Unveiling novel eccentric neighborhood forgotten indices for graphs and gaph operations: A comprehensive exploration of boiling point prediction

Suha Wazzan; Hanan Ahmed; Suha Wazzan; Hanan Ahmed

doi:10.3934/math.2024056

AIMS Mathematics

2024, Volume 9, Issue 1: 1128-1165. doi: 10.3934/math.2024056

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Research article Special Issues

Unveiling novel eccentric neighborhood forgotten indices for graphs and gaph operations: A comprehensive exploration of boiling point prediction

Suha Wazzan ^{1
,
,},
Hanan Ahmed ²

1.
Department of Mathematics, Science Faculty, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Ibb University, Ibb 70270, Yemen

Received: 24 August 2023 Revised: 09 November 2023 Accepted: 13 November 2023 Published: 06 December 2023
MSC : 05C12, 05C90

This paper marks a significant advancement in the field of chemoinformatics with the introduction of two novel topological indices: the forgotten eccentric neighborhood index (FENI) and the modified forgotten eccentric neighborhood index (MFENI). Uniquely developed for predicting the boiling points of various chemical substances, these indices offer groundbreaking tools in understanding and interpreting the thermal properties of compounds. The distinctiveness of our study lies in the in-depth exploration of the discriminative capabilities of FENI and MFENI. Unlike existing indices, they provide a nuanced capture of structural features essential for determining boiling points, a key factor in drug design and chemical analysis. Our comprehensive analyses demonstrate the superior predictive power of FENI and MFENI, highlighting their exceptional potential as innovative tools in the realms of chemoinformatics and pharmaceutical research. Furthermore, this study conducts an extensive investigation into their various properties. We present explicit results on the behavior of these indices in relation to diverse graph types and operations, including join, disjunction, composition and symmetric difference. These findings not only deepen our understanding of FENI and MFENI but also establish their practical versatility across a spectrum of chemical and pharmaceutical applications. Thus the introduction of FENI and MFENI represents a pivotal step forward in the predictive analysis of boiling points, setting a new standard in the field and opening avenues for future research advancements.

Keywords:

eccentric neighborhood forgotten indices,
modified eccentric neighborhood forgotten indices,
boiling point,
graph operations

Citation: Suha Wazzan, Hanan Ahmed. Unveiling novel eccentric neighborhood forgotten indices for graphs and gaph operations: A comprehensive exploration of boiling point prediction[J]. AIMS Mathematics, 2024, 9(1): 1128-1165. doi: 10.3934/math.2024056

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Abstract

1. Introduction

Multiple attribute decision making (MADM) is the necessary context of the decision making science whose aim is to recognize the most exceptional targets among the feasible ones. In real decision making, the person needs to furnish the evaluation of the given choices by various types of evaluation conditions such as crisp numbers, intervals, etc. However, in many cases, it is difficult for a person to opt for a suitable one due to the presence of several kinds of uncertainties in the data, which may occur due to a lack of knowledge or human error. Accordingly, to quantify such risks and to examine the process, a large-scale family of the theory such as fuzzy set (FS) ^[1] and its extensions as intuitionistic FS (IFS) ^[2], cubic intuitionistic fuzzy set ^[3], interval-valued IFS ^[4], linguistic interval-valued IFS ^[5], are appropriated by the researchers. In all these theories, an object is judged by an expert in terms of their two membership degrees such that their sum can't exceed one. Since its presence, various researchers have presented their ideas to illuminate decision-making (DM) problems by using agammaegation operators or information measures ^{[6,7,8,9,10,11]}. For instance, Xu and Yager ^[12] performed the weighted geometric operators for IFSs. Garg ^[9,10] offered interactive weighted agammaegation operators with Einstein t-norm. Ali et al. ^[7] presented a graphical method to rank the intuitionistic fuzzy values using the uncertainty index and entropy. Liu et al. ^[11] presented agammaegation operators based on the centroid transformations of intuitionistic fuzzy value. Garg and Kumar ^[13] presented new similarity measures for IFSs based on the connection number of set pair analysis.

