as the picture fuzzy numbers $(P_{c}FNs)$ over a set $P_{c}$, where k is positive integer.

Definition 3

[29] Let $P=\Big (\alpha _{P_{c}}(x_{i}), \gamma _{P_{c}}(x_{i}), \beta _{P_{c}}(x_{i})\Big )$, $P_{1}=\Big (\alpha ^{1}_{P_{c}}(x_{i}), \gamma ^{1}_{P_{c}}(x_{i}), \beta ^{1}_{P_{c}}(x_{i})\Big )$ and $P_{2}=\Big (\alpha ^{2}_{P_{c}}(x_{i}), \gamma ^{2}_{P_{c}}(x_{i}), \beta ^{2}_{P_{c}}(x_{i})\Big )$ be three $P_{c}FNs$, then arithmetic operations are listed as follows:

• 1. $P_{1} \oplus P_{2}= (\alpha ^{1}_{P_{c}}+\alpha ^{2}_{P_{c}}-\alpha ^{1}_{P_{c}}\times \alpha ^{2}_{P_{c}}, \gamma ^{1}_{P_{c}}\times \gamma ^{2}_{P_{c}}, \beta ^{1}_{P_{c}}\times \beta ^{2}_{P_{c}})$;

• 2. $P_{1}\otimes P_{2}= (\alpha ^{1}_{P_{c}}\times \alpha ^{2}_{P_{c}}, \gamma ^{1}_{P_{c}}+\gamma ^{2}_{P_{c}}-\gamma ^{1}_{P_{c}}\times \gamma ^{2}_{P_{c}}, \beta ^{1}_{P_{c}}+\beta ^{2}_{P_{c}}-\beta ^{1}_{P_{c}}\times \beta ^{2}_{P_{c}})$;

• 3. $\uplambda P = (1-(1-\alpha _{P_{c}})^{\uplambda }, \gamma ^{\uplambda }_{P_{c}}, \beta ^{\uplambda }_{P_{c}})$, where, $\uplambda > 0$;

• 4. $P^{\uplambda }_{p}= (\alpha _{P_{c}}^{\uplambda }, 1-(1-\gamma _{P_{c}})^{\uplambda }, 1-(1-\beta _{P_{c}})^{\uplambda })$, where, $\uplambda > 0$.

Definition 4

[37, 38] A $B_{p}$FS denoted by $B_{p}$ on a universal set $X=\{x_{1}, x_{2},\ldots , x_{n}\}$ is defined as follows:

$$\begin{aligned} B_{p}=\Big \{ \Big \langle x_{i},(\alpha ^{+}_{B_{p}(x_{i})}, \beta ^{-}_{B_{p}}(x_{i}))\Big \rangle \Big | x_{i}\in X, i=1, 2, ..., n\Big \}, \end{aligned}$$

where $\alpha ^{+}_{B_{p}(x_{i})}: X \rightarrow [0,1]$, $\beta ^{-}_{B_{p}}(x_{i}): X \rightarrow [-1,0]$ are named as $P_{v}$BD and $N_{v}$BD of $x_{i}\in X$ to $B_{p}$, respectively.

Definition 5

[10] Let $B_{p}$, $B^{1}_{p}$ and $B^{2}_{p}$ be any three $B_{p}$FSs on $X=\{x_{1}, x_{2},\ldots , x_{n}\}$, then several accumulation operators are listed as follows:

• 1. $B^{1}_{p} \oplus B^{2}_{p}= (\alpha ^{+}_{1} +\alpha ^{+}_{2}-\alpha ^{+}_{1} \times \alpha ^{+}_{2}, -|\beta ^{-}_{1}| \times |\beta ^{-}_{2}|)$;

• 2. $B^{1}_{p} \otimes B^{2}_{p}= (|\alpha ^{-}_{1}| \times |\alpha ^{-}_{2}|, \beta ^{+}_{1} +\beta ^{+}_{2}-\beta ^{+}_{1} \times \beta ^{+}_{2})$;

• 3. $\kappa B_{p} = (1-(1-\alpha ^{+})^{\kappa }, -|\beta ^{-}|)$, where, $\kappa > 0$;

• 4. $B^{\kappa }_{p}= (\alpha ^{+})^{\kappa }, -1+|1+\beta ^{-}|^{\kappa }$, where, $\kappa > 0$;

• 5. $B^{c}_{p}= (1-\alpha ^{+}, |\beta ^{-}-1|$.

Inspired by $B_{p}$FSs and $P_{c}$FSs, we proposed the bipolar picture fuzzy sets (B$P_{c}$FSs) denoted by $\mathcal {P}$ is presented below,

Definition 6

[23] Suppose that $X=\{x_{1}, x_{2},\ldots , x_{n}\}$ is a discourse, then the B$P_{c}$FSs $\mathcal {P}$ on X is presented as:

$$\begin{aligned} \mathcal {P}=\Big \{ \Big \langle x_{i},(\tilde{P}^{+}_{c}(x_{i}), \tilde{P}^{-}_{c}(x_{i}))\Big \rangle \Big | x_{i}\in X, \quad i=1, 2, \ldots , n\Big \}, \end{aligned}$$

here $\tilde{P}^{+}_{c}(x_{i})= (\alpha ^{+}_{\mathcal {P}}(x_{i}),\gamma ^{+}_{\mathcal {P}}(x_{i}), \beta ^{+}_{\mathcal {P}}(x_{i}))$, $\tilde{P}^{-}_{c}(x_{i})= (\alpha ^{-}_{\mathcal {P}}(x_{i}),\gamma ^{-}_{\mathcal {P}}(x_{i}), \beta ^{-}_{\mathcal {P}}(x_{i}))$ satisfy the following condition: $0\le (\alpha ^{+}_{\mathcal {P}}(x_{i})+\gamma ^{+}_{\mathcal {P}}(x_{i})+ \beta ^{+}_{\mathcal {P}}(x_{i})) \le 1$ and $-1\le (\alpha ^{-}_{\mathcal {P}}(x_{i})+\gamma ^{-}_{\mathcal {P}}(x_{i})+ \beta ^{-}_{\mathcal {P}}(x_{i})) \le 0$ for all $x_{i}\in X$.

For the convenient, the duo, $\tilde{p_{k}}(x)= (\tilde{P}^{k+}_{c}(x), \tilde{P}^{k-}_{c}(x))$ is called the B$P_{c}$F number (B$P_{c}$FN) represented by $\tilde{p_{k}}= (\tilde{P}^{k+}_{c}, \tilde{P}^{k-}_{c})$ that fulfills the conditions: $\alpha ^{k+}_{P_{c}}, \gamma ^{k+}_{P_{c}}, \beta ^{k+}_{P_{c}} \in [0,1]$, $\le \alpha ^{k-}_{P_{c}}, \gamma ^{k-}_{P_{c}}, \beta ^{k-}_{P_{c}} \in [-1,0]$, $0\le \alpha ^{k+}_{P_{c}}+ \gamma ^{k+}_{P_{c}}+ \beta ^{k+}_{P_{c}} \le 1$ and $-1\le \alpha ^{k-}_{P_{c}}+ \gamma ^{k-}_{P_{c}}+ \beta ^{k-}_{P_{c}} \le 0$.

