New Dombi aggregation operators on bipolar neutrosophic set with application in multi-attribute decision-making problems

Abstract

In this paper, we investigate two new Dombi aggregation operators on bipolar neutrosophic set namely bipolar neutrosophic Dombi prioritized weighted geometric aggregation (BNDPWGA) and bipolar neutrosophic Dombi prioritized ordered weighted geometric aggregation (BNDPOWGA) by means of Dombi t-norm (TN) and Dombi t-conorm (TCN). We discuss their properties along with proofs and multi-attribute decision making (MADM) methods in detail. New algorithms based on proposed models are presented to solve multi-attribute decision-making (MADM) problems. In contrast, with existing techniques a comparison analysis of proposed methods are also demonstrated to test their validity, accuracy and significance.

Keywords

Bipolar neutrosophic set bipolar neutrosophic Dombi prioritized aggregation operators decision-making environment

1 Introduction

In fuzzy theory, a newly defined model is not accessible unless it reduce (overcome) drawbacks of previously defined related models. Due to vagueness and uncertainties issues in many daily life problems, routine mathematics is not always available. To deal with such issues, various procedures such as hypothesis of probability, rough set hypothesis, and fuzzy set hypotheses have been considered as alternative models. Unfortunately, most of such alternative mathematics has its own down sides and drawbacks. For instance, most of the words like, genuine, lovely, best, renowned are not measurable and are, in fact, ambiguous. The criteria for words like wonderful, best, renowned etc., fluctuate from individual to individual. To handle such type of ambiguous and uncertain information, Zadeh [1] initiated the study of possibility based on participation function, that assigns an enrollment grade in [0, 1], called fuzzy set (FSs). FSs have only one membership degree and cannot handle complex problems. The concept of intuitionistic fuzzy set (IFS) was introduced by Atanassov [2]. The intuitionistic fuzzy set (IFS) includes both membership degree and non-membership degree. Subsequently, the concept of interval-valued intuitionistic fuzzy set (IVIFS) was introduced by Atanassov and Gargov [3]. Note that IVIFS is a generalization of IFS. In decision-making problems, IFS and IVIFS are not good at solving problems regarding inconsistent information. The concept of neutrosophic set (NS) was introduced by Smarandache [4]. The neutrosophic set (NS) includes truth, falsity and indeterminacy membership degree which is used to characterize incomplete, inconsistent and uncertain information. Wang et al. [5, 6] introduced the concept of interval neutrosophic set (INS) and the concept of single-valued neutrosophic set (SVNS) to apply NS in daily life decision-making problems.

When it comes to decision-making problems, we think about positive and negative consequences before taking a decision. Positive information explains why a decision is acceptable, permitted, possible, described, or satisfactory. While a negative information explains why a decision is rejected, forbidden, or impossible [7]. Satisfactory or acceptable perceptions are positive preferences, while unsatisfactory or unacceptable perceptions are negative preferences. Positive preferences are related with desires, while negative preferences relate to the constraints [8]. For example, when a decision maker examines an object, he or she need to explain that why he or she considered a satisfaction or an accessible object. Similarly, one has to explain why he or she considered a dissatisfaction or an unaccessible object [9]. In [10], Zhang introduced the concept of bipolar fuzzy set (BFS) consisting of positive membership degree and negative membership degree to describe the words like satisfaction and accessible. Deli et al. introduced the concept of bipolar neutrosophic set (BNS) which describes fuzzy, bipolar, inconsistent and uncertain information in [11]. In [12], Dey et al. proposed the bipolar neutrosophic TOPSIS (BN-TOPSIS) method to solve MADM problems under bipolar neutrosophic fuzzy environments. Zhang et al. proposed the methods based on the Frank Choquet Bonferroni Mean Operators to solve MADM problems under bipolar neutrosophic fuzzy environments in [9]. Many researchers investigated different kind of operators with application in decision-making problems in [14 –18].

The MADM model refers to making decisions when there are multiple but a finite list of alternatives and attributes. Dombi [13] introduced new triangular norms which are Dombi TN and Dombi TCN. Dombi TN and Dombi TCN showed good flexibility with operational parameters. Until now, Dombi operations have not been extended to aggregate bipolar neutrosophic fuzzy environments.

The compatibility and validity of a newly defined fuzzy operator over different fuzzy environments is always challenging and worthwhile. Also, a comparative study analysis with perviously defined MADM methods is of great intrust. If a newly defined fuzzy operator can be reduced to MADM method with a batter accuracy then this newly defined operator would be more flexible, charming and useful. The motivation behind this work was to introduce new operators based on Dombi aggregation operator with a batter accuracy in contrast to existing operators.

In this work, we extend Dombi operations to aggregate bipolar neutrosophic fuzzy environments. Also, we establish new aggregation operators based on the combination of bipolar neutrosophic numbers(BNNs) and Dombi operations. We have proposed two new bipolar neutrosophic Dombi aggregation operators to aggregate bipolar neutrosophic fuzzy information, and developed MADM methods based on BNDPWGA, and BNDPOWGA operators to solve MADM problems with bipolar neutrosophic fuzzy information. The flexibility and accuracy over a multi-attributed problem has been demonstrated and verified as an alternative multi-criteria decision making tool. This was the true motivation behind this work.

The rest of the paper is arranged as follows. Section 2 reviews some of the basic definitions and concepts which will be used frequently. Section 3 defines Dombi operations of bipolar neutrosophic numbers (BNNs). Section 4 and Section 5 propose new operators (BNDPWGA and BNDPOWGA) together with their properties and comprehensive MADM methods based on proposed bipolar neutrosophic Dombi aggregation operators. Section 6 provides a numerical example for the selection of cultivating crops. Section 7 confers the effect of parameters and a comparative analysis with existing methods.

2 Preliminaries

In this section, we present a brief survey of few fundamentals of different sorts of sets which will be utilized in sequel.

2.1 Bipolar neutrosophic set and bipolar neutrosophic number

Definition 2.1. [11] Let U be a fixed set. Then, BNS N can be defined as follows: $N (u) = {< τ_{N}^{+} (u), ω_{N}^{+} (u), ℧_{N}^{+} (u), τ_{N}^{-} (u), ω_{N}^{-} (u), ℧_{N}^{-} (u) > | u \in U} .$ Where $τ_{N}^{+}, ω_{N}^{+}, ℧_{N}^{+} : U ⟶ [0, 1]$ and $τ_{N}^{-}, ω_{N}^{-}, ℧_{N}^{-} : U ⟶ [- 1, 0] .$ The positive membership degrees $τ_{N}^{+} (u), ω_{N}^{+} (u), ℧_{N}^{+} (u)$ are the truth membership, indeterminacy membership degree and falsity membership degree of an element u ∈ U corresponding to BNS N and the negative membership degrees $τ_{N}^{-} (u), ω_{N}^{-} (u), ℧_{N}^{-} (u)$ denote the truth membership degree, indeterminacy membership degree and falsity membership degree of an element u ∈ U to some implicit counter property corresponding to a BNS N.

In particular, if U has only one element, then $N (u) = < τ_{N}^{+} (u), ω_{N}^{+} (u), ℧_{N}^{+} (u), τ_{N}^{-} (u),$ $ω_{N}^{-} (u), ℧_{N}^{-} (u) >$ is called bipolar neutrosophic number(BNN). For convenience, BNN $N = < τ_{N}^{+} (u), ω_{N}^{+} (u), ℧_{N}^{+} (u), τ_{N}^{-} (u), ω_{N}^{-} (u), ℧_{N}^{-} (u) >$ is also denoted as $N = < τ_{N}^{+}, ω_{N}^{+}, ℧_{N}^{+}, τ_{N}^{-}, ω_{N}^{-}, ℧_{N}^{-} >$ .

Deli et al. [11] defined the algebraic operations of BNNs which are as follows:

Definition 2.2. Let ${\tilde{ℵ}}_{1} = < τ_{1}^{+}, ω_{1}^{+}, ℧_{1}^{+}, τ_{1}^{-},$ $ω_{1}^{-}, ℧_{1}^{-} >$ and ${\tilde{ℵ}}_{2} = < τ_{2}^{+}, ω_{2}^{+}, ℧_{2}^{+}, τ_{2}^{-}, ω_{2}^{-}, ℧_{2}^{-} >$ be two BNNs. Then, algebraic operations of BNNs are defined as follows:

(1) ${\tilde{ℵ}}_{1} \oplus {\tilde{ℵ}}_{2} = < τ_{1}^{+} + τ_{2}^{+} - τ_{1}^{+} τ_{2}^{+}, ω_{1}^{+} ω_{2}^{+}, ℧_{1}^{+} ℧_{2}^{-},$ $- τ_{1}^{-} τ_{2}^{-}, - (- ω_{1}^{-} - ω_{2}^{-} - ω_{1}^{-} ω_{2}^{-}), -$ $(- ℧_{1}^{-} - ℧_{2}^{-} - ℧_{1}^{-} ℧_{2}^{-}) >$ ,

(2) ${\tilde{ℵ}}_{1} \otimes {\tilde{ℵ}}_{2} = < τ_{1}^{+} τ_{2}^{+}, ω_{1}^{+} + ω_{2}^{+} - ω_{1}^{+} ω_{2}^{+}, ℧_{1}^{+} + ℧_{2}^{+} - ℧_{1}^{+} ℧_{2}^{+}, - (- τ_{1}^{-} - τ_{2}^{-} - τ_{1}^{-} τ_{2}^{-}), -$ $ω_{1}^{-} ω_{2}^{-}, - ℧_{1}^{-} ℧_{2}^{-} >$ ,

(3) $ℓ . {\tilde{ℵ}}_{1} = < 1 - (1 - τ_{1}^{+})^{ℓ}, (ω_{1}^{+})^{ℓ}, (℧_{1}^{+})^{ℓ}, (- τ_{1}^{-})^{ℓ},$ $- (1 - (1 - (- ω_{1}^{-})^{ℓ})), - (1 - (1 - (- ℧_{1}^{-})^{ℓ})) > (ℓ > 0)$ ,

(4) ${\tilde{ℵ}}_{1}^{ℓ} = < (τ_{1}^{+})^{ℓ}, 1 - (1 - ω_{1}^{+})^{ℓ}, 1 - (1 - ℧_{1}^{+})^{ℓ},$ $- (1 - (1 - (- τ_{1}^{-})^{ℓ})), - (- ω_{1}^{-})^{ℓ},$ $- (- ℧_{1}^{-})^{ℓ} > (ℓ > 0) .$

For comparing two BNNs, Deli et al. [11] developed a comparison method which consists of the score function, accuracy function and certainty function.

Definition 2.3. [11] Let ${\tilde{ℵ}}_{1} = < τ_{1}^{+}, ω_{1}^{+}, ℧_{1}^{+}, τ_{1}^{-}, ω_{1}^{-}, ℧_{1}^{-} >$ be BNN, then the score function $s (\tilde{ℵ_{1}})$ , accuracy function $a (\tilde{ℵ_{1}})$ and certainty function $c (\tilde{ℵ_{1}})$ are defined as:

$s ({\tilde{ℵ}}_{1}) = \frac{τ_{1}^{+} + 1 - ω_{1}^{+} + 1 - ℧_{1}^{+} + 1 + τ_{1}^{-} - ω_{1}^{-} - ℧_{1}^{-}}{6}$ (1) $a ({\tilde{ℵ}}_{1}) = (τ_{1}^{+} - ℧_{1}^{+}) + (τ_{1}^{-} - ℧_{1}^{-})$ (2) $c ({\tilde{ℵ}}_{1}) = 7 τ_{1}^{+} - ℧_{1}^{-}$ (3) The comparison method of BNNs can be obtained based on Equations (1) - (3) as follows.

Definition 2.4. [11] Let ${\tilde{ℵ}}_{1} = < τ_{1}^{+}, ω_{1}^{+}, ℧_{1}^{+}, τ_{1}^{-}, ω_{1}^{-}, ℧_{1}^{-} >$ and ${\tilde{ℵ}}_{2} = < τ_{2}^{+}, ω_{2}^{+}, ℧_{2}^{+}, τ_{2}^{-}, ω_{2}^{-}, ℧_{2}^{-} >$ be two BNNs, therefore

(1) if $s ({\tilde{ℵ}}_{1}) > s ({\tilde{ℵ}}_{2})$ , then ${\tilde{ℵ}}_{1} > {\tilde{ℵ}}_{2}$ ,

(2) if $s ({\tilde{ℵ}}_{1}) = s ({\tilde{ℵ}}_{2})$ and $a ({\tilde{ℵ}}_{1}) > a ({\tilde{ℵ}}_{2})$ , then ${\tilde{ℵ}}_{1} > {\tilde{ℵ}}_{2}$ ,

(3) if $s ({\tilde{ℵ}}_{1}) = s ({\tilde{ℵ}}_{2})$ , $a ({\tilde{ℵ}}_{1}) = a ({\tilde{ℵ}}_{2})$ and $c ({\tilde{ℵ}}_{1}) > c ({\tilde{ℵ}}_{2})$ , then ${\tilde{ℵ}}_{1} > {\tilde{ℵ}}_{2}$ ,

(4) if $s ({\tilde{ℵ}}_{1}) = s ({\tilde{ℵ}}_{2})$ , $a ({\tilde{ℵ}}_{1}) = a ({\tilde{ℵ}}_{2})$ and $c ({\tilde{ℵ}}_{1}) = c ({\tilde{ℵ}}_{2})$ , then ${\tilde{ℵ}}_{1} \sim {\tilde{ℵ}}_{2}$ .