Among all these different ideas, one is to locate the most appropriate alternative by making utilization of correlation measures which assumes a noteworthy job in statistical and designing applications as it furnishes us with an estimation of the interdependency of two factors. In the statistical investigation, the correlation measures assume as a critical part to degree the linear relationship between two factors, while in the FS hypothesis the correlation measure decides the degree of reliance between two FSs. As a vital content in fuzzy mathematics, the relationship between IFSs has also gained many considerations for their wide application in the real world, such as pattern recognition, decision making, and market expectation. A few strategies to calculate the correlation coefficient between IFSs have been proposed and investigated in recent years. Towards this direction, the correlation coefficients for FSs have been, firstly, characterized by Hung and Wu ^[14]. The correlation coefficients of IFSs have been firstly introduced by Gerstenkorn and Manko ^[15] which is afterward examined by Szmidth and Kacprzyk ^[16]. The research on correlation measure for interval-valued IFSs has been done by Bustince and Burillo ^[17]. Garg ^[18] investigated the correlation measures for Pythagorean FSs. Garg ^[19] proposed the correlation coefficients for the intuitionistic multiplicative sets and connected them to fathom the issues of pattern recognition. Garg and Kumar ^[20] presented the correlation coefficient for IFSs by using the connection number of the set pair analysis.

However, apart from that technique for order preference with respect to the similarity to the ideal solution (TOPSIS), created by Hwang and Yoon ^[21], is a notable technique to fathom DM issues. The objective of this technique is to select the best alternative which obtains the minimum distance from the positive ideal solution. After their presence, various endeavors are made by the analysts to utilize the TOPSIS strategy in fuzzy and IFS environment. For instance, Szmidt and Kacprzyk ^[22] characterized the idea of a distance measure between the IFSs. Hung and Yang ^[23] introduced the similarity measures between the two distinctive IFSs based on Hausdorff distance. Duegenci ^[24] exhibited a distance measure for interval-valued IFS (IVIFS) and their applications to DM with inadequate weight data. Garg ^[25] introduced the TOPSIS method along with an improved score function for IVIFSs. Mohammadi et al. ^[26] presented a gray relational analysis and TOPSIS approach to solving the DM problems. Garg et al. ^[27] proposed an entropy measure under the IFS environment and applied it to illustrate DM issues. Biswas and Kumar ^[28] investigated an integrated TOPSIS method to solve decision-making problems with IVIFS information. Li ^[29] presented a nonlinear programming methodology based on TOPSIS methodology with IVIFS information. Garg and Arora ^[30] extended Li ^[29] approach to the interval-valued intuitionistic fuzzy soft set environment. In Wang and Chen ^[31], Gupta et al. ^[32], authors developed a group decision-making method under the IVIFS environment by integrating extended TOPSIS and linear programming methods. Kumar and Garg ^[33,34] presented TOPSIS methodology to solve MADM problems by utilizing a connection number of the set pair analysis theory. Zhang et al. ^[35] presented a TOPSIS approach based on the covering-based variable precision intuitionistic fuzzy rough set to solve the decision-making problems.