Definition 7

[23] Let $\tilde{p}= (\alpha ^{+}_{P_{c}}, \gamma ^{+}_{P_{c}}, \beta ^{+}_{P_{c}},\alpha ^{-}_{P_{c}},\gamma ^{-}_{P_{c}},\beta ^{-}_{P_{c}})$, $\tilde{p_{1}}= (\alpha ^{1+}_{P_{c}}, \gamma ^{1+}_{P_{c}}, \beta ^{1+}_{P_{c}},\alpha ^{1-}_{P_{c}},\gamma ^{1-}_{P_{c}},\beta ^{1-}_{P_{c}})$ and $\tilde{p_{2}}= (\alpha ^{2+}_{P_{c}}, \gamma ^{2+}_{P_{c}}, \beta ^{2+}_{P_{c}}, \alpha ^{2-}_{P_{c}},\gamma ^{2-}_{P_{c}},\beta ^{2-}_{P_{c}})$ be three B$P_{c}$FNs, then the operational rules are penned as:

• 1. $\tilde{p_{1}} \oplus \tilde{p_{2}}= ((\alpha ^{1+}_{P_{c}}+\alpha ^{2+}_{P_{c}}-\alpha ^{1+}_{P_{c}}\cdot \alpha ^{2+}_{P_{c}}, \gamma ^{1+}_{P_{c}}\cdot \gamma ^{2+}_{P_{c}}, \beta ^{1+}_{P_{c}}\cdot \beta ^{2+}_{P_{c}}), -(\alpha ^{1-}_{P_{c}}+\alpha ^{2-}_{P_{c}}-\alpha ^{1-}_{P_{c}}\cdot \alpha ^{2-}_{P_{c}}), -|\gamma ^{1-}_{P_{c}}|\cdot | \gamma ^{2-}_{P_{c}}|,-|\beta ^{1-}_{P_{c}}|\cdot | \beta ^{2-}_{P_{c}}|)$;

• 2. $\tilde{p_{1}}\otimes \tilde{p_{2}}= (\alpha ^{1+}_{P_{c}}\cdot \alpha ^{2+}_{P_{c}}, \gamma ^{1+}_{P_{c}}+\gamma ^{2+}_{P_{c}}-\gamma ^{1+}_{P_{c}}\cdot \gamma ^{2+}_{P_{c}}, \beta ^{1+}_{P_{c}}+\beta ^{2+}_{P_{c}}-\beta ^{1+}_{P_{c}}\cdot \beta ^{2+}_{P_{c}}), -(|\alpha ^{1-}_{P_{c}}|\cdot |\alpha ^{2-}_{P_{c}}|, -(\gamma ^{1-}_{P_{c}}+\gamma ^{2-}_{P_{c}}-\gamma ^{1-}_{P_{c}}\cdot \gamma ^{2-}_{P_{c}}), -(\beta ^{1-}_{P_{c}}+\beta ^{2-}_{P_{c}}-\beta ^{1-}_{P_{c}}\cdot \beta ^{2-}_{P_{c}}))$;

• 3. $\uplambda \tilde{p} = ((1-(1-\alpha ^{+}_{P_{c}})^{\uplambda }, (\gamma ^{+})^{\uplambda }_{P_{c}}, (\beta ^{+})^{\uplambda }_{P_{c}}), -(1-|(1-\alpha ^{-}_{P_{c}})|^{\uplambda }, -|(\gamma ^{-})^{\uplambda }_{P_{c}}|, -|(\beta ^{-})^{\uplambda }_{P_{c}}|))$, where, $\uplambda > 0$;

• 4. $\tilde{p}^{\uplambda }= ((\alpha ^{+}_{P_{c}})^{\uplambda }, 1-(1-\gamma ^{+}_{P_{c}})^{\uplambda }, 1-(1-\beta ^{+}_{P_{c}})^{\uplambda }), -|(\alpha ^{-}_{P_{c}})^{\uplambda }|, -(1-|(1-\gamma ^{-}_{P_{c}})^{\uplambda }|, -(1-|(1-\beta ^{-}_{P_{c}})^{\uplambda }|))$, where, $\uplambda > 0$.

In order to compare the two B$P_{c}$FN, we presented the score and accuracy functions as follows:

Definition 8

Suppose that $\tilde{p_{i}}= (\tilde{P}^{+}_{ci}, \tilde{P}^{-}_{ci}), (i=1, 2)$ are two B$P_{c}$FNs, then the score functions $\tilde{S^{i}_{f}}$ and accuracy function $\tilde{A^{i}_{f}}$ between two B$P_{c}$FNs are written as:

$$\begin{aligned} \tilde{S^{i}_{f}}(\tilde{p_{i}})= \sum ^{|\tilde{p_{i}}|}_{i=1}\frac{1+\tilde{P}^{+}_{ci}+\tilde{P}^{-}_{ci}}{2}, \end{aligned}$$

and

$$\begin{aligned} \tilde{A^{i}_{f}}(\tilde{p_{i}})= \sum ^{|\tilde{p_{i}}|}_{i=1}\frac{\tilde{P}^{+}_{ci}-\tilde{P}^{-}_{ci}}{2}, \end{aligned}$$

then, we can compare two B$P_{c}$FNs on the basis of following characteristics:

• 1. if $\tilde{S^{1}_{f}}(\tilde{p_{1}})>\tilde{S^{2}_{f}}(\tilde{p_{2}})$, then $\tilde{p_{1}}$ is superior to $\tilde{p_{2}}$ and written as, $\tilde{p_{1}}\succ \tilde{p_{2}}$;

• 2. if $\tilde{S^{1}_{f}}(\tilde{p_{1}})<\tilde{S^{2}_{f}}(\tilde{p_{2}})$, then $\tilde{p_{1}}$ is inferior to $\tilde{p_{2}}$ and denoted as, $\tilde{p_{1}}\prec \tilde{p_{2}}$;

and if $\tilde{S^{1}_{f}}(\tilde{p_{1}})=\tilde{S^{2}_{f}}(\tilde{p_{2}})$, then,

• 1. if $\tilde{A^{1}_{f}}(\tilde{p_{1}})>\tilde{A^{2}_{f}}(\tilde{p_{2}})$, then $\tilde{p_{1}}$ is superior to $\tilde{p_{2}}$ and written as, $\tilde{p_{1}}\succ \tilde{p_{2}}$;

• 2. if $\tilde{A^{1}_{f}}(\tilde{p_{1}})=\tilde{A^{2}_{f}}(\tilde{p_{2}})$, then $\tilde{p_{1}}$ is equivalent to $\tilde{p_{2}}$ and denoted as, $\tilde{p_{1}}\approx \tilde{p_{2}}$.

Definition 9

[24]. The modified LP model is presented as:

$$\begin{aligned} \begin{array}{ll} Maximize: &{} Z=c_{1}y_{1}+c_{2}y_{2}+c_{3}y_{3}+\cdots +c_{n}y_{n} \\ Subject to: &{} a_{11}y_{1}+a_{12}y_{2}+a_{13}y_{3}+\cdots +a_{1n}y_{n}\le b_{1} \\ &{} a_{21}y_{1}+a_{22}y_{2}+a_{23}y_{3}+\cdots +a_{2n}y_{n}\le b_{2} \\ &{} \vdots \\ &{} a_{m1}y_{1}+a_{m2}y_{2}+a_{m3}y_{3}+\cdots +a_{mn}y_{n}\le b_{m} \\ &{} y_{1},y_{2},\ldots ,y_{n}\ge 0, \end{array} \end{aligned}$$

where m represents the number of constraints, and n denotes the cardinality of decision variables $(y_{1},y_{2},\ldots ,y_{n})$, respectively. The solution $(y_{1},y_{2},\ldots ,y_{n})$ is called a feasible solution if it fulfills all the given limitations.

3 B$P_{c}$F Aggregation Operators

In this section, bipolar picture fuzzy weighted average (B$P_{c}$FWA)and geometric (B$P_{c}$FWG) operators are established to accumulate the B$P_{c}$F data.