2.2 Dombi operations

The Dombi product and Dombi sum are special cases of TN and TCN respectively, and are given in the following definition.

Definition 2.5. [13] Let ℵ₁ and ℵ₂ be any two real numbers, then Dombi TN and Dombi TCN are defined in the following expressions:

$\begin{matrix} ℵ_{1} \otimes_{D} ℵ_{2} & = Dom (ℵ_{1}, ℵ_{2}) \\ = \frac{1}{1 + {(\frac{1 - ℵ_{1}}{ℵ_{1}})^{λ} + (\frac{1 - ℵ_{2}}{ℵ_{2}})^{λ}}^{\frac{1}{λ}}} \end{matrix}$ (4) and

$\begin{matrix} ℵ_{1} \oplus_{D} ℵ_{2} & = {Dom}^{*} (ℵ_{1}, ℵ_{2}) \\ = 1 - \frac{1}{1 + {(\frac{ℵ_{1}}{1 - ℵ_{1}})^{λ} + (\frac{ℵ_{2}}{1 - ℵ_{2}})^{λ}}^{\frac{1}{λ}}} \end{matrix}$ (5) where λ ≥ 1 and (ℵ ₁, ℵ ₂) ∈ [0, 1] × [0, 1]. Some special cases can be easily proved.

if λ ⟶ 1, then $ℵ_{1} \oplus_{D} ℵ_{2} ⟶ \frac{ℵ_{1} + ℵ_{2} - 2 ℵ_{1} ℵ_{2}}{1 - ℵ_{1} ℵ_{2}}$ and $ℵ_{1} \otimes_{D} ℵ_{2} ⟶ \frac{ℵ_{1} ℵ_{2}}{ℵ_{1} + ℵ_{2} - ℵ_{1} ℵ_{2}}$ ,

If λ⟶ ∞, then ℵ₁ ⊕ _D ℵ ₂ ⟶ max (ℵ ₁, ℵ ₂) and ℵ₁ ⊗ _D ℵ ₂ ⟶ min (ℵ ₁, ℵ ₂). Dombi sum and Dombi product are reduced to simple max-operator and simple min-operator, respectively.

3 Dombi operations of BNNs

In this section, we discuss the Dombi operations of BNNs and discuss their properties.

Dombi operations of BNNs are defined as follows.

Definition 3.1. Let ${\tilde{ℵ}}_{1} = < τ_{1}^{+}, ω_{1}^{+}, ℧_{1}^{+}, τ_{1}^{-}, ω_{1}^{-}, ℧_{1}^{-} >$ and ${\tilde{ℵ}}_{2} = < τ_{2}^{+}, ω_{2}^{+}, ℧_{2}^{+}, τ_{2}^{-}, ω_{2}^{-}, ℧_{2}^{-} >$ be two BNNs and λ ≥ 1. Then, Dombi sum and Dombi product of two BNNs ${\tilde{ℵ}}_{1}$ and ${\tilde{ℵ}}_{2}$ are denoted by ${\tilde{ℵ}}_{1} \otimes_{D} {\tilde{ℵ}}_{2}$ and ${\tilde{ℵ}}_{1} \oplus_{D} {\tilde{ℵ}}_{2}$ respectively and defined as follows:

(1) ${\tilde{ℵ}}_{1} \oplus_{D} {\tilde{ℵ}}_{2} =$ ${\begin{matrix} 〈 1 - \frac{1}{1 + {(\frac{τ_{1}^{+}}{1 - τ_{1}^{+}})^{λ} + (\frac{τ_{2}^{+}}{1 - τ_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {(\frac{1 - ω_{1}^{+}}{ω_{1}^{+}})^{λ} + (\frac{1 - ω_{2}^{+}}{ω_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {(\frac{1 - ℧_{1}^{+}}{℧_{1}^{+}})^{λ} + (\frac{1 - ℧_{2}^{+}}{℧_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{- 1}{1 + {(\frac{1 + τ_{1}^{-}}{- τ_{1}^{-}})^{λ} + (\frac{1 + τ_{2}^{-}}{- τ_{2}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {(\frac{- ω_{1}^{-}}{1 + ω_{1}^{-}})^{λ} + (\frac{- ω_{2}^{-}}{1 + ω_{2}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {(\frac{- ℧_{1}^{-}}{1 + ℧_{1}^{-}})^{λ} + (\frac{- ℧_{2}^{-}}{1 + ℧_{2}^{-}})^{λ}}^{\frac{1}{λ}}} - 1 〉 \end{matrix}$

(2) ${\tilde{ℵ}}_{1} \otimes_{D} {\tilde{ℵ}}_{2} =$ ${\begin{matrix} 〈 \frac{1}{1 + {(\frac{1 - τ_{1}^{+}}{τ_{1}^{+}})^{λ} + (\frac{1 - τ_{2}^{+}}{τ_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {(\frac{ω_{1}^{+}}{1 - ω_{1}^{+}})^{λ} + (\frac{ω_{2}^{+}}{1 - ω_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {({\frac{℧_{1}^{+}}{1 - ℧_{1}^{+}}}^{λ} + (\frac{℧_{2}^{+}}{1 - ℧_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {(\frac{- τ_{1}^{-}}{1 + τ_{1}^{-}})^{λ} + (\frac{- τ_{2}^{-}}{1 + τ_{2}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {(\frac{1 + ω_{1}^{-}}{- ω_{1}^{-}})^{λ} + (\frac{1 + ω_{2}^{-}}{- ω_{2}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {(\frac{1 + ℧_{1}^{-}}{- ℧_{1}^{-}})^{λ} + (\frac{1 + ℧_{2}^{-}}{- ℧_{2}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

(3) $ℓ ._{D} {\tilde{ℵ}}_{1} =$ ${\begin{matrix} 〈 1 - \frac{1}{1 + {ℓ (\frac{τ_{1}^{+}}{1 - τ_{1}^{+}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {ℓ (\frac{1 - ω_{1}^{+}}{ω_{1}^{+}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {ℓ (\frac{1 - ℧_{1}^{+}}{℧_{1}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{- 1}{1 + {ℓ (\frac{1 + τ_{1}^{-}}{- τ_{1}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {ℓ (\frac{- ω_{1}^{-}}{1 + ω_{1}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {ℓ (\frac{- ℧_{1}^{-}}{1 + ℧_{1}^{-}})^{λ}}^{\frac{1}{λ}}} - 1 〉 (ℓ > 0) \end{matrix}$

(4) ${\tilde{ℵ}}_{1}^{⋀_{D}^{ℓ}} =$ ${\begin{matrix} 〈 \frac{1}{1 + {ℓ (\frac{1 - τ_{1}^{+}}{τ_{1}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {ℓ (\frac{ω_{1}^{+}}{1 - ω_{1}^{+}})^{λ}}^{\frac{1}{q}}}, 1 - \frac{1}{1 + {ℓ (\frac{℧_{1}^{+}}{1 - ℧_{1}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {ℓ (\frac{- τ_{1}^{-}}{1 + τ_{1}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {ℓ (\frac{1 + ω_{1}^{-}}{- ω_{1}^{-}})^{q}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {ℓ (\frac{1 + ℧_{1}^{-}}{- ℧_{1}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 (ℓ > 0) . \end{matrix}$

Theorem 3.2. Let ${\tilde{ℵ}}_{1} = < τ_{1}^{+}, ω_{1}^{+}, ℧_{1}^{+}, τ_{1}^{-}, ω_{1}^{-}, ℧_{1}^{-} >$ and ${\tilde{ℵ}}_{2} = < τ_{2}^{+}, ω_{2}^{+}, ℧_{2}^{+}, τ_{2}^{-}, ω_{2}^{-}, ℧_{2}^{-} >$ be two BNNs and let $\tilde{e} = {\tilde{ℵ}}_{1} \oplus_{D} {\tilde{ℵ}}_{2}$ , $\tilde{f} = {\tilde{ℵ}}_{1} \otimes_{D} {\tilde{ℵ}}_{2}$ , $\tilde{g} = ℓ ._{D} {\tilde{ℵ}}_{1}$ (ℓ >0) and $\tilde{h} = {\tilde{ℵ}}_{1}^{⋀_{D}^{ℓ}}$ (ℓ >0). Therefore, $\tilde{e}$ , $\tilde{f}$ , $\tilde{g}$ and $\tilde{h}$ are also BNNs.

As Theorem 3 can be easily verified. Therefore, the proof is omitted here. The properties of Dombi operations of BNNs are defined as:

Theorem 3.3. Let ${\tilde{ℵ}}_{1} = < τ_{1}^{+}, ω_{1}^{+}, ℧_{1}^{+}, τ_{1}^{-}, ω_{1}^{-}, ℧_{1}^{-} >$ and ${\tilde{ℵ}}_{2} = < τ_{2}^{+}, ω_{2}^{+}, ℧_{2}^{+}, τ_{2}^{-}, ω_{2}^{-}, ℧_{2}^{-} >$ be two BNNs and ℓ, ℓ ₁, ℓ ₂ > 0. Then, the following properties can be proven easily.

(a) ${\tilde{ℵ}}_{1} \oplus_{D} {\tilde{ℵ}}_{2} = {\tilde{ℵ}}_{2} \oplus_{D} {\tilde{ℵ}}_{1}$ ,

(b) ${\tilde{ℵ}}_{1} \otimes_{D} {\tilde{ℵ}}_{2} = {\tilde{ℵ}}_{2} \otimes_{D} {\tilde{ℵ}}_{1}$ ,

(c) $ℓ ._{D} ({\tilde{ℵ}}_{1} \oplus_{D} {\tilde{ℵ}}_{2}) = ℓ ._{D} {\tilde{ℵ}}_{2} \oplus_{D} ℓ ._{D} {\tilde{ℵ}}_{1}$ ,

(d) $({\tilde{ℵ}}_{1} \otimes_{D} {\tilde{ℵ}}_{2})^{⋀_{D}^{ℓ}} = {\tilde{ℵ}}_{1}^{⋀_{D}^{ℓ}} \otimes_{D} {\tilde{ℵ}}_{2}^{⋀_{D}^{ℓ}}$ ,

(e) $(ℓ_{1} + ℓ_{2}) ._{D} {\tilde{ℵ}}_{1} = ℓ_{1} ._{D} {\tilde{ℵ}}_{1} \oplus_{D} ℓ_{2} ._{D} {\tilde{ℵ}}_{1}$ ,

(f) ${\tilde{ℵ}}_{1}^{⋀_{D}^{ℓ_{1} + ℓ_{2}}} = {\tilde{ℵ}}_{1}^{⋀_{D}^{ℓ_{1}}} \otimes_{D} {\tilde{ℵ}}_{1}^{⋀_{D}^{ℓ_{2}}}$ .

4 Bipolar neutrosophic dombi prioritized aggregation operators

In this section, we define bipolar neutrosophic Dombi weighted geometric prioritized aggregation (BNDPWGA) and the bipolar neutrosophic Dombi prioritized ordered weighted geometric aggregation (BNDPOWGA) operators and discussed different properties of these aggregation operators (AOs) in detail.

Definition 4.1. Let C = {C₁, C₂, . . . , C_n} be a collection of attributes and have a prioritization between the attributes followed by the linear ordering C₁ ≻ C₂ ≻ . . . ≻ C_n, which imply that C_η has a higher priority than C_ρ, if η < ρ. The value of C_ρ (u) is the performance of any alternative u under attribute C_η (u), which satisfies C_η ∈ [0, 1], If

$PA (C_{η} (u)) = Σ_{ρ = 1}^{n} υ_{ρ} C_{ρ} (u)$ (6) where $υ_{ρ} = \frac{ℏ_{ρ}}{Σ_{ρ = 1}^{n} ℏ_{ρ}}$ ; $ℏ_{ρ} = Π_{η = 1}^{ρ - 1} C_{η} (u) (ρ = 2, 3, . . ., n)$ , ℏ₁ = 1. Then, PA is called the prioritized averaging operator.

