# Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

_{c}(x), V

_{c}(x), F

_{c}(x) >, < t

_{c}(x), v

_{c}(x), f

_{c}(x)> | x ∈ 𝓩},

_{c}(x), V

_{c}(x), F

_{c}(x) > is an INS [4] in 𝓩, and the intervals ${T}_{c}(x)=[{T}_{c}{}^{L}(x),\text{}{T}_{c}{}^{U}(x)]\subseteq [0,1],\text{}{V}_{c}(x)=[{V}_{c}{}^{\mathrm{L}}(x),\text{}{V}_{c}{}^{\mathrm{U}}(x)]\subseteq [0,1]$, and ${F}_{c}(x)=[{F}_{c}{}^{\mathrm{L}}(x),\text{}{F}_{c}{}^{\mathrm{U}}(x)]\subseteq [0,1]$ for x ∈ 𝓩 represent respectively the truth, indeterminacy, and falsity membership functions; then < t

_{c}(x), v

_{c}(x), f

_{c}(x) > is an SVNS [3,5] in 𝓩, and t

_{c}(x), v

_{c}(x), f

_{c}(x) ∈ [0,1] for x ∈ 𝓩 represent the truth, indeterminacy, and falsity membership functions, respectively.

_{c}(x), V

_{c}(x), F

_{c}(x) >, < t

_{c}(x), v

_{c}(x), f

_{c}(x) >) in an NCS G a neutrosophic cubic number (NCN) [20]; for convenience, we denoted it as $g=(<[{T}^{\mathrm{L}},\text{}{T}^{\mathrm{U}}],\text{}[{V}^{\mathrm{L}},\text{}{V}^{\mathrm{U}}],\text{}[{F}^{\mathrm{L}},\text{}{F}^{\mathrm{U}}],t,\text{}v,\text{}f)$, where t, v, f ∈ [0,1] and $[{T}^{\mathrm{L}},\text{}{T}^{\mathrm{U}}],\text{}[{V}^{\mathrm{L}},\text{}{V}^{\mathrm{U}}],\text{}[{F}^{\mathrm{L}},\text{}{F}^{\mathrm{U}}]\subseteq [0,1]$ satisfy the condition $0\le {T}^{\mathrm{U}}+\text{}{V}^{\mathrm{U}}+\text{}{F}^{\mathrm{U}}\le 3$ and 0 ≤ t + v + f ≤ 3.

_{c}(x), V

_{c}(x), F

_{c}(x) >, < t

_{c}(x), v

_{c}(x), f

_{c}(x) > | x ∈ 𝓩} is called an internal NCS if ${T}_{c}{}^{L}(x)$ ≤ t

_{c}(x) ≤ ${T}_{c}{}^{U}(x)$, ${V}_{c}{}^{L}(x)$ ≤ v

_{c}(x) ≤ ${V}_{c}{}^{U}(x)$, and ${F}_{c}{}^{L}(x)$ ≤ f

_{c}(x) ≤ ${F}_{c}{}^{U}(x)$ for x ∈ 𝓩; and an NCS G is called an external NCS if t

_{c}(x) ∉$({T}_{c}{}^{L}(x),\text{}{T}_{c}{}^{U}(x))$, v

_{c}(x) ∉ $({V}_{c}{}^{L}(x),\text{}{V}_{c}{}^{U}(x))$, and f

_{c}(x) ∉$({F}_{c}{}^{L}(x),\text{}{F}_{c}{}^{U}(x))$ for x ∈ 𝓩 [18,19].