Although the prevailing theories are widely used by the researchers, they have restrictions because of their inadequacy over the parameterizations tool and consequently the decision-maker(s) can't give an accurate decision. To overwhelm these disadvantages, Molodtsov ^[36] collaborated the soft set (SS) theory in which ratings are given on specific parameters. Maji et al. ^[37,38] extended this theory by joining it with existing FS and IFS theoretical approach and developed with the idea of fuzzy soft set (FSS) and intuitionistic fuzzy soft set (IFSS) respectively. In IFSS theory, the preferences of the experts are given over a set of parameters. For instance, in case an individual needs to buy a laptop then according to the IFS theory the information is captured corresponding to a single parameter on two classifications- one is acceptance degree and the other is degree of rejection, whereas in IFSS, a deep insight about the specifications of laptop are considered such as "price", "processor", "RAM", "screen size". In IFSS theory, the information is gathered based on more than one parameter which makes it more generalized than IFS theory. Since their appearance, some basic properties ^[39], entropy measures ^[40,41], distance and similarity measures ^[42,43] are proposed by several researchers using the soft set features. To agammaegate the IFSS information, Arora and Garg ^[44,45], firstly, established an averaging and geometric agammaegation, as well as, the prioritized averaging and geometric agammaegation operators to solve the MADM problems by agammaegating the different intuitionistic fuzzy soft numbers (IFSNs). Later on, Garg and Arora ^[46] presented the concept of generalized and group generalized IFSS and connected it to illuminate MADM problems. Feng et al. ^[47] presented an application of the decision-making problems using the generalized intuitionistic fuzzy soft sets. Arora and Garg ^[48] describes an approach for solving decision-making problems with correlation measures having dual hesitant fuzzy soft set information. Garg and Arora ^[49] presented some generalized Archimedean t-norm based agammaegation operators for IFSS information. ^[50,51] defined the similarity measures for the pairs of the IFSSs. Petchimuthu et al. ^[52] defined the generalized product of the fuzzy soft matrices to solve the decision-making problems. Zhan et al. ^[53] presented the relationships among rough sets, soft sets, and hemirings and hence introduced the concept of soft rough hemirings. Garg and Arora ^[54] presented a Bonferroni mean agammaegation operators by considering the influence of the two parameters into the analysis. Zhan et al. ^[55] introduced the Z-soft fuzzy rough set employing three uncertain models: soft sets, rough sets, and fuzzy sets. Garg and Arora ^[56] investigated distance measures for dual hesitant fuzzy FSSs and applied them to solve the MADM problems. Hu et al. ^[57] presented a similarity measure of IFSSs and applied it to solve the group decision-making model for medical diagnosis. Currently, research on the hypothetical and application perspectives of SSs and its different extensions are advancing quickly.

Considering the versatility of IFSS and the importance of the TOPSIS method, this paper extends the TOPSIS strategy to the IFSS environment, where the components are given in terms of intuitionistic fuzzy soft numbers. To address it completely, in the present paper, we develop novel correlation coefficients to measure the degree of dependency of two or more IFSSs. The several properties of the proposed measure are investigated. Further, in contrast to the classical TOPSIS strategy which is based on a separate degree, this paper applies the proposed correlation measure to set up the comparative index of the closeness coefficient. The presented method has been illustrated with a numerical example and the advantages, the superiorities of the method is shown over the classical TOPSIS method. To reach the target precisely, we extended the given TOPSIS approach for solving the MADM problems. Holding all the above tips in mind, the main objective of the present work is listed as

(ⅰ) to define some new correlation measures for given numbers under the IFSS environment.

(ⅱ) to develop an algorithm to determine the MADM problems based on the extended TOPSIS approach.

(ⅲ) to test the presented approach with a numerical example.

The correlation measures for IFSS are defined for the pairs of IFSSs which are also utilized to calculate the interrelation as well as the extent of dependency of one element over the other. Since the existing intuitionistic fuzzy set is a special case of the considered IFSS and hence the proposed measure is more generalized than the existing ones. Apart from this, the major assets of the presented TOPSIS method over the others as in the basic TOPSIS method, different distance, and similarity measures are usually involved to compute the closeness coefficient. However, in our presented TOPSIS method, the correlation coefficient is utilized to calculate the closeness coefficient. The main advantage of taking the correlation coefficient is that it preserves the linear relationship among the elements under consideration.

The rest of the text is summarized as. Section 2 presents a basic concept related to soft sets, FSS and IFSS. In Section 3, we define new correlation measures for the IFSSs and studied their properties. In Section 4, an extended TOPSIS method is presented based on the proposed correlation coefficient. In Section 5, we present an algorithm, based on the proposed TOPSIS method, to solve the MADM problem and illustrate with a numerical example. Further, the outcomes of the proposed algorithm are compared with the existing approaches to justify the proficiency of the proposed work. Finally, a concrete conclusion is given in Section 6.

2. Preliminaries

In this section, we discuss some basic terms associated with soft set theory. Let $\mathbf{E}$ be a set of parameters and $\mathbf{U}$ be the set of experts.

