# Multicriteria Decision-Making Methods Using Bipolar Neutrosophic Hamacher Geometric Aggregation Operators

Authors: Muhammad Jamil; Farkhanda Afzal; Deeba Afzal; Dhana Kumari Thapa; Ayesha Maqbool
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The study presents a novel conception of aggregation operators (AOs) based on bipolar neutrosophic sets by using Hamacher operations and their application in modeling real-life multicriteria decision-making problems. The neutrosophic set represents incomplete, inconsistent, and indeterminate information effectively. For better understanding in this paper, we have explained all essential definitions and their respective derived neutrosophic sets (NSs) and generalization bipolar neutrosophic sets (BNSs). The primary focus of our work is Hamacher aggregation operators like BN Hamacher weighted geometric (BNHWG), BN Hamacher ordered weighted geometric (BNHOWG), and BN Hamacher hybrid geometric (BNHHG) and their required properties. The proposed scheme provides decision-makers with a comprehensive view of the complexities and vagueness in multicriteria decision-making. As compared to existing methods, these techniques provide comprehensive, increasingly exact, and precise results. Finally, we applied different types of newly introduced AOs and numerical representation on a practical example to demonstrate the effectiveness of the proposed method. Our proposed model and its application have shown improved utility and applicability in the complex decision-making process.


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## Body

## 1. Introduction

In the current modern age of community decision-making, data is frequently inadequate, imprecise, and incompatible. Zadeh anticipated the theory of a fuzzy set [1], which deals with vagueness and has applications in a diversity of fields. It does, however, have a flaw in that it can only express a membership value and cannot state any information about nonmembership. To overcome this, Atanassov [2] set up the fundamental concept of intuitionistic fuzzy set (IFS) and its theory to sum up the initiative idea of fuzzy sets. Each and every element of IFS is represented by a pair of membership value (truth-membership) Iχ as well as nonmembership value (falsity-membership) ƒχ and satisfies the conditions Iχ,ƒχ∈0,1 along with 0≤Iχ+ƒχ≤1. IFSs can only handle incomplete data; they cannot handle indeterminate or unreliable data sets. Smarandache [3] developed the novel neutrosophic set (NS), which added extra indeterminacy membership value Ιχ with IFS. NS is capable of dealing with knowledge that is incomplete, indeterminate, and contradictory very effectively. When Iχ+Ιχ+ƒχ<1, it represents the information indeterminate. When Iχ+Ιχ+ƒχ>1, it shows that this represents the inconsistent information under a neutrosophic environment.A single-valued neutrosophic set (SVNS) deals with real-life problems as developed by Wang et al. [4], along with conditions Iχ,Ιχ,ƒχ∈0,1 as well as 0≤Iχ+Ιχ+ƒχ≤3. Dubois et al. [5] defined the correlation coefficient as well as suggested a method for comparing SVNS. The interval-valued neutrosophic set developed by Wang and others [6] broadens the truth, indeterminacy, and false membership range of the value between 0 and 1.Hamacher’st-norms/t-conorms [7] are more flexible than algebraic as well as Einstein t-norms/t-conorms. Many academics have developed the Hamacher operations to address issues involving numerous multicriteria fuzzy decision-making [8–11]. There has not been much research done on Hamacher operations and their applicability to bipolar neutrosophic numbers since the beginning of this field. We developed bipolar neutrosophic Hamacher geometric AOs for multicriteria decision-making by extending Hamacher operations to bipolar neutrosophic sets.Aggregation operators (AOs) are of great consequence for researchers to attract their attention. Many scientists [12–16] have made a significant contribution toward theory development of IFS since its inception. Based on IFS, Xu and Yager [14] develop the concept of different IF aggregation operators (AOs). They as well used AOs to make decisions related to real life. Einstein aggregation operators (AOs) were developed by Wang and Liu and Chen [17, 18]. Jamil and others [9, 19] develop aggregation operators (AOs) based on bipolar neutrosophic values along with application to group decision-making issues. The bipolar fuzzy set [20–22] has come up at the same time as a different approach in the direction of dealing with ambiguity related to decision-making problems. BFS has both positive and negative membership degrees. The bipolar fuzzy set’s membership degree varies from -1 to 1. BFS is very useful in a variety of study domains, including decision-making [6, 18, 23–25]. Wang et al. [10] define bipolar averaging as well as geometric fuzzy aggregation operators (AOs). Deli et al. [26, 27] offered the bipolar neutrosophic set by means of fundamental operations along with the comparison method. Fan and others [28] develop Heronian mean aggregation operators (AOs).Despite the fact that there is a variety of literature on the topic, the following points about the BNS and Hamacher operations motivated the researcher to conduct a systematic as well as in-depth investigation into the decision analysis. Our most important tools are stated below:(1)
SVNSs make it easier to deal with uncertain details. This set incorporates the generality of previous sets like classical set, FS, and IFS(2)
Bipolar fuzzy sets are extremely useful for dealing with unpredictable real-world situations and are useful in dealing with both positive and negative membership values(3)
The main and foremost intention of the current study include(a)
suggesting various bipolar neutrosophic Hamacher AOs and their related properties to our study(b)
based on BNN, establishing a multicriteria decision-making (MCDM) approach toward real-life problems(c)
giving a descriptive numerical example of MCDM programThe rest of the research is structured as follows: In Section2, there are essential definitions as well as their related properties. In Section 3, we introduced BNHWG aggregation operators. In Section 4, these novel AOs are applied to multicriteria decision-making in addition to that of a numerical example. Section 5, at last, proposed a comparative study along with concluding remarks.


