Abstract:
A correlation coefficient is a statistical measure that helps determine the degree to which changes in one variable predict changes in another. Wang’s single-valued neutrosophic sets have been further developed into pentapartitioned neutrosophic sets. In this article, we analyze the characteristics of pentapartitioned neutrosophic Pythagorean sets with improved correlation coefficients. We have also used the same approach in multiple-attribute decision-making methodologies, including one involving a pentapartitioned neutrosophic Pythagorean environment. Finally, we applied the above technique to a multiple-attribute group decision-making problem.
Introduction
Fuzzy sets were introduced by Zadeh [23] in 1965, allowing membership values within the interval [0,1]; fuzzy set theory is an extension of classical set theory. Fuzzy sets help deal with uncertainty, vagueness, and imprecision that cannot be represented within classical Cantorian sets. As an extension of Zadeh’s fuzzy set theory, the intuitionistic fuzzy set (IFS) was introduced by Atanassov [1] in 1986; it consists of degrees of membership and non-membership that lie within the interval [0,1]. IFS theory is widely used in areas such as logic programming, decision-making problems, and medical diagnosis.
Florentin Smarandache [15] introduced the idea of a neutrosophic set in 1995, providing information about neutral states by introducing an additional component referred to as indeterminacy within the set. Thus, the neutrosophic set was formulated to include the truth-membership function (T), indeterminacy-membership function (I), and falsity-membership function (F), respectively. Neutrosophic sets deal with non-normal intervals of ]−0 1+[. Since neutrosophic sets deal with indeterminacy effectively, they play an important role in several application areas, including information technology, decision networks, multicriteria decision-making, electronic database systems, and diagnosis.
To process incomplete or imperfect data and address uncertainty in real-world problems, Wang [16] (2010) introduced the idea of single-valued neutrosophic sets (SVNS), an extension of intuitionistic fuzzy sets that became an active research topic. Rama Malik et al. [14] proposed the idea of pentapartitioned single-valued neutrosophic sets based on Belnap’s five-valued logic and Smarandache’s five-numerical-valued logic. In PSVNS, indeterminacy is divided into three functions referred to as ‘Contradiction’ (both true and false), ‘Ignorance’ (neither true nor false), and ‘Unknown’ membership, so that PSVNS has five components, T, C, U, F, and G, that also lie within the non-normal unit interval ]−0 1+[. Further, R. Radha and A. Stanis Arul Mary [7] defined a new hybrid model of pentapartitioned neutrosophic Pythagorean sets (PNPS) in 2021. The correlation coefficient is an effective mathematical tool for testing the strength of the relationship between two variables. Many researchers have studied different correlation coefficients for sets such as fuzzy sets, IFS, SVNS, and QSVNS. In 1999, D.A. Chiang and N. P. Lin [3] proposed the correlation of fuzzy sets under fuzzy settings. Later, D.H. Hong [4] (2006) defined fuzzy measures for a correlation coefficient of fuzzy numbers under Tw (the weakest t-norm), based on fuzzy arithmetic operations.
Correlation coefficients play an important role in several real-world problems, such as multiple-attribute group decision-making, cluster analysis, decision-making problems, pattern recognition, and diagnosis. Therefore, several authors have focused on defining correlation coefficients to resolve real-world issues in multicriteria decision-making strategies. Jun Ye [19] defined improved correlation coefficients of single-valued neutrosophic sets and interval neutrosophic sets for multiple-attribute decision-making to overcome the drawbacks of the correlation coefficients of single-valued neutrosophic sets (SVNSs) defined in [22]. In this paper, we apply an improved correlation coefficient to pentapartitioned neutrosophic Pythagorean sets and study it through an example.
Preliminaries
2.1 Definition [15]
Let X be a universe. A neutrosophic set A on X can be defined as follows.
2.2 Definition [7]
Let X be a universe. A pentapartitioned neutrosophic Pythagorean set A, with T, F, C, and U as dependent neutrosophic components and I as an independent component for A on X, is an object of the following form.