Definition 2.1. ^[36] A pair $(\mathbf{F}, \mathbf{E})$ is called as soft set, if $\mathbf{F}$ is a map defined as $\mathbf{F}: \mathbf{E}\to \mathbf{K}^\mathbf{U}$ where $\mathbf{K}^\mathbf{U}$ is a set of all subsets of $\mathbf{U}$ .

Definition 2.2. ^[36] Let $A, B\subset \mathbf{E}$ and $(\mathbf{F}, A)$ , $(\mathbf{G}, B)$ be two soft sets over $\mathbf{U}$ . Then, the basic operations over them are stated as

1) $(\mathbf{F}, A) \subseteq (\mathbf{G}, B)$ if $A\subseteq B$ and $\mathbf{F}(e) \subseteq \mathbf{G}(e)$ , $\forall e\in A$ .

2) $(\mathbf{F}, A) = (\mathbf{G}, B)$ if $(\mathbf{F}, A)\subseteq(\mathbf{G}, B)$ and $(\mathbf{G}, B)\subseteq(\mathbf{F}, A)$ .

3) Complement: $(\mathbf{F}, A)^c = (\mathbf{F}^c, A)$ , where $\mathbf{F}^c: A \to \mathbf{K}^\mathbf{U}$ defined as $\mathbf{F}^c(e) = \mathbf{U}-\mathbf{F}(e)$ , $\forall e \in A$ .

Definition 2.3. ^[37] A map $\mathbf{F}: \mathbf{E}\to \mathbf{F}^\mathbf{U}$ is called fuzzy soft set defined as

$\begin{eqnarray} \mathbf{F}_{u_i}(e_j) = \left\{(u_i, \zeta_j(u_i)) \mid u_i \in \mathbf{U} \right\}, \end{eqnarray}$

(2.1)

where $\mathbf{F}^\mathbf{U}$ be a set of all fuzzy subsets of $\mathbf{U}$ and $\zeta_j(u_i)$ is acceptance degree of an expert $u_i$ over parameter $e_j\in \mathbf{E}$ .

Definition 2.4. ^[37] For $A, B\subset \mathbf{E}$ and $(\mathbf{F}, A)$ , $(\mathbf{G}, B)$ be any two fuzzy soft sets over $\mathbf{U}$ , then

1) $(\mathbf{F}, A)\subseteq (\mathbf{G}, B)$ if, $A\subset B$ and $\mathbf{F}(e) \leq \mathbf{G}(e)$ for each $e\in A$ .

2) $(\mathbf{F}, A) = (\mathbf{G}, B)$ are equal if $(\mathbf{F}, A) \subseteq (\mathbf{G}, B)$ and $(\mathbf{G}, B) \subseteq (\mathbf{F}, A)$ .

3) Complement: $(\mathbf{F}^c, A)$ where for each $e\in A$ , $\mathbf{F}^c(e) = 1-\mathbf{F}(e)$ .

Definition 2.5. ^[38] A mapping $\mathbf{F}: \mathbf{E}\rightarrow \mathbf{IF}^\mathbf{U}$ is called as intuitionistic fuzzy soft set defined as

$\begin{eqnarray} \mathbf{F}_{u_i}(e_j) = \left\{(u_i, \zeta_j(u_i), \vartheta_j(u_i)) \mid u_i \in \mathbf{U} \right\}, \end{eqnarray}$

(2.2)

where $\mathbf{IF}^\mathbf{U}$ is the intuitionistic fuzzy subsets of $\mathbf{U}$ and $\zeta_j$ and $\vartheta_j$ are "acceptance degree" and "rejection degree" respectively, with $0\leq \zeta_j(u_i), \vartheta_j(u_i), \zeta_j(u_i) + \vartheta_j(u_i) \leq 1$ for all $u_i\in \mathbf{U}$ . For simplicity, we denote the pair of $\mathbf{F}_{u_i}(e_j)$ as $\beta_{ij} = (\zeta_{ij}, \vartheta_{ij})$ or $(\mathbf{F}, \mathbf{E}) = (\zeta_{ij}, \vartheta_{ij})$ and called as an intuitionistic fuzzy soft number (IFSN).

