Correlation Coefficient between Dynamic Single Valued Neutrosophic Multisets and Its Multiple Attribute Decision-Making Method

Based on dynamic information collected from different time intervals in some real situations, this paper firstly proposes a dynamic single valued neutrosophic multiset (DSVNM) to express dynamic information and operational relations of DSVNMs. Then, a correlation coefficient between DSVNMs and a weighted correlation coefficient between DSVNMs are presented to measure the correlation degrees between DSVNMs, and their properties are investigated. Based on the weighted correlation coefficient of DSVNMs, a multiple attribute decision-making method is established under a DSVNM environment, in which the evaluation values of alternatives with respect to attributes are collected from different time intervals and are represented by the form of DSVNMs. The ranking order of alternatives is performed through the weighted correlation coefficient between an alternative and the ideal alternative, which is considered by the attribute weights and the time weights, and thus the best one(s) can also be determined. Finally, a practical example shows the application of the proposed method.


Introduction
The theory of neutrosophic sets presented by Smarandache [1] is a powerful technique to handle incomplete, indeterminate and inconsistent information in the real world.As the generalization of a classic set, fuzzy set [2], intuitionistic fuzzy set [3], and interval-valued intuitionistic fuzzy set [4], a neutrosophic set can independently express a truth-membership degree, an indeterminacy-membership degree, and a falsity-membership degree.All the factors described by the neutrosophic set are very suitable for human thinking due to the imperfection of knowledge that humans receive or observe from the external world.For example, consider the given proposition "Movie X would be a hit."In this situation, the human brain certainly cannot generate precise answers in terms of yes or no, because indeterminacy is the sector of unawareness of a proposition's value between truth and falsehood.Obviously, the neutrosophic components are very suitable for the representation of indeterminate and inconsistent information.

Some Concepts of SVNSs
Smarandache [1] originally presented the concept of a neutrosophic set from a philosophical point of view.To easily use it in real applications, Wang et al. [6] introduced the concept of SVNS as a subclass of the neutrosophic set and gave the following definition.
Definition 1 [6].Let X be a universal set.A SVNS A in X is characterized by a truth-membership function µ A (x), an indeterminacy-membership function τ A (x) and a falsity-membership function ν A (x).Then, a SVNS A can be denoted by the following form: 1] for each x in X.Therefore, the sum of µ A (x), τ A (x) and ν A (x) satisfies the condition 0 ≤ µ A (x) + τ A (x) + ν A (x) ≤ 3.

For two SVNSs
there are the following relations [6]: (1) Complement: Then, Ye [8] defined a correlation coefficient between A and B as follows: The correlation coefficient between A and B satisfies the following properties [8]:

Dynamic Single Valued Neutrosophic Multiset
This section proposes a dynamic single valued neutrosophic multiset and its operational relations.

Correlation Coefficient of DSVNMs
Correlation coefficients are usually used in science and engineering applications.They play an important role in decision-making, pattern recognition, clustering analysis, and so on.In regards to this, this section proposes a correlation coefficient of DSVNMs and a weighted correlation coefficient of DSVNMs.
Based on DSVNMs constructed by dynamic truth-membership degrees, dynamic indeterminacymembership degrees, and dynamic falsity-membership degrees corresponding to t = {t 1 , t 2 , . . ., t q }, we can give the following definition of a correlation coefficient between DSVNMs.
∈ X} be any two DSVNMs in t = {t 1 , t 2 , . . ., t q } and X = (x 1 , x 2 , . . ., x n ).Then, a correlation coefficient between A(t) and B(t) is defined as: Theorem 1.The correlation coefficient between A and B satisfies the following properties: According to the Cauchy-Schwarz inequality: where (x 1 , x 2 , . . ., x n ) ∈ R n and (y 1 , y 2 , . . ., y n ) ∈ R n .Then, we can obtain the following inequality: According to the above inequality, there is the following inequality: Hence, there is: (P3) It is straightforward.
Theorem 2. The correlation coefficient ρ w (A(t), B(t)) also satisfies the following three properties: By the previous similar proof method in Theorem 1, we can prove the properties (P1)-(P3) (omitted).

Correlation Coefficient for Multiple Attribute Decision-Making
In this section, we apply the weighted correlation coefficient of DSVNMs to multiple attribute decision-making problems with DSVNM information.
The weighted correlation coefficient between an alternative g i (t) (i = 1, 2, . . ., m) and the ideal solution g*(t) is calculated by use of the following formula: The bigger the value of ρ w (g i (t), g*(t)), the better the alternative g i .Then, we rank the alternatives and select the best one(s) according to the values of weighted correlation coefficients.
By applying Equation ( 4), we can obtain the values of the weighted correlation coefficient between each alternative and the ideal alternative as follows: ρ w (g 1 (t), g*(t)) = 0.6285, ρ w (g 2 (t), g*(t)) = 0.9122, ρ w (g 3 (t), g*(t)) = 0.7758 and ρ w (g 4 (t), g*(t)) = 0.9025.According to the above values of weighted correlation coefficients, we can give the ranking order of the four alternatives: The bigger the value of ρw(gi(t), g*(t)), the better the alternative gi.Then, we rank the alternatives and select the best one(s) according to the values of weighted correlation coefficients.
Therefore, the alternative A2 is the best choice.
The example clearly indicates that the proposed decision-making method is simple and effective under the DSVNM environment, based on the weighted correlation coefficient of DSVNMs for dealing with multiple attribute decision-making problems with DSVNM information, since such a decision-making method can represent and handle the dynamic evaluation data given The bigger the value of ρw(gi(t), g*(t)), the better the alternative gi.Then, we rank the alternatives and select the best one(s) according to the values of weighted correlation coefficients.