4.1 Bipolar neutrosophic dombi prioritized weighted geometric aggregation operator

Definition 4.2. Let ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > (α = 1,, 2, 3, . . ., r)$ be a family of BNNs. A mapping BNDPWGA: U^r → U is called BNDPWGA operator, if it satisfies

$BNDPWGA ({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = ⨂_{α = 1}^{r} {\tilde{ℵ}}_{α}^{⋀^{υ_{α}}} = {\tilde{ℵ}}_{1}^{⋀_{D}^{υ_{1}}} \otimes_{D} {\tilde{ℵ}}_{2}^{⋀_{D}^{υ_{2}}} \otimes_{D} . . . \otimes_{D} {\tilde{ℵ}}_{r}^{⋀_{D}^{υ_{r}}}$ (7) where $υ_{α} = \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}}$ , $ℏ_{α} = \prod_{k = 1}^{α - 1} s ({\tilde{ℵ}}_{k})$ (α = 2, 3, . . . , r), ℏ₁ = 1 and $s ({\tilde{ℵ}}_{k})$ is the value of score for ${\tilde{ℵ}}_{α}$ . where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T is the weight vector of ${\tilde{ℵ}}_{α}$ (α = 1, 2, . . . , r), ψ_α ∈ [0, 1] and Σ_α=1ψ_α = 1.

Theorem 4.3. Let ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > (α = 1, 2, 3, . . ., r)$ be a family of BNNs and ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of ${\tilde{ℵ}}_{α}$ , ψ_α ∈ [0, 1] and Σ_α=1ψ_α = 1. Then, the value aggregated by using BNDPWGA operator is still a BNN, which is calculated by using the following formula BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = {\tilde{ℵ}}_{1}^{⋀_{D}^{υ_{1}}} \otimes_{D} {\tilde{ℵ}}_{2}^{⋀_{D}^{υ_{2}}} \otimes_{D} . . . \otimes_{D} {\tilde{ℵ}}_{r}^{⋀_{D}^{υ_{r}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 - τ_{α}^{+}}{τ_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{j = 1}^{r} ℏ_{α} ψ_{α}} (\frac{ω_{α}^{+}}{1 - ω_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{℧_{α}^{+}}{1 - ℧_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{α}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ω_{α}^{-}}{- ω_{α}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{α}^{-}}{- ℧_{α}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$ (8)

Proof. If r = 2 based on Definition 4.1 for the Dombi operations of BNNs, the following result can be obtained:

BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}) = {\tilde{ℵ}}_{1}^{⋀_{D}^{υ_{1}}} \otimes_{D} {\tilde{ℵ}}_{2}^{⋀_{D}^{υ_{2}}} =$ ${\begin{matrix} 〈 \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 - t_{1}^{+}}{τ_{1}^{+}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 - τ_{2}^{+}}{τ_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{ω_{1}^{+}}{1 - ω_{1}^{+}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{ω_{2}^{+}}{1 - ω_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ 1 - \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{℧_{1}^{+}}{1 - ℧_{1}^{+}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{℧_{2}^{+}}{1 - ℧_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{- τ_{1}^{-}}{1 + τ_{1}^{-}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{- τ_{2}^{-}}{1 + τ_{2}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \\ \frac{- 1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ω_{1}^{-}}{- ω_{1}^{-}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ω_{2}^{-}}{- ω_{2}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ℧_{1}^{-}}{- ℧_{1}^{-}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ℧_{2}^{-}}{- ℧_{2}^{-}})^{λ}}^{\frac{1}{λ}}} 〉, \end{matrix}$

= ${\begin{matrix} 〈 \frac{1}{1 + {Σ_{α = 1}^{2} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{2} ℏ_{α} ψ_{α}} (\frac{1 - τ_{α}^{+}}{τ_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{2} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{2} ℏ_{α} ψ_{α}} (\frac{ω_{α}^{+}}{1 - ω_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ 1 - \frac{1}{1 + {Σ_{α = 1}^{2} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{2} ℏ_{α} ψ_{α}} (Σ_{α = 1}^{2} \frac{℧_{α}^{+}}{1 - ℧_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {Σ_{α = 1}^{2} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{2} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{α}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \\ \frac{- 1}{1 + {Σ_{α = 1}^{2} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{2} ℏ_{α} ψ_{α}} (\frac{1 + ω_{α}^{-}}{- ω_{α}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{α = 1}^{2} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{2} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{α}^{-}}{- ℧_{α}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 . \end{matrix}$

If r = s, based on equation (8), then we have got the following equation:

BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{s}) = ⨂_{α = 1}^{s} {\tilde{ℵ}}_{α}^{⋀^{υ_{α}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{1 - τ_{α}^{+}}{τ_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{ω_{α}^{+}}{1 - ω_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{℧_{α}^{+}}{1 - ℧_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{α}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{1 + ω_{α}^{-}}{- ω_{α}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{α}^{-}}{- ℧_{α}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 . \end{matrix}$

If r = s + 1, then there is following result:

BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{s}, {\tilde{ℵ}}_{s + 1}) = ⨂_{α = 1}^{s} {\tilde{ℵ}}_{α}^{⋀^{υ_{α}}} ⨂ {\tilde{ℵ}}_{s + 1}^{⋀^{υ_{s + 1}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{1 - τ_{α}^{+}}{τ_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{ω_{α}^{+}}{1 - ω_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{℧_{α}^{+}}{1 - ℧_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{α}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{1 + ω_{α}^{-}}{- ω_{α}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {Σ_{α = 1}^{s} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{α}^{-}}{- ℧_{α}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

⊗_D ${\begin{matrix} 〈 \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{1 - τ_{s + 1}^{+}}{τ_{s + 1}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{ω_{s + 1}^{+}}{1 - ω_{s + 1}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{℧_{s + 1}^{+}}{1 - ℧_{s + 1}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{s + 1}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{1 + ω_{s + 1}^{-}}{- ω_{s + 1}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{s + 1}^{-}}{- ℧_{s + 1}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

= ${\begin{matrix} 〈 \frac{1}{1 + {Σ_{α = 1}^{s + 1} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{1 - τ_{α}^{+}}{τ_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{s + 1} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{ω_{α}^{+}}{1 - ω_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{s + 1} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{℧_{α}^{+}}{1 - ℧_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{α = 1}^{s + 1} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{α}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {Σ_{α = 1}^{s + 1} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{1 + ω_{α}^{-}}{- ω_{α}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {Σ_{α = 1}^{s + 1} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{s + 1} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{α}^{-}}{- ℧_{α}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

Hence, Theorem (4.3) is true for r = s + 1. Thus, equation (8) holds for all r. □

The BNDPWGA operator also has the following properties:

(1) Idomopotency: Let all the BNNs be $\tilde{b_{α}} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > = \tilde{ℵ}$ (α = 1, 2, 3, . . . , r), where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of ${\tilde{ℵ}}_{α}, ψ_{α} \in [0, 1]$ and Σ_α=1ψ_α = 1. Then, BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = \tilde{ℵ}$ .

BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r})$ ≥ BNDPWGA $({\tilde{ℵ}}_{1}^{,}, {\tilde{ℵ}}_{2}^{,}, . . ., {\tilde{ℵ}}_{r}^{,})$ .

(3) Boundedness: Let ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > (α = 1,, 2, 3, . . ., r)$ be a family of BNNs, where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of ${\tilde{ℵ}}_{α}$ , ψ_α ∈ [0, 1] and Σ_α=1ψ_α = 1. Therefore, we have BNDPWGA $({\tilde{ℵ}}^{-}, {\tilde{ℵ}}^{-}, . . ., {\tilde{ℵ}}^{-})$

$\leq BNDPWGA ({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r})$ $\leq BNDPWGA ({\tilde{ℵ}}^{+}, {\tilde{ℵ}}^{+}, . . ., {\tilde{ℵ}}^{+})$ ,

where ${\tilde{ℵ}}^{-} = < τ_{{\tilde{ℵ}}^{-}}^{+}, ω_{{\tilde{ℵ}}^{-}}^{+}, ℧_{{\tilde{ℵ}}^{-}}^{+}, τ_{{\tilde{ℵ}}^{-}}^{-}, ω_{{\tilde{ℵ}}^{-}}^{-}, ℧_{{\tilde{ℵ}}^{-}}^{-} >$

= ${\begin{matrix} < \min (τ_{1}^{+}, τ_{2}^{+}, . . ., τ_{r}^{+}), \max (ω_{1}^{+}, ω_{2}^{+}, . . ., ω_{r}^{+}), \max (℧_{1}^{+}, ℧_{2}^{+}, . . ., ℧_{r}^{+}), \\ \max (τ_{1}^{-}, τ_{2}^{-}, . . ., τ_{r}^{-}), \min (ω_{α}^{-}, ω_{α}^{-}, . . ., ω_{α}^{-}), \min (℧_{1}^{-}, ℧_{2}^{-}, . . ., ℧_{r}^{-}) > \end{matrix}$

and ${\tilde{ℵ}}^{+} = < τ_{{\tilde{ℵ}}^{+}}^{+}, ω_{{\tilde{ℵ}}^{+}}^{+}, ℧_{{\tilde{ℵ}}^{+}}^{+}, τ_{{\tilde{ℵ}}^{+}}^{-}, ω_{{\tilde{ℵ}}^{+}}^{-}, ℧_{{\tilde{ℵ}}^{+}}^{-} >$

= ${\begin{matrix} < \max (τ_{1}^{+}, τ_{2}^{+}, . . ., τ_{r}^{+}), \min (ω_{1}^{+}, ω_{2}^{+}, . . ., ω_{r}^{+}), \min (℧_{1}^{+}, ℧_{2}^{+}, . . ., ℧_{r}^{+}), \\ \min (τ_{1}^{-}, τ_{2}^{-}, . . ., τ_{r}^{-}), \max (ω_{α}^{-}, ω_{α}^{-}, . . . ω_{α}^{-}), \max (℧_{1}^{-}, ℧_{2}^{-}, . . ., ℧_{r}^{-}) > . \end{matrix}$

Proof.

(1) Since ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > = \tilde{ℵ} (α = 1,, 2, 3, . . ., r)$ . Then, the following result can be obtained by using equation (8).

BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = ⨂_{α = 1}^{r} {\tilde{ℵ}}_{α}^{⋀^{υ_{α}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 - τ_{α}^{+}}{τ_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{ω_{α}^{+}}{1 - ω_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{℧_{α}^{+}}{1 - ℧_{α}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{α}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ω_{α}^{-}}{- ω_{α}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{α}^{-}}{- ℧_{α}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

$= {\begin{matrix} 〈 \frac{1}{1 + {(\frac{1 - τ^{+}}{τ^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {(\frac{ω^{+}}{1 - ω^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {(\frac{℧^{+}}{1 - ℧^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {(\frac{- τ^{-}}{1 + τ^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {(\frac{1 + ω^{-}}{- ω^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {(\frac{1 + ℧^{-}}{- ℧^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

$= < τ^{+}, ω^{+}, ℧^{+}, τ^{-}, ω^{-}, ℧^{-} > = \tilde{ℵ} .$

Hence, BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = \tilde{ℵ}$ holds.

(2) The property is obvious based on the equation (8).