Definition 10

Let $\tilde{p_{k}}= (\tilde{P}^{+}_{c}, \tilde{P}^{-}_{c})$, where, $k=1, 2,3, \ldots , n$ be a set of B$P_{c}$FNs, a B$P_{c}$FWA operator is defined as:

$$\begin{aligned}&BP_{c}FWA( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{n}})= \oplus ^{n}_{k=1}w_{k}\cdot \tilde{p_{k}}.\\&BP_{c}FWA( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{n}})= ((1- \varPi ^{n}_{k=1}(1-\alpha ^{+}_{P_{c}})^{w_{k}},\\&\varPi ^{n}_{k=1}(\gamma ^{+}_{P_{c}})^{w_{k}}, \varPi ^{n}_{k=1}(\beta ^{+}_{P_{c}})^{w_{k}}), \\&-(1-|\varPi ^{n}_{k=1}(1-\alpha ^{-}_{P_{c}})^{w_{k}}|, -\varPi ^{n}_{k=1}|(\gamma ^{-}_{P_{c}})^{w_{k}}|,\\&-\varPi ^{n}_{k=1}|(\beta ^{-}_{P_{c}})^{w_{k}}|)), \end{aligned}$$

where $w_{k}= (w_{1}, w_{2}, \ldots , w_{n})^{T}$ is the weight’s vector that connected $\tilde{p_{k}}$ and sustaining the limitations: $w_{k}>0$ and $\sum ^{n} _{k=1}w_{k} = 1$.

Definition 11

Let $\tilde{p_{k}}= (\tilde{P}^{+}_{c}, \tilde{P}^{-}_{c})$, where, $k=1, 2,3,\ldots , n$ be a set of B$P_{c}$FNs, a B$P_{c}$FWG operator is presented as:

$$\begin{aligned}&BP_{c}FWG( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{n}})= \otimes ^{n}_{k=1} (\tilde{p_{k}})^{w_{k}},\\&=\varPi ^{n}_{k=1}(\alpha ^{+}_{P_{c}})^{w_{k}}, 1-\varPi ^{n}_{k=1}(1-\gamma ^{+}_{P_{c}})^{w_{k}},\\&\quad 1-\varPi ^{n}_{k=1}(1-(\beta ^{+}_{P_{c}})^{w_{k}}),\\&\quad -\varPi ^{n}_{k=1}|(\alpha ^{-}_{P_{c}})^{w_{k}}|, -(1-\varPi ^{n}_{k=1}|(1-\gamma ^{-}_{P_{c}})^{w_{k}}|),\\&\quad -(1-\varPi ^{n}_{k=1}|1-(\beta ^{-}_{P_{c}})^{w_{k}}|)), \end{aligned}$$

where $w_{k}= (w_{1}, w_{2},\ldots , w_{n})^{T}$ is the weight’s vector that connected $\tilde{p_{k}}$ and sustaining the limitations: $w_{k}>0$ and $\sum ^{n} _{k=1}w_{k} = 1$.

Theorem 1

The B$P_{c}$FWG operator returns a B$P_{c}$FN with $BP_{c}FWG( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{n}})= \otimes ^{n}_{k=1} (\tilde{p_{k}})^{w_{k}}.$

Proof

We can prove the Theorem 1 by using mathematical induction on n as follows: (1) for $n = 2$, we get

$$\begin{aligned}&(\tilde{p_{1}})^{w_{1}}= ((\alpha ^{1+}_{P_{c}})^{w_{1}}, 1-(1-\gamma ^{1+}_{P_{c}})^{w_{1}}, 1-(1-(\beta ^{1+}_{P_{c}})^{w_{1}}, \\&-|(\alpha ^{1-}_{P_{c}})^{w_{1}}|, -(1-|(1-\gamma ^{1-}_{P_{c}})^{w_{1}}|), -(1-|1-(\beta ^{1-}_{P_{c}})^{w_{1}}|)),\\&(\tilde{p_{2}})^{w_{2}}= ((\alpha ^{2+}_{P_{c}})^{w_{2}}, 1-(1-\gamma ^{2+}_{P_{c}})^{w_{2}}, 1-(1-(\beta ^{2+}_{P_{c}})^{w_{2}}), \\&-(|(\alpha ^{2-}_{P_{c}})^{w_{2}}|, -(1-|(1-\gamma ^{2-}_{P_{c}})^{w_{2}}|), -(1-|1-(\beta ^{2-}_{P_{c}})^{w_{2}}|)).\\&(\tilde{p_{1}})^{w_{1}}(\tilde{p_{2}})^{w_{2}}= (\alpha ^{1+}_{P_{c}})^{w_{1}}(\alpha ^{2+}_{P_{c}})^{w_{2}}, 1-(1-\gamma ^{1+}_{P_{c}})^{w_{1}}\\&(1-\gamma ^{2+}_{P_{c}})^{w_{2}}, 1-(1-(\beta ^{1+}_{P_{c}})^{w_{1}})(1-(\beta ^{2+}_{P_{c}})^{w_{2}},\\&-(|(\alpha ^{1-}_{P_{c}})^{w_{1}}|(|(\alpha ^{2-}_{P_{c}})^{w_{2}}|, -(1-|(1-\gamma ^{1-}_{P_{c}})^{w_{1}}|)(1-|\gamma ^{2-}_{P_{c}})^{w_{2}}|),\\&-(1-|1-(\beta ^{1-}_{P_{c}})^{w_{1}}|)(|1-(\beta ^{2-}_{P_{c}})^{w_{2}}|). \end{aligned}$$

Thus Theorem 1 holds for $n = 2$. Suppose that it holds for $n = i$, where $i<k$, that is

$$\begin{aligned}&BP_{c}FWG( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{i}})= \varPi ^{n}_{k=1}(\alpha ^{i+}_{P_{c}})^{w_{i}},1-\varPi ^{n}_{k=1}(1-\gamma ^{i+}_{P_{c}})^{w_{i}},\\&1-\varPi ^{n}_{k=1}(1-(\beta ^{i+}_{P_{c}})^{w_{i}}),-\varPi ^{n}_{k=1}(\alpha ^{i-}_{P_{c}})^{w_{i}},\\&-(1-\varPi ^{n}_{k=1}(1-\gamma ^{i-}_{P_{c}})^{w_{i}}), -(1-\varPi ^{n}_{k=1}(1-(\beta ^{i-}_{P_{c}})^{w_{i}}))), \end{aligned}$$

then, when $n = i+1$, by the operational rules in Theorem 1, we get

$$\begin{aligned}&\varPi ^{i+1}_{k=1}(\tilde{p_{k}})^{w_{k}}= \varPi ^{i}_{k=1}(\tilde{p_{k}})^{w_{k}}\cdot (\tilde{p_{i+1}})^{w_{i+1}},\\&= (\varPi ^{i}_{k=1}(\alpha ^{i+}_{P_{c}})^{w_{i}},1-\varPi ^{i}_{k=1}(1-\gamma ^{i+}_{P_{c}})^{w_{i}},\\&\quad 1-\varPi ^{i}_{k=1}(1-(\beta ^{i+}_{P_{c}})^{w_{i}}), -\varPi ^{i}_{k=1}(\alpha ^{i-}_{P_{c}})^{w_{i}}, \\&\quad -(1-\varPi ^{i}_{k=1}(1-\gamma ^{i-}_{P_{c}})^{w_{i}}),\\&\quad -(1-\varPi ^{i}_{k=1}(1-(\beta ^{i-}_{P_{c}})^{w_{i}})) \cdot ((\alpha ^{(i+1)+}_{P_{c}})^{w_{i+1}},\\&\quad 1-(1-\gamma ^{(i+1)+}_{P_{c}})^{w_{i+1}}, 1-(1-(\beta ^{(i+1)+}_{P_{c}})^{w_{i+1}}),\\&\quad -(\alpha ^{(i+1)-}_{P_{c}})^{w_{i+1}}, -(1-(1-\gamma ^{(i+1)-}_{P_{c}})^{w_{i+1}}), -(1-(1-(\beta ^{(i+1)-}_{P_{c}})^{w_{i+1}})),\\&= \varPi ^{i+1}_{k=1}(\alpha ^{k+}_{P_{c}})^{w_{k}},\\&\quad 1-\varPi ^{i+1}_{k=1}(1-\gamma ^{i+}_{P_{c}})^{w_{i}},1-\varPi ^{i+1}_{k=1}(1-\beta ^{k+}_{P_{c}})^{w_{k}}),\\&\quad -(\varPi ^{i+1}_{k=1}(\alpha ^{k-}_{P_{c}})^{w_{k}}, -(1-\varPi ^{i+1}_{k=1}(1-\gamma ^{k-}_{P_{c}})^{w_{k}}),\\&\quad -(1-\varPi ^{i+1}_{k=1}(1-(\beta ^{k-}_{P_{c}})^{w_{k}})),\\ \end{aligned}$$