- (1)
- ${({g}_{1})}^{C}=\text{}([{F}_{1}{}^{\mathrm{L}},\text{}{F}_{1}{}^{\mathrm{U}}],\text{}[1-{V}_{1}{}^{\mathrm{U}},\text{}1-{V}_{1}{}^{\mathrm{L}}],\text{}[{T}_{1}{}^{\mathrm{L}},\text{}{T}_{1}{}^{\mathrm{U}}],{f}_{1},\text{}1-{v}_{1},\text{}{t}_{1})$ (complement of g
_{1}); - (2)
- $\begin{array}{ll}{g}_{1}\oplus {g}_{2}=& (<[{T}_{1}{}^{\mathrm{L}}+{T}_{2}{}^{\mathrm{L}}-{T}_{1}{}^{\mathrm{L}}{T}_{2}{}^{\mathrm{L}},\text{}{T}_{1}{}^{\mathrm{U}}+\text{}{T}_{2}{}^{\mathrm{U}}-{T}_{1}{}^{\mathrm{U}}{T}_{2}{}^{\mathrm{U}}\text{}],\text{}[{V}_{1}{}^{\mathrm{L}}{V}_{2}{}^{\mathrm{L}},\text{}{V}_{1}{}^{\mathrm{U}}{V}_{2}{}^{\mathrm{U}}],\text{}[{F}_{1}{}^{\mathrm{L}}{F}_{1}{}^{\mathrm{L}},\text{}{F}_{1}{}^{\mathrm{U}}{F}_{2}{}^{\mathrm{U}}],\\ & {t}_{1}+{t}_{2}-{t}_{1}{t}_{2},{v}_{1}{v}_{2},\text{}{f}_{1}{f}_{2});\end{array}$
- (3)
- $\begin{array}{ll}{g}_{1}\otimes {g}_{2}=& (<[{T}_{1}{}^{\mathrm{L}}{T}_{2}{}^{\mathrm{L}},\text{}{T}_{1}{}^{U}{T}_{2}{}^{\mathrm{U}}\text{}],\text{}[{V}_{1}{}^{\mathrm{L}}+{V}_{2}{}^{\mathrm{L}}-{V}_{1}{}^{\mathrm{L}}{V}_{2}{}^{\mathrm{L}},\text{}{V}_{1}{}^{\mathrm{U}}+{V}_{2}{}^{\mathrm{U}}-{V}_{1}{}^{\mathrm{U}}{V}_{2}{}^{\mathrm{U}}],\text{}[{F}_{1}{}^{\mathrm{L}}+{F}_{1}{}^{\mathrm{L}}-{F}_{1}{}^{\mathrm{L}}{F}_{1}{}^{\mathrm{L}},\text{}{F}_{1}{}^{\mathrm{U}}\\ & +{F}_{2}{}^{\mathrm{U}}-{F}_{1}{}^{\mathrm{U}}{F}_{2}{}^{\mathrm{U}}],{t}_{1}{t}_{2},{v}_{1}+{v}_{2}-{v}_{1}{v}_{2},\text{}{f}_{1}+{f}_{2}-{f}_{1}{f}_{2});\end{array}$
- (4)
- $\begin{array}{ll}\lambda {g}_{1}=& (<[1-(1-{T}_{1}{}^{\mathrm{L}}{)}^{\lambda},\text{}1-(1-{T}_{1}{}^{\mathrm{U}}{)}^{\lambda}\text{}],\text{}[({V}_{1}{}^{\mathrm{L}}{)}^{\lambda},\text{}{({V}_{1}{}^{\mathrm{U}})}^{\lambda}],\text{}[({F}_{1}{}^{\mathrm{L}}{)}^{\lambda},{({F}_{1}{}^{\mathrm{U}})}^{\lambda}],1-(1-{t}_{1}{)}^{\lambda},\text{}({v}_{1}{)}^{\lambda},\text{}\\ & ({f}_{1}){\text{}}^{\lambda})\text{}\mathrm{for}\text{}\lambda 0;\end{array}$
- (5)
- $\begin{array}{l}{({g}_{1})}^{\lambda}=(<[{({{T}_{1}}^{\mathrm{L}})}^{\lambda},{({{T}_{1}}^{\mathrm{U}})}^{\lambda}\text{}],\text{}[\text{}1-{(1-{{V}_{1}}^{\mathrm{L}})}^{\lambda},1-{(1-{{V}_{1}}^{\mathrm{U}})}^{\lambda}],\text{}1-{(1-{{F}_{1}}^{\mathrm{L}})}^{\lambda},1-{(1-{F}_{1}{}^{\mathrm{U}})}^{\lambda}],{({t}_{1})}^{\lambda},\text{}\\ 1-{(1-{v}_{1})}^{\lambda},\text{}1-{(1-{f}_{1})}^{\lambda})\text{}\mathrm{for}\text{}\lambda 0;\end{array}$