## 2. Preliminaries

We have given a basic definition of the neutrosophic set in the present segment. Different fuzzy sets along with BNS, score, accuracy as well as certainty functions, and Einstein operation are defined.Definition 1 (see [3]).
ConsiderR to represent a universal set with the neutrosophic set stated below:
(1)N=χ,Iχ,Ιχ,ƒχχ∈R.
The truth-membership is represented by the functionI:N⟶Q, indeterminacy-membership is represented by the function Ι:N⟶Q, and falsity-membership is represented by the function ƒ:N⟶Q, where Q=0−,1+. There is no limitation on the summation of Iχ,Ιχ, and ƒχ, 0−≤Iχ+Ιχ+ƒχ≤3+.Since applying NS to real-life science as well as business fields is difficult, Ye [29] suggested the idea of SVNS as stated.Definition 2 (see [4]).
ConsiderP to represent a fixed set; the single-valued neutrosophic set (SVNS) of A is stated as
(2)ANS=χ,Iχ,Ιχ,ƒχχ∈P.
The truth-membership isI:ANS⟶L, indeterminacy-membership is Ι:ANS⟶L, and falsity-membership is ƒ:ANS⟶L, where L=0,1. The condition of SVNS is 0≤Iχ+Ιχ+ƒχ≤3.Example 1.
LetP=χ1,χ2,χ3, then
(3)A=χ1,0.1,0.3,0.8,χ2,0.3,0.6,0.5,χ3,0.4,0.1,0.7is a SVNS subset of universal set P.Fundamental operations between two single-valued neutrosophic sets are(4)ANS=χ,IANSχ,ΙANSχ,ƒANSχχ∈P,BNS=χ,IBNSχ,ΙBNSχ,ƒBNSχχ∈Pgiven as the following:
(i)
The subsetANS⊆BNS represented as(5)IANSχ≤IBNSχ,ΙANSχ≥ΙBNSχ,ƒANSχ≥ƒBNSχ(ii)
ANS=BNS represented as(6)IANSχ=IBNSχ,ΙANSχ=ΙBNSχ,ƒANSχ=ƒBNSχ(iii)
The complementANS′ is(7)ANS′=χ,ƒANSχ,1−ΙANSχ,IANSχχ∈P(iv)
ANS∩BNS represented as(8)ANS∩BNS=χ,minIANSχ,IBNSχ,maxΙANSχ,ΙBNSχ,maxƒANSχ,ƒBNSχχ∈P(v)
The union is defined by(9)ANS∪BNS=χ,maxIANSχ,IBNSχ,minΙANSχ,ΙBNSχ,minƒANSχ,ƒBNSχχ∈PDefinition 3 (see [5]).
Consider two single-valued neutrosophic numbers (SVNNs)u1=I1,Ι1,ƒ1 and u2=I2,Ι2,ƒ2. Thus, the different basic operations between two SVNNs are defined below:
(10)u1+u2=I1+I2−I1I2,Ι1Ι2,ƒ1ƒ2,u1⋅u2=I1+I2,Ι1+Ι2−Ι1Ι2,ƒ1+ƒ2−ƒ1ƒ2,λu1=1−1−I1λ,Ι1λ,ƒ1λ,u1λ=I1λ,1−1−Ι1λ,1−1−ƒ1λ,where λ>0.Definition 4 (see [5]).
Letu1=I1,Ι1,ƒ1 represent a SVNN. The score function for SVNN is su1 stated as
(11)su1=I1+1−Ι1+1−ƒ13.Definition 5 (see [5]).
The accuracy function for a SVNNu1=I1,Ι1,ƒ1 is denoted by u1 defined below:
(12)au1=I1−ƒ1.Definition 6 (see [5]).
The certainty function for a SVNNu1=I1,Ι1,ƒ1 is denoted by cu1 defined below:
(13)cu1=I1.Definition 7 (see [5]).
Consider two SVNNsu1=I1,Ι1,ƒ1 and u2=I2,Ι2,ƒ2. Then, the relationship between two SVNNs is stated as below:
(i)
Ifsu1>su2, then u1 is greater than u2, denoted by u1>u2(ii)
Ifsu1=su2 and au1>au2, then u1 is greater than u2, denoted by u1>u2(iii)
Ifsu1=su2, au1=au2, and cu1>cu2, then u1 is greater than u2, denoted by u1>u2(iv)
Ifsu1=su2, au1=au2, and cu1=cu2, then u1 is equal to u2, denoted by u1=u2Definition 8 (see [21]).
ConsiderP as a universal set, and then, the bipolar fuzzy set will be stated as below:
(14)F=χ,IF+χ,ƒF−χχ∈P,where IF+χ:F⟶L is the positive membership degree and ƒF−χ:F⟶K is the negative membership degree; here, L=1,0, K=−1,0.Definition 9 (see [27]).
ConsiderA as a bipolar neutrosophic set (BNS) within universal set P stated below:
(15)A=χ,I+χ,Ι+χ,ƒ+χ,I−χ,Ι−χ,ƒ−χχ∈P.
LetI+χ,Ι+χ,ƒ+χ=BN+ and I−χ,Ι−χ,ƒ−χ=BN−, where I+χ,Ι+χ,ƒ+χ represent the positive membership function for the truth value, indeterminate value, and false value of an element χ∈P and I−χ,Ι−χ,ƒ−χ represent the negative membership function for the truth value, indeterminate value, and false value for an element χ∈P. Then, BN+:A⟶L and BN−:A⟶K, where L=1,0, K=−1,0. There is a condition that 0≤I+χ+Ι+χ+ƒ+χ+I−χ+Ι−χ+ƒ−χ≤6.Example 2.
LetP=χ1,χ2,χ3, then
(16)A=χ1,0.4,0.3,0.5,−0.6,−0.7,−0.8,χ2,0.6,0.8,0.2,−0.5,−0.8,−0.4,χ3,0.7,0.5,0.9,−0.6,−0.5,−0.4is a BNS subset of universal set P.Fundamental operations [27] for BNSs are shown below.ConsiderA1=χ,I1+χ,Ι1+χ,ƒ1+χ,I1−χ,Ι1−χ,ƒ1−χχ∈P and A2=χ,I2+χ,Ι2+χ,ƒ2+χ,I2−χ,Ι2−χ,ƒ2−χχ∈P as two BNSs.
(i)
ThenA1⊆A2 if and only if(17)I1+χ≤I2+χ,Ι1+χ≤Ι2+χ,ƒ1+χ≥ƒ2+χ,I1−χ≥I2−χ,Ι1−χ≥Ι2−χ,ƒ1−χ≤ƒ2−χ(ii)
A1=A2 if and only if(18)I1+χ=I2+χ,Ι1+χ=Ι2+χ,ƒ1+χ=ƒ2+χ,I1−χ=I2−χ,Ι1−χ=Ι2−χ,ƒ1−χ=ƒ2−χ(iii)
The union is defined as below:(19)A1∪A2=maxI1+χ,I2+χ,Ι1+χ+Ι2+χ2,minƒ1+χ,ƒ2+χ,minI1−χ,I2−χ,Ι1−χ+Ι2−χ2,maxƒ1−χ,ƒ2−χ(iv)
The intersection is defined as(20)A1∩A2=minI1+χ,I2+χ,Ι1+χ+Ι2+χ2,maxƒ1+χ,ƒ2+χ,maxI1−χ,I2−χ,Ι1−χ+Ι2−χ2,minƒ1−χ,ƒ2−χ(v)
LetA=χ,I+χ,Ι+χ,ƒ+χ,I−χ,Ι−χ,ƒ−χ∣χ∈P and be a BNS. Then, the complement A′ is defined as(21)IA′+χ=1+−IA+χ,ΙA′+χ=1+−ΙA+χ,ƒA′+χ=1+−ƒA+χ,IA′−χ=1−−IA−χ,ΙA′−χ=1−−ΙA−χ,ƒA′−χ=1−−ƒA−χ.Definition 10 (see [27]).
Let us consider two bipolar neutrosophic numbers (BNNs)u1=I1+,Ι1+,ƒ1+,I1−,Ι1−,ƒ1− and u2=I2+,Ι2+,ƒ2+,I2−,Ι2−,ƒ2−. Then, the basic operations between two BNNs are stated below:
(22)u1+u2=I1++I2+−I1+I2+,Ι1+Ι2+,ƒ1+ƒ2+,−I1−I2−,−−Ι1−−Ι2−−Ι1−Ι2−,−−ƒ1−−ƒ2−−ƒ1−ƒ2−,u1⋅u2=I1+I2+,Ι1++Ι2+−Ι1+Ι2+,ƒ1++ƒ2+−ƒ1+ƒ2+,−−I1−−I2−−I1−I2−,−Ι1−Ι2−,−ƒ1−ƒ2−,λu1=1−1−I1+λ,Ι1+λ,ƒ1+λ,−−I1−λ,−−Ι1−λ,−1−1−−ƒ1−λ,u1λ=I1+λ,1−1−Ι1+λ,1−1−ƒ1+λ,−1−1−−I1−λ,−−Ι1−λ,−−ƒ1−λ,where λ>0.Definition 11 (see [27]).
The score function for a BNNu=I+,Ι+,ƒ+,I−,Ι−,ƒ− is denoted by Su defined as below:
(23)Su=16I++1−Ι++1−ƒ++1+I−−Ι−−ƒ−.Definition 12 (see [27]).
The accuracy function for a BNNu=I+,Ι+,ƒ+,I−,Ι−,ƒ− is denoted by au defined as below:
(24)au=I+−ƒ++I−−ƒ−.Definition 13 (see [27]).
Letu=I+,Ι+,ƒ+,I−,Ι−,ƒ− represent a BNN, then certainty function for BNN u is stated as
(25)cu=I+−ƒ−.Definition 14 (see [27]).
Supposeu1=I1+,Ι1+,ƒ1+,I1−,Ι1−,ƒ1− and u2=I2+,Ι2+,ƒ2+,I2−,Ι2−,ƒ2− are bipolar neutrosophic numbers (BNNs), so the comparison method among BNNs is stated below:
(i)
IfSu1≻Su2, then u1 is greater than u2, denoted by u1≻u2(ii)
IfSu1=Su2 and au1≻au2, then u1 is superior to u2, denoted by u1≻u2(iii)
If conditionSu1=Su2,au1=au2, and cu1≻cu2, then u1 is greater than u2, denoted by u1≻u2(iv)
If conditionSu1=Su2,au1=au2, and cu1=cu2, then u1 is equal to u2 denoted by u1=u2Hamacher [7] proposed a more generalized t-norm and t-conorm. Hamacher product ⊗ represent a t-norm as well as Hamacher sum ⊕ represent a t-conorm, where
(26)TÞ,ß=Þ⊗ß=Þßγ+1−γÞ+ß−Þß,γ>0,T″Þ,ß=Þ⊕ß=Þ+ß−Þß−1−γÞß1−1−γÞß,γ>0.Whenγ=1, the Hamacher t-norm as well as t-conorm will be reduced to algebraic t-norm as well as t-conorm, respectively,
(27)TÞ,ß=Þ⊗ß=Þß,T″Þ,ß=Þ⊕ß=Þ+ß−Þß.Whenγ=2, the Hamacher t-norm as well as t-conorm will be equivalent to the Einstein t-norm as well as t-conorm, respectively [21]:
(28)TÞ,ß=Þ⊗ß=Þß1+1−Þ1−ß,T″Þ,ß=Þ⊕ß=Þ+ß1+Þß.Now, we are introducing the Hamacher operation of bipolar neutrosophic set; at the same time, the concept of Hamacher sum, Hamacher product, Hamacher scalar multiple, and Hamacher exponential operation for BNN is stated below.Definition 15.
Letu=I+,Ι+,ƒ+,I−,Ι−,ƒ−, u1=I1+,Ι1+,ƒ1+,I1−,Ι1−,ƒ1−, and u2=I2+,Ι2+,ƒ2+,I2−,Ι2−,ƒ2− be three bipolar neutrosophic values and λ≻0 represent any of the real values; then, basic Hamacher operations are γ≻0:
(29)u1⊕u2=I1++I2+−I1+I2+−1−γI1+I2+1−1−γI1+I2+,Ι1+Ι2+γ+1−γΙ1++Ι2+−Ι1+Ι2+,ƒ1+ƒ2+γ+1−γƒ1++ƒ2+−ƒ1+ƒ2+,−I1−I2−γ+1−γI1−+I2−−I1−I2−,−−Ι1−−Ι2−−Ι1−Ι2−−1−γΙ1−Ι2−1−1−γΙ1−Ι2−,−−ƒ1−−ƒ2−−ƒ1−ƒ2−−1−γƒ1−ƒ2−1−1−γƒ1−ƒ2−,u1⊗u2=I1+I2+γ+1−γI1++I2+−I1+I2+,Ι1++Ι2+−Ι1+Ι2+−1−γΙ1+Ι2+1−1−γΙ1+Ι2+,ƒ1++ƒ2+−ƒ1+ƒ2+−1−γƒ1+ƒ2+1−1−γƒ1+ƒ2+,−−I1−−I2−−I1−I2−−1−γI1−I2−1−1−γI1−I2−,−Ι1−Ι2−γ+1−γΙ1−+Ι2−−Ι1−Ι2−,−ƒ1−ƒ2−γ+1−γƒ1−+ƒ2−−ƒ1−ƒ2−,λu=1+γ−1I+λ−1−I+λ1+γ−1I+λ+γ−11−I+λ,γΙ+λ1+γ−11−Ι+λ+γ−1Ι+λ,γƒ+λ1+γ−11−ƒ+λ+γ−1ƒ+λ,−γI−λ1+γ−11+I−λ+γ−1I−λ,−γΙ−λ1+γ−11+Ι−λ+γ−1Ι−λ,−1+γ−1ƒ−λ−1+ƒ−λ1+γ−1ƒ−λ+γ−11+ƒ−λ,uλ=I+λ1+γ−11−I+λ+γ−1I+λ,1+γ−1Ι+λ−1−Ι+λ1+γ−1Ι+λ+γ−11−Ι+λ,1+γ−1ƒ+λ−1−ƒ+λ1+γ−1ƒ+λ+γ−11−ƒ+λ,−1+γ−1I−λ−1+I−λ1+γ−1I−λ+γ−11+I−λ,−γΙ−λ1+γ−11+Ι−λ+γ−1Ι−λ,−γƒ−λ1+γ−11+ƒ−λ+γ−1ƒ−λ.


## 3. Bipolar Neutrosophic Hamacher AOs

In this section, we develop a number of basic properties for the bipolar neutrosophic Hamacher weighted geometric aggregation operator, bipolar neutrosophic Hamacher ordered weighted geometric aggregation operator, and bipolar neutrosophic Hamacher hybrid geometric aggregation operator.