(3) Let ${\tilde{ℵ}}^{-} = < τ_{{\tilde{ℵ}}^{-}}^{+}, ω_{{\tilde{ℵ}}^{-}}^{+}, ℧_{{\tilde{ℵ}}^{-}}^{+}, τ_{{\tilde{ℵ}}^{-}}^{-}, ω_{{\tilde{ℵ}}^{-}}^{-}, ℧_{{\tilde{ℵ}}^{-}}^{-}$ and ${\tilde{ℵ}}^{+} = < τ_{{\tilde{ℵ}}^{+}}^{+}, ω_{{\tilde{ℵ}}^{+}}^{+}, ℧_{{\tilde{ℵ}}^{+}}^{+}, τ_{{\tilde{ℵ}}^{+}}^{-}, ω_{{\tilde{ℵ}}^{+}}^{-}, ℧_{{\tilde{ℵ}}^{+}}^{-} >$ . There are the following inequalities:

$\frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 - τ_{{\tilde{ℵ}}^{-}}^{+}}{1_{{\tilde{ℵ}}^{-}}^{+}})^{λ}}^{\frac{1}{λ}}} \leq \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 - τ_{α}^{+}}{τ_{α}^{+}})^{λ}}^{\frac{1}{λ}}} \leq \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 - τ_{\tilde{ℵ^{+}}}^{+}}{τ_{{\tilde{ℵ}}^{+}}^{+}})^{λ}}^{\frac{1}{λ}}}$ ,

$1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{ω_{{\tilde{ℵ}}^{+}}^{+}}{1 - ω_{{\tilde{ℵ}}^{+}}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{ω_{α}^{+}}{1 - ω_{α}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{ω_{{\tilde{ℵ}}^{-}}^{+}}{1 - ω_{\tilde{ℵ^{-}}}^{+}})^{λ}}^{\frac{1}{λ}}}$ ,

$1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{℧_{{\tilde{ℵ}}^{+}}^{+}}{1 - ℧_{{\tilde{ℵ}}^{+}}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{℧_{α}^{+}}{1 - ℧_{α}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{℧_{{\tilde{ℵ}}^{-}}^{+}}{1 - ℧_{\tilde{ℵ^{-}}}^{+}})^{λ}}^{\frac{1}{λ}}}$ ,

$\frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{- τ_{\tilde{ℵ^{+}}}^{-}}{1 + τ_{{\tilde{ℵ}}^{+}}^{-}})^{λ}}^{\frac{1}{λ}}} - 1 \leq \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{- τ_{α}^{-}}{1 + τ_{α}^{-}})^{λ}}^{\frac{1}{λ}}} - 1 \leq \frac{1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} ({\frac{- τ_{\tilde{ℵ^{-}}}^{-}}{1 + τ_{\tilde{ℵ^{-}}}^{-}}}^{λ}}^{\frac{1}{λ}}} - 1$ ,

$\frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ω_{{\tilde{ℵ}}^{-}}^{-}}{- ω_{{\tilde{ℵ}}^{-}}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ω_{α}^{-}}{- ω_{α}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ω_{{\tilde{ℵ}}^{+}}^{-}}{- ω_{{\tilde{ℵ}}^{+}}^{-}})^{λ}}^{\frac{1}{λ}}}$ ,

$\frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{{\tilde{ℵ}}^{-}}^{-}}{- ℧_{{\tilde{ℵ}}^{-}}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{α}^{-}}{- ℧_{α}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{α = 1}^{r} \frac{ℏ_{α} ψ_{α}}{Σ_{α = 1}^{r} ℏ_{α} ψ_{α}} (\frac{1 + ℧_{{\tilde{ℵ}}^{+}}^{-}}{- ℧_{{\tilde{ℵ}}^{+}}^{-}})^{λ}}^{\frac{1}{λ}}}$ .

Hence, BNDPWGA $({\tilde{ℵ}}^{-}, {\tilde{ℵ}}^{-}, . . ., {\tilde{ℵ}}^{-}) \leq$ BNDPWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) \leq$ BNDPWGA $({\tilde{ℵ}}^{+}, {\tilde{ℵ}}^{+}, . . ., {\tilde{ℵ}}^{+})$ holds. □

Example 4.4. Let ${\tilde{ℵ}}_{1} = < 0.6, 0.7, 0.3, - 0.6, - 0.3, - 0.5 >$ , ${\tilde{ℵ}}_{2} = < 0.5, 0.4, 0.6, - 0.6, - 0.7, - 0.3 >$ , ${\tilde{ℵ}}_{3} = < 0.6, 0.7, 0.4, - 0.9, - 0.7, - 0.7 >$ and ${\tilde{ℵ}}_{4} = < 0.2, 0.6, 0.8, - 0.6, - 0.3, - 0.9 >$ be four BNNs and let the weight vector of BNNs ${\tilde{ℵ}}_{α} (α = 1, 2, 3, 4)$ be $ψ = (\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{8})^{T}$ . Now, by Definition 4.1, ℏ₁ = 1, $ℏ_{2} = s ({\tilde{ℵ}}_{1}) = 0.4667$ , $ℏ_{3} = s ({\tilde{ℵ}}_{1}) s ({\tilde{ℵ}}_{2}) = 0.2256$ and $ℏ_{4} = s ({\tilde{ℵ}}_{1}) s ({\tilde{ℵ}}_{2}) s ({\tilde{ℵ}}_{3}) = 0.1128$ . $ψ_{1} = \frac{1}{2}$ , $ψ_{2} = \frac{1}{4}$ , $ψ_{3} = \frac{1}{8}$ and $ψ_{4} = \frac{1}{8}$ are the weight of ${\tilde{ℵ}}_{α} (α = 1, 2, 3, 4)$ such that Σ_α=1ψ_α = 1. By Definition 4.2, $\frac{ℏ_{1} ψ_{1}}{Σ_{α = 1}^{4} ℏ_{α} ψ_{α}} = 0.7588$ , $\frac{ℏ_{2} ψ_{2}}{Σ_{α = 1}^{4} ℏ_{α} ψ_{α}} = 0.1771$ , $\frac{ℏ_{3} ψ_{3}}{Σ_{α = 1}^{4} ℏ_{α} ψ_{α}} = 0.0428$ , $\frac{ℏ_{4} ψ_{4}}{Σ_{α = 1}^{4} ℏ_{α} ψ_{α}} = 0.0214$ . Then, by Theorem 4.3, for λ = 3

$BNDPWGA ({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, {\tilde{ℵ}}_{3}, {\tilde{ℵ}}_{4}) =$ $\begin{matrix} {\begin{matrix} 〈 \frac{1}{1 + {0.7588 (\frac{1 - 0.6}{0.6})^{3} + 0.1771 (\frac{1 - 0.5}{0.5})^{3} + 0.0428 (\frac{1 - 0.6}{0.6})^{3} + 0.0214 (\frac{1 - 0.2}{0.2})^{3}}^{\frac{1}{3}}}, \\ 1 - \\ \frac{1}{1 + {0.7588 (\frac{0.7}{1 - 0.7})^{3} + 0.1771 (\frac{0.4}{1 - 0.4})^{3} + 0.0428 (\frac{0.7}{1 - 0.7})^{3} + 0.0214 (\frac{0.6}{1 - 0.6})^{3}}^{\frac{1}{3}}}, \\ 1 - \\ \frac{1}{1 + {0.7588 (\frac{0.3}{1 - 0.3})^{3} + 0.1771 (\frac{0.6}{1 - 0.6})^{3} + 0.0428 (\frac{0.4}{1 - 0.4})^{3} + 0.0214 (\frac{0.8}{1 - 0.8})^{3}}^{\frac{1}{3}}}, \\ \frac{1}{1 + {0.7588 (\frac{0.6}{1 - 0.6})^{3} + 0.1771 (\frac{0.6}{1 - 0.6})^{3} + 0.0428 (\frac{0.9}{1 - 0.9})^{3} + 0.0214 (\frac{0.6}{1 - 0.6})^{3}}^{\frac{1}{3}}} \\ - 1, \\ \frac{- 1}{1 + {0.7588 (\frac{1 - 0.3}{0.3})^{3} + 0.1771 (\frac{1 - 0.7}{0.7})^{3} + 0.0428 (\frac{1 - 0.7}{0.7})^{3} + 0.0214 (\frac{1 - 0.3}{0.3})^{3}}^{\frac{1}{3}}}, \\ \frac{- 1}{1 + {0.7588 (\frac{1 - 0.5}{0.5})^{3} + 0.1771 (\frac{1 - 0.3}{0.3})^{3} + 0.0428 (\frac{1 - 0.7}{0.7})^{3} + 0.0214 (\frac{1 - 0.9}{0.9})^{3}}^{\frac{1}{3}}} 〉, \end{matrix} \end{matrix}$ $BNDPWGA ({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, {\tilde{ℵ}}_{3}, {\tilde{ℵ}}_{4}) = 〈 0.4519, 0.6852, 0.5591, - 0.7649, - 0.3175, - 0.4092 〉 .$

4.2 Bipolar neutrosophic Dombi prioritized ordered weighted geometric aggregation operator

In this section, we propose bipolar neutrosophic Dombi prioritized ordered weighted geometric aggregation operator (BNDPOWGA) which is more useful to pervious defined bipolar neutrosophic Dombi prioritized weighted geometric aggregation operator (BNDPWGA). Because in this operator one more parameter added, called an ordered parameter. This mean that BNDPOWGA is more informative than (BNDPWGA).

Definition 4.5. Let ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > (α = 1, 2, 3, . . ., r)$ be a family of BNNs. A mapping BNDPOWGA: U^r → U is called BNDPOWGA operator, if it satisfies

$BNDPOWGA ({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = ⨂_{β = 1}^{r} {\tilde{ℵ}}_{σ (β)}^{⋀^{υ_{β}}} = {\tilde{ℵ}}_{σ (1)}^{⋀_{D}^{υ_{1}}} \otimes_{D} {\tilde{ℵ}}_{σ (2)}^{⋀_{D}^{υ_{2}}} \otimes_{D} . . . \otimes_{D} {\tilde{ℵ}}_{σ (r)}^{⋀_{D}^{v_{r}}}$ (9) where $υ_{β} = \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}}$ , $ℏ_{β} = \prod_{α = 1}^{β - 1} s ({\tilde{ℵ}}_{α})$ (α = 2, 3, . . . , r), ℏ₁ = 1 and $s ({\tilde{ℵ}}_{α})$ is the value of score for ${\tilde{ℵ}}_{α}$ . where σ is permutation that orders the elements: ${\tilde{ℵ}}_{σ (1)} \geq {\tilde{ℵ}}_{σ (2)} \geq . . . \geq {\tilde{ℵ}}_{σ (r)}$ . where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T is the weight vector of ${\tilde{ℵ}}_{α}$ (α = 1, 2, . . . , r), ψ_α ∈ [0, 1] and Σ_α=1ψ_α = 1.

Theorem 4.6. Let ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > (α = 1, 2, 3, . . ., r)$ be a family of BNNs and ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of ${\tilde{ℵ}}_{α}$ , ψ_α ∈ [0, 1] and Σ_α=1ψ_α = 1. Then, the value aggregated by using BNDPOWGA operator is still a BNN, which is calculated by using the following formula

BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = {\tilde{ℵ}}_{σ (1)}^{⋀_{D}^{υ_{1}}} \otimes_{D} {\tilde{ℵ}}_{σ (2)}^{⋀_{D}^{υ_{2}}} \otimes_{D} . . . \otimes_{D} {\tilde{ℵ}}_{σ (r)}^{⋀_{D}^{υ_{r}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 - τ_{β}^{+}}{τ_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{ω_{β}^{+}}{1 - ω_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{℧_{β}^{+}}{1 - ℧_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{- τ_{β}^{-}}{1 + τ_{β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ω_{β}^{-}}{- ω_{β}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{β}^{-}}{- ℧_{β}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$ (10)

Proof. If r = 2 based on Definition 4.2 for the Dombi operations of BNNs, the following result can be obtained:

BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}) = {\tilde{ℵ}}_{σ (1)}^{⋀_{D}^{υ_{1}}} \otimes_{D} {\tilde{ℵ}}_{σ (2)}^{⋀_{D}^{υ_{2}}} =$ ${\begin{matrix} 〈 \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 - τ_{1}^{+}}{τ_{1}^{+}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 - τ_{2}^{+}}{τ_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{ω_{1}^{+}}{1 - ω_{1}^{+}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{ω_{2}^{+}}{1 - ω_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ 1 - \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{℧_{1}^{+}}{1 - ℧_{1}^{+}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{℧_{2}^{+}}{1 - ℧_{2}^{+}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{- τ_{1}^{-}}{1 + τ_{1}^{-}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{- τ_{2}^{-}}{1 + τ_{2}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \\ \frac{- 1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ω_{1}^{-}}{- ω_{1}^{-}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ω_{2}^{-}}{- ω_{2}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {\frac{ℏ_{1} ψ_{1}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ℧_{1}^{-}}{- ℧_{1}^{-}})^{λ} + \frac{ℏ_{2} ψ_{2}}{ℏ_{1} ψ_{1} + ℏ_{2} ψ_{2}} (\frac{1 + ℧_{2}^{-}}{- ℧_{2}^{-}})^{λ}}^{\frac{1}{λ}}} 〉, \end{matrix}$

= ${\begin{matrix} 〈 \frac{1}{1 + {Σ_{β = 1}^{2} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{2} ℏ_{β} ψ_{β}} (\frac{1 - τ_{β}^{+}}{τ_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{2} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{2} ℏ_{β} ψ_{β}} (\frac{ω_{β}^{+}}{1 - ω_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{2} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{2} ℏ_{β} ψ_{β}} (Σ_{β = 1}^{2} \frac{℧_{β}^{+}}{1 - ℧_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{β = 1}^{2} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{2} ℏ_{β} ψ_{β}} (\frac{- τ_{β}^{-}}{1 + τ_{β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{β = 1}^{2} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{2} ℏ_{β} ψ_{β}} (\frac{1 + ω_{β}^{-}}{- ω_{β}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{β = 1}^{2} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{2} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{β}^{-}}{- ℧_{β}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 . \end{matrix}$

If r = s, based on equation (10), then we have got the following equation:

BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{s}) = ⨂_{β = 1}^{s} {\tilde{ℵ}}_{β}^{⋀^{υ_{β}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{1 - τ_{β}^{+}}{τ_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{ω_{β}^{+}}{1 - ω_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{℧_{β}^{+}}{1 - ℧_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{- τ_{β}^{-}}{1 + τ_{β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{1 + ω_{β}^{-}}{- ω_{β}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{β}^{-}}{- ℧_{β}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 . \end{matrix}$

If r = s + 1, then there is following result:

BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{s}, {\tilde{ℵ}}_{s + 1}) = ⨂_{β = 1}^{s} {\tilde{ℵ}}_{σ (β)}^{⋀^{υ_{β}}} ⨂ {\tilde{ℵ}}_{σ (s + 1)}^{⋀^{υ_{s + 1}}}$

$= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{1 - τ_{β}^{+}}{τ_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{ω_{β}^{+}}{1 - ω_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{℧_{β}^{+}}{1 - ℧_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{- τ_{β}^{-}}{1 + τ_{β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{1 + ω_{β}^{-}}{- ω_{β}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {Σ_{β = 1}^{s} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{β}^{-}}{- ℧_{β}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

⊗_D ${\begin{matrix} 〈 \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{1 - τ_{s + 1}^{+}}{τ_{s + 1}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{ω_{s + 1}^{+}}{1 - ω_{s + 1}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{℧_{s + 1}^{+}}{1 - ℧_{s + 1}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{- τ_{s + 1}^{-}}{1 + τ_{s + 1}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{1 + ω_{s + 1}^{-}}{- ω_{s + 1}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {\frac{ℏ_{s + 1} ψ_{s + 1}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{s + 1}^{-}}{- ℧_{s + 1}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

= ${\begin{matrix} 〈 \frac{1}{1 + {Σ_{β = 1}^{s + 1} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{1 - τ_{β}^{+}}{τ_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{s + 1} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{ω_{β}^{+}}{1 - ω_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{s + 1} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{℧_{β}^{+}}{1 - ℧_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{β = 1}^{s + 1} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{- τ_{β}^{-}}{1 + τ_{β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{1}{1 + {Σ_{β = 1}^{s + 1} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{1 + ω_{β}^{-}}{- ω_{β}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{1}{1 + {Σ_{β = 1}^{s + 1} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{s + 1} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{β}^{-}}{- ℧_{β}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

Hence, Theorem 4.6 is true for r = s + 1. Thus, Equation (10) holds for all r. □

The BNDPOWGA operator also has the following properties:

(1) Idompotency: Let all the BNNs be ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > = \tilde{ℵ}$ (α = 1, 2, 3, . . . , r), where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of ${\tilde{ℵ}}_{α}, ψ_{α} \in [0, 1]$ and Σ_α=1ψ_α = 1.

Then, BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = \tilde{ℵ}$ .

(2) Monotonicity: Let ${\tilde{ℵ}}_{α} (α = 1, 2, . . ., r)$ and ${\tilde{ℵ}}_{α}^{,} (α = 1, 2, . . ., r)$ be two families of BNNs, where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of ${\tilde{ℵ}}_{α}$ and ${\tilde{ℵ}}_{α}^{,}$ , ψ_α ∈ [0, 1] and Σ_α=1ψ_α = 1. For all α,if ${\tilde{ℵ}}_{α} \geq {\tilde{ℵ}}_{α}^{,}$ , then BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r})$ ≥ BNDPOWGA $({\tilde{ℵ}}_{1}^{,}, {\tilde{ℵ}}_{2}^{,}, . . ., {\tilde{ℵ}}_{r}^{,})$ .

(3) Boundedness: Let ${\tilde{ℵ}}_{α} = < τ_{α}^{+}, ω_{α}^{+}, ℧_{α}^{+}, τ_{α}^{-}, ω_{α}^{-}, ℧_{α}^{-} > (α = 1,, 2, 3, . . ., r)$ be a family of BNNs, where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of ${\tilde{ℵ}}_{α}$ , ψ_α ∈ [0, 1] and Σ_α=1ψ_α = 1. Therefore, we have BNDPOWGA $({\tilde{ℵ}}^{-}, {\tilde{ℵ}}^{-}, . . ., {\tilde{ℵ}}^{-})$ ≤ BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r})$ ≤ BNDPOWGA $({\tilde{ℵ}}^{+}, {\tilde{ℵ}}^{+}, . . ., {\tilde{ℵ}}^{+})$ ,

and ${\tilde{ℵ}}^{+} = < τ_{{\tilde{ℵ}}^{+}}^{+}, ω_{{\tilde{ℵ}}^{+}}^{+}, ℧_{{\tilde{ℵ}}^{+}}^{+}, t_{{\tilde{ℵ}}^{+}}^{-}, ω_{{\tilde{ℵ}}^{+}}^{-}, ℧_{{\tilde{ℵ}}^{+}}^{-} >$

= ${\begin{matrix} < \max (τ_{1}^{+}, τ_{2}^{+}, . . ., τ_{r}^{+}), \min (ω_{1}^{+}, ω_{2}^{+}, . . ., ω_{r}^{+}), \min (℧_{1}^{+}, ℧_{2}^{+}, . . ., ℧_{r}^{+}), \\ \min (τ_{1}^{-}, τ_{2}^{-}, . . ., τ_{r}^{-}), \max (ω_{α}^{-}, ω_{α}^{-}, . . ., ω_{α}^{-}), \max (℧_{1}^{-}, ℧_{2}^{-}, . . ., ℧_{r}^{-}) > . \end{matrix}$

Proof.

BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = ⨂_{β = 1}^{r} {\tilde{ℵ}}_{β}^{⋀^{υ_{β}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 - τ_{β}^{+}}{τ_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{ω_{β}^{+}}{1 - ω_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{℧_{β}^{+}}{1 - ℧_{β}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{- τ_{β}^{-}}{1 + τ_{β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ω_{β}^{-}}{- ω_{β}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{β}^{-}}{- ℧_{β}^{-}})^{λ}}^{\frac{1}{λ}}} 〉 \end{matrix}$

Hence, BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r}) = \tilde{ℵ}$ holds.

(2) The property is obvious based on the equation (10).

(3) Let ${\tilde{ℵ}}^{-} = < τ_{{\tilde{ℵ}}^{-}}^{+}, ω_{{\tilde{ℵ}}^{-}}^{+}, ℧_{{\tilde{ℵ}}^{-}}^{+}, τ_{{\tilde{ℵ}}^{-}}^{-}, ω_{{\tilde{ℵ}}^{-}}^{-}, ℧_{{\tilde{ℵ}}^{-}}^{-} >$ and ${\tilde{ℵ}}^{+} = < τ_{{\tilde{ℵ}}^{+}}^{+}, ω_{{\tilde{ℵ}}^{+}}^{+}, ℧_{{\tilde{ℵ}}^{+}}^{+}, τ_{{\tilde{ℵ}}^{+}}^{-}, ω_{{\tilde{ℵ}}^{+}}^{-}, ℧_{{\tilde{ℵ}}^{+}}^{-} >$ . There are the following inequalities:

$\frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 - τ_{{\tilde{ℵ}}^{-}}^{+}}{1_{{\tilde{ℵ}}^{-}}^{+}})^{λ}}^{\frac{1}{λ}}} \leq \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 - τ_{β}^{+}}{τ_{β}^{+}})^{λ}}^{\frac{1}{λ}}} \leq \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 - τ_{\tilde{ℵ^{+}}}^{+}}{τ_{{\tilde{ℵ}}^{+}}^{+}})^{λ}}^{\frac{1}{λ}}}$ ,

$1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{ω_{{\tilde{ℵ}}^{+}}^{+}}{1 - ω_{{\tilde{ℵ}}^{+}}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{ω_{β}^{+}}{1 - ω_{β}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{ω_{{\tilde{ℵ}}^{-}}^{+}}{1 - ω_{\tilde{ℵ^{-}}}^{+}})^{λ}}^{\frac{1}{λ}}}$ ,

$1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{℧_{{\tilde{ℵ}}^{+}}^{+}}{1 - ℧_{{\tilde{ℵ}}^{+}}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{℧_{β}^{+}}{1 - ℧_{β}^{+}})^{λ}}^{\frac{1}{λ}}} \leq 1 - \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{℧_{{\tilde{ℵ}}^{-}}^{+}}{1 - ℧_{\tilde{ℵ^{-}}}^{+}})^{λ}}^{\frac{1}{λ}}}$ ,

$\frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{- τ_{\tilde{ℵ^{+}}}^{-}}{1 + τ_{{\tilde{ℵ}}^{+}}^{-}})^{λ}}^{\frac{1}{λ}}} - 1 \leq \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{- τ_{β}^{-}}{1 + τ_{β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1 \leq \frac{1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{- τ_{\tilde{ℵ^{-}}}^{-}}{1 + τ_{\tilde{ℵ^{-}}}^{-}})^{λ}}^{\frac{1}{λ}}} - 1$ ,

$\frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ω_{{\tilde{ℵ}}^{-}}^{-}}{- ω_{{\tilde{ℵ}}^{-}}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ω_{β}^{-}}{- ω_{β}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ω_{{\tilde{ℵ}}^{+}}^{-}}{- ω_{{\tilde{ℵ}}^{+}}^{-}})^{λ}}^{\frac{1}{λ}}}$ ,

$\frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{{\tilde{ℵ}}^{-}}^{-}}{- ℧_{{\tilde{ℵ}}^{-}}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{β}^{-}}{- ℧_{β}^{-}})^{λ}}^{\frac{1}{λ}}} \leq \frac{- 1}{1 + {Σ_{β = 1}^{r} \frac{ℏ_{β} ψ_{β}}{Σ_{β = 1}^{r} ℏ_{β} ψ_{β}} (\frac{1 + ℧_{{\tilde{ℵ}}^{+}}^{-}}{- ℧_{{\tilde{ℵ}}^{+}}^{-}})^{λ}}^{\frac{1}{λ}}}$ .

Hence, BNDPOWGA $({\tilde{ℵ}}^{-}, {\tilde{ℵ}}^{-}, . . ., {\tilde{ℵ}}^{-})$ ≤ BNDPOWGA $({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, . . ., {\tilde{ℵ}}_{r})$ ≤ BNDPOWGA $({\tilde{ℵ}}^{+}, {\tilde{ℵ}}^{+}, . . ., {\tilde{ℵ}}^{+})$ holds. □

Example 4.7 Let ${\tilde{ℵ}}_{1} = < 0.6, 0.7, 0.3, - 0.6, - 0.3, - 0.5 >$ , ${\tilde{ℵ}}_{2} = < 0.5, 0.4, 0.6, - 0.6, - 0.7, - 0.3 >$ , ${\tilde{ℵ}}_{3} = < 0.6, 0.7, 0.4, - 0.9, - 0.7, - 0.7 >$ and ${\tilde{ℵ}}_{4} = < 0.2, 0.6, 0.8, - 0.6, - 0.3, - 0.9 >$ be four BNNs and let the weight vector of BNNs ${\tilde{ℵ}}_{α} (α = 1, 2, 3, 4)$ be $ψ = (\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{8})^{T}$ . Now, by Definition 4.2, ℏ₁ = 1, $ℏ_{2} = s ({\tilde{ℵ}}_{1}) = 0.4667$ , $ℏ_{3} = s ({\tilde{ℵ}}_{1}) s ({\tilde{ℵ}}_{2}) = 0.2256$ and $ℏ_{4} = s ({\tilde{ℵ}}_{1}) s ({\tilde{ℵ}}_{2}) s ({\tilde{ℵ}}_{3}) = 0.1128$ . $ψ_{1} = \frac{1}{2}$ , $ψ_{2} = \frac{1}{4}$ , $ψ_{3} = \frac{1}{8}$ and $ψ_{4} = \frac{1}{8}$ are the weights of ${\tilde{ℵ}}_{α} (α = 1, 2, 3, 4)$ such that Σ_α=1ψ_α = 1. By Definition 4.2, $\frac{ℏ_{1} ψ_{1}}{Σ_{β = 1}^{4} ℏ_{β} ψ_{β}} = 0.7588$ , $\frac{ℏ_{2} ψ_{2}}{Σ_{β = 1}^{4} ℏ_{β} ψ_{β}} = 0.1771$ , $\frac{ℏ_{3} ψ_{3}}{Σ_{β = 1}^{4} ℏ_{β} ψ_{β}} = 0.0428$ , $\frac{ℏ_{4} ψ_{4}}{Σ_{β = 1}^{4} ℏ_{β} ψ_{β}} = 0.0214$ . Then, by Theorem 4.2, for λ = 3