**Definition**

**2.**

- (i)
- If Ψ(g
_{1}) > Ψ(g_{2}), then g_{1}≻ g_{2}; - (ii)
- If Ψ(g
_{1}) = Ψ(g_{2}) and Γ(g_{1}) > Γ(g_{2}), then g_{1}≻ g_{2}; - (iii)
- If Ψ(g
_{1}) = Ψ(g_{2}) and Γ(g_{1}) = Γ(g_{2}), then g_{1}~ g_{2}.

_{1}, g

_{2}, …, g

_{n})

_{1}, g

_{2}, …, g

_{n})

_{i}$\in $ (i = 1, 2, …, n), satisfying ${\sum}_{i=1}^{n}{\xi}_{i}=1$.

**Case**

**1.**

_{1}= (< [0.001, 0.002], [0, 0], [0, 0] >, <0.001, 0, 0 >) and g

_{2}= (< [0, 1], [0, 0], [0, 0] >, <1, 0, 0 >) be two NCNs, with their weights ζ

_{1}= 0.9 and ζ

_{2}= 0.1, respectively.

_{1}, g

_{2}) = (< [0.001, 1], [0, 0], [0, 0] >, <1, 0, 0 >) and NCWGA (g

_{1}, g

_{2}) = (< [0, 0.004], [0, 0], [0, 0] >, <0.002, 0, 0 >).

**Case**

**2.**

_{1}= (< [0.001, 0.002], [0, 0], [0, 0] >, <0.001, 0, 0 >) and g

_{2}= (< [0, 1], [0, 0], [0, 0] >, <1, 0, 0 >) with their weights ζ

_{1}= 0.1 and ζ

_{2}= 0.9, respectively.

_{1}, g

_{2}) = (< [0, 1], [0, 0], [0, 0] >, <1, 0, 0 >) and NCWGA (g

_{1}, g

_{2}) = (< [0, 0.537], [0, 0], [0, 0] >, <0.501, 0, 0 >).

_{1}, g

_{2}) operator tend to the maximum value, while the aggregated results of NCWGA (g

_{1}, g

_{2}) operator tend to the maximum weight value. It is obvious that the NCWAA and NCWGA operators may cause unreasonable results of NCNs in some cases. In order to overcome the drawbacks, it is necessary to improve the NCWAA and NCWGA operators provided in [31]. Hence, in the next section, a new NCHWAGA is proposed by extending the hybrid arithmetic and geometric aggregation operators presented in [34,35].

## 3. Hybrid Arithmetic and Geometric Aggregation Operators of NCNs

#### 3.1. NCHWAGA Operator

**Definition**

**3.**

_{i}(i = 1, 2, …, n) is the weight of g

_{i}(i = 1, 2, …, n), satisfying ζ

_{i}$\in $ [0, 1] and ${\sum}_{i=1}^{n}{\xi}_{i}=1$.

**Theorem**

**1.**

_{i}(i = 1, 2, …, n) be the corresponding weight of g

_{i}(i = 1, 2, …, n), satisfying ζ

_{i}$\in $ [0, 1] and ${\sum}_{i=1}^{n}{\xi}_{i}=1$. Then, the aggregated value of the NCHWAGA operator is also an NCN, which can be calculated by:

**Proof.**

- (i)
- Idempotency: If g
_{i}= g for i = 1, 2, …, n, then NCHWAGA (g_{1}, g_{2}, …, g_{n}) = g. - (ii)
- Boundedness: If g
_{min}= min (g_{1}, g_{2}, …, g_{n}) and g_{max}= max (g_{1}, g_{2}, …, g_{n}) for i = 1, 2, …, n, then g_{min}≤ NCHWAGA (g_{1}, g_{2}, …, g_{n}) ≤ g_{max}. - (iii)
- Monotonicity: If g
_{i}≤ g_{i}^{*}for i = 1, 2, …, n, then NCHWAGA (g_{1}, g_{2}, …, g_{n}) ≤ NCHWAGA (g_{1}^{*}, g_{2}^{*}, …, g_{n}^{*}).

