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Another form of correlation coefficient between single valued neutrosophic sets and its multiple attribute decision-making method.

1 Introduction

To handle the indeterminate information and inconsistent information which exist commonly in real situations, Smarandache [1] firstly presented a neutrosophic set from philosophical point of view, which is a powerful general formal framework and generalized the concept of the classic set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, interval-valued intuitionistic fuzzy set, paraconsistent set, dialetheist set, paradoxist set, and tautological set [1, 2]. In the neutrosophic set, a truth-membership, an indeterminacy-membership, and a falsity-membership are represented independently. Its functions [T.sub.A](x), [I.sub.A](x) and [F.sub.A](x) are real standard or nonstandard subsets of ][sup.-]0,[1.sup.+] [i.e., [T.sub.A](x): X [right arrow]][sup.-]0, [1.sup.+][, [I.sub.A](x): X [right arrow] ][sup.-]0, [1.sup.+][, and [F.sub.A](x): X [right arrow] ][sup.-]0, [1.sup.+] Obviously, it will be difficult to apply in real scientific and engineering areas. Therefore, Wang et al. [3] proposed the concept of a single valued neutrosophic set (SVNS), which is the subclass of a neutrosophic set, and provided the set-theoretic operators and various properties of SVNSs. Thus, SVNSs can be applied in real scientific and engineering fields and give us an additional possibility to represent uncertainty, imprecise, incomplete, and inconsistent information which exist in real world. However, the correlation coefficient is one of the most frequently used tools in engineering applications. Therefore, Hanafy et al. [4] introduced the correlation of neutrosophic data. Then, Ye [5] presented the correlation coefficient of SVNSs based on the extension of the correlation coefficient of intuitionistic fuzzy sets and proved that the cosine similarity measure of SVNSs is a special case of the correlation coefficient of SVNSs, and then applied it to single valued neutrosophic multicriteria decision-making problems. Hanafy et al. [6] presented the centroid-based correlation coefficient of neutrosophic sets and investigated its properties. Recently, S. Broumi and F. Smarandache [8] Correlation coefficient of interval neutrosophic set and investigated its properties.

In this paper, we propose another form of correlation coefficient between SVNSs and investigate its properties. Then, a multiple attribute decision-making method using the correlation coefficient of SVNSs is established under single valued neutrosophic environment. To do so, the rest of the paper is organized as follows. Section 2 briefly describes some concepts of SVNSs. In Section 3, we develop another form of correlation coefficient between SVNSs and investigate its properties. Section 4 establishes a multiple attribute decision-making method using the correlation coefficient of SVNSs under single valued neutrosophic environment. In Section 5, two illustrative examples are presented to demonstrate the applications of the developed approach. Section 6 contains a conclusion and future research.

2 Some concepts of SVNSs

Smarandache [1] firstly presented the concept of a neutrosophic set from philosophical point of view and gave the following definition of a neutrosophic set.

Definition 1 [1]. Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function [T.sub.A](x), an indeterminacy-membership function [I.sub.A](x), and a falsity-membership function [F.sub.A](x). The functions [T.sub.A](x), [I.sub.A](x) and [F.sub.A](x) are real standard or nonstandard subsets of ][sup.-]0, [1.sup.+] i.e., [T.sub.A](x): X [right arrow]][sup.-]0, [1.sup.+][, [I.sub.A](x): X [right arrow]][sup.-]0, [1.sup.+] and [F.sub.A](x): X [right arrow] ][sup.-]0, [1.sup.+][. There is no restriction on the sum of [T.sub.A](x), [I.sub.A](x) and [F.sub.A](x), so [sup.-]0 [less than or equal to] sup [T.sub.A](x) + sup [I.sub.A](x) + sup [F.sub.A](x) [less than or equal to] [3.sup.+].

Obviously, it is difficult to apply in practical problems. Therefore, Wang et al. [3] introduced the concept of a SVNS, which is an instance of a neutrosophic set, to apply in real scientific and engineering applications. In the following, we introduce the definition of a SVNS [3]. Definition 2 [3]. Let X be a space of points (objects) with generic elements in X denoted by x. A SVNS A in X is characterized by a truth-membership function [T.sub.A](x), an indeterminacy-membership function [I.sub.A](x), and a falsity-membership function [F.sub.A](x) for each point x in X, [T.sub.A](x), [I.sub.A](x), [F.sub.A](x) [member of] [0, 1]. Thus, A SVNS A can be expressed as

A = {{x, [T.sub.A](x), [I.sub.A](x), [F.sub.A](x)>|x [member of] X}.