which reveals that Theorem 1 holds for $n=i+1$. Hence, we can say that Theorem 1 satisfies for all n. Then clearly,

$$\begin{aligned}&(\varPi ^{n}_{k=1}(\alpha ^{k+}_{P_{c}})^{w_{k}},1-\varPi ^{n}_{k=1}(1-\gamma ^{k+}_{P_{c}})^{w_{k}},\\&1-\varPi ^{n}_{k=1}(1-(\beta ^{k+}_{P_{c}})^{w_{k}})\in [0,1], \end{aligned}$$

and

$$\begin{aligned}&\varPi ^{n}_{k=1}(\alpha ^{k+}_{P_{c}})^{w_{k}}+1-\varPi ^{n}_{k=1}(1-\gamma ^{k+}_{P_{c}})^{w_{k}}\\&+1-\varPi ^{n}_{k=1}(1-(\beta ^{k+}_{P_{c}})^{w_{k}})\le 1, \end{aligned}$$

also,

$$\begin{aligned}&(-\varPi ^{n}_{k=1}(\alpha ^{k-}_{P_{c}})^{w_{k}}),-(1-\varPi ^{n}_{k=1}(1-\gamma ^{k-}_{P_{c}})^{w_{k}}),\\&-(1-\varPi ^{n}_{k=1}(1-(\beta ^{k-}_{P_{c}})^{w_{k}})\in [-1,0], \end{aligned}$$

and

$$\begin{aligned}&(-\varPi ^{n}_{k=1}(\alpha ^{k-}_{P_{c}})^{w_{k}})-(1-\varPi ^{n}_{k=1}(1-\gamma ^{k-}_{P_{c}})^{w_{k}})\\&-(1-\varPi ^{n}_{k=1}(1-(\beta ^{k-}_{P_{c}})^{w_{k}})\le -1 \end{aligned}$$

Hence, $BP_{c}FWG( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{i}})$ form a $BP_{c}FN$.

Theorem 2

Let $\tilde{p_{k}}= (\alpha ^{k+}, \gamma ^{k+}, \beta ^{k+}, \alpha ^{k-}, \gamma ^{k-}, \beta ^{k-})$ be a collection $BP_{c}FNs$, then the $BP_{c}FWG$ operator hold the following properties:

• 1. Idempotent,

• 2. Monotonic,

• 3. Bounded,

• 4. Commutative.

Proof

• 1. Let $\tilde{p_{1}}=\tilde{p_{2}}= ,\ldots , = \tilde{p_{n}} = \tilde{p}= (\alpha ^{+}, \gamma ^{+}, \beta ^{+}, \alpha ^{-}, \gamma ^{-}, \beta ^{-})$, then,

  $$\begin{aligned}&BP_{c}FWG( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{n}})= (\varPi ^{n}_{k=1}(\alpha ^{k+}_{P_{c}})^{w_{k}},\\&\quad 1-\varPi ^{n}_{k=1}(1-\gamma ^{k+}_{P_{c}})^{w_{k}}, 1-\varPi ^{n}_{k=1}(1-(\beta ^{k+}_{P_{c}})^{w_{k}},\\&\quad -\varPi ^{n}_{k=1}(\alpha ^{k-}_{P_{c}})^{w_{k}}), -(1-\varPi ^{n}_{k=1}(1-\gamma ^{k-}_{P_{c}})^{w_{k}}),\\&\quad -(1-\varPi ^{n}_{k=1}(1-(\beta ^{k-}_{P_{c}})^{w_{k}}),\\&=(\varPi ^{n}_{k=1}(\alpha ^{+}_{P_{c}})^{w_{k}}, 1-\varPi ^{n}_{k=1}(1-\gamma ^{+}_{P_{c}})^{w_{k}},\\&\quad 1-\varPi ^{n}_{k=1}(1-(\beta ^{+}_{P_{c}})^{w_{k}}, -\varPi ^{n}_{k=1}(\alpha ^{-}_{P_{c}})^{w_{k}}),\\&\quad -(1-\varPi ^{n}_{k=1}(1-\gamma ^{-}_{P_{c}})^{w_{k}}), -(1-\varPi ^{n}_{k=1}(1-(\beta ^{-}_{P_{c}})^{w_{k}}),\\&=((\alpha ^{+}_{P_{c}})^{\sum ^{n} _{k=1}w_{n}}, 1- (1-\gamma ^{+}_{P_{c}})^{\sum ^{n} _{k=1}w_{n}},\\&\quad 1-(1-(\beta ^{+}_{P_{c}})^{\sum ^{n} _{k=1}w_{n}}, -(\alpha ^{-}_{P_{c}})^{\sum ^{n} _{k=1}w_{n}}),\\&\quad -(1-(1-\gamma ^{-}_{P_{c}})^{\sum ^{n} _{k=1}w_{n}}), -(1-(1-(\beta ^{-}_{P_{c}})^{\sum ^{n} _{k=1}w_{n}}),\\&=((\alpha ^{+}_{P_{c}}),1- (1-\gamma ^{+}_{P_{c}}), 1-(1-(\beta ^{+}_{P_{c}}),\\&\quad -(\alpha ^{-}_{P_{c}})),-(1-(1-\gamma ^{-}_{P_{c}})), -(1-(1-(\beta ^{-}_{P_{c}})). \end{aligned}$$

  Since $\sum _{k=1}^{n}w_{n} = 1$, then we get, $BP_{c}FWG( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{n}})=(\alpha ^{+}_{P_{c}},\gamma ^{+}_{P_{c}}, \beta ^{+}_{P_{c}}, -\alpha ^{-}_{P_{c}},-\gamma ^{-}_{P_{c}}, -\beta ^{-}_{P_{c}})= \tilde{p}$, which is required.

• 2. Let $\alpha ^{+} _{i_{j}}\ge \alpha ^{+} _{\theta _{j}},\gamma ^{+} _{i_{j}}\ge \gamma ^{+} _{\theta _{j}}$, $\beta ^{+} _{i_{j}}\le \beta ^{+}_{\theta _{j}}, \alpha ^{-} _{i_{j}}= \alpha ^{-} _{\theta _{j}}, \gamma ^{-} _{i_{j}}= \gamma ^{-} _{\theta _{j}}$ and $\beta ^{-} _{i_{j}}= \beta ^{-}_{\theta _{j}}.$ Consider

  $$\begin{aligned}&\Rightarrow \alpha ^{+} _{i_{j}}\ge \alpha ^{+} _{\theta _{j}} \\&\Rightarrow (\alpha ^{+} _{i_{j}})^{w_{k}}\ge (\alpha ^{+} _{\theta _{j}})^{w_{k}} \\&\Rightarrow \varPi ^{n}_{k=1}(\alpha ^{+} _{i_{j}})^{w_{k}}\ge \varPi ^{n}_{k=1}(\alpha ^{+} _{\theta _{j}})^{w_{k}} \end{aligned}$$

  also,

  $$\begin{aligned}&\gamma ^{+} _{i_{j}}\ge \gamma ^{+} _{\theta _{j}}\\&\Rightarrow 1-\gamma ^{+} _{i_{j}}\le 1-\gamma ^{+} _{\theta _{j}}\\&\Rightarrow \varPi ^{n}_{k=1}(1-\gamma ^{+} _{i_{j}})^{w^{k}} \le \varPi ^{n}_{k=1}(1-\gamma ^{+} _{\theta _{j}})^{w_{k}}\\&\Rightarrow 1- \varPi ^{n}_{k=1}(1- \gamma ^{+} _{i_{j}})^{w_{k}}\ge \varPi ^{n}_{k=1}1-(1- \gamma ^{+}_{\theta _{j}})^{w_{k}} \end{aligned}$$