### 3.1. Bipolar Neutrosophic Hamacher Weighted Geometric AO

Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ− be a family of BNNs, where ℓ∈1,2,3,⋯,n.Definition 16.
The (BNHWG) aggregation operator is defined as follows:(30)BNHWGνu1,u2,u3,⋯,un=⊗ℓ=1nνℓuℓ=ν1u1⊗ν2u2⊗⋯⊗νnun,where ν=ν1,ν2,ν3,⋯,νnT represent the associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=1.Theorem 17.
The BNHWG operators give in return a bipolar neutrosophic value (BNV) with(31)BNHWGνu1,u2,⋯,un=γЂ1~Ђ2~+γ−1Ђ1~,ζ1~−ζ2~ζ1~+γ−1ζ2~,f1~−f2~f1~+γ−1f2~,−Ђ3~−Ђ4~Ђ3~+γ−1Ђ4~,−γζ3~ζ4~+γ−1ζ3~,−γf3~f4~+γ−1f3~,where
(32)Ђ1~=∏ℓ=1nIℓ+νℓ,Ђ2~=∏ℓ=1n1+γ−11−Iℓ+νℓ,Ђ3~=∏ℓ=1n1+γ−1Iℓ−νℓ,Ђ4~=∏ℓ=1n1+Iℓ−νℓ,ζ1~=∏ℓ=1n1+γ−ΙΙℓ+νℓ,ζ2~=∏ℓ=1n1−Ιℓ+νℓ,ζ3~=∏ℓ=1nΙℓ−νℓ,ζ4~=∏ℓ=1n1+γ−11+Ιℓ−νℓ,f1~=∏ℓ=1n1+γ−1ƒℓ+νℓ,f2~=∏ℓ=1n1−ƒℓ+νℓ,f3~=∏ℓ=1nƒℓ−νℓ,f4~=∏ℓ=1n1+γ−11+ƒℓ−νℓ,where ν=ν1,ν2,ν3,⋯,νnT represent the associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=11,γ>0.Proof.
Now by mathematical induction.
Forn=2,
(33)ν1u1=γI1+ν11+γ−11−I1+ν1+γ−1I1+ν11+γ−1Ι1+ν1−1−Ι1+ν11+γ−1Ι1+ν1+γ−11−Ι1+ν1,1+γ−1ƒ1+ν1−1−ƒ1+ν11+γ−1ƒ1+ν1+γ−11−ƒ1+ν1,−1+γ−1I1−ν1−1+I1−ν11+γ−1I1−ν1+γ−11+I1−ν1,−γΙ1−ν11+γ−11+Ι1−ν1+γ−1Ι1−ν1,−γƒ1−ν11+γ−11+ƒ1−ν1+γ−1ƒ1−ν1,ν2u2=γI2+ν21+γ−11−I2+ν2+γ−1I2+ν21+γ−1Ι2+ν2−1−Ι2+ν21+γ−1Ι2+ν2+γ−11−Ι2+ν2,1+γ−1ƒ2+ν2−1−ƒ2+ν21+γ−1ƒ2+ν2+γ−11−ƒ2+ν2,−1+γ−1I2−ν2−1+I2−ν21+γ−1I2−ν2+γ−11+I2−ν2,−γΙ2−ν21+γ−11+Ι2−ν2+γ−1Ι2−ν2,−γƒ2−ν21+γ−11+ƒ2−ν2+γ−1ƒ2−ν2,and for
(34)ν1u1=γI1+ν11+γ−11−I1+ν1+γ−1I1+ν11+γ−1Ι1+ν1−1−Ι1+ν11+γ−1Ι1+ν1+γ−11−Ι1+ν1,1+γ−1ƒ1+ν1−1−ƒ1+ν11+γ−1ƒ1+ν1+γ−11−ƒ1+ν1,−1+γ−1I1−ν1−1+I1−ν11+γ−1I1−ν1+γ−11+I1−ν1,−γΙ1−ν11+γ−11+Ι1−ν1+γ−1Ι1−ν1,−γƒ1−ν11+γ−11+ƒ1−ν1+γ−1ƒ1−ν1.
So, satisfied forn=1, put n=r for equation (34),
(35)BNHWGνu1,u2,⋯,ur=γ∏ℓ=1rIℓ+νℓ∏ℓ=1r1+γ−11−Iℓ+νℓ+γ−1∏ℓ=1rIℓ+νℓ,∏ℓ=1r1+γ−1Ιℓ+νℓ−∏ℓ=1r1−Ιℓ+νℓ∏ℓ=1r1+γ−1Ιℓ+νℓ+γ−1∏ℓ=1r1−Ιℓ+νℓ,∏ℓ=1r1+γ−1ƒℓ+νℓ−∏ℓ=1r1−ƒℓ+νℓ∏ℓ=1r1+γ−1ƒℓ+νℓ+γ−1∏ℓ=1r1−ƒℓ+νℓ,−∏ℓ=1r1−γ−1Iℓ−νℓ−∏ℓ=1r1−Iℓ−νℓ∏ℓ=1r1−γ−1Iℓ−νℓ+γ−1∏ℓ=1r1+Iℓ−νℓ,−γ∏ℓ=1rΙℓ−νℓ∏ℓ=1r1+γ−11+Ιℓ−νℓ+γ−1∏ℓ=1rΙℓ−νℓ,−γ∏ℓ=1rƒℓ−νℓ∏ℓ=1r1+γ−11+ƒℓ−νℓ+γ−1∏ℓ=1rƒℓ−νℓ.
If equation (34) is true n=r, then we show that (34) is true for n=r+1; thus,
(36)BNHWGνu1,u2,⋯,ur=γ∏ℓ=1rIℓ+νℓ∏ℓ=1r1+γ−11−Iℓ+νℓ+γ−1∏ℓ=1rIℓ+νℓ,∏ℓ=1r1+γ−1Ιℓ+νℓ−∏ℓ=1r1−Ιℓ+νℓ∏ℓ=1r1+γ−1Ιℓ+νℓ+γ−1∏ℓ=1r1−Ιℓ+νℓ,∏ℓ=1r1+γ−1ƒℓ+νℓ−∏ℓ=1r1−ƒℓ+νℓ∏ℓ=1r1+γ−1ƒℓ+νℓ+γ−1∏ℓ=1r1−ƒℓ+νℓ,−∏ℓ=1r1+γ−1Iℓ−νℓ−∏ℓ=1r1+Iℓ−νℓ∏ℓ=1r1+γ−1Iℓ−νℓ+γ−1∏ℓ=1r1+Iℓ−νℓ,−γ∏ℓ=1rΙℓ−νℓ∏ℓ=1r1+γ−11+Ιℓ−νℓ+γ−1∏ℓ=1rΙℓ−νℓ,−γ∏ℓ=1rƒℓ−νℓ∏ℓ=1r1+γ−11+ƒℓ−νℓ+γ−1∏ℓ=1rƒℓ−νℓ⊗γIr+1+νr+11+γ−11−Ir+1+νr+1+γ−1Ir+1+νr+1,1+γ−1Ιr+1+νr+1−1−Ιr+1+νr+11+γ−1Ιr+1+νr+1+γ−11−Ιr+1+νr+1,1+γ−1ƒr+1+νr+1−1−ƒr+1+νr+11+γ−1ƒr+1+νr+1+γ−11−ƒr+1+νr+1,−1+γ−1Ir+1−νr+1−1+Ir+1−νr+11+γ−1Ir+1−νr+1+γ−11+Ir+1−νr+1,−γΙr+1−νr+11+γ−11+Ιr+1−νr+1+γ−1Ιr+1−νr+1,−γƒr+1−νr+11+γ−11+ƒr+1−νr+1+γ−1ƒr+1−νr+1=γ∏ℓ=1r+1Iℓ+νℓ∏ℓ=1r+11+γ−11−Iℓ+νℓ+γ−1∏ℓ=1r+1Iℓ+νℓ,∏ℓ=1r+11+γ−1Ιℓ+νℓ−∏ℓ=1r+11−Ιℓ+νℓ∏ℓ=1r+11+γ−1Ιℓ+νℓ+γ−1∏ℓ=1r+11−Ιℓ+νℓ,∏ℓ=1r+11+γ−1ƒℓ+νℓ−∏ℓ=1r+11−ƒℓ+νℓ∏ℓ=1r+11+γ−1ƒℓ+νℓ+γ−1∏ℓ=1r+11−ƒℓ+νℓ,−∏ℓ=1r+11+γ−1Iℓ−νℓ−∏ℓ=1r+11+Iℓ−νℓ∏ℓ=1r+11+γ−1Iℓ−νℓ+γ−1∏ℓ=1r+11+Iℓ−νℓ,−γ∏ℓ=1r+1Ιℓ−νℓ∏ℓ=1r+11+γ−11+Ιℓ−νℓ+γ−1∏ℓ=1r+1Ιℓ−νℓ,−γ∏ℓ=1r+1ƒℓ−νℓ∏ℓ=1r+11+γ−11+ƒℓ−νℓ+γ−1∏ℓ=1r+1ƒℓ−νℓ.
Thus, equation (34) is true for n=r+1. Hence, equation (34) is true for all n.☐Theorem 18 (idempotency).
Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n be a set of equal bipolar neutrosophic values, that is, uℓ=u:
(37)BNHWGνu1,u2,⋯,un=u.Theorem 19 (bounded).
Supposeu−=minℓuℓ,u+=maxℓuℓ,
(38)u−≤BNHWGνu1,u2,⋯,un≤u+.Theorem 20 (monotonicity).
Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n and uℓ′=I′ℓ+,Ι′ℓ+,ƒ′ℓ+,I′ℓ−,Ι′ℓ−,ƒ′ℓ−ℓ=1,2,⋯,n be two collections of bipolar neutrosophic values. If uℓ≤uℓ′, then
(39)BNHWGνu1,u2,⋯,un≤BNHWGνu1′,u2′,u3′,⋯,un′.Now, we will look at two particular examples of the BNHWG operator.(i)
Ifγ=1, then the BNHWG operator is equivalent to the basic algebraic operator bipolar neutrosophic weighted geometric (BNWG) operator:(40)BNHWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=∏ℓ=1nIℓ+νℓ,1−∏ℓ=1n1−Ιℓ+νℓ,1−∏ℓ=1n1−ƒℓ+νℓ,−1−∏ℓ=1n1+Iℓ−νℓ,−∏ℓ=1nΙℓ−νℓ,−∏ℓ=1nƒℓ−νℓ(ii)
Ifγ=2, then the bipolar neutrosophic Einstein weighted geometric (BNEWG) operator is equivalent to the BNHWG operator:(41)BNHWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=2∏ℓ=1nIℓ+νℓ∏ℓ=1n2−Iℓ+νℓ+∏ℓ=1nIℓ+νℓ,∏ℓ=1n1+Ιℓ+νℓ−∏ℓ=1n1−Ιℓ+νℓ∏ℓ=1n1+Ιℓ+νℓ+∏ℓ=1n1−Ιℓ+νℓ,∏ℓ=1n1+ƒℓ+νℓ−∏ℓ=1n1−ƒℓ+νℓ∏ℓ=1n1+ƒℓ+νℓ+∏ℓ=1n1−ƒℓ+νℓ,−∏ℓ=1n1+Iℓ−νℓ−∏ℓ=1n1+Iℓ−νℓ∏ℓ=1n1+Iℓ−νℓ+∏ℓ=1n1+Iℓ−νℓ,−−2∏ℓ=1nΙℓ−νℓ∏ℓ=1n2+Ιℓ−νℓ+∏ℓ=1nΙℓ−νℓ,−2∏ℓ=1nƒℓ−νℓ∏ℓ=1n2+ƒℓ−νℓ+∏ℓ=1nƒℓ−νℓ