Definition 2.6. ^[45] For an IFSN $(\mathbf{F}, \mathbf{E}) = (\zeta, \vartheta)$ , a score function is defined as

$\begin{eqnarray} \text{Sc}(\mathbf{F}, \mathbf{E}) = \zeta-\vartheta, \end{eqnarray}$

(2.3)

while an accuracy function is

$\begin{eqnarray} H(\mathbf{F}, \mathbf{E}) = \zeta + \vartheta \end{eqnarray}$

(2.4)

Definition 2.7. ^[45] An order relation to compare the two IFSNs $(\mathbf{F}, \mathbf{E})$ and $(\mathbf{G}, \mathbf{E})$ , denoted by $(\mathbf{F}, \mathbf{E}) \succ (\mathbf{G}, \mathbf{E})$ , holds if either of the condition holds:

1) $\text{Sc}(\mathbf{F}, \mathbf{E}) > \text{Sc}(\mathbf{G}, \mathbf{E})$ ;

2) $\text{Sc}(\mathbf{F}, \mathbf{E}) = \text{Sc}(\mathbf{G}, \mathbf{E})$ and $H(\mathbf{F}, \mathbf{E}) > H(\mathbf{G}, \mathbf{E})$ .

Here " $\succ$ " represent "preferred to".

3. Correlation coefficients for IFSSs

Let $\mathbf{E} = \{e_1, e_2, \ldots, e_m\}$ be a set of parameters and $\mathbf{U} = \{u_1, u_2, \ldots, u_n\}$ be the set of experts. In this section, we propose some correlation coefficients under the IFSS environment which are summarized as follows:

Definition 3.1. Let $(\mathbf{J}, \mathbf{E}) = \{u_i, (\zeta_{\mathbf{J}_{j}}(u_{i}), \vartheta_{\mathbf{J}_{j}}(u_{i})) \mid u_i \in \mathbf{U}\}$ and $(\mathbf{K}, \mathbf{E}) = \{u_i, (\zeta_{\mathbf{K}_{j}}(u_{i}), \vartheta_{\mathbf{K}_{j}}(u_{i})) \mid u_i \in \mathbf{U}\}$ be two IFSSs defined over a set of parameters $\mathbf{E} = \{e_1, e_2, \ldots, e_m\}$ . Then, the informational intuitionistic energies of two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ are described as:

$\begin{eqnarray} E_{\text{IFSS}}(\mathbf{J}, \mathbf{E})& = &\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\zeta_{\mathbf{J}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{J}_{j}}^{2}(u_{i})\right) \end{eqnarray}$

(3.1)

$\begin{eqnarray} E_{\text{IFSS}}(\mathbf{K}, \mathbf{E})& = &\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right) \end{eqnarray}$

(3.2)

The correlation of two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ can be described as:

$\begin{eqnarray} C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big) = \sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left( \zeta_{\mathbf{J}_{j}}(u_{i})\zeta_{\mathbf{K}_{j}}(u_{i})+ \vartheta_{\mathbf{J}_{j}}(u_{i})\vartheta_{\mathbf{K}_{j}}(u_{i})\right) \end{eqnarray}$

(3.3)

It can be observed that the following properties hold for the correlation of IFSSs.

(ⅰ) $C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{J}, \mathbf{E})\big) = E_{IFSS}(\mathbf{J}, \mathbf{E});$

(ⅱ) $C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big) = C_{IFSS}\big((\mathbf{K}, \mathbf{E}), (\mathbf{J}, \mathbf{E})\big).$

Definition 3.2. The correlation coefficient between two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ is given by $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ and is described as:

$\begin{eqnarray} \rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)& = &\frac{C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big)}{\sqrt{E_{\text{IFSS}}(\mathbf{J}, \mathbf{E}).E_{\text{IFSS}}(\mathbf{K}, \mathbf{E})}}\\ & = & \frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left( \zeta_{\mathbf{J}_{j}}(u_{i})\zeta_{\mathbf{K}_{j}}(u_{i})+ \vartheta_{\mathbf{J}_{j}}(u_{i})\vartheta_{\mathbf{K}_{j}}(u_{i})\right)}{\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{J}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{J}_{j}}^{2}(u_{i})\right)}\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)}} \end{eqnarray}$