  now take,

  $$\begin{aligned}&\beta ^{+} _{i_{j}}\le \beta ^{+}_{\theta _{j}}\\&\Rightarrow \varPi ^{n}_{k=1} (\beta ^{+} _{i_{j}})^{w_{k}}\le \varPi ^{n}_{k=1}(\beta ^{+}_{\theta _{j}})^{w_{k}}\\&\Rightarrow 1-\varPi ^{n}_{k=1}(\beta ^{+} _{i_{j}})^{w_{k}}\ge 1- \varPi ^{n}_{k=1}(\beta ^{+}_{\theta _{j}})^{w_{k}}\\&\Rightarrow 1-( 1-\varPi ^{n}_{k=1}(\beta ^{+} _{i_{j}})^{w_{k}})\le 1-( 1- \varPi ^{n}_{k=1}(\beta ^{+}_{\theta _{j}})^{w_{k}}). \end{aligned}$$

• 3. Let $\tilde{b}_{i}~=(\alpha ^{+} _{i},\gamma ^{+} _{i},\beta ^{+} _{i}, \alpha ^{-} _{i},\gamma ^{-} _{i},\beta ^{-} _{i})$ with $i=1,~2, \ldots ,~k$, $\tilde{b}_{p}=(\alpha ^{+} _{\max i},\gamma ^{+} _{\max i},\beta ^{+} _{\max i}, \alpha ^{-} _{\max i},\gamma ^{-} _{\max i_{j}},\beta ^{-} _{\max i})$ and $\tilde{b}_{n}=(\alpha ^{+} _{\min i},\gamma ^{+} _{\min i},\beta ^{+} _{\min i}, \alpha ^{-} _{\min i},\gamma ^{-} _{\min i},\beta ^{-} _{\min i})$ be a set of $BP_{c}FNs$, then we have to prove that $BP_{c}FWG$ is bounded, that is, $\tilde{b}_{n}<BP_{c}FWG(\tilde{b}_{1},\tilde{b}_{2},\ldots ,\tilde{b}_{k})< \tilde{b}_{p}.$ It follows from properties (1) and (2) as, $BP_{c}FWG(\tilde{b}_{1},\tilde{b}_{2},\ldots ,\tilde{b}_{k})\ge BP_{c}FWG(\tilde{b}_{n},\tilde{b}_{n},\ldots ,\tilde{b}_{n})=\tilde{b}_{n},$ $BP_{c}FWG(\tilde{b}_{1},\tilde{b}_{2},\ldots ,\tilde{b}_{k})\le BP_{c}FWG(\tilde{b}_{p},\tilde{b}_{p},\ldots ,\tilde{b}_{p})=\tilde{b}_{p},$ $\Rightarrow \tilde{b}_{n}<BP_{c}FWG(\tilde{b}_{1},\tilde{b}_{2},\ldots , \tilde{b}_{k})<\tilde{b}_{p}.$

• 4. Let, $\tilde{a}_{i}=(\alpha ^{+} _{i},\gamma ^{+} _{i},\beta ^{+} _{i}, \alpha ^{-} _{i},\gamma ^{-} _{i},\beta ^{-} _{i})~$and$~ \tilde{b}_{\theta }=(\alpha ^{+} _{\theta },\gamma ^{+} _{\theta },\beta ^{+}_{\theta }, \alpha ^{-} _{\theta },\gamma ^{-} _{\theta },\beta ^{-}_{\theta })$ for all $i,~\theta =1,~2,~...,~k$ be two sets of $BP_{c}FNs$, then have to prove that $BP_{c}FWG$ is commutative, that is, $BP_{c}FWG(\tilde{a}_{1},\tilde{a}_{2},\ldots ,\tilde{a}_{k})= BP_{c}FWG(\tilde{b}_{1},\tilde{b}_{2},\ldots ,\tilde{b}_{k}).$ Since $\tilde{b}_{\theta }$ is any permutation of $\tilde{a}_{i}$, then $\otimes ^{k}_{i=1}(\tilde{a}_{i})^{w_{i}}= \otimes ^{k}_{\theta =1}(\tilde{b}_{\theta })^{w_{\theta }}.$ Hence, $BP_{c}FWG(\tilde{a}_{1},\tilde{a}_{2},\ldots ,\tilde{a} _{k})~=BP_{c}FWG(\tilde{b}_{1},\tilde{b}_{2},\ldots ,\tilde{b}_{k}).$

Theorem 3

The B$P_{c}$FWA operator returns a B$P_{c}$FN with $BP_{c}FWA( \tilde{p_{1}}, \tilde{p_{2}},\ldots , \tilde{p_{n}})= \oplus ^{n}_{k=1}w_{k}\cdot \tilde{p_{k}}.$

Proof

The proof of this Theorem is obvious.

Theorem 4

Let $\tilde{p_{k}}= (\alpha ^{k+}, \gamma ^{k+}, \beta ^{k+}, \alpha ^{k-}, \gamma ^{k-}, \beta ^{k-})$ be a collection $BP_{c}FNs$, then the $BP_{c}FWA$ operator satisfy the following properties:

• 1. Idempotent,

• 2. Monotonic,

• 3. Bounded,

• 4. Commutative.

Proof

We can prove it by adopting the same steps as Theorem 2.

4 MCDM Approach for B$P_{c}$F Environment

By considering B$P_{c}$F aggregation operators established in Sect. 3, an MCDM model is presented to handle the B$P_{c}$F information. Suppose that,

$Q= \{Q_{1}, Q_{2}, \ldots , Q_{n}\}$ and $V= \{V_{1}, V_{2}, \ldots , V_{m}\}$ are the discrete collection of alternatives and criteria, respectively. If the DMs gave the various esteems to the alternative $Q_{i} (i = 1, 2, \ldots , n)$ under the criteria $V_{j}(j = 1, 2,\ldots , m)$. A bipolar picture fuzzy decision matrix $B_{pc}= [b_{ij}]_{n\times m}$ is constructed on the basis of B$P_{c}$F information. Since the weights of the criteria have an excessive impact, thereby a weighing vector of criteria is provided as $w= (w_{1}, w_{2}, w_{3}, \ldots , w_{j})^{T}$, where $\sum _{j=1}^{m}w_{j} =1, j = 1, 2,\ldots , m$ and $w_{j}> 0$ can be evaluated by implementing the LP model described in Definition 9. The MCDM model based on proposed B$P_{c}$F aggregation operators has the following steps.

Step 1. Based on the B$P_{c}$F information provided by DM form a bipolar picture fuzzy decision matrix denoted by $B_{pc}= [b_{ij}]_{n\times m}$.