Here, is the truth membership, is contradiction membership, is ignorance membership, is the false membership, and IA () is an unknown membership.
2.3 Definition [14]
Let P be a non-empty set. A pentapartitioned neutrosophic set A over P characterizes each element p in P by a truth-membership function, a contradiction-membership function, an ignorance-membership function, an unknown-membership function, and a false-membership function, such that for each p in P.
2.4 Definition [7]
The complement of a pentapartitioned neutrosophic Pythagorean set (F, A) on X is denoted by and is defined as
(x)=
2.5 Definition [7]
Let A = and B = be pentapartitioned neutrosophic Pythagorean sets. Then
A B = <
A B =
Improved Correlation Coefficients
Based on the concept of the correlation coefficient of PNPSs, we define improved correlation coefficients of PNPSs in the following section.
3.1 Definition
Let P and Q be any two PNPSs in the universe of discourse R = { r1, r2, r3,…, rn }, then the improved correlation coefficient between P and Q is defined as follows.
K (P, Q) =) ) ) ) ) ]
(3.1)
3.2 Theorem
For any two PNPSs P and Q in the universe of discourse R = { r1, r2, r3,…, rn }, the improved correlation coefficient K (P, Q) satisfies the following properties.
K (P, Q) = K (Q, P);
0 ;
K (P, Q) = 1 iff P =Q.
Proof
It is obvious and straightforward.
Here, 0 1, 0 1, 0 1, 0 0 1, 1 1,
1 1, 1 1, 1 1, 1 1, Therefore the following inequation satisfies
(1 5. Hence, we have 0
(3) If K (P, Q) = 1, then we get (1 = 5. Since 0 (1 1, 0 1, 0 1, 0 1 and 0 1, there are (1 1, 1, 1, 1 and 1. And also since 0 1, 1, 0 1, 0 1 and 0 1, 1 1, 1, 1, 1, 1. We get and 1 = 1 = 1. This implies, Hence, and for any and k = 1,2,3….n. Hence, P = Q.
Conversely, assume that P = Q, this implies, and for any and k = 1,2,3….n. Thus, Hence, we get K (P, Q) = 1.
The defined improved correlation coefficient formula also satisfies the properties in the above theorem.
3.3 Example
Let A = { r, 0,0,0,0} and B = { r, 0.4,0.2,0.5,0.1,0.2} be any two PNPS s in R. Therefore, by equation (3.1); we get K(A, B) =0.871.56. It shows that the above-defined improved correlation coefficient overcomes the disadvantages of the correlation coefficient.
In the following, we define a weighted correlation coefficient between PNPSs because differences among the elements are taken into account.
Let the weight of each element be given for k = 1,2…n; then the weighted correlation coefficient between the PNPSs A and B is defined as follows.
(A, B) = ) ) ) ) ) ]
(3.2)
If w = (1/n,1/n,1/n,….1/n) T, then equation (4) reduces to equation (3). (A, B) also satisfies the three properties in the above theorem.
3.4 Theorem
Let be the weight for each element (k = 1,2,…n), [0,1] and then the weighted correlation coefficient between the PNPS s A and B, which is denoted by (A, B) defined in equation ( ) satisfies the following properties.
(A, B) = (B, A);
(A, B) ;
(A, B) = 1 iff A = B.
The proof is similar to that of the properties in Theorem 3.1.
Decision Making using the improved correlation coefficient of PNPS s
Multiple attribute decision-making (MADM) problems refer to making decisions when several attributes are involved in real-life problems. For example, one may buy a vehicle by analysing the attributes given in terms of price, style, safety, comfort, etc.
Here we consider a multiple-attribute decision-making problem with pentapartitioned neutrosophic Pythagorean information, and the characteristic of an alternative (i = 1,2,…m) on an attribute (j = 1,2…n) is represented by the following PNPS s:
= {(, \ }
Where and
0 for and I = 1,2,…m.