Then, the sum of [T.sub.A](x), [I.sub.A](x) and [F.sub.A](x) satisfies the condition 0 [less than or equal to] [T.sub.A](x) + [I.sub.A](x) + [F.sub.A](x) [less than or equal to] 3. Definition 3 [3]. The complement of a SVNS A is denoted by [A.sup.c] and is defined as

[A.sup.c] = {<x, [F.sub.A](x), 1 - [I.sub.A](x),[T.sub.A](x)>| x [member of] X}.

Definition 4 [3]. A SVNS A is contained in the other SVNS B, A [subset or equal to] B if and only if [T.sub.A](x) [less than or equal to] [T.sub.B](x), [I.sub.A](x) [greater than or equal to] [I.sub.B](x), and [F.sub.A](x) [greater than or equal to] [F.sub.B](x) for every x in X.

Definition 5 [3]. Two SVNSs A and B are equal, written as A = B, if and only if A [subset or equal to] B and B [subset or equal to] A.

3 Correlation coefficient of SVNSs

Motivated by another correlation coefficient between intuitionistic fuzzy sets [7], this section proposes another form of correlation coefficient between SVNSs as a generalization of the correlation coefficient of intuitionistic fuzzy sets [7].

Definition 6. For any two SVNSs A and B in the universe of discourse X = {[x.sub.1], [x.sub.2], ... , [x.sub.n]}, another form of correlation coefficient between two SVNSs A and B is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Theorem 1. The correlation coefficient N(A, B) satisfies the following properties:

(1) N(A, B) = N(B, A);

(2) 0 [less than or equal to] N(A, B) [less than or equal to] 1;

(3) N(A, B) = 1, if A = B.

Proof. (1) It is straightforward.

(2) The inequality N(A, B) [greater than or equal to] 0 is obvious. Thus, we only prove the inequality N(A, B) [less than or equal to] 1.

N(A, B) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

According to the Cauchy-Schwarz inequality:

[([x.sub.1] * [y.sub.1] + [x.sub.2] * [y.sub.2] + ... + [x.sub.n] * [y.sub.n]).sup.2] [less than or equal to] ([x.sup.2.sub.1] + [x.sup.2.sub.2] + ... + [x.sup.2.sub.n] * ([y.sup.2.sub.1] + [y.sup.2.sub.2] + ... + [y.sup.2.sub.n]'

where ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] [R.sup.n] and ([y.sub.1], [y.sub.2], ..., [y.sub.n]) [member of] [R.sup.n], we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, N(A,B) [less than or equal to][[N(A,A)].sup.1/2] * [[N(B,B)].sup.1/2].

Then, N(A,B) [less than or equal to] max{N(A, A),N(B,B)}.

Therefore, N(A, B) [less than or equal to] 1.

(3) If A = B, there are [T.sub.A](xi) = [T.sub.B]([x.sub.i]), [I.sub.A]([x.sub.i]) = [I.sub.B]([x.sub.i]), and [F.sub.A]([x.sub.i]) = [F.sub.B]([x.sub.i]) for any [x.sub.i] [member of] X and i = 1, 2, ..., n. Thus, there are N(A, B) = 1.

In practical applications, the differences of importance are considered in the elements in the universe. Therefore, we need to take the weights of the elements [x.sub.i](i = 1, 2, ... ,n) into account. Let wi be the weight for each element [x.sub.i](i = 1, 2, ... ,n), [w.sub.i] [member of] [0, 1], and [[summation].sup.n.sub.i=1] [w.sub.i] = 1, then we have the following weighted correlation coefficient between the SVNSs A and B:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

If w = [(1/n, 1/n, ... ,1/n).sup.T], then Eq. (2) reduce to Eq. (1). Note that W(A, B) also satisfy the three properties of Theorem 1.

Theorem 2. Let [w.sub.i] be the weight for each element [x.sub.i](i = 1, 2, ..., n), [w.sub.i] [member of] [0, 1], and [[summation].sup.n.sub.i=1] [w.sub.i] = 1, then the weighted correlation coefficient W(A, B) defined in Eq. (2) also satisfies the following properties:

(1) W(A, B) = W(B, A);

(2) 0 [less than or equal to] W(A, B) [less than or equal to] 1;

(3) W(A, B) = 1, if A = B.

Since the process to prove these properties is similar to that in Theorem 1, we do not repeat it here.

4 Decision-making method using the correlation coefficient of SVNSs

This section proposes a single valued neutrosophic multiple attribute decision-making method using the proposed correlation coefficient of SVNSs.