### 3.2. Bipolar Neutrosophic Hamacher Ordered Weighted Geometric AO

Definition 21.
The bipolar neutrosophic Hamacher ordered weighted geometric (BNHOWG) aggregation operator is defined as(42)BNHOWGνu1,u2,⋯,un=⊗ℓ=1nνℓuσℓ=ν1uσ1⊗ν2uσ2⊗ν3uσ3⊗⋯⊗νnuσn,where σ1,⋯,σn−1,σn represent permutation of 1,2,3,⋯,n−1,n with uσℓ−1≥uσℓℓ=2,⋯,n−1,n and ν=ν1,ν2,ν3,⋯,νnT represent associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=1,γ>0.Theorem 22.
The BNHOWG operators give in return a BNV with(43)BNHOWGνu1,u2,⋯,un=γЂ∂1~Ђ∂2~+γ−1Ђ∂1~,ζ∂1~−ζ∂2~ζ∂1~+γ−1ζ∂2~,f∂1~−f∂2~f∂1~+γ−1f∂2~,−Ђ∂3~−Ђ∂4~Ђ∂3~+γ−1Ђ∂4~,−γζ∂3~ζ∂4~+γ−1ζ∂3~,−γf∂3~f∂4~+γ−1f∂3~,where
(44)Ђ∂1~=∏ℓ=1nIσℓ+νℓ,Ђ∂2~=∏ℓ=1n1+γ−11−Iσℓ+νℓ,Ђ∂3~=∏ℓ=1n1+γ−1Iσℓ−νℓ,Ђ∂4~=∏ℓ=1n1+Iσℓ−νℓ,ζ∂1~=∏ℓ=1n1+γ−1Ισℓ+νℓ,ζ∂2~=∏ℓ=1n1−Ισℓ+νℓ,ζ∂3~=∏ℓ=1nΙσℓ−νℓ,ζ∂4~=∏ℓ=1n1+γ−11+Ισℓ−νℓ,f∂1~=∏ℓ=1n1+γ−1ƒσℓ+νℓ,f∂2~=∏ℓ=1n1−ƒσℓ+νℓ,f∂3~=∏ℓ=1nƒσℓ−νℓ,f∂4~=∏ℓ=1n1+γ−11+ƒσℓ−νℓ,where σ1,⋯,σn−1,σn represent permutation of 1,2,3,⋯,n−1,n with uσℓ−1≥uσℓℓ=2,⋯,n−1,n and ν=ν1,ν2,ν3,⋯,νnT represent associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=1,γ>0.Proof.
The theorem is straightforward.☐Theorem 23 (idempotency).
Consideruℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n to represent a set of equal BNN that is uℓ=u:
(45)BNHOWGνu1,u2,⋯,un=u.Theorem 24 (bounded).
Supposeu−=minℓuℓ,u+=maxℓuℓ,
(46)u−≤BNHOWGνu1,u2,⋯,un≤u+.Theorem 25 (monotonicity).
Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n and uℓ′=I′ℓ+,Ι′ℓ+,ƒ′ℓ+,I′ℓ−,Ι′ℓ−,ƒ′ℓ−ℓ=1,2,⋯,n be a set of two bipolar neutrosophic values. If uℓ≤uℓ′, then
(47)BNHOWGνu1,u2,⋯,un≤BNHOWGνu1′,u2′,⋯,un′.At the moment, we will look at two particular examples related to BNHOWG AO.(i)
The bipolar neutrosophic ordered weighted geometric (BNOWG) operator is equivalent to the BNHOWG operator ifγ=1:(48)BNOWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=∏ℓ=1nIσℓ+νℓ,1−∏ℓ=1n1−Ισℓ+νℓ,1−∏ℓ=1n1−ƒσℓ+νℓ,−1−∏ℓ=1n1+Iσℓ−νℓ,−∏ℓ=1nΙσℓ−νℓ,−∏ℓ=1nƒσℓ−νℓ(ii)
The bipolar neutrosophic Einstein ordered weighted geometric (BNEOWG) operator is equivalent to the BNHOWG operator, ifγ=2:(49)BNOWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=2∏ℓ=1nIσℓ+νℓ∏ℓ=1n2−Iσℓ+νℓ+∏ℓ=1nIσℓ+νℓ,∏ℓ=1n1+Ισℓ+νℓ−∏ℓ=1n1−Ισℓ+νℓ∏ℓ=1n1+Ισℓ+νℓ+∏ℓ=1n1−Ισℓ+νℓ,∏ℓ=1n1+ƒσℓ+νℓ−∏ℓ=1n1−ƒσℓ+νℓ∏ℓ=1n1+ƒσℓ+νℓ+∏ℓ=1n1−ƒσℓ+νℓ,−∏ℓ=1n1+Iσℓ−νℓ−∏ℓ=1n1+Iσℓ−νℓ∏ℓ=1n1+Iσℓ−νℓ+∏ℓ=1n1+Iσℓ−νℓ,−−2∏ℓ=1nΙσℓ−νℓ∏ℓ=1n2+Ισℓ−νℓ+∏ℓ=1nΙΙσℓ−νℓ,−2∏ℓ=1nƒσℓ−νℓ∏ℓ=1n2+ƒσℓ−νℓ+∏ℓ=1nƒσℓ−νℓ


### 3.3. Bipolar Neutrosophic Hamacher Hybrid Geometric AO

Definition 26.
The bipolar neutrosophic Hamacher hybrid weighted geometric aggregation (BNHHG) operator is defined as(50)BNHHGw,νu1,u2,⋯,un=⊗ℓ=1nνℓuσℓ=ν1u̇σ1⊗ν2u̇σ2⊗ν3u̇σ3⊗⋯⊗νnu̇σn,where w=w1,w2,w3,⋯,wn−1,wn represent associated weighting vector for uℓ along with wℓ∈0,1, ∑ℓ=1nwℓ=1,u̇σℓ represent the ℓth biggest element of the BN arguments u̇ℓu̇ℓ=nvℓuℓ, ν=ν1,ν2,ν3,⋯,νn−1,νn represent the weighting vector of bipolar neutrosophic argument uℓ with ωℓ∈0,1, ∑ℓ=1nωℓ=1. Here, n represent the balancing coefficient.Note that the BNHHG operator reduces to the BNHWG operator ifw=1/n,1/n,⋯,1/nT as well as the BNHOWG operator if v=1/n,1/n,⋯,1/n.Theorem 27.
The BNHHGoperators give in return a BNV with(51)BNHHWGw,vu1,u2,⋯,un=γЂ̇∂1~Ђ̇∂2~+γ−1Ђ̇∂1~,ζ̇∂1~−ζ̇∂2~ζ̇∂1~+γ−1ζ̇∂2~,ḟ∂1~−ḟ∂2~ḟ∂1~+γ−1ḟ∂2~,Ђ̇∂3~−Ђ̇∂4~Ђ̇∂3~+γ−1Ђ̇∂4~,γζ̇∂3~ζ̇∂4~+γ−1ζ̇∂3~,−γḟ∂3~ḟ∂4~+γ−1ḟ∂3~,where
(52)Ђ̇∂1~=∏ℓ=1nİσℓ+νℓ,Ђ̇∂2~=∏ℓ=1n1+γ−11−İσℓ+νℓ,Ђ̇∂3~=∏ℓ=1n1+γ−1İσℓ−νℓ,Ђ̇∂4~=∏ℓ=1n1+İσℓ−νℓ,ζ̇∂1~=∏ℓ=1n1+γ−1Ι̇σℓ+νℓ,ζ̇∂2~=∏ℓ=1n1−Ι̇σℓ+νℓ,ζ̇∂3~=∏ℓ=1nΙ̇σℓ−νℓ,ζ̇∂4~=∏ℓ=1n1+γ−11+Ι̇σℓ−νℓ,ḟ∂1~=∏ℓ=1n1+γ−1ƒ̇σℓ+νℓ,ḟ∂2~=∏ℓ=1n1−ƒ̇σℓ+νℓ,ḟ∂3~=∏ℓ=1nƒ̇σℓ−νℓ,ḟ∂4~=∏ℓ=1n1+γ−11+ƒ̇σℓ−νℓ,where w=w1,w2,w3,⋯,wn−1,wn represent the associated weighting vector for uℓ along with wℓ∈0,1, ∑ℓ=1nwℓ=1, u̇σℓ represent the ℓth biggest element of the BN arguments u̇ℓu̇ℓ=nvℓuℓ, ν=ν1,ν2,ν3,⋯,νn−1,νn represent the weighting vector of bipolar neutrosophic argument uℓ with ωℓ∈0,1, ∑ℓ=1nwℓ=1. Here, n represent the balancing coefficient, γ>0.Proof.
The proof is followed from Theorem17.☐Now, we will look at two particular cases of the BNHOWG operator.(i)
The bipolar neutrosophic hybrid geometric (BNHG) operator is equivalent to the BNHHG operator, ifγ=1:(53)BNHHGw,νu1,u2,⋯,un=⊗ℓ=1nνℓu̇ℓ=∏ℓ=1nİσℓ+νℓ,1−∏ℓ=1n1−Ι̇σℓ+νℓ,1−∏ℓ=1n1−ƒ̇σℓ+νℓ,−1−∏ℓ=1n1+İσℓ−νℓ,−∏ℓ=1nΙ̇σℓ−νℓ,−∏ℓ=1nƒ̇σℓ−νℓ(ii)
The bipolar neutrosophic Einstein hybrid geometric (BNHG) operator is equivalent to the BNHHG operator, ifγ=2:(54)BNEHGw,νu1,u2,⋯,un=⊗ℓ=1nνℓu̇ℓ=2∏ℓ=1nİσℓ+νℓ∏ℓ=1n2−İσℓ+νℓ+∏ℓ=1nİσℓ+νℓ,∏ℓ=1n1+Ι̇σℓ+νℓ−∏ℓ=1n1−Ι̇σℓ+νℓ∏ℓ=1n1+Ι̇σℓ+νℓ+∏ℓ=1n1−Ι̇σℓ+νℓ,∏ℓ=1n1+ƒ̇σℓ+νℓ−∏ℓ=1n1−ƒ̇σℓ+νℓ∏ℓ=1n1+ƒ̇σℓ+νℓ+∏ℓ=1n1−ƒ̇σℓ+νℓ,−∏ℓ=1n1+İσℓ−νℓ−∏ℓ=1n1+İσℓ−νℓ∏ℓ=1n1+İσℓ−νℓ+∏ℓ=1n1+İσℓ−νℓ,−−2∏ℓ=1nΙ̇σℓ−νℓ∏ℓ=1n2+Ι̇σℓ−νℓ+∏ℓ=1nΙ̇σℓ−νℓ,−2∏ℓ=1nƒ̇σℓ−νℓ∏ℓ=1n2+ƒ̇σℓ−νℓ+∏ℓ=1nƒ̇σℓ−νℓ