Step 2. In order to obtain the objective function, the following steps of the TOPSIS technique are adopted:

• Step 2 (a) Determine the B$P_{c}$F positive ideal solution ($B_{p}$FPIS) denoted by $B^{+}_{pc}$ and B$P_{c}$F negative ideal solution ($B_{p}$FNIS)represented by $B^{-}_{pc}$, respectively.

  $$\begin{aligned}&B^{+}_{pc}= \left( \begin{array}{c} \underset{j}{\max }(\alpha ^{+}_{ij}),\underset{j}{\max }(\gamma ^{+}_{ij}),\underset{j}{\max }(\beta ^{+}_{ij})), \underset{j}{\min }(\alpha ^{-}_{ij}),\underset{j}{\min }(\gamma ^{-}_{ij}),\underset{j}{\min }(\beta ^{-}_{ij})) \end{array} \right) \\&B^{-}_{pc}= \left( \begin{array}{c} (\underset{j}{\min }(\alpha ^{+}_{ij}),\underset{j}{\min }(\gamma ^{+}_{ij}),\underset{j}{\min }(\beta ^{+}_{ij})), \underset{j}{\max }(\alpha ^{-}_{ij}),\underset{j}{\max }(\gamma ^{-}_{ij}),\underset{j}{\max }(\beta ^{-}_{ij})) \end{array} \right) . \end{aligned}$$

• Step 2 (b) For any two B$P_{c}$FSs L and Q, the weighted distance measure presented by Sindhu et al. [23] is given below, compute the similarity measure $\tilde{S}_{pc}(L,Q)$ as: $\tilde{S}_{pc}(L,Q)= 1-D^{w}_{pc}(L,Q)$, where

  $$\begin{aligned} \tiny D^{w}_{pc}(L,Q)=\sum ^{n}_{i=1} w_{j}\left( \begin{array}{c} \left[ \begin{array}{c} |\alpha _{B_{Pc}}^{l+}(x_{i})-\alpha _{B_{Pc}}^{q+}(x_{i})|+|\gamma _{B_{Pc}}^{l+}(x_{i})-\gamma _{B_{Pc}}^{q+}(x_{i})|+|\beta _{B_{Pc}}^{l+}(x_{i})-\beta _{B_{Pc}}^{q+}(x_{i})| \\ +|\alpha _{B_{Pc}}^{l-}(x_{i})-\alpha _{B_{Pc}}^{q-}(x_{i})|+|\gamma _{B_{Pc}}^{l-}(x_{i})-\gamma _{B_{Pc}}^{q-}(x_{i})|+|\beta _{B_{Pc}}^{l-}(x_{i})-\beta _{B_{Pc}}^{q-}(x_{i})| \end{array} \right] + \\ \max \left[ \begin{array}{c} |\alpha _{B_{Pc}}^{l+}(x_{i})-\alpha _{B_{Pc}}^{q+}(x_{i})|,|\gamma _{B_{Pc}}^{l+}(x_{i})-\gamma _{B_{Pc}}^{q+}(x_{i})|,|\beta _{B_{Pc}}^{l+}(x_{i})-\beta _{B_{Pc}}^{q+}(x_{i})| \\ ,|\alpha _{B_{Pc}}^{l-}(x_{i})-\alpha _{B_{Pc}}^{q-}(x_{i})|,|\gamma _{B_{Pc}}^{l-}(x_{i})-\gamma _{B_{Pc}}^{q-}(x_{i})|,|\beta _{B_{Pc}}^{l-}(x_{i})-\beta _{B_{Pc}}^{q-}(x_{i})| \end{array} \right] \end{array} \right) . \end{aligned}$$

• Step 2 (c) Evaluate the degree of similarity as: $\tilde{S}^{+}_{Pci}(B_{i},\Delta ^{+})=1- D^{w}_{Pc}(B_{i},B^{+}_{pc})$, $\tilde{S}^{-}_{Pci}(B_{i},\Delta ^{-})=1- D^{w}_{Pc}(B_{i},B^{-}_{pc})$, where, $1 \le i \le n$.

• Step 2 (d) Compute the objective function Z as: $Z = \sum ^{n}_{i=1}(\tilde{S}^{+}_{Pci}(B_{i},B^{+}_{pc})-\tilde{S}^{-}_{Pci}(B_{i},B^{-}_{pc})).$

Step 3. With the help of LP model as described in Definition 9, evaluate the weights of criteria by maximizing the objective function Z under the given constraints.

Step 4. Applying the B$P_{c}$FWA and B$P_{c}$FWG operators to process the information in matrix $B_{pc}= [b_{ij}]_{n\times m}$ to accumulate the information of the alternatives $Q_{i}(i = 1, 2, \ldots , n)$.

Step 5. Based on Definition 8, evaluate the score values $\tilde{S^{i}_{f}}$ of $Q_{i}(i = 1, 2, \ldots , n)$.

Step 6. Arrange all the alternatives $Q_{i} (i = 1, 2,\ldots , n)$ from highest to lowest values of $\tilde{S^{i}_{f}}$ obtained in Step 5 and then rank them to choose the best one. The highest and lowest values of $\tilde{S^{i}_{f}}$ indicate the best and worst alternatives, respectively.

5 Practical Example

From the beginning, it has been a big problem for the investors to invest their money in something that can make maximum profit. To do this, investors have to resort to different companies (alternatives). Suppose that an investment company wants to invest a sum of money in a favourable field to make the maximum possible profit. The investment company has formed a committee that will select one of the best of the proposed four companies (alternatives):(1) a car company ($B_{1}$); (2) a food company ($B_{2}$); (3) a computer company ($B_{3}$); and (4) an arms company ($B_{4}$). The committee must decide by considering the following three beneficial criteria: (1) the risk; (2) the growth; and (3) the customer satisfaction. The weights of the criteria are completely unknown and can find out by utilizing objective function computed with the help of TOPSIS under some constraints as given in Step 3. The four possible companies (alternatives) are to be evaluated under the above three criteria in the form of B$P_{c}$FNs, as shown in the following simplified bipolar picture fuzzy decision matrix (B$P_{c}$FDM) denoted by $B_{pc} = [b_{ij}]_{4\times 3}$.

$$\begin{aligned} \tiny B_{pc} = \left( \begin{array}{ccc} \langle 0.5, 0.4, 0.1, -0.3, -0.4, -0.2 \rangle &{} \langle 0.6, 0.3, 0.1, -0.5, -0.3, -0.2\rangle &{} \langle 0.4, 0.4, 0.2, -0.2, -0.4, -0.4\rangle \\ \langle 0.4, 0.4, 0.2, -0.4, -0.3, -0.2\rangle &{} \langle 0.4, 0.1, 0.3, -0.4, -0.3, -0.1\rangle &{} \langle 0.5, 0.5, 0.0, -0.1, -0.2, -0.5\rangle \\ \langle 0.6, 0.2, 0.1, -0.3, -0.2, -0.1\rangle &{} \langle 0.7, 0.2, 0.1, -0.4, -0.3, -0.2\rangle &{} \langle 0.3, 0.4, 0.2, -0.2, -0.3, -0.4\rangle \\ \langle 0.6, 0.3, 0.1, -0.4, -0.2, -0.1\rangle &{} \langle 0.4, 0.1, 0.4, -0.3, -0.2, -0.3\rangle &{} \langle 0.4, 0.5, 0.1, -0.1, -0.4, -0.4\rangle \\ \end{array} \right) \\ \end{aligned}$$

Step 1. Formulate the information given by the DM as a B$P_{c}$FDM,

$B_{pc} = [b_{ij}]_{4\times 3}$.

Step 2. Based on TOPSIS, by using positive and negative ideal solutions given below: $B^{+}_{pc}= \{\langle 0.6, 0.4, 0.2, -0.3, -0.2, -0.1\rangle \langle 0.7, 0.3, 0.4, -0.3, -0.2, -0.1\rangle \langle 0.5, 0.5, 0.2, -0.1, -0.2, -0.4\rangle \}$ $B^{-}_{pc}=\{\langle 0.4, 0.2, 0.1, -0.4, -0.4, -0.2\rangle \langle 0.4, 0.1, 0.1, -0.5, -0.3, -0.3\rangle \langle 0.4, 0.4, 0, -0.2, -0.4, -0.5\rangle \},$ we get the objective function as: $-0.4w_{1}+ 0.8w_{2}$.