$BNDPOWGA ({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, {\tilde{ℵ}}_{3}, {\tilde{ℵ}}_{4}) =$ $\begin{matrix} {\begin{matrix} 〈 \frac{1}{1 + {0.7588 (\frac{1 - 0.6}{0.6})^{3} + 0.1771 (\frac{1 - 0.5}{0.5})^{3} + 0.0428 (\frac{1 - 0.6}{0.6})^{3} + 0.0214 (\frac{1 - 0.2}{0.2})^{3}}^{\frac{1}{3}}}, \\ 1 - \\ \frac{1}{1 + {0.7588 (\frac{0.7}{1 - 0.7})^{3} + 0.1771 (\frac{0.4}{1 - 0.4})^{3} + 0.0428 (\frac{0.7}{1 - 0.7})^{3} + 0.0214 (\frac{0.6}{1 - 0.6})^{3}}^{\frac{1}{3}}}, \\ 1 - \\ \frac{1}{1 + {0.7588 (\frac{0.4}{1 - 0.4})^{3} + 0.1771 (\frac{0.6}{1 - 0.6})^{3} + 0.0428 (\frac{0.3}{1 - 0.3})^{3} + 0.0214 (\frac{0.8}{1 - 0.8})^{3}}^{\frac{1}{3}}}, \\ \frac{1}{1 + {0.7588 (\frac{0.9}{1 + 0.9})^{3} + 0.1771 (\frac{0.6}{1 + 0.6})^{3} + 0.0428 (\frac{0.6}{1 + 0.6})^{3} + 0.0214 (\frac{0.6}{1 + 0.6})^{3}}^{\frac{1}{3}}} \\ - 1, \\ \frac{- 1}{1 + {0.7588 (\frac{1 - 0.7}{0.7})^{3} + 0.1771 (\frac{1 - 0.7}{0.7})^{3} + 0.0428 (\frac{1 - 0.3}{0.3})^{3} + 0.0214 (\frac{1 - 0.3}{0.3})^{3}}^{\frac{1}{3}}}, \\ \frac{- 1}{1 + {0.7588 (\frac{1 - 0.7}{0.7})^{3} + 0.1771 (\frac{1 - 0.3}{0.3})^{3} + 0.0428 (\frac{1 - 0.5}{0.5})^{3} + 0.0214 (\frac{1 - 0.9}{0.9})^{3}}^{\frac{1}{3}}} 〉, \end{matrix} \end{matrix}$ $BNDPOWGA ({\tilde{ℵ}}_{1}, {\tilde{ℵ}}_{2}, {\tilde{ℵ}}_{3}, {\tilde{ℵ}}_{4}) = 〈 0.4519, 0.6852, 0.5651, - 0.8915, - 0.5098, - 0.4292 〉$ .

5 Model for MADM using bipolar neutrosophic information

In this section, three comprehensive MADM methods are extended based on the proposed BNDPWGA and BNDPOWGA operators.

For MADM model with bipolar neutrosophic fuzzy information, let A = {A₁, A₂, . . . , A_r} be a set of alternatives and C = {C₁, C₂, . . . , C_r} be a set of attributes. For BNDPWGA and BNDPOWGA operators, there is a prioritization between the attributes expressed by the linear ordering C₁ ≻ C₂ ≻ . . . ≻ C_r, indicates attribute C_η has a higher priority than C_ρ, if η < ρ. Let ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T be the weight vector of attributes such that ψ_β > 0, Σ_β=1ψ_β = 1 (β = 1, 2, . . . , r) and ψ_β refers to the weight of attribute C_β. Suppose that $N = ({\tilde{ℵ}}_{α β})_{s \times r} = (τ_{α β}^{+}, ω_{α β}^{+}, ℧_{α β}^{+}, τ_{α β}^{-}, ω_{α β}^{-}, ℧_{α β}^{-})_{s \times r} (α = 1, 2, . . ., s) (β = 1, 2, . . ., r)$ is BNN decision matrix, where $τ_{α β}^{+}, ω_{α β}^{+}, ℧_{α β}^{+}$ indicates the truth membership degree, indeterminacy membership degree and falsity membership degree of alternative A_α under attribute C_β with respect to positive preferences and $τ_{α β}^{-}, ω_{α β}^{-}, ℧_{α β}^{-}$ indicates the truth membership degree, indeterminacy membership degree and falsity membership degree of alternative A_α under attribute C_β with respect to negative preferences. We have conditions $τ_{α β}^{+}$ , $ω_{α β}^{+}$ , $℧_{α β}^{+} \in [0, 1]$ and $τ_{α β}^{-}$ , $ω_{α β}^{-}$ , $℧_{α β}^{-} \in [- 1, 0]$ such that $0 \leq τ_{α β}^{+} + ω_{α β}^{+} + ℧_{α β}^{+} - τ_{α β}^{-} - ω_{α β}^{-} - ℧_{α β}^{-} \leq 6$ for (α = 1, 2, . . . , s) (β = 1, 2, . . . , r).

An algorithm is constructed based on proposed bipolar neutrosophic Dombi aggregation operators which solves MADM problems.

Algorithm

Step 1 Collect information on the bipolar neutrosophic evaluation.

Step 2 Calculate score and the accuracy values of collected information.

The score values $s ({\tilde{ℵ}}_{α β})$ and accuracy values $a ({\tilde{ℵ}}_{α β})$ of alternatives A_α can be calculated by using Equations (1) and (2).

Step 3 Reordering the value of each attribute by using the comparison method.

Step 4 Derive the collective BNN ${\tilde{ℵ}}_{α}$ (α = 1, 2, . . . , s) for the alternative A_α (α = 1, 2, . . . , s).

By using method (1), calculate the values of ℏ_αβ (α = 1, 2, . . . , s) (β, δ = 2, 3, . . . , r) as follows: $ℏ_{α β} = \prod_{α δ = 1}^{α β - 1} s ({\tilde{ℵ}}_{α δ}) (α = 1, 2, . . ., s) (β = 2, 3, . . ., r),$ (11) $ℏ_{α 1} = 1 (α = 1, 2, . . ., s),$ (12) and utilize BNDPWGA operator to calculate the collective BNN for each alternative, then

${\tilde{ℵ}}_{α} =$ BNDPWGA $({\tilde{ℵ}}_{α 1}, {\tilde{ℵ}}_{α 2}, . . ., {\tilde{ℵ}}_{α r}) = ⨂_{β = 1}^{r} {\tilde{ℵ}}_{α β}^{⋀^{υ_{α β}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{β}^{r} \frac{ℏ_{α β} ψ_{α β}}{Σ_{β}^{r} ℏ_{α β} ψ_{α β}} (\frac{1 - τ_{α β}^{+}}{τ_{α β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β}^{r} \frac{ℏ_{α β} ψ_{α β}}{Σ_{β}^{r} ℏ_{α β} ψ_{α β}} (\frac{ω_{α β}^{+}}{1 - ω_{α β}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{β}^{r} \frac{ℏ_{α β} ψ_{α β}}{Σ_{β}^{r} ℏ_{α β} ψ_{α β}} (\frac{℧_{α β}^{+}}{1 - ℧_{α β}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{β}^{r} \frac{ℏ_{α β} ψ_{α β}}{Σ_{β}^{r} ℏ_{α β} ψ_{α β}} (\frac{- τ_{α β}^{-}}{1 + τ_{α β}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{β}^{r} \frac{ℏ_{α β} ψ_{α β}}{Σ_{β}^{r} ℏ_{α β} ψ_{α β}} (\frac{1 + ω_{α β}^{-}}{- ω_{α β}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{β}^{r} \frac{ℏ_{α β} ψ_{α β}}{Σ_{β}^{r} ℏ_{α β} ψ_{α β}} (\frac{1 + ℧_{α β}^{-}}{- ℧_{α β}^{-}})^{λ}}^{\frac{1}{λ}}} 〉, \end{matrix}$ (13) where $υ_{α β} = \frac{ℏ_{α β} ψ_{α β}}{Σ_{j = 1}^{n} ℏ_{α β} ψ_{α β}}$ and ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T is the weight vector (α = 1, 2, . . . , s) such that ψ_β ∈ [0, 1] and Σ_β=1ψ_β = 1.

By using method (2) calculate the values of ℏ_αγ (α = 1, 2, . . . , s) (γ = 1, 2, . . . , r) as follows: $ℏ_{α γ} = \prod_{α δ = 1}^{α δ - 1} s ({\tilde{ℵ}}_{α δ}) (α = 1, 2, . . ., s) (β = 2, 3, . . ., r),$ (14) $ℏ_{α 1} = 1 (α = 1, 2, . . ., s),$ (15) and utilize BNDPOWGA operator to calculate the collective BNN for each alternative, then ${\tilde{ℵ}}_{α} =$ BNDPOWGA $({\tilde{ℵ}}_{α 1}, {\tilde{ℵ}}_{α 2}, . . ., {\tilde{ℵ}}_{α r}) = ⨂_{γ = 1}^{r} {\tilde{ℵ}}_{α γ}^{⋀^{υ_{α γ}}}$ $= {\begin{matrix} 〈 \frac{1}{1 + {Σ_{γ = 1}^{r} \frac{ℏ_{α γ} ψ_{α γ}}{Σ_{γ = 1}^{r} ℏ_{α γ} ψ_{α γ}} (\frac{1 - τ_{α γ}^{+}}{τ_{α γ}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{γ = 1}^{r} \frac{ℏ_{α γ} ψ_{α γ}}{Σ_{γ = 1}^{r} ℏ_{α γ} ψ_{α γ}} (\frac{ω_{α γ}^{+}}{1 - ω_{α γ}^{+}})^{λ}}^{\frac{1}{λ}}}, 1 - \frac{1}{1 + {Σ_{γ = 1}^{r} \frac{ℏ_{α γ} ψ_{α γ}}{Σ_{γ = 1}^{r} ℏ_{α γ} ψ_{α γ}} (\frac{℧_{α γ}^{+}}{1 - ℧_{α γ}^{+}})^{λ}}^{\frac{1}{λ}}}, \\ \frac{1}{1 + {Σ_{γ = 1}^{r} \frac{ℏ_{α γ} ψ_{α γ}}{Σ_{γ = 1}^{r} ℏ_{α γ} ψ_{α γ}} (\frac{- τ_{α γ}^{-}}{1 + τ_{α γ}^{-}})^{λ}}^{\frac{1}{λ}}} - 1, \frac{- 1}{1 + {Σ_{γ = 1}^{r} \frac{ℏ_{α γ} ψ_{α γ}}{Σ_{γ = 1}^{r} ℏ_{α γ} ψ_{α γ}} (\frac{1 + ω_{α γ}^{-}}{- ω_{α γ}^{-}})^{λ}}^{\frac{1}{λ}}}, \frac{- 1}{1 + {Σ_{γ = 1}^{r} \frac{ℏ_{α γ} ψ_{α γ}}{Σ_{γ = 1}^{r} ℏ_{α γ} ψ_{α γ}} (\frac{1 + ℧_{α γ}^{-}}{- ℧_{α γ}^{-}})^{λ}}^{\frac{1}{λ}}} 〉, \end{matrix}$ (16) where $υ_{α γ} = \frac{ℏ_{α γ} ψ_{α γ}}{Σ_{γ = 1}^{r} ℏ_{α γ} ψ_{α γ}}$ and σ is permutation that orders the elements: ${\tilde{ℵ}}_{σ (α 1)} \geq {\tilde{ℵ}}_{σ (α 2)} \geq . . . \geq {\tilde{ℵ}}_{σ (α r)}$ . where ψ = (ψ₁, ψ₂, . . . , ψ_r) ^T is the weight vector such that ψ_γ ∈ [0, 1] and Σ_γ=1ψ_γ = 1.

Step 5 Calculate the score values $s ({\tilde{ℵ}}_{α})$ (α = 1, 2, . . . , s) of BNNs ${\tilde{ℵ}}_{α}$ (α = 1, 2, . . . , s) to rank all the alternatives A_α (α = 1, 2, . . . , s) and then select favorable one(s). If score values ${\tilde{ℵ}}_{α}$ and ${\tilde{ℵ}}_{β}$ of BNNs are equal, then we calculate accuracy values $a ({\tilde{ℵ}}_{α})$ and $a ({\tilde{ℵ}}_{β})$ of BNNs ${\tilde{ℵ}}_{α}$ and ${\tilde{ℵ}}_{β}$ , respectively and then rank the alternatives A_α and A_β as accuracy values $a ({\tilde{ℵ}}_{α})$ and $a ({\tilde{ℵ}}_{β})$ .

Step 6 Rank all the alternatives A_α (α = 1, 2, . . . , s) and select favorable one(s).

Step 7 End.

The flow chart of proposed algorithm is given below in Figure 1.

Fig. 1

Flow Chart.