Let A = {[A.sub.1], [A.sub.2], ... ,[A.sub.m]} be a set of alternatives and C = {[C.sub.1], [C.sub.2], ... ,[C.sub.n]} be a set of attributes. Assume that the weight of an attribute [C.sub.j] (j = 1, 2, ... ,n), entered by the decision-maker, is [w.sub.j], [w.sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1] [x.sub.j] = 1. In this case, the characteristic of an alternative [A.sub.i] (i = 1, 2, ... ,m) with respect to an attribute [C.sub.j](j = 1, 2,., n) is represented by a SVNS form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For convenience, the values of the three functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are denoted by a single valued neutrosophic value (SVNV) [a.sub.ij] = <[t.sub.ij], [i.sub.ij], [f.sub.ij]> (i = 1, 2, ..., m; j = 1, 2, ..., n), which is usually derived from the evaluation of an alternative [A.sub.i] with respect to an attribute [C.sub.j] by the expert or decision maker. Thus, we can establish a single valued neutrosophic decision matrix D = [([a.sub.ij]).sub.mxn]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the decision-making method, the concept of ideal point has been used to help identify the best alternative in the decision set. The ideal alternative provides a useful theoretical construct against which to evaluate alternatives. Generally, the evaluation attributes can be categorized into two kinds, benefit attributes and cost attributes. Let H be a collection of benefit attributes and L be a collection of cost attributes. An ideal SVNV can be defined by an ideal element for a benefit attribute in the ideal alternative [A.sup.*] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for j [member of] H

while an ideal SVNV can be defined by an ideal element for a cost attribute in the ideal alternative [A.sup.*] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, by applying Eq. (2) the weighted correlation coefficient between an alternative [A.sub.i] (i = 1, 2, m) and the ideal alternative [A.sup.*] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Then, the bigger the measure value W([A.sub.i], [A.sup.*]) (i = 1, 2, ..., m) is, the better the alternative [A.sub.i] is, because the alternative [A.sub.i] is close to the ideal alternative [A.sup.*]. Through the weighted correlation coefficient between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best one can be easily identified as well.

5 Illustrative examples

In this section, two illustrative examples for the multiple attribute decision-making problems are provided to demonstrate the application of the proposed decision-making method.

5.1 Example 1

Now, we discuss the decision-making problem adapted from the literature [5]. 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) [A.sub.1] is a car company; (2) [A.sub.2] is a food company; (3) [A.sub.3] is a computer company; (4) [A.sub.4] is an arms company. The investment company must take a decision according to the three attributes: (1) [C.sub.1] is the risk; (2) [C.sub.2] is the growth; (3) [C.sub.3] is the environmental impact, where [C.sub.1] and [C.sub.2] are benefit attributes and [C.sub.3] is a cost attribute. The weight vector of the three attributes is given by w = [(0.35, 0.25, 0.4).sup.T]. The four possible alternatives are to be evaluated under the above three attributes by the form of SVNVs.

For the evaluation of an alternative [A.sub.i] with respect to an attribute [C.sub.j] (i =1, 2, 3, 4; j =1, 2, 3), it is obtained from the questionnaire of a domain expert. For example, when we ask the opinion of an expert about an alternative [A.sub.1] with respect to an attribute [C.sub.1], he or she may say that the possibility in which the statement is good is 0.4 and the statement is poor is 0.3 and the degree in which he or she is not sure is 0.2. For the neutrosophic notation, it can be expressed as [a.sub.11] = <0.4, 0.2, 0.3>. Thus, when the four possible alternatives with respect to the above three attributes are evaluated by the expert, we can obtain the following single valued neutrosophic decision matrix D:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, we utilize the developed approach to obtain the most desirable alternative(s).

From the single valued neutrosophic decision matrix, we can obtain the following ideal alternative:

[A.sup.*] = {{[C.sub.1], 0.7,0.1,0.1>, {[C.sub.2], 0.6,0.1,0.2>, {[C.sub.3], 0.5,0.3,0.8>}.

By using Eq. (3), we can obtain the values of the weighted correlation coefficient W([A.sub.i], [A.sup.*]) (i =1, 2, 3, 4):

W([A.sub.1], [A.sup.*]) = 0.8016, W([A.sub.2], [A.sup.*]) = 0.9510, W([A.sub.3], [A.sup.*]) = 0.8588, and W([A.sub.4], [A.sup.*]) = 0.9664.

Thus, the ranking order of the four alternatives is [A.sub.4] [??] [A.sub.2] [??] [A.sub.3] [??] [A.sub.1]. Therefore, the alternative [A.sub.4] is the best choice among the four alternatives.

5.2 Example 2

A multi-criteria decision making problem is concerned with a manufacturing company which wants to select the best global supplier according to the core competencies of suppliers. Now suppose that there are a set of four suppliers A = {[A.sub.1], [A.sub.2], [A.sub.3], [A.sub.4]} whose core competencies are evaluated by means of the four attributes: (1) [C.sub.1] is the level of technology innovation; (2) [C.sub.2] is the control ability of flow; (3) [C.sub.3] is the ability of management; (4) C4 is the level of service, where [C.sub.1], [C.sub.2] and [C.sub.2] are all benefit attributes. Assume that the weight vector for the four attributes is w = [(0.3, 0.25, 0.25, 0.2).sup.T].