Step 3. By applying the LP model, the criteria’s weights under some constraints are computed as:

$$\begin{aligned} \begin{array}{ll} Maximize: &{} Z= -0.4w_{1}+ 0.8w_{2} \\ Subject to: &{} 10w_{1}+ 8w_{2}+ 12w_{3} \ge 10, \\ &{} 10w_{1} + 8w_{2}+ 12w_{3}\le 10.5, \\ &{} 8w_{1}+ 11w_{2}+ 7w_{3}\ge 8, \\ &{} 8w_{1} + 11w_{2}+ 7w_{3}\le 8.5, \\ &{} 12w_{1}+ 15w_{2}+ 12w_{3} \ge 12, \\ &{} 12w_{1} + 15w_{2}+ 12w_{3}\le 12.5, \\ &{} w_{1} + w_{2}+ w_{3}= 1, \\ &{} w_{1},w_{2},w_{3}\ge 0, \\ \end{array} \end{aligned}$$

$w_{1}=0.4000$, $w_{2}=0.3960$ and $w_{3}=0.2000$.

Step 4. Applying the B$P_{c}$FWG operators to accumulate the information given in $B_{pc}= [b_{ij}]_{4\times 3}$, we get,

$$\begin{aligned} R=\left( \begin{array}{c} \langle 0.5154, 0.0024, 0.1159, -0.3403, -0.0024, -0.2312 \rangle \\ \langle 0.4198 , 0.0010, 0, -0.3043, -0.0008, -0.1837 \rangle \\ \langle 0.5563, 0.0007, 0.1159, -0.3115, -0.0008, -0.1752 \rangle \\ \langle 0.5563, 0.0007, 0.1159, -0.2715, -0.0007, -0.2058 \rangle \\ \end{array} \right) \end{aligned}$$

Step 5. Based on Definition 8, the score values $\tilde{S^{i}_{f}}$ of R are obtained as: $\tilde{S^{1}_{f}}=0.5299$, $\tilde{S^{2}_{f}}=0.4660$, $\tilde{S^{3}_{f}}=0.0927$, $\tilde{S^{4}_{f}}=0.0975$.

Step 6. According to the score values of $\tilde{S^{i}_{f}}$ obtained in Step 5, the arrangement of the alternatives is, $B_{1}\succ B_{2}\succ B_{4} \succ B_{3}$, hence, $B_{1}$ (alternative) is the best one.

5.1 Computation for B$P_{c}$FWA Operator

In the current subsection, Steps 4 to 6 are repeated for B$P_{c}$FWA operator:

Step$\acute{4}$. Applying the B$P_{c}$FWA operators to accumulate the information given in $B_{pc}= [b_{ij}]_{4\times 3}$, we have,

$$\begin{aligned} \acute{R}=\left( \begin{array}{c} \langle 0.5240, 0.3582, 0.1159, -0.3698, -0.3582, -0.2312\rangle \\ \langle 0.4203, 0.2424, 0, -0.3480, -0.2780, -0.1837\rangle \\ \langle 0.5993, 0.2312, 0.1159, -0.3227, -0.2563, -0.1752\rangle \\ \langle 0.4888, 0.2161, 0.1747, -0.3069, -0.2312, -0.2058\rangle \\ \end{array} \right) \end{aligned}$$

Step$\acute{5}$. Based on Definition 8, the score values $\tilde{S^{i}_{f}}$ of R are obtained as: $\tilde{S^{1}_{f}}= 0.5194$, $\tilde{S^{2}_{f}}=0.4265$, $\tilde{S^{3}_{f}}=0.0961$, $\tilde{S^{4}_{f}}=0.0679$.

Step$\acute{6}$. According to the score values of $\tilde{S^{i}_{f}}$ obtained in Step 5, the arrangement of the alternatives is, $B_{1}\succ B_{2}\succ B_{3} \succ B_{4}$, that is, the best alternative is $B_{1}$.

6 Comparative Analysis

6.1 Comparative Analysis with B$P_{c}$F-TOPSIS

A comparative study of the proposed B$P_{c}$F MCDM approach with other MCDM techniques like the B$P_{c}$F-TOPSIS presented by Sindhu et al. [23], and VIKOR [17] technique is penned in this section. The MCDM problem provided in Sect. 5 is solved by B$P_{c}$F-TOPSIS to compare the outcomes obtained from these methods. When solving the problem given in Sect. 5 by using B$P_{c}$F-TOPSIS, the steps for evaluating the weights of criteria are the same as that presented in Step 2 of the proposed MCDM model. So, we can repeat the first three steps of the proposed MCDM approach and then move to the next step to achieve the best alternative as:

Step 4. Evaluate the degree of similarity $\tilde{S}^{+}_{Pci}$ and $\tilde{S}^{-}_{Pci}$ by using the formulae as described in Step 2(c) between each alternative and the elements of $B^{+}_{pc}$ and $B^{-}_{pc}$, respectively, we get,

$\tilde{S}^{+}_{Pc1 }= 0.8608$; $\tilde{S}^{+}_{Pc2} = 0.7962$; $\tilde{S}^{+}_{Pc3} = 0.7859$ $\tilde{S}^{+}_{Pc4} = 0.7961$,

and $\tilde{S}^{-}_{Pc1 }= 0.7810$; $\tilde{S}^{-}_{Pc1 }= 0.8306$; $\tilde{S}^{-}_{Pc1 }= 0.8310$; $\tilde{S}^{-}_{Pc1 }= 0.8258$.

Step 5. The relative closeness $R_{Ci}$ of alternative $B_{i}$ with respect to the $B_{P}$FPIS $B^{+}_{pc}$ is obtained on the basis of following formula as:

$$R_{Ci}= \frac{\tilde{S}^{+}_{Pci}}{\tilde{S}^{+}_{Pci}+\tilde{S}^{-}_{Pci}}.$$

$R_{C1} = 0.5243$; $R_{C2} =0.4894$; $R_{C3} =0.4861$; $R_{C4} =0.4908.$ The ranking order is obtained as: $B_{1}\succ B_{4}\succ B_{2}\succ B_{3}\succ ,$ that is $B_{1}$ is the best alternative which matches with the proposed MCDM model perfectly.

6.2 Comparative Analysis with B$P_{c}$F VIKOR

In the present subsection, based on VIKOR technique, an MCDM investigation approach named B$P_{c}$F- VIKOR is developed and implemented to solve the MCDM problems. The MCDM problem provided in Sect. 5 resolved with the help of B$P_{c}$F- VIKOR approach by considering the following steps:

Step 1. Repeat first three Steps of the proposed MCDM model.

Step 4. For any two B$P_{c}$FSs L and Q, the distance measure presented by Sindhu et al. [23] is given below,

$$\begin{aligned} \tiny D_{pc}(L,Q)=\frac{1}{n}\sum ^{n}_{i=1} \left( \begin{array}{c} \left[ \begin{array}{c} |\alpha _{B_{Pc}}^{l+}(x_{i})-\alpha _{B_{Pc}}^{q+}(x_{i})|+|\gamma _{B_{Pc}}^{l+}(x_{i})-\gamma _{B_{Pc}}^{q+}(x_{i})|+|\beta _{B_{Pc}}^{l+}(x_{i})-\beta _{B_{Pc}}^{q+}(x_{i})| \\ +|\alpha _{B_{Pc}}^{l-}(x_{i})-\alpha _{B_{Pc}}^{q-}(x_{i})|+|\gamma _{B_{Pc}}^{l-}(x_{i})-\gamma _{B_{Pc}}^{q-}(x_{i})|+|\beta _{B_{Pc}}^{l-}(x_{i})-\beta _{B_{Pc}}^{q-}(x_{i})| \end{array} \right] + \\ \max \left[ \begin{array}{c} |\alpha _{B_{Pc}}^{l+}(x_{i})-\alpha _{B_{Pc}}^{q+}(x_{i})|,|\gamma _{B_{Pc}}^{l+}(x_{i})-\gamma _{B_{Pc}}^{q+}(x_{i})|,|\beta _{B_{Pc}}^{l+}(x_{i})-\beta _{B_{Pc}}^{q+}(x_{i})| \\ ,|\alpha _{B_{Pc}}^{l-}(x_{i})-\alpha _{B_{Pc}}^{q-}(x_{i})|,|\gamma _{B_{Pc}}^{l-}(x_{i})-\gamma _{B_{Pc}}^{q-}(x_{i})|,|\beta _{B_{Pc}}^{l-}(x_{i})-\beta _{B_{Pc}}^{q-}(x_{i})| \end{array} \right] \end{array} \right) . \end{aligned}$$