#### 3.2. Numerical Example

**For Case 1:**Let two NCNs g

_{1}= (< [0.001, 0.002], [0, 0], [0, 0] >, <0.001, 0, 0 >) and g

_{2}= (< [0, 1], [0, 0], [0, 0] >, <1, 0, 0 >) with their weights ζ

_{1}= 0.9 and ζ

_{2}= 0.1, by Equation (6), we obtain NCHWAGA (g

_{1}, g

_{2}) = (< [0, 0.061], [0, 0], [0, 0] >, <0.045, 0, 0 >), which is between NCWAA (g

_{1}, g

_{2}) = (< [0.001, 1], [0, 0], [0, 0] >, <1, 0, 0 >) and NCWGA (g

_{1}, g

_{2}) = (< [0, 0.004], [0, 0], [0, 0] >, <0.002, 0, 0 >).

**For Case 2:**Also take two NCNs g

_{1}= (< [0.001, 0.002], [0, 0], [0, 0] >, <0.001, 0, 0 >) and g

_{2}= (< [0, 1], [0, 0], [0, 0] >, <1, 0, 0 >) with their weights ζ

_{1}= 0.1 and ζ

_{2}= 0.9, then, by Equation (6), we get NCHWAGA (g

_{1}, g

_{2}) = (< [0, 0.733], [0, 0], [0, 0] >, <0.708, 0, 0 >), which is between NCWAA (g

_{1}, g

_{2}) = (< [0, 1], [0, 0], [0, 0] >, <1, 0, 0 >) and NCWGA (g

_{1}, g

_{2}) = (< [0, 0.537], [0, 0], [0, 0] >, <0.501, 0, 0 >).

## 4. MADM Method Using the NCHWAGA Operator

_{1}, G

_{2}, …, G

_{k}} is a set of k alternatives and P = {P

_{1}, P

_{2}, …, P

_{n}} is a set of attributes. Suppose that the weight vector of P is ${\omega}_{P}=\text{{}{\omega}_{{P}_{1}},\text{}{\omega}_{{P}_{2}},\dots ,\text{}{\omega}_{{P}_{n}}\text{}}$ with ${\omega}_{{P}_{j}}\in [0,1]$ and ${\sum}_{j=1}^{n}{\omega}_{{P}_{j}}=1$. The evaluation value of an alternative G

_{i}under an attribute P

_{j}can be expressed using an NCN ${g}_{ij}=(<[{T}_{ij}^{L},\text{}{T}_{Ij}^{U}],\text{}[{V}_{ij}^{L},\text{}{V}_{ij}^{U}],\text{}[{F}_{ij}^{L},\text{}{F}_{ij}^{U}],{t}_{ij},{v}_{ij},\text{}{f}_{ij})$ (i = 1, 2, …, k; j = 1, 2, …, n), where $[{T}_{ij}^{L},\text{}{T}_{ij}^{U}],\text{}[{V}_{ij}^{L},\text{}{V}_{ij}^{U}],\text{}[{F}_{ij}^{L},\text{}{F}_{ij}^{U}]\text{}\subseteq \text{}[0,1],\text{}\mathrm{and}\text{}{t}_{ij},\text{}{v}_{ij},\text{}{f}_{ij}\in [0,1]$. Then, we can construct a decision matrix G = (g

_{ij})

_{k}

_{×n}with the NCN information, and provide the following MADM procedures based on the proposed NCHWAGA operator:

**Step 1.**Calculate the aggregated value of g_{i}for each alternative G_{i}(i = 1, 2, …, k) using the NCHWAGA operator:$$\begin{array}{l}{g}_{i}=NCHWAGA({g}_{i1},{g}_{i2},\dots ,{g}_{in})={\left({\displaystyle \sum _{j=1}^{n}{\omega}_{{p}_{j}}}{g}_{ij}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{g}_{ij}^{{\omega}_{p}{}_{{}_{j}}}}\right)}^{(1-\rho )}\\ =\left(\begin{array}{l}\langle \begin{array}{l}\text{}\left[{\left(1-{\displaystyle \prod _{j=1}^{n}{(1-{T}_{ij}^{\mathrm{L}})}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{\left({T}_{ij}^{\mathrm{L}}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )},\text{}{\left(1-{\displaystyle \prod _{j=1}^{n}{(1-{T}_{i}^{\mathrm{U}})}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{\left({T}_{ij}^{\mathrm{U}}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )}\right]\text{},\\ \left[1-{\left(1-{\displaystyle \prod _{j=1}^{n}{\left({V}_{ij}^{\mathrm{L}}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{(1-{V}_{ij}^{\mathrm{L}})}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )},\text{}1-{\left(1-{\displaystyle \prod _{j=1}^{n}{\left({V}_{ij}^{\mathrm{U}}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{(1-{V}_{ij}^{\mathrm{U}})}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )}\right]\text{},\\ \text{}\left[1-{\left(1-{\displaystyle \prod _{j=1}^{n}{\left({F}_{ij}^{\mathrm{L}}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{(1-{F}_{ij}^{\mathrm{L}})}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )},\text{}1-{\left(1-{\displaystyle \prod _{j=1}^{n}{\left({F}_{j}^{\mathrm{U}}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{(1-{F}_{ij}^{\mathrm{U}})}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )}\right]\end{array}\rangle \\ \langle {\left(1-{\displaystyle \prod _{j=1}^{n}{(1-{t}_{ij})}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{\left({t}_{ij}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )},\text{}1-{\left(1-{\displaystyle \prod _{j=1}^{n}{\left({v}_{ij}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{(1-{v}_{ij})}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )},\text{}1-{\left(1-{\displaystyle \prod _{j=1}^{n}{\left({f}_{ij}\right)}^{{\omega}_{{p}_{j}}}}\right)}^{\rho}{\left({\displaystyle \prod _{j=1}^{n}{(1-{f}_{ij})}^{{\omega}_{{p}_{j}}}}\right)}^{(1-\rho )}\rangle \end{array}\right)\end{array}$$**Step 2.**Obtain the score values of Ψ(x) (the accuracy degrees of Γ(x) if necessary) of the collective NCN g_{i}(i = 1, 2, …, k) by Equations (1) and (2).**Step 3.**Rank all the alternatives corresponding to the values of Ψ(x) and Γ(x), and select the best one(s) based on the largest value.**Step 4.**End.

## 5. Illustrative Example and Comparison Analysis

_{i}(i = 1, 2, 3, 4). G

_{1}, G

_{2}, G

_{3}and G

_{4}represent a textile company, an automobile company, a computer company, and a software company, respectively. The four alternatives need to be evaluated according to the three attributes P

_{j}(j = 1, 2, 3). P

_{1}, P

_{2}and P

_{3}represent respectively the risk, the growth, and the environmental impact. Corresponding to the three attributes, the weight vector is ${\omega}_{P}=(0.32,\text{}0.38,\text{}0.3)$. When the decision maker evaluates the four alternatives G

_{i}(i = 1, 2, 3, 4) based on the three attributes P

_{j}(j = 1, 2, 3) with the NCN information, the decision matrix can be established as shown in Table 1.