The proposed decision making method is applied to solve this problem for selecting suppliers.

For the evaluation of an alternative Ai (i =1, 2, 3, 4) with respect to a criterion [C.sub.j](j =1, 2, 3, 4), it is obtained from the questionnaire of a domain expert. For example, when we ask the opinion of an expert about an alternative [A.sub.1] with respect to a criterion [C.sub.1], he or she may say that the possibility in which the statement is good is 0.5 and the statement is poor is 0.3 and the degree in which he or she is not sure is 0.1. For the neutrosophic notation, it can be expressed as [a.sub.11] = <0.5, 0.1, 0.3>. Thus, when the four possible alternatives with respect to the above four attributes are evaluated by the similar method from the expert, we can obtain the following single valued neutrosophic decision matrix D:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, we employ the developed approach to obtain the most desirable alternative(s).

From the single valued neutrosophic decision matrix, we can obtain the following ideal alternative:

[A.sup.*] = {{[C.sub.1], 0.6,0.1,0.1>, {[C.sub.2], 0.5,0.1,0.3>, {[C.sub.3], 0.9,0.0,0.1>, {[C.sub.4], 0.7,0.2,0.1>}

By applying Eq. (3), we can obtain the values of the weighted correlation coefficient W([A.sub.i], [A.sup.*]) (i =1, 2, 3, 4):

W([A.sub.1], [A.sup.*]) = 0.7998, W([A.sub.2], [A.sup.*]) = 0.8756, W([A.sub.3], [A.sup.*]) = 0.7580, and W([A.sub.4], [A.sup.*]) = 0.7532.

Thus, the ranking order of the four alternatives is [A.sub.2] [??] [A.sub.1] [??] [A.sub.3] [??] [A.sub.4]. Therefore, the alternative [A.sub.2] is the best choice among the four alternatives.

From the two examples, we can see that the proposed single valued neutrosophic multiple attribute decision-making method is more suitable for real scientific and engineering applications because it can handle not only incomplete information but also the indeterminate information and inconsistent information which exist commonly in real situations.

6 Conclusion

In this paper, we proposed another form of the correlation coefficient between SVNSs. Then a multiple attribute decision-making method has been established in single valued neutrosophic setting by means of the weighted correlation coefficient between each alternative and the ideal alternative. Through the correlation coefficient, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, two illustrative examples illustrated the applications of the developed approach. Then the technique proposed in this paper is suitable for handling decision-making problems with single value neutrosophic information and can provide a useful way for decision-makers. In the future, we shall continue working in the applications of the correlation coefficient between SVNSs to other domains, such as data analysis and classification, pattern recognition, and medical diagnosis.

Received: Nov. 1st, 2013. Accepted: Nov. 20, 2013

References

[1] F. Smarandache. A unifying field in logics. neutrosophy: Neutrosophic probability, set and logic, American Research Press, Rehoboth, 1999.

[2] F. Smarandache, Neutrosophic set. A generalization of the intuitionistic fuzzy set. International Journal of Pure and Applied Mathematics, 24 (2005), 287-297.

[3] H. Wang, F. Smarandache, Y.Q. Zhang, and R. Sunderraman. Single valued neutrosophic sets, Multispace and Multistructure 4 (2010) 410-413.

[4] I.M. Hanafy, A.A. Salama, and K. Mahfouz. Correlation of neutrosophic Data, International Refereed Journal of Engineering and Science, 1(2) (2012), 39-43.

[5] J. Ye. Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment, International Journal of General Systems, 42(4) (2013), 386-394.

[6] M. Hanafy, A. A. Salama, and K. M. Mahfouz. Correlation Coefficients of Neutrosophic Sets by Centroid Method, International Journal of Probability and Statistics, 2(1) (2013), 9-12

[7] Z.S. Xu, J. Chen, and J.J. Wu. Clustering algorithm for intuitionistic fuzzy sets, Information Sciences, 178 (2008), 3775-3790.

[8] S. Broumi, F. Smarandache, "Correlation Coefficient of Interval Neutrosophic set", Periodical of Applied Mechanics and Materials, Vol. 436, 2013, with the title Engineering Decisions and Scientific Research in Aerospace, Robotics, Biomechanics, Mechanical Engineering and Manufacturing; Proceedings of the International Conference ICMERA, Bucharest, October (2013).

Jun Ye

Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang 312000, P.R. China

E-mail: yehjun@aliyun.com
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Author:Ye, Jun
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Date:Dec 1, 2013
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