Practical Example
A practical example about investment alternatives for a multiple attribute decision-making problem adapted from Ye [8] is used to demonstrate the applications of the proposed decision-making method under a DSVNM environment.There is an investment company, which wants to invest a sum of money in the best option.There is a panel with four possible alternatives to invest the money: (1) g1 is a car company; (2) g2 is a food company; (3) g3 is a computer company; (4) g4 is an arms company.The investment company must take a decision according to the three attributes: (1) x1 is the risk factor; (2) x2 is the growth factor; (3) x3 is the environmental factor.Let us consider the evaluations of the alternatives on the attributes given by decision makers or experts in the time sequence t = {t1, t2, t3}.Assume that the weighting vector of the attributes is given by w = (0.35, 0.25, 0.40) T and the weighting vector of times is given by ω(t) = (0.25, 0.35, 0.40) T .The four possible alternatives of gi (i = 1, 2, 3, 4) regarding the three attributes of xj (j = 1, 2, 3) are evaluated by decision makers, and then the evaluation values are represented by using DSVNMEs, which are given as the following DSVNM decision matrix D(t): (0.4, 0.5, 0.3), (0.1, 0.2, 0.3), (0.3, 0.2, 0.3) (0.6, 0.4, 0.5), (0. Then, the developed approach is utilized to give the ranking order of the alternatives and the best one(s).
Therefore, the alternative A2 is the best choice.
The example clearly indicates that the proposed decision-making method is simple and effective under the DSVNM environment, based on the weighted correlation coefficient of DSVNMs for dealing with multiple attribute decision-making problems with DSVNM information, since such a decision-making method can represent and handle the dynamic evaluation data given The bigger the value of ρw(gi(t), g*(t)), the better the alternative gi.Then, we rank the alternatives and select the best one(s) according to the values of weighted correlation coefficients.

Practical Example
A practical example about investment alternatives for a multiple attribute decision-making problem adapted from Ye [8] is used to demonstrate the applications of the proposed decision-making method under a DSVNM environment.There is an investment company, which wants to invest a sum of money in the best option.There is a panel with four possible alternatives to invest the money: (1) g1 is a car company; (2) g2 is a food company; (3) g3 is a computer company; (4) g4 is an arms company.The investment company must take a decision according to the three attributes: (1) x1 is the risk factor; (2) x2 is the growth factor; (3) x3 is the environmental factor.Let us consider the evaluations of the alternatives on the attributes given by decision makers or experts in the time sequence t = {t1, t2, t3}.Assume that the weighting vector of the attributes is given by w = (0.35, 0.25, 0.40) T and the weighting vector of times is given by ω(t) = (0.25, 0.35, 0.40) T .The four possible alternatives of gi (i = 1, 2, 3, 4) regarding the three attributes of xj (j = 1, 2, 3) are evaluated by decision makers, and then the evaluation values are represented by using DSVNMEs, which are given as the following DSVNM decision matrix D(t): (0.4, 0.5, 0.3), (0.1, 0.2, 0.3), (0. Then, the developed approach is utilized to give the ranking order of the alternatives and the best one(s).
Therefore, the alternative A2 is the best choice.
The example clearly indicates that the proposed decision-making method is simple and effective under the DSVNM environment, based on the weighted correlation coefficient of DSVNMs for dealing with multiple attribute decision-making problems with DSVNM information, since such a decision-making method can represent and handle the dynamic evaluation data given A 1 , which is in accordance with the one of [8].Therefore, the alternative A 2 is the best choice.
The example clearly indicates that the proposed decision-making method is simple and effective under the DSVNM environment, based on the weighted correlation coefficient of DSVNMs for dealing with multiple attribute decision-making problems with DSVNM information, since such a decision-making method can represent and handle the dynamic evaluation data given by experts or decision makers at different time intervals, while existing various neutrosophic decision-making methods cannot do this.

Conclusions
Based on dynamic information collected from different time intervals in some real situations, this paper proposed a DSVNM to express dynamic information and the operational relations of DSVNMs.The DSVNM is a dynamic set encompassing a time sequence, where its truth-membership degrees, indeterminacy-membership degrees, and falsity-membership degrees are represented by time sequences.Therefore, DSVNM has desirable characteristics and advantages of its own for handling dynamic problems according to a time sequence in real applications, whereas existing neutrosophic sets cannot deal with them.
Then, we proposed the correlation coefficient of DSVNMs and weighted correlation coefficient of DSVNMs and investigated their properties.Based on the weighted correlation coefficient of DSVNMs, the multiple attribute decision-making method was proposed under a DSVNM environment, in which the evaluated values of alternatives regarding attributes take the form of DSVNMEs.Through the weighted correlation coefficient between each alternative and the ideal alternative, one can rank alternatives and choose the best one(s).Finally, a practical example about investment alternatives was given to demonstrate the practicality and effectiveness of the developed approach.The proposed method is simple and effective under the DSVNM decision-making environment.
In the future, we shall extend DSVNMs to interval/bipolar neutrosophic sets and develop dynamic interval/bipolar neutrosophic decision-making and medical diagnosis methods in different time intervals.