6 Numerical example

A numerical example of the selection of cultivating crops taken from Deli et al. [10] is provided here. Furthermore, parametric analysis and comparative analysis confirm the flexibility and effectiveness of the proposed methods. In order to increase the production of agriculture, an agriculture department considers selecting a crop for cultivating in the farm. Four cultivating crops A₁, A₂, A₃, and A₄ are selected for further evaluation through preliminary screening. The agriculture department decided to invite four group experts to evaluate information. The expert group consists of investment experts, land experts, weather experts and labour experts. The four cultivating crops are evaluated by experts on the basis of four attributes or criteria: cost (C₁), nutrition of land (C₂), effect of weather (C₃), labor (C₄). For the proposed methods BNDPWGA and BNDPOWGA operators, the prioritization relation for the attributes is given as: C₁ ≻ C₂ ≻ C₃ ≻ C₄. These attributes are interactive and interlinked.

Step 1 Collect information on bipolar neutrosophic evaluation. The information collected from expert discussion on evaluation is given in Table 1.

Table 1
Bipolar neutrosophic evaluation information

C ₁ C ₂ C ₃ C ₄

A ₁ 〈0.5, 0.7, 0.2, - 0.7, - 0.3, - 0.6〉〈0.4, 0.4, 0.5, - 0.7, - 0.8, - 0.4〉〈0.7, 0.7, 0.5, - 0.8, - 0.7, - 0.6〉〈0.1, 0.5, 0.7, - 0.5, - 0.2, - 0.8〉

A ₂ 〈0.9, 0.7, 0.5, - 0.7, - 0.7, - 0.1〉〈0.7, 0.6, 0.8, - 0.7, - 0.5, - 0.1〉〈0.9, 0.4, 0.6, - 0.1, - 0.7, - 0.5〉〈0.5, 0.2, 0.7, - 0.5, - 0.1, - 0.9〉

A ₃ 〈0.3, 0.4, 0.2, - 0.6, - 0.3, - 0.7〉〈0.2, 0.2, 0.2, - 0.4, - 0.7, - 0.4〉〈0.9, 0.5, 0.5, - 0.6, - 0.5, - 0.2〉〈0.7, 0.5, 0.3, - 0.4, - 0.2, - 0.2〉

A ₄ 〈0.9, 0.7, 0.2, - 0.8, - 0.6, - 0.1〉〈0.3, 0.5, 0.2, - 0.5, - 0.5, - 0.2〉〈0.5, 0.4, 0.5, - 0.1, - 0.7, - 0.2〉〈0.4, 0.2, 0.8, - 0.5, - 0.5, - 0.6〉

	C ₁	C ₂	C ₃	C ₄
A ₁	〈0.5, 0.7, 0.2, - 0.7, - 0.3, - 0.6〉	〈0.4, 0.4, 0.5, - 0.7, - 0.8, - 0.4〉	〈0.7, 0.7, 0.5, - 0.8, - 0.7, - 0.6〉	〈0.1, 0.5, 0.7, - 0.5, - 0.2, - 0.8〉
A ₂	〈0.9, 0.7, 0.5, - 0.7, - 0.7, - 0.1〉	〈0.7, 0.6, 0.8, - 0.7, - 0.5, - 0.1〉	〈0.9, 0.4, 0.6, - 0.1, - 0.7, - 0.5〉	〈0.5, 0.2, 0.7, - 0.5, - 0.1, - 0.9〉
A ₃	〈0.3, 0.4, 0.2, - 0.6, - 0.3, - 0.7〉	〈0.2, 0.2, 0.2, - 0.4, - 0.7, - 0.4〉	〈0.9, 0.5, 0.5, - 0.6, - 0.5, - 0.2〉	〈0.7, 0.5, 0.3, - 0.4, - 0.2, - 0.2〉
A ₄	〈0.9, 0.7, 0.2, - 0.8, - 0.6, - 0.1〉	〈0.3, 0.5, 0.2, - 0.5, - 0.5, - 0.2〉	〈0.5, 0.4, 0.5, - 0.1, - 0.7, - 0.2〉	〈0.4, 0.2, 0.8, - 0.5, - 0.5, - 0.6〉

Step 2 Calculate score and accuracy values of collected information.

For each alternative A_α under attribute C_β, the score values $s ({\tilde{ℵ}}_{α β})$ and accuracy values $a ({\tilde{ℵ}}_{α β})$ can be calculated based on equations (1) and (2). The score values $s ({\tilde{ℵ}}_{α β})$ and accuracy values $a ({\tilde{ℵ}}_{α β})$ are shown in Tables 2 and 3, respectively.

Table 2

Score values $s ({\tilde{ℵ}}_{α β})$

	C ₁	C ₂	C ₃	C ₄
A ₁	0.4667	0.5000	0.5000	0.4000
A ₂	0.4667	0.3667	0.6667	0.5167
A ₃	0.5167	0.5833	0.5000	0.4833
A ₄	0.4833	0.4667	0.5667	0.5000

Table 3

Accuracy values $a ({\tilde{ℵ}}_{α β})$

	C ₁	C ₂	C ₃	C ₄
A ₁	0.2000	-0.4000	0	-0.3000
A ₂	-0.2000	-0.7000	0.7000	0.2000
A ₃	0.2000	0	0	0.2000
A ₄	0	-0.2000	0.1000	-0.3000

Step 3 Reordering information on evaluation under each attribute.

Step 4 Derive the collective BNN ${\tilde{ℵ}}_{α}$ (α = 1, 2, . . . , s) for the alternative A_α (α = 1, 2, . . . , s).

Method (1) Calculate the values of ℏ_αβ (α = 1, 2, . . . , s) (β = 1, 2, . . . , r) using equations (11) and (12) as follows: $ℏ_{α β} = (\begin{matrix} 1.0000 & 0.4667 & 0.2333 & 0.1167 \\ 1.0000 & 0.4667 & 0.1711 & 0.1141 \\ 1.0000 & 0.5167 & 0.3014 & 0.1507 \\ 1.0000 & 0.4833 & 0.2256 & 0.1278 \end{matrix})$ and utilize BNDPWGA operator using equation (13) and supporting λ = 7 to calculate the collective BNN for each alternative, then ${\tilde{ℵ}}_{1} = 〈 0.1608, 0.6933, 0.5760, - 0.7305, - 0.2840, - 0.4601 〉;$ ${\tilde{ℵ}}_{2} = 〈 0.6325, 0.6923, 0.7578, - 0.6983, - 0.1610, - 0.1007 〉;$ ${\tilde{ℵ}}_{3} = 〈 0.2387, 0.4263, 0.3979, - 0.5917, - 0.2807, - 0.2631 〉;$ ${\tilde{ℵ}}_{4} = 〈 0.3533, 0.6914, 0.7013, - 0.7934, - 0.5494, - 0.1037 〉 .$

By using method (2) calculate the values of ℏ_αβ (α = 1, 2, . . . , s) (β = 1, 2, . . . , r) using equations (14) and (15) as follows: $ℏ_{α β} = (\begin{matrix} 1.0000 & 0.4667 & 0.2333 & 0.1167 \\ 1.0000 & 0.4667 & 0.1711 & 0.1141 \\ 1.0000 & 0.5167 & 0.3014 & 0.1507 \\ 1.0000 & 0.4833 & 0.2256 & 0.1278 \end{matrix})$

and utilize BNDPOWGA operator using equation (13) and supporting λ = 7 to calculate the collective BNN for each alternative, then ${\tilde{ℵ}}_{1} = 〈 0.1608, 0.6933, 0.5788, - 0.7937, - 0.2999, - 0.4601 〉;$ ${\tilde{ℵ}}_{2} = 〈 0.5612, 0.5900, 0.7046, - 0.6067, - 0.1244, - 0.1441 〉;$ ${\tilde{ℵ}}_{3} = 〈 0.4097, 0.5978, 0.7582, - 0.7181, - 0.5555, - 0.1441 〉;$ ${\tilde{ℵ}}_{4} = 〈 0.2072, 0.4162, 0.3979, - 0.5511, - 0.2903, - 0.2629 〉 .$

Step 5 Calculate the score values $s ({\tilde{ℵ}}_{α})$ (α = 1, 2, . . . , s) of BNNs ${\tilde{ℵ}}_{α}$ (α = 1, 2, . . . , s) for each alternatives A_α (α = 1, 2, . . . , s). The score values $s ({\tilde{ℵ}}_{α})$ is calculated by using equation (1).

Method 1 The following score values are obtained by using the BNDPWGA operator.

$s ({\tilde{ℵ}}_{1}) = 0.3175$ ; $s ({\tilde{ℵ}}_{2}) = 0.2910$ ; $s ({\tilde{ℵ}}_{3}) = 0.3944$ ; $s ({\tilde{ℵ}}_{4}) = 0.3034$ .

Method 2 The following score values are obtained by using the BNDPOWGA operator.

$s ({\tilde{ℵ}}_{1}) = 0.3091$ ; $s ({\tilde{ℵ}}_{2}) = 0.3214$ ; $s ({\tilde{ℵ}}_{3}) = 0.3992$ ; $s ({\tilde{ℵ}}_{4}) = 0.3398$ .

Step 6 Rank all the alternatives A_α (α = 1, 2, . . . , s) and select favorable one(s).

The alternative can be ranked in descending order based on the comparison method, and favorable alternative can be selected.

Method 1 The ranking order based on score values is obtained by using BNDPWGA operator: A₃ ≻ A₁ ≻ A₄ ≻ A₂. Thus, A₃ is favorable.

Method 2 The ranking order based on score values is obtained by using BNDPOWGA operator: A₃ ≻ A₄ ≻ A₂ ≻ A₁. Thus, A₃ is favorable.

Step 7 End.

7 Parametric analysis and comparative analysis

This section describes effect of parametric λ on decision making results and comparison between proposed methods and existing methods.

7.1 Analysis on the effect of parameter λ on decision making results

This subsection discusses the effect of parameter λ in detail.

First, effect of parameter λ on the proposed operators is as follows.

Table 5 shows that the corresponding ranking orders with respect to the BNDPWGA operator are changed as the value of λ changing from 1 to 10.

Table 4
Reordering bipolar neutrosophic evaluation information

C ₁ C ₂ C ₃ C ₄

A ₁ 〈0.7, 0.7, 0.5, - 0.8, - 0.7, - 0.6〉〈0.4, 0.4, 0.5, - 0.7, - 0.8, - 0.4〉〈0.5, 0.7, 0.2, - 0.7, - 0.3, - 0.6〉〈0.1, 0.5, 0.7, - 0.5, - 0.2, - 0.8〉

A ₂ 〈0.9, 0.4, 0.6, - 0.1, - 0.7, - 0.5〉〈0.5, 0.2, 0.7, - 0.5, - 0.1, - 0.9〉〈0.9, 0.7, 0.5, - 0.7, - 0.7, - 0.1〉〈0.7, 0.6, 0.8, - 0.7, - 0.5, - 0.1〉

A ₃ 〈0.2, 0.2, 0.2, - 0.4, - 0.7, - 0.4〉〈0.3, 0.4, 0.2, - 0.6, - 0.3, - 0.7〉〈0.9, 0.5, 0.5, - 0.6, - 0.5, - 0.2〉〈0.7, 0.5, 0.3, - 0.4, - 0.2, - 0.2〉

A ₄ 〈0.5, 0.4, 0.5, - 0.1, - 0.7, - 0.2〉〈0.4, 0.2, 0.8, - 0.5, - 0.5, - 0.6〉〈0.9, 0.7, 0.2, - 0.8, - 0.6, - 0.1〉〈0.3, 0.5, 0.2, - 0.5, - 0.5, - 0.2〉

	C ₁	C ₂	C ₃	C ₄
A ₁	〈0.7, 0.7, 0.5, - 0.8, - 0.7, - 0.6〉	〈0.4, 0.4, 0.5, - 0.7, - 0.8, - 0.4〉	〈0.5, 0.7, 0.2, - 0.7, - 0.3, - 0.6〉	〈0.1, 0.5, 0.7, - 0.5, - 0.2, - 0.8〉
A ₂	〈0.9, 0.4, 0.6, - 0.1, - 0.7, - 0.5〉	〈0.5, 0.2, 0.7, - 0.5, - 0.1, - 0.9〉	〈0.9, 0.7, 0.5, - 0.7, - 0.7, - 0.1〉	〈0.7, 0.6, 0.8, - 0.7, - 0.5, - 0.1〉
A ₃	〈0.2, 0.2, 0.2, - 0.4, - 0.7, - 0.4〉	〈0.3, 0.4, 0.2, - 0.6, - 0.3, - 0.7〉	〈0.9, 0.5, 0.5, - 0.6, - 0.5, - 0.2〉	〈0.7, 0.5, 0.3, - 0.4, - 0.2, - 0.2〉
A ₄	〈0.5, 0.4, 0.5, - 0.1, - 0.7, - 0.2〉	〈0.4, 0.2, 0.8, - 0.5, - 0.5, - 0.6〉	〈0.9, 0.7, 0.2, - 0.8, - 0.6, - 0.1〉	〈0.3, 0.5, 0.2, - 0.5, - 0.5, - 0.2〉