## 3.1. Bipolar Neutrosophic Hamacher Weighted Geometric AO

Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ− be a family of BNNs, where ℓ∈1,2,3,⋯,n.Definition 16.
The (BNHWG) aggregation operator is defined as follows:(30)BNHWGνu1,u2,u3,⋯,un=⊗ℓ=1nνℓuℓ=ν1u1⊗ν2u2⊗⋯⊗νnun,where ν=ν1,ν2,ν3,⋯,νnT represent the associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=1.Theorem 17.
The BNHWG operators give in return a bipolar neutrosophic value (BNV) with(31)BNHWGνu1,u2,⋯,un=γЂ1~Ђ2~+γ−1Ђ1~,ζ1~−ζ2~ζ1~+γ−1ζ2~,f1~−f2~f1~+γ−1f2~,−Ђ3~−Ђ4~Ђ3~+γ−1Ђ4~,−γζ3~ζ4~+γ−1ζ3~,−γf3~f4~+γ−1f3~,where
(32)Ђ1~=∏ℓ=1nIℓ+νℓ,Ђ2~=∏ℓ=1n1+γ−11−Iℓ+νℓ,Ђ3~=∏ℓ=1n1+γ−1Iℓ−νℓ,Ђ4~=∏ℓ=1n1+Iℓ−νℓ,ζ1~=∏ℓ=1n1+γ−ΙΙℓ+νℓ,ζ2~=∏ℓ=1n1−Ιℓ+νℓ,ζ3~=∏ℓ=1nΙℓ−νℓ,ζ4~=∏ℓ=1n1+γ−11+Ιℓ−νℓ,f1~=∏ℓ=1n1+γ−1ƒℓ+νℓ,f2~=∏ℓ=1n1−ƒℓ+νℓ,f3~=∏ℓ=1nƒℓ−νℓ,f4~=∏ℓ=1n1+γ−11+ƒℓ−νℓ,where ν=ν1,ν2,ν3,⋯,νnT represent the associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=11,γ>0.Proof.
Now by mathematical induction.
Forn=2,
(33)ν1u1=γI1+ν11+γ−11−I1+ν1+γ−1I1+ν11+γ−1Ι1+ν1−1−Ι1+ν11+γ−1Ι1+ν1+γ−11−Ι1+ν1,1+γ−1ƒ1+ν1−1−ƒ1+ν11+γ−1ƒ1+ν1+γ−11−ƒ1+ν1,−1+γ−1I1−ν1−1+I1−ν11+γ−1I1−ν1+γ−11+I1−ν1,−γΙ1−ν11+γ−11+Ι1−ν1+γ−1Ι1−ν1,−γƒ1−ν11+γ−11+ƒ1−ν1+γ−1ƒ1−ν1,ν2u2=γI2+ν21+γ−11−I2+ν2+γ−1I2+ν21+γ−1Ι2+ν2−1−Ι2+ν21+γ−1Ι2+ν2+γ−11−Ι2+ν2,1+γ−1ƒ2+ν2−1−ƒ2+ν21+γ−1ƒ2+ν2+γ−11−ƒ2+ν2,−1+γ−1I2−ν2−1+I2−ν21+γ−1I2−ν2+γ−11+I2−ν2,−γΙ2−ν21+γ−11+Ι2−ν2+γ−1Ι2−ν2,−γƒ2−ν21+γ−11+ƒ2−ν2+γ−1ƒ2−ν2,and for
(34)ν1u1=γI1+ν11+γ−11−I1+ν1+γ−1I1+ν11+γ−1Ι1+ν1−1−Ι1+ν11+γ−1Ι1+ν1+γ−11−Ι1+ν1,1+γ−1ƒ1+ν1−1−ƒ1+ν11+γ−1ƒ1+ν1+γ−11−ƒ1+ν1,−1+γ−1I1−ν1−1+I1−ν11+γ−1I1−ν1+γ−11+I1−ν1,−γΙ1−ν11+γ−11+Ι1−ν1+γ−1Ι1−ν1,−γƒ1−ν11+γ−11+ƒ1−ν1+γ−1ƒ1−ν1.
So, satisfied forn=1, put n=r for equation (34),
(35)BNHWGνu1,u2,⋯,ur=γ∏ℓ=1rIℓ+νℓ∏ℓ=1r1+γ−11−Iℓ+νℓ+γ−1∏ℓ=1rIℓ+νℓ,∏ℓ=1r1+γ−1Ιℓ+νℓ−∏ℓ=1r1−Ιℓ+νℓ∏ℓ=1r1+γ−1Ιℓ+νℓ+γ−1∏ℓ=1r1−Ιℓ+νℓ,∏ℓ=1r1+γ−1ƒℓ+νℓ−∏ℓ=1r1−ƒℓ+νℓ∏ℓ=1r1+γ−1ƒℓ+νℓ+γ−1∏ℓ=1r1−ƒℓ+νℓ,−∏ℓ=1r1−γ−1Iℓ−νℓ−∏ℓ=1r1−Iℓ−νℓ∏ℓ=1r1−γ−1Iℓ−νℓ+γ−1∏ℓ=1r1+Iℓ−νℓ,−γ∏ℓ=1rΙℓ−νℓ∏ℓ=1r1+γ−11+Ιℓ−νℓ+γ−1∏ℓ=1rΙℓ−νℓ,−γ∏ℓ=1rƒℓ−νℓ∏ℓ=1r1+γ−11+ƒℓ−νℓ+γ−1∏ℓ=1rƒℓ−νℓ.
If equation (34) is true n=r, then we show that (34) is true for n=r+1; thus,
(36)BNHWGνu1,u2,⋯,ur=γ∏ℓ=1rIℓ+νℓ∏ℓ=1r1+γ−11−Iℓ+νℓ+γ−1∏ℓ=1rIℓ+νℓ,∏ℓ=1r1+γ−1Ιℓ+νℓ−∏ℓ=1r1−Ιℓ+νℓ∏ℓ=1r1+γ−1Ιℓ+νℓ+γ−1∏ℓ=1r1−Ιℓ+νℓ,∏ℓ=1r1+γ−1ƒℓ+νℓ−∏ℓ=1r1−ƒℓ+νℓ∏ℓ=1r1+γ−1ƒℓ+νℓ+γ−1∏ℓ=1r1−ƒℓ+νℓ,−∏ℓ=1r1+γ−1Iℓ−νℓ−∏ℓ=1r1+Iℓ−νℓ∏ℓ=1r1+γ−1Iℓ−νℓ+γ−1∏ℓ=1r1+Iℓ−νℓ,−γ∏ℓ=1rΙℓ−νℓ∏ℓ=1r1+γ−11+Ιℓ−νℓ+γ−1∏ℓ=1rΙℓ−νℓ,−γ∏ℓ=1rƒℓ−νℓ∏ℓ=1r1+γ−11+ƒℓ−νℓ+γ−1∏ℓ=1rƒℓ−νℓ⊗γIr+1+νr+11+γ−11−Ir+1+νr+1+γ−1Ir+1+νr+1,1+γ−1Ιr+1+νr+1−1−Ιr+1+νr+11+γ−1Ιr+1+νr+1+γ−11−Ιr+1+νr+1,1+γ−1ƒr+1+νr+1−1−ƒr+1+νr+11+γ−1ƒr+1+νr+1+γ−11−ƒr+1+νr+1,−1+γ−1Ir+1−νr+1−1+Ir+1−νr+11+γ−1Ir+1−νr+1+γ−11+Ir+1−νr+1,−γΙr+1−νr+11+γ−11+Ιr+1−νr+1+γ−1Ιr+1−νr+1,−γƒr+1−νr+11+γ−11+ƒr+1−νr+1+γ−1ƒr+1−νr+1=γ∏ℓ=1r+1Iℓ+νℓ∏ℓ=1r+11+γ−11−Iℓ+νℓ+γ−1∏ℓ=1r+1Iℓ+νℓ,∏ℓ=1r+11+γ−1Ιℓ+νℓ−∏ℓ=1r+11−Ιℓ+νℓ∏ℓ=1r+11+γ−1Ιℓ+νℓ+γ−1∏ℓ=1r+11−Ιℓ+νℓ,∏ℓ=1r+11+γ−1ƒℓ+νℓ−∏ℓ=1r+11−ƒℓ+νℓ∏ℓ=1r+11+γ−1ƒℓ+νℓ+γ−1∏ℓ=1r+11−ƒℓ+νℓ,−∏ℓ=1r+11+γ−1Iℓ−νℓ−∏ℓ=1r+11+Iℓ−νℓ∏ℓ=1r+11+γ−1Iℓ−νℓ+γ−1∏ℓ=1r+11+Iℓ−νℓ,−γ∏ℓ=1r+1Ιℓ−νℓ∏ℓ=1r+11+γ−11+Ιℓ−νℓ+γ−1∏ℓ=1r+1Ιℓ−νℓ,−γ∏ℓ=1r+1ƒℓ−νℓ∏ℓ=1r+11+γ−11+ƒℓ−νℓ+γ−1∏ℓ=1r+1ƒℓ−νℓ.
Thus, equation (34) is true for n=r+1. Hence, equation (34) is true for all n.☐Theorem 18 (idempotency).
Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n be a set of equal bipolar neutrosophic values, that is, uℓ=u:
(37)BNHWGνu1,u2,⋯,un=u.Theorem 19 (bounded).
Supposeu−=minℓuℓ,u+=maxℓuℓ,
(38)u−≤BNHWGνu1,u2,⋯,un≤u+.Theorem 20 (monotonicity).
Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n and uℓ′=I′ℓ+,Ι′ℓ+,ƒ′ℓ+,I′ℓ−,Ι′ℓ−,ƒ′ℓ−ℓ=1,2,⋯,n be two collections of bipolar neutrosophic values. If uℓ≤uℓ′, then
(39)BNHWGνu1,u2,⋯,un≤BNHWGνu1′,u2′,u3′,⋯,un′.Now, we will look at two particular examples of the BNHWG operator.(i)
Ifγ=1, then the BNHWG operator is equivalent to the basic algebraic operator bipolar neutrosophic weighted geometric (BNWG) operator:(40)BNHWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=∏ℓ=1nIℓ+νℓ,1−∏ℓ=1n1−Ιℓ+νℓ,1−∏ℓ=1n1−ƒℓ+νℓ,−1−∏ℓ=1n1+Iℓ−νℓ,−∏ℓ=1nΙℓ−νℓ,−∏ℓ=1nƒℓ−νℓ(ii)
Ifγ=2, then the bipolar neutrosophic Einstein weighted geometric (BNEWG) operator is equivalent to the BNHWG operator:(41)BNHWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=2∏ℓ=1nIℓ+νℓ∏ℓ=1n2−Iℓ+νℓ+∏ℓ=1nIℓ+νℓ,∏ℓ=1n1+Ιℓ+νℓ−∏ℓ=1n1−Ιℓ+νℓ∏ℓ=1n1+Ιℓ+νℓ+∏ℓ=1n1−Ιℓ+νℓ,∏ℓ=1n1+ƒℓ+νℓ−∏ℓ=1n1−ƒℓ+νℓ∏ℓ=1n1+ƒℓ+νℓ+∏ℓ=1n1−ƒℓ+νℓ,−∏ℓ=1n1+Iℓ−νℓ−∏ℓ=1n1+Iℓ−νℓ∏ℓ=1n1+Iℓ−νℓ+∏ℓ=1n1+Iℓ−νℓ,−−2∏ℓ=1nΙℓ−νℓ∏ℓ=1n2+Ιℓ−νℓ+∏ℓ=1nΙℓ−νℓ,−2∏ℓ=1nƒℓ−νℓ∏ℓ=1n2+ƒℓ−νℓ+∏ℓ=1nƒℓ−νℓ