**Step 1.**By Equation (7) for ρ = 0.5, we calculate the aggregated value of the collective NCN g_{i}for the each alternative G_{i}(i = 1, 2, 3, 4) as follows:g _{1}= (< [0.5302, 0.6645], [0.1272, 0.3000], [0.1669, 0.3355] >, <0.3430, 0.4709, 0.2306>)g _{2}= (< [0.6000, 0.7335], [0.1523, 0.2563], [0.1669, 0.2685] >, <0.6628, 0.2525, 0.2346>)g _{3}= (< [0.4677, 0.6307], [0.2000, 0.3000], [0.2264, 0.3672] >, <0.6000, 0.2365, 0.3025>)g _{4}= (< [0.6328, 0.7335], [0.1523, 0.2563], [0.1272, 0.2665] >, <0.7335, 0.1523, 0.2000>)**Step 2.**By Equation (1), we calculate the score values of Ψ(g_{i}) for the alternatives G_{i}(i = 1, 2, 3, 4) as the follows:Ψ(g_{1}) = 0.6563, Ψ(g_{2}) = 0.7405, Ψ(g_{3}) = 0.6740, Ψ(g_{4}) = 0.7717.**Step 3.**According to Ψ(g_{4}) > Ψ(g_{2}) > Ψ(g_{3}) > Ψ(g_{1}), the ranking of the alternatives is G_{4}≻ G_{2}≻ G_{3}≻ G_{1}. So, the alternative G_{4}is the best one.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GRA | Grey relational analysis |

INSs | Interval neutrosophic sets |

MADM | Multi-attribute decision-making |

MCGDM | Multi-criteria group decision making |

MVNSs | Multi-valued neutrosophic sets |

NCHWAGA | Neutrosophic cubic hybrid weighted arithmetic and geometric aggregation |

NCSs | Neutrosophic cubic sets |

NCWAA | Neutrosophic cubic weighted arithmetic average |

NCWGA | Neutrosophic cubic geometric weighted average |

SNSs | Simplified neutrosophic sets |

SVNSs | Single-valued neutrosophic sets |

WAA | Weighted arithmetic average |

WGA | Geometric average |

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Alternative | Attribute (P_{1}) | Attribute (P_{2}) | Attribute (P_{3}) |
---|---|---|---|

G_{1} | (< [0.5, 0.6], [0.1, 0.3], [0.2, 0.4]>, <0.2, 0.6, 0.3>) | (< [0.5, 0.6], [0.1, 0.3], [0.2, 0.4]>, <0.2, 0.6, 0.3>) | (< [0.6, 0.8], [0.2, 0.3], [0.1, 0.2]>, <0.7, 0.2, 0.1>) |

G_{2} | (< [0.6, 0.8], [0.1, 0.2], [0.2, 0.3]>, <0.7, 0.1, 0.2>) | (< [0.6, 0.7], [0.1, 0.2], [0.2, 0.3]>, <0.6, 0.3, 0.4>) | (< [0.6, 0.7], [0.3, 0.4], [0.1, 0.2]>, <0.7, 0.4, 0.2>) |

G_{3} | (< [0.4, 0.6], [0.2, 0.3], [0.1, 0.3]>, <0.6, 0.2, 0.2>) | (< [0.5, 0.6], [0.2, 0.3], [0.3, 0.4]>, <0.6, 0.3, 0.4>) | (< [0.5, 0.7], [0.2, 0.3], [0.3, 0.4]>, <0.6, 0.2, 0.3>) |

G_{4} | (< [0.7, 0.8], [0.1, 0.2], [0.1, 0.2]>, <0.8, 0.1, 0.2>) | (< [0.6, 0.7], [0.1, 0.2], [0.1, 0.3]>, <0.7, 0.1, 0.2>) | (< [0.6, 0.7], [0.3, 0.4], [0.2, 0.3]>, <0.7, 0.3, 0.2>) |

**Table 2.**Decision results based on the neutrosophic cubic hybrid weighted arithmetic and geometric aggregation (NCHWAGA) operator and cosine similarity measures.

