## Chunfang Liu*## |

C _{ 1 } | C _{ 2 } | C _{ 3 } | |

A _{ 1 } | {{[0.3,0.4],[0.4,0.4], | {{[0.4,0.5],[0.5,0.6]}, | {{[0.2,0.3]}, |

[0.4,0.5]}, | {[0.2,0.3]}, | {[0.1,0.2]}, | |

{[0.1,0.2]},{[0.3,0.4]}} | {[0.3,0.3],[0.3,0.4]}} | {[0.4,0.5],[0.5,0.6]}} | |

| {{[0.6,0.7]},{[0.1,0.2]}, | {{[0.5,0.7]}, {[0.1,0.2]}, | {{[0.6,0.7]},{[0.1,0.2]}, |

{[0.2,0.3]}} | {[0.2,0.3]}} | {[0.1,0.2]}, | |

| {{[0.3,0.4], [0.5,0.6]}, | {{[0.5,0.6]}, | {{[0.5,0.6]}, |

{[0.2,0.4]}, | {[0.2,0.3]}, | {[0.1,0.2], [0.2,0.3]}, | |

{[0.2,0.3]}} | {[0.3,0.4]}} | {[0.2,0.3]}} | |

| {{[0.7,0.8]}, {[0,0.1]}, | {{[0.6,0.7]}, {[0,0.1]}, | {{[0.3,0.5]}, [0.2,0.3]}, |

{[0.1,0.2]}} | {[0.2,0.3]}} | {[0.1,0.2],[0.3,0.4]}} |

According to Section 4, the decision-making process is as follows.

First, I calculate the novel correlation coefficient [TeX:] $$\lambda _ { \mathrm { IVNHFS } } \left( A _ { i } , A ^ { * } \right) ( i = 1,2,3,4 )$$ according to Eq. (1), (2), (3) as follows:

Since

[TeX:] $$\lambda _ { \mathrm { IVNHFS } } \left( A _ { 2 } , A ^ { * } \right) \geq \lambda _ { \mathrm { IVNHFS } } \left( A _ { 4 } , A ^ { * } \right) \geq \lambda _ { \mathrm { IVNHFS } } \left( A _ { 3 } , A ^ { * } \right) \geq \lambda _ { \mathrm { INNHFS } } \left( A _ { 1 } , A ^ { * } \right)$$

Then, I rank the alternative as [TeX:] $$A _ { 2 } > A _ { 4 } > A _ { 3 } > A _ { 1 } . \text { The best alternative is } A _ { 2 }$$ , that is, a decoration company is the best alternative.

5.2 AnalysisIn this part, I have proposed a new method to deal with the MADM problem described by IVNHFS information. In the MADM problems, I adopt the novel correlation coefficient of IVNHFSs to rank the alternatives and obtain the best one. Moreover, the calculation of the method is simpler and more practical. In order to show the feasibility of my approach, I will compare the method of [ 19 ] and [ 25 ] with the proposed method in this paper.

Aggregation Operator Method [ 19 ]: They use the interval-valued neutrosophic hesitant fuzzy generalized hybrid weighted aggregation (INHFGHWA) operators to deal with the MADM problems. First, they aggregate the data of different attributes of the alternative based on Eq. (47) in Reference [ 19 ]. Then, they calculate the score functions of the alternatives as follows:

[TeX:] $$s \left( A _ { 1 } \right) = 1.0273 , s \left( A _ { 2 } \right) = 1.402 , s \left( A _ { 3 } \right) = 1.1339 , s \left( A _ { 4 } \right) = 1.3969.$$

So we rank the alternatives as [TeX:] $$A _ { 2 } > A _ { 4 } > A _ { 3 } > A _ { 1 } , \text { then } A _ { 2 }$$ is the best alternative.

Method of [ 25 ]：They use the interval-valued neutrosophic hesitant fuzzy correlation coefficient to deal with the MADM problems. They use Eq. (7) in [ 25 ] to calculate the correlation coefficient as follows:

[TeX:] $$\lambda _ { \mathrm { IVNHFS } } \left( A _ { 1 } , A ^ { * } \right) = 0.4766 , \lambda _ { \mathrm { IVNHFS } } \left( A _ { 2 } , A ^ { * } \right) = 0.9285,$$

[TeX:] $$\lambda _ { \mathrm { IVNHFS } } \left( A _ { 3 } , A ^ { * } \right) = 0.6823 , \lambda _ { \mathrm { IVNHFS } } \left( A _ { 4 } , A ^ { * } \right) = 0.9053$$

Since [TeX:] $$\lambda _ { \mathrm { INNHFS } } \left( A _ { 2 } , A ^ { * } \right) \geq \lambda _ { \mathrm { IVNHFS } } \left( A _ { 4 } , A ^ { * } \right) \geq \lambda _ { \mathrm { IVNHFS } } \left( A _ { 3 } , A ^ { * } \right) \geq \lambda _ { \mathrm { IVNHFS } } \left( A _ { 1 } , A ^ { * } \right)$$ , and I rank the alternative as [TeX:] $$A _ { 2 } > A _ { 4 } > A _ { 3 } > A _ { 1 }$$ . The most desirable one is * A _{ 2 } * , which is agreement with the result of the proposed method.

From the above analysis, I find that three ways have the same result that the best alternative is * A _{ 2 } * . First, let us consider the method of [ 19 ], the decision making method based on the INHFGHWA operators used the score functions to rank the alternatives. It dealt with the problem using the aggregation operators and fully used the decision information values that the expert gave. I found that the calculation is very complex. In [ 25 ], the author gave several correlation coefficient formulas, the calculation of the correlation coefficient is complex. With regard to my proposed method, the ideal element is defined first, my method is based on the novel formula. I used the IVNHFS to get a new IVNS, and got the novel correlation coefficient of IVNHFSs, the calculation is simple and effective. In a word, the proposed method in my paper gave a new approach to calculate the correlation coefficient and provide an alternative perspective in dealing with MADM problems.

In the study, a novel correlation coefficient of IVNHFSs is proposed and its properties are discussed. On the basis of the novel correlation coefficient, I have developed a method to cope with the MADM problem within the framework of IVNHFS. The process of calculation is simple and provides a new idea for solving decision-making problems under the environment of IVNHFS. In the end, a practical numerical example is given to verify the feasibility of the developed MADM method. As we know, similarity measure, correlation coefficient and entropy are important topics in fuzzy set theory. In the future, I will continue to study the similarity measure of neutrosophic HFSs

She received M.S. degree in Harbin Institute of Technology, China, in 2003 and Ph.D. degree in Harbin Engineering University, China, in 2017. She is currently an associate professor at College of Science, Northeast Forestry University. Her research interests includes system engineering, intelligent control and fuzzy systems.

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