Based on the above distance measure, evaluate the values of $\alpha _{i}$, $\beta _{i}$ and $\eta _{i}$ as:

$$\begin{aligned} \alpha _{i}= & {} \sum ^{n}_{j=1}w_{j}\frac{D_{Pc}(B^{+}_{pc},b_{ij})}{D_{Pc}(B^{+}_{pc},B^{-}_{pc})},\\ \beta _{i}= & {} \underset{j}{\max }[w_{j}\frac{D_{Pc}(B^{+}_{pc},b_{ij})}{D_{Pc}(B^{+}_{pc},B^{-}_{pc})}],\\ \eta _{i}= & {} \frac{v(\alpha _{i}-\alpha ^{\star })}{(\alpha ^{-}-\alpha ^{\star })}+\frac{(1-v)(\beta _{i}-\beta ^{\star })}{(\beta ^{-}-\beta ^{\star })}, \end{aligned}$$

where, $\alpha ^{\star }=\underset{i}{\min }\alpha _{i}, \alpha ^{-}=\underset{i}{\max }\alpha _{i}$ and $\beta ^{\star }=\underset{i}{\min }\beta _{i}, \beta ^{-}=\underset{i}{\max }\beta _{i}$, and v is the weight of the strategy of the majority of the criteria, and its esteems always lie in the interval [0, 1]., and generally, the value of v cab be assumed as $v=0.5.$ The values of $\alpha _{i}$, $\beta _{i}$, and $\eta _{i}$, where $i=1, 2, 3, 4$ obtained by the above formulae are penned in Table 1.

Table 1 Results obtained $\alpha _{i}$, $\beta _{i}$, and $\eta _{i}$

Arrange the alternatives according to the values of $\alpha _{i}$, $\beta _{i}$, and $\eta _{i}$ from lower to higher-order. From these values of $\alpha _{i}$, $\beta _{i}$, and $\eta _{i}$, we get three ranking arrangements that are further used to suggest the compromise solution of the options. The term $\eta _{i}$ is called the measure of separation of $B_{i}$ from the superior option, which represents that the minimum value of $\eta _{i}$ gives the superior option. The compromise solution of the alternative $B_{1}$ is computed when it has a minimum value of $\eta _{i}$ and satisfied the following two conditions:

Condition 1. An acceptable advantage for decision-making: $\eta (B^{2})-\eta (B^{1})\ge \frac{1}{m},$ where m is the number of alternatives, $B^{1}$ and $B^{2}$ are the first two alternatives in $\eta _{i}.$

Condition 2. Acceptable stability for decision-making: This condition describes that if the alternative $B^{1}$ has superior ranking according to the values of $\eta _{i}$ then it must also be the superior according to the values obtained by $\alpha _{i}$ and/or $\beta _{i}$: If any one of the two conditions is not fulfilled, a collection of compromise solutions are gotten as follows:

• $B^{1}$ and $B^{2}$ if only the Condition 2. is not satisfied, or

• If the Condition 1. is not satisfied, then this compromise solution contains the alternatives $B^{1}, B^{2}, \ldots , B^{m}$ is determined as: $\eta (B^{K})-\eta (B^{1})\ge \frac{1}{m}$ for largest K.

Step 5. Since the Condition 1. and Condition 2. are satisfied, and hence the alternative $B^{1}$ is the superior one which is illustrated graphically in Fig. 1.

Fig. 1

Ranking order of alternatives

The outcomes obtained by B$P_{c}$F- TOPSIS and B$P_{c}$F- VIKOR are coincides completely with the proposed MCDM approach which shows the authentication and reliability of our model.

6.3 Time Complexity

In this subsection, a brief and comprehensive comparative have been performed by using time complexity (TC) analysis. TC is measured by the number of fundamental operations which is executed for given information as a function of input size k (k is the number of alternatives). Generally, AHP presented by Saaty [19] is used to find out the weights of criteria in MCDM processes. However, it has an issue in time complexity, when the number of criteria increases in MCDM problems then AHP takes much time to execute. To overcome this problem, we introduced the LP technique to evaluate the weights of criteria which takes less time to execute the big data calculations. By using MATLAB, we have $\frac{1}{4}$seconds, $\frac{2}{5}$seconds and $\frac{23}{50}$seconds TC values (run time) obtained by proposed MCDM approach, B$P_{c}$F- TOPSIS and B$P_{c}$F- VIKOR, respectively. Figure 2 reveals the comparison based on TC between the proposed, B$P_{c}$F- TOPSIS and B$P_{c}$F- VIKOR approaches.

Fig. 2

Comparison of time complexity

The analysis indicates that our proposed MCDM approach take minimum TC value that is our approach is better than B$P_{c}$F- TOPSIS and B$P_{c}$F- VIKOR approaches.

7 Sensitivity Analysis

Generally, the data for MCDM problems are unclear and imprecise so, there is a need for an instrument that provides us with an effective decision that’s why we use the SA in this regard. SA can be applied to investigate the variation of outcomes by changing the weights of criteria. In this section, weighted SA is used to examine the effect on the outcomes achieved by the proposed model after changing the weights of criteria. A formula introduced by Alireza [5] is implemented to get the new weight vector for the criteria and then investigate the behavior of the outcomes attained by the proposed model. We altered the weights of criteria individually by adding 0.2 and determine the effect on the outcomes. The results obtained by altering the values of weights of criteria are illustrated in Tables 2 and 3.

Table 2 Results obtained for altering the weights of criteria

Table 3 Results obtained for altering the weights of criteria

Figures 3 and 4 show the slight fluctuation among the alternatives after altering the weights of criteria by adding 0.2 in each weight, it reveals that a slight change is occurred in the numeric values of score function, however, the order of ranking remains same which empower our proposed model.

Fig. 3

Ranking comparison with distinct weights of criteria

Fig. 4

Ranking comparison with distinct weights of criteria

8 Discussion and Conclusions

The main purpose of this work is to make it easy for DMs to make decisions. AOs are presented for B$P_{c}$FSs which accumulate more information in a better way. The features and characteristics of AOs like B$P_{c}$FWA and B$P_{c}$FWG operators are discussed comprehensively. An MCDM approach is proposed to deal with the uncertain, vague and incomplete information under the framework of B$P_{c}$FSs. However, existing techniques [26, 33] deal with the intuitionistic fuzzy and picture fuzzy environment rather than B$P_{c}$F environment. Consequently, the suggested MCDM approach based on B$P_{c}$FWG and B$P_{c}$FWA operators reveals the trustworthiness in the field of rational manipulation. The results obtained by $P_{c}$FWG, B$P_{c}$FWA, B$P_{c}$F-TOPSIS, and B$P_{c}$F-VIKOR are penned in Table 4.

Table 4 Comparison with proposed MCDM model, B$P_{c}$F-TOPSIS and B$P_{c}$F-VIKOR

Figure 5 reveals a comparative analysis of proposed MCDM approach with other techniques graphically.

Fig. 5

Ranking order of alternatives

AOs play a vital role, to sum up, the information in the decision-making procedure, and therefore, the current article presented a couple of novel AOs for B$P_{c}$FSs, named as B$P_{c}$FWA and B$P_{c}$FWG operators. Various features of the endorsed operators are presented and then, we have used these operators to remedy MCDM problems. There has always been a problem for DMs to allocate the weights to criteria. To overcome this problem, we have used the LP model to find out the weights of criteria so that favouritism can be eliminated. Based on B$P_{c}$FWA and B$P_{c}$FWG operators, an MCDM model is presented to resolve a money investment problem for validity and effectiveness. These basic AOs can help us to develop generalized weighted AOs, Bonferroni, and Hamy mean for B$P_{c}$F environment in future.