Table 5

Ranking orders with parameter of BNDPWGA operator

λ	$s ({\tilde{ℵ}}_{1})$	s $({\tilde{ℵ}}_{2})$	$s ({\tilde{ℵ}}_{3})$	$s ({\tilde{ℵ}}_{4})$	BNDPWGA
1	0.4426	0.4250	0.4947	0.4380	A₃ ≻ A₁ ≻ A₄ ≻ A₂
2	0.4090	0.3784	0.4653	0.3852	A₃ ≻ A₁ ≻ A₄ ≻ A₂
3	0.3773	0.3423	0.4422	0.3499	A₃ ≻ A₁ ≻ A₄ ≻ A₂
4	0.3547	0.3210	0.4249	0.3299	A₃ ≻ A₁ ≻ A₄ ≻ A₂
5	0.3388	0.3073	0.4121	0.3176	A₃ ≻ A₁ ≻ A₄ ≻ A₂
6	0.3269	0.2979	0.4023	0.3093	A₃ ≻ A₁ ≻ A₄ ≻ A₂
7	0.3175	0.2910	0.3944	0.3034	A₃ ≻ A₁ ≻ A₄ ≻ A₂
8	0.3099	0.2857	0.3880	0.2988	A₃ ≻ A₁ ≻ A₄ ≻ A₂
9	0.3036	0.2817	0.3826	0.2953	A₃ ≻ A₁ ≻ A₄ ≻ A₂
10	0.2984	0.2784	0.3781	0.2924	A₃ ≻ A₁ ≻ A₄ ≻ A₂

Table 5 shows that ranking order is stable and the corresponding favorable alternative remains identical, when the value of λ is changed for BNDPWGA operator. For, 1 ≤ λ ≤ 10 the corresponding ranking order is A₃ ≻ A₁ ≻ A₄ ≻ A₂, then favorable one is A₃. As a result, favorable stable alternative is A₃. The behavior of BNDPWGA operator is shown in Figure 2.

Fig. 2

BNDPWGA operator

Table 6 shows that the corresponding ranking orders with respect to the BNDPOWGA operator are changed as the value of λ changing from 1 to 10.

Table 6

Ranking orders with parameter of BNDPOWGA operator

λ	$s ({\tilde{ℵ}}_{1})$	$s ({\tilde{ℵ}}_{2})$	$s ({\tilde{ℵ}}_{3})$	$s ({\tilde{ℵ}}_{4})$	BNDPOWGA
1	0.4655	0.5416	0.5292	0.5041	A₂ ≻ A₃ ≻ A₄ ≻ A₁
2	0.4159	0.4545	0.4947	0.4522	A₃ ≻ A₄ ≻ A₂ ≻ A₁
3	0.3747	0.4031	0.4647	0.4140	A₃ ≻ A₄ ≻ A₂ ≻ A₁
4	0.3490	0.3710	0.4408	0.3894	A₃ ≻ A₄ ≻ A₂ ≻ A₁
5	0.3317	0.3491	0.4229	0.3663	A₃ ≻ A₄ ≻ A₂ ≻ A₁
6	0.3189	0.3332	0.4094	0.3512	A₃ ≻ A₄ ≻ A₂ ≻ A₁
7	0.3091	0.3214	0.3398	0.3992	A₃ ≻ A₄ ≻ A₂ ≻ A₁
8	0.3015	0.3123	0.3912	0.3308	A₃ ≻ A₄ ≻ A₂ ≻ A₁
9	0.2955	0.3052	0.3849	0.3237	A₃ ≻ A₄ ≻ A₂ ≻ A₁
10	0.2907	0.2994	0.3798	0.3179	A₃ ≻ A₄ ≻ A₂ ≻ A₁

Table 6 shows that the ranking order is different when the value of λ is changed for BNDPOWGA operator. For, 1 ≤ λ ≤ 2, the corresponding ranking orders are A₂ ≻ A₃ ≻ A₄ ≻ A₁ and A₃ ≻ A₂ ≻ A₄ ≻ A₁. It follows that the favorable alternatives are A₂ and A₃, respectively. For, 3 ≤ λ ≤ 10, the corresponding ranking order is A₃ ≻ A₄ ≻ A₂ ≻ A₁. As a result, favorable one is A₃. The behavior of BNDPOWGA operator is shown in Figure 3.

Fig. 3

BNDPOWGA operator

7.2 Comparative analysis

In this subsection, a comparative analysis of the proposed methods based on proposed bipolar neutrosophic Dombi prioritized aggregation operators with existing methods will be discussed.

Table 7
Ranking orders obtained by different methods

Methods Rankings

A_ψ operator [11] A₃ ≻ A₄ ≻ A₂ ≻ A₁

G_ψ operator [11] A₃ ≻ A₄ ≻ A₂ ≻ A₁

BN-TOPSIS method(ψ₁) [12] A₄ ≻ A₂ ≻ A₃ ≻ A₁

BN-TOPSIS method(ψ₂) [12] A₃ ≻ A₂ ≻ A₄ ≻ A₁

FBNCWBM operator (s, t = 1) [9] A₃ ≻ A₄ ≻ A₂ ≻ A₁

FBNCGBM operator (s, t = 1) [9] A₃ ≻ A₄ ≻ A₂ ≻ A₁

BNDPWGA operator (λ = 7) A₃ ≻ A₁ ≻ A₄ ≻ A₂

BNDPOWGA operator (λ = 7) A₃ ≻ A₄ ≻ A₂ ≻ A₁

Methods	Rankings
A_ψ operator [11]	A₃ ≻ A₄ ≻ A₂ ≻ A₁
G_ψ operator [11]	A₃ ≻ A₄ ≻ A₂ ≻ A₁
BN-TOPSIS method(ψ₁) [12]	A₄ ≻ A₂ ≻ A₃ ≻ A₁
BN-TOPSIS method(ψ₂) [12]	A₃ ≻ A₂ ≻ A₄ ≻ A₁
FBNCWBM operator (s, t = 1) [9]	A₃ ≻ A₄ ≻ A₂ ≻ A₁
FBNCGBM operator (s, t = 1) [9]	A₃ ≻ A₄ ≻ A₂ ≻ A₁
BNDPWGA operator (λ = 7)	A₃ ≻ A₁ ≻ A₄ ≻ A₂
BNDPOWGA operator (λ = 7)	A₃ ≻ A₄ ≻ A₂ ≻ A₁

The ranking orders obtained by the proposed methods show that A₃ is favorable alternative.

Table 8

Characteristic comparison of different methods

Methods	Flexible measure easier	prioritized
A_ψ operator [11]	No	No
G_ψ operator [11]	No	No
BN-TOPSIS method(ψ₁) [12]	No	No
BN-TOPSIS method(ψ₂) [12]	No	No
FBNCWBM operator (s, t = 1) [9]	Yes	No
FBNCGBM operator (s, t = 1) [9]	Yes	No
BNDPWGA operator (λ = 7)	Yes	Yes
BNDPOWGA operator (λ = 7)	Yes	Yes

In contrast to A_ψ, G_ψ, BN-TOPSIS(ψ₁) and BN-TOPSIS(ψ₂), FBNCWBM (s, t = 1) and FBNCGBM (s, t = 1) methods, the proposed method based on proposed BNDPWGA operator considers the prioritized relationship among the attributes by establishing the prioritized aggregation operator. In some practical MADM problems, the prioritization relationship exists among attributes. Then, DMs can use the proposed method based on the proposed BNDPWGA operator to solve MADM problems which have prioritization relationship among attributes. In contrast to A_ψ, G_ψ, BN-TOPSIS(ψ₁) and BN-TOPSIS(ψ₂), FBNCWBM (s, t = 1) and FBNCGBM (s, t = 1) methods, the proposed method based on proposed BNDPOWGA operator considers the prioritized relationship among the attributes by establishing the prioritized aggregation operator, and the interaction and interrelationship among attributes by using ordered weighted geometric aggregation operator. In some practical MADM problems, the prioritization relationship, interaction and interrelationship exist among the attributes. Then, DMs can use the proposed method based on the proposed BNDPOWGA operator to solve the MADM problems which have the prioritization relationship, interaction and interrelationship among the attributes. Thus, the proposed methods based on the proposed bipolar neutrosophic Dombi prioritized aggregation operators are more reliable and flexible. The DMs can use these proposed methods based on the proposed bipolar neutrosophic Dombi prioritized aggregation operators according to their requirements in practical MADM problems.

8 Conclusion

BNSs describe fuzzy, bipolar, inconsistent and uncertain information. BNDPWGA and BNDPOWGA operators were proposed based on Dombi operations to make sure that the bipolar neutrosophic Dombi prioritized aggregation operators are reliable and flexible. Furthermore, a numerical example was given to verify proposed methods. The reliability and flexibility of the proposed methods were further illustrated through a parameter analysis and a comparative analysis with existing methods. The contribution of this paper is as follows. First, BNSs were used to present the decision-making information based on evaluation. Secondly, Dombi operations were put forward to bipolar neutrosophic fuzzy environments. Thirdly, BNDPWGA and BNDPOWGA operators are proposed under the bipolar neutrosophic fuzzy environment. Fourthly, MADM methods based on proposed bipolar neutrosophic Dombi prioritized aggregation operators were developed. Finally, the flexibility and reliability of proposed methods were verified by a numerical example. In future, BNDPWGA and BNDPOWGA operators can be put forward to other fuzzy environments such as bipolar neutrosophic soft expert sets and bipolar interval neutrosophic sets. Also it could be seen for MAGDM with multi-granular hesitant fuzzy linguistic term sets by means of different kinds of fuzzy environments.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets Syst 20(1) (1986), 87–96.

Atanassov

K.T.

, Gargov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst 33(1) (1989), 343–349.

Smarandache

, A unifying field in logics: neutrosophic logic, Multiple-valued Logic 8(3) (1999), 489–503.

Wang

, Madiraju

, Zhang

, Sunderrraman

, Interval neutrosophic sets, Mathematics 1 (2004), 274–277.

Wang

, Samarandache

, Zhang

, Sunderraman

, Single valued neutrosophic sets, Rev Air Force Acad 17 (2010), 10–14.

Gruber

, Seigel

E.H.

, Purcell

A.L.

, Earls

H.A.

, Cooper

, Barrett

L.F.

, Unseen positive and negative affective information inuences social perception in bipolar I disorder and healthy adults, J Affect Disord 192 (2015), 191–198.

Bistarelli

, Pini

M.S.

, Venable

K.B.

, Modelling and solving bipolar preference problems, Ercim Workshop on constraints Lisbona Giugno 23(4) (2015), 545–575.

Wang

, Zhang

, Wang

, Frank Choquet Bonferroni Mean Operators of Bipolar Neutrosophic Sets and Their Application to Multi-criteria Decision-making Problems, In Fuzzy Syst 20(1) (2018), 13–28.

10.

Zhang

W.R.

, Bipolar fuzzy setss and relations a computational framework for cognitive modeling and multiagent decision analysis, In the workshop on fuzzy information processing society bi conference (1994), 305–309.

11.

Deli

, Ali

, Smarandache

, Bipolar neutrosophic sets and their application based on multicreteria decision making problems, International Conference on Advanced Mechatronic Systems (2015), 249–254.

12.

Dey

, Pramanik

, Giri

B.C.

, Toposis for solving multi-attribute decision making problems under bipolar neutrosophic environment, Pons asbl Brussels European Union (2016), 65–77.

13.

Dombi

J.A.

, General class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets Syst 8 (1982), 149–163.

14.

Yager

R.R.

, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008), 263–274.

15.

Yager

R.R.

, Kacprzyk

, The Ordered Weighted Averaging Operators, Theory and Application (1997), 15–28.

16.

Zeng

, Shoaib

, Ali

, Abbas

, Nadeem

M.S.

, Complex Vague Graphs and Their Application in Decision-Making Problems, IEEE Access 8 (2020), 174094–174104.

17.

Mahmood

M.K.

, Zeng

, Gulfam

, Ali

, Jin

, Bipolar Neutrosophic Dombi Aggregation Operators With Application in MultiAttribute Decision Making Problems, IEEE Access 8 (2020), 156600–156614.

18.

Z.S.

, Da

Q.L.

, The ordered weighted geometric averaging operators, International Journal of Intelligent Systems 17(7) (2002), 709–716.