To make it convenient, we are considering the following five functions in terms of pentapartitioned neutrosophic Pythagorean value (PNPV)
Here, the values are usually derived from the evaluation of an alternative with respect to criteria by the expert or decision maker. Therefore, we obtain a pentapartitioned neutrosophic Pythagorean decision matrix.
In the case of an ideal alternative, an ideal PNPV can be defined by
= ( = (1,1,0,0,0)(j = 1,2…n) in the decision making method,
For i = 1,2….m and j = 1,2….n.
By using the above weighted correlation coefficient, we can derive the ranking order of all alternatives and choose the best one among them.
4.1 Example
This section presents an example of a multiple-attribute decision-making problem with the given alternatives corresponding to criteria under the pentapartitioned neutrosophic Pythagorean environment.
For this example, the three potential alternatives are evaluated under four different attributes. The types of intellectual property rights are the alternatives, and the various cybercrimes are the attributes of this example. The three potential alternatives are copyright, patent rights, and trademarks, and the four different attributes are infringement, piracy, cybersquatting, and hacking. The evaluation of an alternative with respect to an attribute is obtained from a domain expert’s questionnaire. According to the attributes, we derive the ranking order of all alternatives, and based on this ranking order, the decision-maker selects the best one.
By assigning the weight vector of the above attributes as w = (0.2,0.35,0.25,0.2), the alternatives are evaluated under the four attributes in the form of PNPSs; in general, the evaluation of an alternative Ai with respect to the attributes Cj (i=1,2,3, j=1,2,3,4) will be done by the questionnaire of a domain expert. In particular, while asking the opinion about an alternative A1 with respect to an attribute C1, the possibility that he (or) she says the statement is true is 0.4, both true and false is 0.3, neither true nor false is 0.2, false is 0.1, and unknown is 0.4. It can be denoted in neutrosophic notation as d11 = (0.4,0.3,0.4,0.2,0.1).
| \ | ||||
| [0.4,0.3,0.4,0.2,0.1] | [0.5.0.4,0.5,0.3,0.2] | [0.4,0.1,0.4,0.1,0.1] | [0.6,0.2,0.6,0.3,0.2] | |
| [0.4,0.2,0.6,0.1,0.2] | [0.3,0.3,0.5,0.2,0.1] | [0.1,0.4,0.2,0.3,0.2] | [0.5,0.3,0.6,0.1,0.1] | |
| [0.3,0.4,0.4,0.3,0.4] | [0.5,0.1,0.6,0.2,0.1] | [0.4,0.5,0.4,0.3,0.2] | [0.3,0.2,0.5,0.2,0.2] |
Then, by using the proposed method, we obtain the most desirable alternative. We can get the values of the correlation coefficient Mw (Ai, A∗) (i = 1,2,3) by using Equation (3.3).
Hence, Mw (A1, A∗) = 0.586276, Mw (A2, A∗) = 0.5640, Mw (A3, A∗) = 0.56921.
Thus, the ranking order of the three potential alternatives is A1>A3>A2. Therefore, we can say that alternative A1, copyright, has more cyber-related problems involving original literary, dramatic, musical, artistic, cinematographic, sound-recording, and computer-program works than the other intellectual-property-right alternatives. The decision-making method provided in this paper is more judicious and robust.
Conclusion
In this paper, we have outlined the improved correlation coefficient of PNPSs, which is applicable in cases where the correlation coefficient of PNPSs is undefined or not meaningful. We have also studied its properties. Decision-making is a process that plays a significant role in real-world issues. A common step in decision-making is recognizing a problem or opportunity and deciding how to address it. Here, we have discussed a decision-making technique using the improved correlation of PNPSs. An illustrative example is provided for a multiple-attribute decision-making problem involving several alternatives evaluated according to varied criteria. Therefore, our proposed improved correlation of PNPSs helps identify the most appropriate alternative for the decision-maker based on the given criteria.
Funding: “This research received no external funding.”
Acknowledgements: I would like to express my special thanks and gratitude to S.P. Rhea and R. Kathiresan for their guidance and constant support in completing my paper.
Conflicts of Interest: “The authors declare no conflict of interest.”
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