## 3.2. Bipolar Neutrosophic Hamacher Ordered Weighted Geometric AO

Definition 21.
The bipolar neutrosophic Hamacher ordered weighted geometric (BNHOWG) aggregation operator is defined as(42)BNHOWGνu1,u2,⋯,un=⊗ℓ=1nνℓuσℓ=ν1uσ1⊗ν2uσ2⊗ν3uσ3⊗⋯⊗νnuσn,where σ1,⋯,σn−1,σn represent permutation of 1,2,3,⋯,n−1,n with uσℓ−1≥uσℓℓ=2,⋯,n−1,n and ν=ν1,ν2,ν3,⋯,νnT represent associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=1,γ>0.Theorem 22.
The BNHOWG operators give in return a BNV with(43)BNHOWGνu1,u2,⋯,un=γЂ∂1~Ђ∂2~+γ−1Ђ∂1~,ζ∂1~−ζ∂2~ζ∂1~+γ−1ζ∂2~,f∂1~−f∂2~f∂1~+γ−1f∂2~,−Ђ∂3~−Ђ∂4~Ђ∂3~+γ−1Ђ∂4~,−γζ∂3~ζ∂4~+γ−1ζ∂3~,−γf∂3~f∂4~+γ−1f∂3~,where
(44)Ђ∂1~=∏ℓ=1nIσℓ+νℓ,Ђ∂2~=∏ℓ=1n1+γ−11−Iσℓ+νℓ,Ђ∂3~=∏ℓ=1n1+γ−1Iσℓ−νℓ,Ђ∂4~=∏ℓ=1n1+Iσℓ−νℓ,ζ∂1~=∏ℓ=1n1+γ−1Ισℓ+νℓ,ζ∂2~=∏ℓ=1n1−Ισℓ+νℓ,ζ∂3~=∏ℓ=1nΙσℓ−νℓ,ζ∂4~=∏ℓ=1n1+γ−11+Ισℓ−νℓ,f∂1~=∏ℓ=1n1+γ−1ƒσℓ+νℓ,f∂2~=∏ℓ=1n1−ƒσℓ+νℓ,f∂3~=∏ℓ=1nƒσℓ−νℓ,f∂4~=∏ℓ=1n1+γ−11+ƒσℓ−νℓ,where σ1,⋯,σn−1,σn represent permutation of 1,2,3,⋯,n−1,n with uσℓ−1≥uσℓℓ=2,⋯,n−1,n and ν=ν1,ν2,ν3,⋯,νnT represent associated weighting vector for uℓ, such that νℓ>0 and ∑ℓ=1nνℓ=1,γ>0.Proof.
The theorem is straightforward.☐Theorem 23 (idempotency).
Consideruℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n to represent a set of equal BNN that is uℓ=u:
(45)BNHOWGνu1,u2,⋯,un=u.Theorem 24 (bounded).
Supposeu−=minℓuℓ,u+=maxℓuℓ,
(46)u−≤BNHOWGνu1,u2,⋯,un≤u+.Theorem 25 (monotonicity).
Letuℓ=Iℓ+,Ιℓ+,ƒℓ+,Iℓ−,Ιℓ−,ƒℓ−ℓ=1,2,⋯,n and uℓ′=I′ℓ+,Ι′ℓ+,ƒ′ℓ+,I′ℓ−,Ι′ℓ−,ƒ′ℓ−ℓ=1,2,⋯,n be a set of two bipolar neutrosophic values. If uℓ≤uℓ′, then
(47)BNHOWGνu1,u2,⋯,un≤BNHOWGνu1′,u2′,⋯,un′.At the moment, we will look at two particular examples related to BNHOWG AO.(i)
The bipolar neutrosophic ordered weighted geometric (BNOWG) operator is equivalent to the BNHOWG operator ifγ=1:(48)BNOWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=∏ℓ=1nIσℓ+νℓ,1−∏ℓ=1n1−Ισℓ+νℓ,1−∏ℓ=1n1−ƒσℓ+νℓ,−1−∏ℓ=1n1+Iσℓ−νℓ,−∏ℓ=1nΙσℓ−νℓ,−∏ℓ=1nƒσℓ−νℓ(ii)
The bipolar neutrosophic Einstein ordered weighted geometric (BNEOWG) operator is equivalent to the BNHOWG operator, ifγ=2:(49)BNOWGνu1,u2,⋯,un=⊗ℓ=1nνℓuℓ=2∏ℓ=1nIσℓ+νℓ∏ℓ=1n2−Iσℓ+νℓ+∏ℓ=1nIσℓ+νℓ,∏ℓ=1n1+Ισℓ+νℓ−∏ℓ=1n1−Ισℓ+νℓ∏ℓ=1n1+Ισℓ+νℓ+∏ℓ=1n1−Ισℓ+νℓ,∏ℓ=1n1+ƒσℓ+νℓ−∏ℓ=1n1−ƒσℓ+νℓ∏ℓ=1n1+ƒσℓ+νℓ+∏ℓ=1n1−ƒσℓ+νℓ,−∏ℓ=1n1+Iσℓ−νℓ−∏ℓ=1n1+Iσℓ−νℓ∏ℓ=1n1+Iσℓ−νℓ+∏ℓ=1n1+Iσℓ−νℓ,−−2∏ℓ=1nΙσℓ−νℓ∏ℓ=1n2+Ισℓ−νℓ+∏ℓ=1nΙΙσℓ−νℓ,−2∏ℓ=1nƒσℓ−νℓ∏ℓ=1n2+ƒσℓ−νℓ+∏ℓ=1nƒσℓ−νℓ


## 3.3. Bipolar Neutrosophic Hamacher Hybrid Geometric AO

Definition 26.
The bipolar neutrosophic Hamacher hybrid weighted geometric aggregation (BNHHG) operator is defined as(50)BNHHGw,νu1,u2,⋯,un=⊗ℓ=1nνℓuσℓ=ν1u̇σ1⊗ν2u̇σ2⊗ν3u̇σ3⊗⋯⊗νnu̇σn,where w=w1,w2,w3,⋯,wn−1,wn represent associated weighting vector for uℓ along with wℓ∈0,1, ∑ℓ=1nwℓ=1,u̇σℓ represent the ℓth biggest element of the BN arguments u̇ℓu̇ℓ=nvℓuℓ, ν=ν1,ν2,ν3,⋯,νn−1,νn represent the weighting vector of bipolar neutrosophic argument uℓ with ωℓ∈0,1, ∑ℓ=1nωℓ=1. Here, n represent the balancing coefficient.Note that the BNHHG operator reduces to the BNHWG operator ifw=1/n,1/n,⋯,1/nT as well as the BNHOWG operator if v=1/n,1/n,⋯,1/n.Theorem 27.
The BNHHGoperators give in return a BNV with(51)BNHHWGw,vu1,u2,⋯,un=γЂ̇∂1~Ђ̇∂2~+γ−1Ђ̇∂1~,ζ̇∂1~−ζ̇∂2~ζ̇∂1~+γ−1ζ̇∂2~,ḟ∂1~−ḟ∂2~ḟ∂1~+γ−1ḟ∂2~,Ђ̇∂3~−Ђ̇∂4~Ђ̇∂3~+γ−1Ђ̇∂4~,γζ̇∂3~ζ̇∂4~+γ−1ζ̇∂3~,−γḟ∂3~ḟ∂4~+γ−1ḟ∂3~,where
(52)Ђ̇∂1~=∏ℓ=1nİσℓ+νℓ,Ђ̇∂2~=∏ℓ=1n1+γ−11−İσℓ+νℓ,Ђ̇∂3~=∏ℓ=1n1+γ−1İσℓ−νℓ,Ђ̇∂4~=∏ℓ=1n1+İσℓ−νℓ,ζ̇∂1~=∏ℓ=1n1+γ−1Ι̇σℓ+νℓ,ζ̇∂2~=∏ℓ=1n1−Ι̇σℓ+νℓ,ζ̇∂3~=∏ℓ=1nΙ̇σℓ−νℓ,ζ̇∂4~=∏ℓ=1n1+γ−11+Ι̇σℓ−νℓ,ḟ∂1~=∏ℓ=1n1+γ−1ƒ̇σℓ+νℓ,ḟ∂2~=∏ℓ=1n1−ƒ̇σℓ+νℓ,ḟ∂3~=∏ℓ=1nƒ̇σℓ−νℓ,ḟ∂4~=∏ℓ=1n1+γ−11+ƒ̇σℓ−νℓ,where w=w1,w2,w3,⋯,wn−1,wn represent the associated weighting vector for uℓ along with wℓ∈0,1, ∑ℓ=1nwℓ=1, u̇σℓ represent the ℓth biggest element of the BN arguments u̇ℓu̇ℓ=nvℓuℓ, ν=ν1,ν2,ν3,⋯,νn−1,νn represent the weighting vector of bipolar neutrosophic argument uℓ with ωℓ∈0,1, ∑ℓ=1nwℓ=1. Here, n represent the balancing coefficient, γ>0.Proof.
The proof is followed from Theorem17.☐Now, we will look at two particular cases of the BNHOWG operator.(i)
The bipolar neutrosophic hybrid geometric (BNHG) operator is equivalent to the BNHHG operator, ifγ=1:(53)BNHHGw,νu1,u2,⋯,un=⊗ℓ=1nνℓu̇ℓ=∏ℓ=1nİσℓ+νℓ,1−∏ℓ=1n1−Ι̇σℓ+νℓ,1−∏ℓ=1n1−ƒ̇σℓ+νℓ,−1−∏ℓ=1n1+İσℓ−νℓ,−∏ℓ=1nΙ̇σℓ−νℓ,−∏ℓ=1nƒ̇σℓ−νℓ(ii)
The bipolar neutrosophic Einstein hybrid geometric (BNHG) operator is equivalent to the BNHHG operator, ifγ=2:(54)BNEHGw,νu1,u2,⋯,un=⊗ℓ=1nνℓu̇ℓ=2∏ℓ=1nİσℓ+νℓ∏ℓ=1n2−İσℓ+νℓ+∏ℓ=1nİσℓ+νℓ,∏ℓ=1n1+Ι̇σℓ+νℓ−∏ℓ=1n1−Ι̇σℓ+νℓ∏ℓ=1n1+Ι̇σℓ+νℓ+∏ℓ=1n1−Ι̇σℓ+νℓ,∏ℓ=1n1+ƒ̇σℓ+νℓ−∏ℓ=1n1−ƒ̇σℓ+νℓ∏ℓ=1n1+ƒ̇σℓ+νℓ+∏ℓ=1n1−ƒ̇σℓ+νℓ,−∏ℓ=1n1+İσℓ−νℓ−∏ℓ=1n1+İσℓ−νℓ∏ℓ=1n1+İσℓ−νℓ+∏ℓ=1n1+İσℓ−νℓ,−−2∏ℓ=1nΙ̇σℓ−νℓ∏ℓ=1n2+Ι̇σℓ−νℓ+∏ℓ=1nΙ̇σℓ−νℓ,−2∏ℓ=1nƒ̇σℓ−νℓ∏ℓ=1n2+ƒ̇σℓ−νℓ+∏ℓ=1nƒ̇σℓ−νℓ


## 4. Multicriteria Decision-Making Based on Bipolar Neutrosophic Einstein AOs

This section comprises multicriteria application based on bipolar neutrosophic Hamacher aggregation operators with crisp numbers as attribute weights and BNNs as attribute values.