MADM Method | Score Values (Cosine Measures Value) | Ranking Order | The Best Alternative |
---|---|---|---|

NCHWAGA (ρ = 0.5) | 0.6563, 0.7405, 0.6740, 0.7717 | G_{4} ≻ G_{2} ≻ G_{1} ≻ G_{3} | G_{4} |

Cosine Measure S_{w1} [20] | 0.9564, 0.9855, 0.9596, 0.9945 | G_{4} ≻ G_{2} ≻ G_{1} ≻ G_{3} | G_{4} |

Cosine Measure S_{w2} [20] | 0.9769, 0.9944, 0.9795, 0.9972 | G_{4} ≻ G_{2} ≻ G_{1} ≻ G_{3} | G_{4} |

Cosine Measure S_{w3} [20] | 0.9892, 0.9959, 0.9897, 0.9989 | G_{4} ≻ G_{2} ≻ G_{1} ≻ G_{3} | G_{4} |

Aggregation Operator | Aggregated Result | Score Value | Ranking Order | The Best Alternative |
---|---|---|---|---|

NCHWAGA (ρ = 0.5) | g_{1} = (< [0.5302, 0.6645], [0.1272, 0.3000], [0.1669, 0.3355] >, < 0.3430, 0.4709, 0.2306 >) | Ψ(g_{1}) = 0.6563 | G_{4} ≻ G_{2} ≻ G_{1} ≻ G_{3} | G_{4} |

g_{2} = (< [0.6000, 0.7335], [0.1523, 0.2563], [0.1669, 0.2685] >, <0.6628, 0.2525, 0.2346>) | Ψ(g_{2}) = 0.7405 | |||

g_{3} = (< [0.4677, 0.6307], [0.2000, 0.3000], [0.2264, 0.3672] >, <0.6000, 0.2365, 0.3025>) | Ψ(g_{3}) = 0.6740 | |||

g_{4} = (< [0.6328, 0.7335], [0.1523, 0.2563], [0.1272, 0.2665] >, <0.7335, 0.1523, 0.2000>) | Ψ(g_{4}) = 0.7717 | |||

NCWAA [31] | g_{1} = (< [0.5324, 0.6751], [ 0.1231, 0.3000], [0.1625, 0.3249] >, < 0.4039, 0.4315, 0.2158 >), | Ψ(g_{1}) = 0.6726 | G_{4} ≻ G_{2} ≻ G_{3} ≻ G_{1} | G_{4} |

g_{2} = (< [0.6000, 0.7365], [0.1390, 0.2462], [0.1625, 0.2656] >, <0.6653, 0.2301, 0.2114>) | Ψ(g_{2}) = 0.7497 | |||

g_{3} = (< [0.4700, 0.6331], [0.2000, 0.3000], [0.2111, 0.3648] >, <0.6000, 0.2333, 0.2939>) | Ψ(g_{3}) = 0.6778 | |||

g_{4} = (< [0.6352, 0.7365], [0.1390, 0.2462], [0.1231, 0.2635] >, <0.7365, 0.1390, 0.2000>) | Ψ(g_{4}) = 0.7775 | |||

NCWGA [31] | g_{1} = (< [0.5281, 0.6541], [ 0.1312, 0.3000], [0.1712, 0.3459] >, < 0.2912, 0.5075, 0.2452 >) | Ψ(g_{1}) = 0.6414 | G_{4} ≻ G_{2} ≻ G_{3} ≻ G_{1} | G_{4} |

g_{2} = (< [0.6000, 0.7306], [0.1654, 0.2661], [0.1712, 0.2714] >, <0.6602, 0.2757, 0.2571>) | Ψ(g_{2}) = 0.7315 | |||

g_{3} = (< [0.4655, 0.6284], [0.2000, 0.3000], [0.2414, 0.3697] >, <0.6000, 0.2396, 0.3110>) | Ψ(g_{3}) = 0.6703 | |||

g_{4} = (< [0.6303, 0.7306], [0.1654, 0.2661], [0.1312, 0.2694] >, <0.7306, 0.1654, 0.2000>) | Ψ(g_{4}) = 0.7660 |

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**MDPI and ACS Style**

Shi, L.; Yuan, Y.
Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM. *Symmetry* **2019**, *11*, 278.
https://doi.org/10.3390/sym11020278

**AMA Style**

Shi L, Yuan Y.
Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM. *Symmetry*. 2019; 11(2):278.
https://doi.org/10.3390/sym11020278

**Chicago/Turabian Style**

Shi, Lilian, and Yue Yuan.
2019. "Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM" *Symmetry* 11, no. 2: 278.
https://doi.org/10.3390/sym11020278