(3.4)

Theorem 3.1. The correlation coefficient between two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ stated in Eq. (3.4), fulfill the properties as follows:

(P1) $0\leq \rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1;$

(P2) $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) = \rho_{1}\left((\mathbf{K}, \mathbf{E}), (\mathbf{J}, \mathbf{E})\right);$

(P3) If $(\mathbf{J}, \mathbf{E}) = (\mathbf{K}, \mathbf{E}),$ i.e. if $\forall i, j, \zeta_{\mathbf{J}_{j}}(u_{i}) = \zeta_{\mathbf{K}_{j}}(u_{i})$ and $\vartheta_{\mathbf{J}_{j}}(u_{i}) = \vartheta_{\mathbf{K}_{j}}(u_{i}),$ then $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) = 1.$

Proof. For two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ , we have

(P1) The inequality $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)\geq 0$ is obvious, so we just need to prove $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1.$

$\begin{eqnarray*} && C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big)\\ & = & \sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left( \zeta_{\mathbf{J}_{j}}(u_{i})\zeta_{\mathbf{K}_{j}}(u_{i})+ \vartheta_{\mathbf{J}_{j}}(u_{i})\vartheta_{\mathbf{K}_{j}}(u_{i})\right)\\ & = & \sum\limits_{j = 1}^{m}\left(\zeta_{\mathbf{J}_{j}}(u_{1})\zeta_{\mathbf{K}_{j}}(u_{1})+ \vartheta_{\mathbf{J}_{j}}(u_{1})\vartheta_{\mathbf{K}_{j}}(u_{1})\right)+ \sum\limits_{j = 1}^{m}\left(\zeta_{\mathbf{J}_{j}}(u_{2})\zeta_{\mathbf{K}_{j}}(u_{2})+ \vartheta_{\mathbf{J}_{j}}(u_{2})\vartheta_{\mathbf{K}_{j}}(u_{2})\right) \\ &&+ \ldots\ldots + \sum\limits_{j = 1}^{m}\left(\zeta_{\mathbf{J}_{j}}(u_{n})\zeta_{\mathbf{K}_{j}}(u_{n})+ \vartheta_{\mathbf{J}_{j}}(u_{n})\vartheta_{\mathbf{K}_{j}}(u_{n})\right)\\ & = &\left\{ \begin{aligned} &\Big(\zeta_{\mathbf{J}_{1}}(u_{1})\zeta_{\mathbf{K}_{1}}(u_{1})+ \vartheta_{\mathbf{J}_{1}}(u_{1})\vartheta_{\mathbf{K}_{1}}(u_{1})\Big)+\Big(\zeta_{\mathbf{J}_{2}}(u_{1})\zeta_{\mathbf{K}_{2}}(u_{1})+ \vartheta_{\mathbf{J}_{2}}(u_{1})\vartheta_{\mathbf{K}_{2}}(u_{1})\Big)\\ &+\ldots\ldots +\Big(\zeta_{\mathbf{J}_{m}}(u_{1})\zeta_{\mathbf{K}_{m}}(u_{1})+ \vartheta_{\mathbf{J}_{m}}(u_{1})\vartheta_{\mathbf{K}_{m}}(u_{1})\Big) \end{aligned} \right\}\\ &+& \left\{ \begin{aligned} &\Big(\zeta_{\mathbf{J}_{1}}(u_{2})\zeta_{\mathbf{K}_{1}}(u_{2})+ \vartheta_{\mathbf{J}_{1}}(u_{2})\vartheta_{\mathbf{K}_{1}}(u_{2})\Big)+\Big(\zeta_{\mathbf{J}_{2}}(u_{2})\zeta_{\mathbf{K}_{2}}(u_{2})+ \vartheta_{\mathbf{J}_{2}}(u_{2})\vartheta_{\mathbf{K}_{2}}(u_{2})\Big)\\ &+\ldots\ldots +\Big(\zeta_{\mathbf{J}_{m}}(u_{2})\zeta_{\mathbf{K}_{m}}(u_{2})+ \vartheta_{\mathbf{J}_{m}}(u_{2})\vartheta_{\mathbf{K}_{m}}(u_{2})\Big)\Big\} \end{aligned} \right\}\\ &+& \; \vdots\\ && \; \vdots\\ &+& \left\{ \begin{aligned} &\Big(\zeta_{\mathbf{J}_{1}}(u_{n})\zeta_{\mathbf{K}_{1}}(u_{n})+ \vartheta_{\mathbf{J}_{1}}(u_{n})\vartheta_{\mathbf{K}_{1}}(u_{n})\Big)+\Big(\zeta_{\mathbf{J}_{2}}(u_{n})\zeta_{\mathbf{K}_{2}}(u_{n})+ \vartheta_{\mathbf{J}_{2}}(u_{n})\vartheta_{\mathbf{K}_{2}}(u_{n})\Big)\\ &+\ldots\ldots +\Big(\zeta_{\mathbf{J}_{m}}(u_{n})\zeta_{\mathbf{K}_{m}}(u_{n})+ \vartheta_{\mathbf{J}_{m}}(u_{n})\vartheta_{\mathbf{K}_{m}}(u_{n})\Big)\Big\} \end{aligned} \right\}\\ & = & \sum\limits_{j = 1}^{m}\Big(\zeta_{\mathbf{J}_{j}}(u_{1})\zeta_{\mathbf{K}_{j}}(u_{1})+ \zeta_{\mathbf{J}_{j}}(u_{2})\zeta_{\mathbf{K}_{j}}(u_{2})+\ldots+ \zeta_{\mathbf{J}_{j}}(u_{n})\zeta_{\mathbf{K}_{j}}(u_{n})\Big) \\ &+& \sum\limits_{j = 1}^{m}\Big(\vartheta_{\mathbf{J}_{j}}(u_{1})\vartheta_{\mathbf{K}_{j}}(u_{1})+ \vartheta_{\mathbf{J}_{j}}(u_{2})\vartheta_{\mathbf{K}_{j}}(u_{2})+\ldots+ \vartheta_{\mathbf{J}_{j}}(u_{n})\vartheta_{\mathbf{K}_{j}}(u_{n})\Big) \end{eqnarray*}$