### 4.1. Algorithm

LetM=M1,M2,⋯,Mm be the set of m alternatives, L=L1,L2,⋯,Ln be the set or collection of n criterions, D=D1,D2,⋯,Dk−1,Dk represent finite k decision-makers. Suppose v=v1,v2,⋯,vnT is the weighted vector for decision-makers D¯ss=1,2,3,⋯,k−1,k, with vℓ∈0,1 and ∑ℓ=1nvℓ=1. Let w=w1,w2,⋯,wn−1,wnT represent the associated weighting vector of the attribute set L=L1,L2,⋯,Ln such that wℓ∈0,1 and ∑ℓ=1nwℓ=1. The decision-maker evaluates an alternative criterion; values are defined by BN values. Let ui⌢j⌢s=Ii⌢j⌢+,Ιi⌢j⌢+,ƒi⌢j⌢+,Ii⌢j⌢−,Ιi⌢j⌢−,ƒi⌢j⌢−m×n represent decision-making matrices supplied by the decision-maker and ui⌢j⌢s represent a BNN for Li⌢ criterion-associated alternatives. We have the condition Ii⌢j⌢+,Ιi⌢j⌢+,ƒi⌢j⌢+,Ii⌢j⌢−,Ιi⌢j⌢− as well as fi⌢j⌢−∈0,1 with condition 0≤Ii⌢j⌢+,Ιi⌢j⌢+,ƒi⌢j⌢+,Ii⌢j⌢−,Ιi⌢j⌢−,ƒi⌢j⌢−≤6 where i⌢=1,2,3,⋯,m as well as j⌢=1,2,3,⋯,n−1,n.Step 1.
Construction of matrices (Tables1–3) for decision D¯s=ui⌢j⌢sm×ns=1,2,3,⋯,k−1,k.Step 2.
Computing (Table4) BNHWGvri⌢1,ri⌢2,⋯,ri⌢n for every i⌢=1,2,⋯,m−1,m:
(55)ri⌢=Ii⌢+,Ιi⌢+,ƒi⌢+,Ii⌢−,Ιi⌢−,fi⌢−=BNHWGvri⌢1,ri⌢2,⋯,ri⌢n=⊗ℓ=1nνℓriℓ=γЂ1~Ђ2~+γ−1Ђ1~,ζ1~−ζ2~ζ1~+γ−1ζ2~,f1~−f2~f1~+γ−1f2~,−Ђ3~−Ђ4~Ђ3~+γ−1Ђ4~,−γζ3~ζ4~+γ−1ζ3~,−γf3~f4~+γ−1f3~.Step 3.
Compute the score valuesSri⌢i⌢=1,2,⋯,m−1,m.Step 4.
Ranking of various software systems forBNHWGνui⌢1,ui⌢2,⋯,ui⌢n as per their scores.Step 5.
Choose the most excellent possible alternative(s).Figure 1
Flow chart for MCDM.Table 1
BNN decision matrixD1.
L1L2L3L4M10.6,0.50,0.4,−0.5,−0.4,−0.30.3,0.6,0.1,−0.4,−0.5,−0.40.4,0.5,0.30,−0.1,−0.6,−0.70.3,0.7,0.2,−0.4,−0.5,−0.6M20.7,0.5,0.40,−0.1,−0.9,−0.30.2,0.4,0.9,−0.4,−0.8,−0.40.2,0.4,0.7,−0.5,−0.3,−0.90.2,0.8,0.5,−0.2,−0.3,−0.5M30.6,0.2,0.4,−0.9,−0.2,−0.30.9,0.4,0.3,−0.6,−0.3,−0.20.5,0.3,0.7,−0.40,−0.6,−0.90.4,0.6,0.5,−0.3,−0.9,−0.8M40.1,0.5,0.4,−0.70,−0.6,−0.30.80,0.5,0.3,−0.7,−0.1,−0.40.5,0.8,0.7,−0.6,−0.40,−0.60.4,0.6,0.5,−0.6,−0.4,−0.7Table 2
BNN decision matrixD2.
L1L2L3L4M10.1,0.6,0.3,−0.7,−0.9,−0.30.4,0.5,0.4,−0.3,−0.7,−0.90.5,0.60,0.2,−0.70,−0.2,−0.80.2,0.5,0.5,−0.8,−0.3,−0.9M20.50,0.30,0.7,−0.5,−0.2,−0.40.4,0.8,0.7,−0.3,−0.6,−0.50.50,0.30,0.8,−0.3,−0.4,−0.60.5,0.7,0.4,−0.6,−0.5,−0.4M30.5,0.3,0.60,−0.70,−0.5,−0.60.3,0.8,0.10,−0.7,−0.4,−0.70.4,0.6,0.20,−0.5,−0.7,−0.30.5,0.7,0.2,−0.4,−0.3,−0.8M40.4,0.6,0.3,−0.8,−0.4,−0.50.4,0.4,0.5,−0.2,−0.5,−0.60.6,0.4,0.9,−0.5,−0.40,−0.700.8,0.2,0.4,−0.7,−0.3,−0.5Table 3
BNN decision matrixD3.
L1L2L3L4M10.3,0.70,0.5,−0.8,−0.2,−0.40.5,0.3,0.4,−0.6,−0.1,−0.90.5,0.4,0.6,−0.7,−0.2,−0.50.4,0.6,0.5,−0.2,−0.8,−0.5M20.1,0.60,0.2,−0.6,−0.7,−0.40.3,0.9,0.60,−0.8,−0.3,−0.50.8,0.5,0.6,−0.2,−0.7,−0.10.5,0.8,0.4,−0.5,−0.3,−0.6M30.40,0.50,0.6,−0.1,−0.5,−0.30.5,0.3,0.4,−0.6,−0.7,−0.40.6,0.7,0.3,−0.6,−0.1,−0.70.3,0.7,0.5,−0.2,−0.7,−0.6M40.70,0.4,0.8,−0.3,−0.7,−0.60.7,0.4,0.5,−0.3,−0.5,−0.30.3,0.7,0.4,−0.8,−0.3,−0.50.3,0.5,0.9,−08,−0.3,−0.5Table 4
Collective BNN decision matrixR.
(a)L1L2M10.3062,0.6274,0.4331,−0.7102,−0.3488,−0.34720.4127,0.4403,0.3155,−0.4896,−0.2545,−0.7288M20.2653,0.5178,0.3816,−0.4515,−0.6161,−0.36760.2828,0.7943,0.7450,−0.6281,−0.4779,−0.4683M30.4748,0.3777,0.5457,−0.5819,−0.3868,−0.37920.5552,0.4634,0.3138,−0.6218,−0.4968,−0.3720M40.3738,0.4740,0.6281,−0.5613,−0.6027,−0.54860.6599,0.4312,0.4441,−0.4264,−0.3216,−0.3792(b)L3L4M10.4683,0.4740,0.4462,−0.5630,−0.2856,−0.61320.3214,0.6148,0.4181,−0.4204,−0.5867,−0.6015M20.5064,0.4331,0.6786,−0.3168,−0.4968,−0.30840.3868,0.7827,0.4312,−0.4415,−0.3339,−0.5260M30.5260,0.5818,0.4264,−0.5252,−0.2717,−0.65630.3640,0.6724,0.4462,−0.2719,−0.6563,−0.6967M40.4064,0.6900,0.6449,−0.6999,−0.3472,−0.56690.4071,0.4801,0.7555,−0.7309,−0.3276,−0.5559Now, we give a numerical example as follows.


### 4.2. Illustrative Example

We consider a Medicine Business Company which needs to employ a professional manager. The organization creates a working group of three decision-makers for this reason, with weighting vectorsv=0.3,0.2,0.5T. There are several considerations to consider when choosing the most knowledgeable manager, but in this case, the committee only considers the four criteria mentioned below, with weighting vector ω=0.2,0.4,0.1,0.3T. Following the first screening exam, four managers Mii=1,2,3,4 will proceed to the next round of the procedure. The committee must make a decision based on the four attributes listed below.
(1)
L1: salary(2)
L2: experience(3)
L3: working skill(4)
L4: dealing with publicStep 1.
Construct the decision matrices.Step 2.
ComputeBNHWGνu1,u2,u3,⋯,un for every i⌢=1,2,⋯,m−1,m:
(56)ri⌢=Ii⌢+,Ιi⌢+,ƒi⌢+,Ii⌢−,Ιi⌢−,fi⌢−=BNHWGvri⌢1,ri⌢2,⋯,ri⌢n=⊕ℓ=1nνℓriℓ=γЂ1~Ђ2~+γ−1Ђ1~,ζ1~−ζ2~ζ1~+γ−1ζ2~,f1~−f2~f1~+γ−1f2~,−Ђ3~−Ђ4~Ђ3~+γ−1Ђ4~,−γζ3~ζ4~+γ−1ζ3~,−γf3~f4~+γ−1f3~.Step 3.
Now, we calculate the score function, forγ=3:
(57)r1=0.3667,0.5380,0.3836,−0.5280,−0.3603,−0.5948,r2=0.3274,0.7190,0.5915,−0.5117,−0.4581,−0.4457,r3=0.4748,0.5286,0.4137,−0.5088,−0.4915,−0.4883,r4=0.4936,0.4839,0.6088,−0.5856,−0.3742,−0.4805,Sr1=0.4787,Sr2=0.4015,Sr3=0.5006,Sr4=0.4450.Step 4.
By calculating the scores forγ=3, we have
(58)M3<M1<M4<M2.Step 5.
Thus, the best option isM3.


## 4.1. Algorithm

LetM=M1,M2,⋯,Mm be the set of m alternatives, L=L1,L2,⋯,Ln be the set or collection of n criterions, D=D1,D2,⋯,Dk−1,Dk represent finite k decision-makers. Suppose v=v1,v2,⋯,vnT is the weighted vector for decision-makers D¯ss=1,2,3,⋯,k−1,k, with vℓ∈0,1 and ∑ℓ=1nvℓ=1. Let w=w1,w2,⋯,wn−1,wnT represent the associated weighting vector of the attribute set L=L1,L2,⋯,Ln such that wℓ∈0,1 and ∑ℓ=1nwℓ=1. The decision-maker evaluates an alternative criterion; values are defined by BN values. Let ui⌢j⌢s=Ii⌢j⌢+,Ιi⌢j⌢+,ƒi⌢j⌢+,Ii⌢j⌢−,Ιi⌢j⌢−,ƒi⌢j⌢−m×n represent decision-making matrices supplied by the decision-maker and ui⌢j⌢s represent a BNN for Li⌢ criterion-associated alternatives. We have the condition Ii⌢j⌢+,Ιi⌢j⌢+,ƒi⌢j⌢+,Ii⌢j⌢−,Ιi⌢j⌢− as well as fi⌢j⌢−∈0,1 with condition 0≤Ii⌢j⌢+,Ιi⌢j⌢+,ƒi⌢j⌢+,Ii⌢j⌢−,Ιi⌢j⌢−,ƒi⌢j⌢−≤6 where i⌢=1,2,3,⋯,m as well as j⌢=1,2,3,⋯,n−1,n.Step 1.
Construction of matrices (Tables1–3) for decision D¯s=ui⌢j⌢sm×ns=1,2,3,⋯,k−1,k.Step 2.
Computing (Table4) BNHWGvri⌢1,ri⌢2,⋯,ri⌢n for every i⌢=1,2,⋯,m−1,m:
(55)ri⌢=Ii⌢+,Ιi⌢+,ƒi⌢+,Ii⌢−,Ιi⌢−,fi⌢−=BNHWGvri⌢1,ri⌢2,⋯,ri⌢n=⊗ℓ=1nνℓriℓ=γЂ1~Ђ2~+γ−1Ђ1~,ζ1~−ζ2~ζ1~+γ−1ζ2~,f1~−f2~f1~+γ−1f2~,−Ђ3~−Ђ4~Ђ3~+γ−1Ђ4~,−γζ3~ζ4~+γ−1ζ3~,−γf3~f4~+γ−1f3~.Step 3.
Compute the score valuesSri⌢i⌢=1,2,⋯,m−1,m.Step 4.
Ranking of various software systems forBNHWGνui⌢1,ui⌢2,⋯,ui⌢n as per their scores.Step 5.
Choose the most excellent possible alternative(s).Figure 1
Flow chart for MCDM.Table 1
BNN decision matrixD1.
L1L2L3L4M10.6,0.50,0.4,−0.5,−0.4,−0.30.3,0.6,0.1,−0.4,−0.5,−0.40.4,0.5,0.30,−0.1,−0.6,−0.70.3,0.7,0.2,−0.4,−0.5,−0.6M20.7,0.5,0.40,−0.1,−0.9,−0.30.2,0.4,0.9,−0.4,−0.8,−0.40.2,0.4,0.7,−0.5,−0.3,−0.90.2,0.8,0.5,−0.2,−0.3,−0.5M30.6,0.2,0.4,−0.9,−0.2,−0.30.9,0.4,0.3,−0.6,−0.3,−0.20.5,0.3,0.7,−0.40,−0.6,−0.90.4,0.6,0.5,−0.3,−0.9,−0.8M40.1,0.5,0.4,−0.70,−0.6,−0.30.80,0.5,0.3,−0.7,−0.1,−0.40.5,0.8,0.7,−0.6,−0.40,−0.60.4,0.6,0.5,−0.6,−0.4,−0.7Table 2
BNN decision matrixD2.
L1L2L3L4M10.1,0.6,0.3,−0.7,−0.9,−0.30.4,0.5,0.4,−0.3,−0.7,−0.90.5,0.60,0.2,−0.70,−0.2,−0.80.2,0.5,0.5,−0.8,−0.3,−0.9M20.50,0.30,0.7,−0.5,−0.2,−0.40.4,0.8,0.7,−0.3,−0.6,−0.50.50,0.30,0.8,−0.3,−0.4,−0.60.5,0.7,0.4,−0.6,−0.5,−0.4M30.5,0.3,0.60,−0.70,−0.5,−0.60.3,0.8,0.10,−0.7,−0.4,−0.70.4,0.6,0.20,−0.5,−0.7,−0.30.5,0.7,0.2,−0.4,−0.3,−0.8M40.4,0.6,0.3,−0.8,−0.4,−0.50.4,0.4,0.5,−0.2,−0.5,−0.60.6,0.4,0.9,−0.5,−0.40,−0.700.8,0.2,0.4,−0.7,−0.3,−0.5Table 3
BNN decision matrixD3.
L1L2L3L4M10.3,0.70,0.5,−0.8,−0.2,−0.40.5,0.3,0.4,−0.6,−0.1,−0.90.5,0.4,0.6,−0.7,−0.2,−0.50.4,0.6,0.5,−0.2,−0.8,−0.5M20.1,0.60,0.2,−0.6,−0.7,−0.40.3,0.9,0.60,−0.8,−0.3,−0.50.8,0.5,0.6,−0.2,−0.7,−0.10.5,0.8,0.4,−0.5,−0.3,−0.6M30.40,0.50,0.6,−0.1,−0.5,−0.30.5,0.3,0.4,−0.6,−0.7,−0.40.6,0.7,0.3,−0.6,−0.1,−0.70.3,0.7,0.5,−0.2,−0.7,−0.6M40.70,0.4,0.8,−0.3,−0.7,−0.60.7,0.4,0.5,−0.3,−0.5,−0.30.3,0.7,0.4,−0.8,−0.3,−0.50.3,0.5,0.9,−08,−0.3,−0.5Table 4
Collective BNN decision matrixR.
(a)L1L2M10.3062,0.6274,0.4331,−0.7102,−0.3488,−0.34720.4127,0.4403,0.3155,−0.4896,−0.2545,−0.7288M20.2653,0.5178,0.3816,−0.4515,−0.6161,−0.36760.2828,0.7943,0.7450,−0.6281,−0.4779,−0.4683M30.4748,0.3777,0.5457,−0.5819,−0.3868,−0.37920.5552,0.4634,0.3138,−0.6218,−0.4968,−0.3720M40.3738,0.4740,0.6281,−0.5613,−0.6027,−0.54860.6599,0.4312,0.4441,−0.4264,−0.3216,−0.3792(b)L3L4M10.4683,0.4740,0.4462,−0.5630,−0.2856,−0.61320.3214,0.6148,0.4181,−0.4204,−0.5867,−0.6015M20.5064,0.4331,0.6786,−0.3168,−0.4968,−0.30840.3868,0.7827,0.4312,−0.4415,−0.3339,−0.5260M30.5260,0.5818,0.4264,−0.5252,−0.2717,−0.65630.3640,0.6724,0.4462,−0.2719,−0.6563,−0.6967M40.4064,0.6900,0.6449,−0.6999,−0.3472,−0.56690.4071,0.4801,0.7555,−0.7309,−0.3276,−0.5559Now, we give a numerical example as follows.


## 4.2. Illustrative Example

We consider a Medicine Business Company which needs to employ a professional manager. The organization creates a working group of three decision-makers for this reason, with weighting vectorsv=0.3,0.2,0.5T. There are several considerations to consider when choosing the most knowledgeable manager, but in this case, the committee only considers the four criteria mentioned below, with weighting vector ω=0.2,0.4,0.1,0.3T. Following the first screening exam, four managers Mii=1,2,3,4 will proceed to the next round of the procedure. The committee must make a decision based on the four attributes listed below.
(1)
L1: salary(2)
L2: experience(3)
L3: working skill(4)
L4: dealing with publicStep 1.
Construct the decision matrices.Step 2.
ComputeBNHWGνu1,u2,u3,⋯,un for every i⌢=1,2,⋯,m−1,m:
(56)ri⌢=Ii⌢+,Ιi⌢+,ƒi⌢+,Ii⌢−,Ιi⌢−,fi⌢−=BNHWGvri⌢1,ri⌢2,⋯,ri⌢n=⊕ℓ=1nνℓriℓ=γЂ1~Ђ2~+γ−1Ђ1~,ζ1~−ζ2~ζ1~+γ−1ζ2~,f1~−f2~f1~+γ−1f2~,−Ђ3~−Ђ4~Ђ3~+γ−1Ђ4~,−γζ3~ζ4~+γ−1ζ3~,−γf3~f4~+γ−1f3~.Step 3.
Now, we calculate the score function, forγ=3:
(57)r1=0.3667,0.5380,0.3836,−0.5280,−0.3603,−0.5948,r2=0.3274,0.7190,0.5915,−0.5117,−0.4581,−0.4457,r3=0.4748,0.5286,0.4137,−0.5088,−0.4915,−0.4883,r4=0.4936,0.4839,0.6088,−0.5856,−0.3742,−0.4805,Sr1=0.4787,Sr2=0.4015,Sr3=0.5006,Sr4=0.4450.Step 4.
By calculating the scores forγ=3, we have
(58)M3<M1<M4<M2.Step 5.
Thus, the best option isM3.


## 5. Comparison

So far, the researchers have used a variety of decision-making techniques. These tools are the following: Chen [18] utilize FSs, and afterward, Atanassov [2] utilize IFSs; BFSs is used by Dubois et al. [5], and Zavadskas et al. [23] made use of NSs; Deli et al. [27] made use of bipolar neutrosophic soft sets, while Deli et al. [26] made use of BNSs as well as a variety of other research decisions. In this paper, we use Hamacher operations to apply bipolarity to neutrosophic sets.The advantage of our proposed methods is that this paper’s aggregation operators are more general and versatile.M3 is, however, the most attractive investment firm.


## 6. Conclusion

The aim of this paper is to investigate various BNS AOs and apply Hamachert-norms/t-conorms to multicriteria community decision-making, with BNS values as the criteria. We proposed bipolar neutrosophic Hamacher aggregation operators motivated by Hamacher operations. To begin with, we discussed BN Hamacher aggregation operators and their required properties. These AOs are BNHWA, BNHOWA, and BNHHWG along with their cases. Finally, we presented a framework for making multicriteria decision-making. An illustrative example applied to our proposed AO is selecting the best manager for Medicine Company. When we take γ=1, our proposed AO gives results similar to that of Deli et al. [27], but if γ=2, results vary (Table 5). The findings in the current manuscript demonstrate that our intended approaches are more accurate as well as realistic in practice. We plan to expand the anticipated model extended to other domains and their application in our future research, like pattern recognition and risk analysis.Table 5
Ranking for all alternatives.
γOperatorsRanking1BNHWG aggregation operator [27]M3<M1<M4<M22BNHWG aggregation operatorM3<M1<M2<M43BNHWG aggregation operatorM3<M1<M4<M2



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*Source: 5052867-2022-01-27.xml*