By applying Cauchy-Schwarz inequality:

$\begin{eqnarray*} \left(x_{1}y_{1}+x_{2}y_{2}+\ldots+x_{n}y_{n}\right)^{2}\leq \left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)\left(y_{1}^{2}+y_{2}^{2}+\ldots+y_{n}^{2}\right) \end{eqnarray*}$

we have

$\begin{eqnarray*} \left(C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big)\right)^{2}&\leq& \left\{ \begin{aligned} &\sum\limits_{j = 1}^{m}\Big(\zeta_{\mathbf{J}_{j}}^{2}(u_{1})+ \zeta_{\mathbf{J}_{j}}^{2}(u_{2})+\ldots+ \zeta_{\mathbf{J}_{j}}^{2}(u_{n})\Big)\\ & +\sum\limits_{j = 1}^{m}\Big(\vartheta_{\mathbf{J}_{j}}^{2}(u_{1})+ \vartheta_{\mathbf{J}_{j}}^{2}(u_{2})+\ldots+ \vartheta_{\mathbf{J}_{j}}^{2}(u_{n})\Big) \end{aligned} \right\}\\ && \times \left\{ \begin{aligned} &\sum\limits_{j = 1}^{m}\Big(\zeta_{\mathbf{K}_{j}}^{2}(u_{1})+ \zeta_{\mathbf{K}_{j}}^{2}(u_{2})+\ldots+ \zeta_{\mathbf{K}_{j}}^{2}(u_{n})\Big)\\ & +\sum\limits_{j = 1}^{m}\Big(\vartheta_{\mathbf{K}_{j}}^{2}(u_{1})+ \vartheta_{\mathbf{K}_{j}}^{2}(u_{2})+\ldots+ \vartheta_{\mathbf{K}_{j}}^{2}(u_{n})\Big) \end{aligned} \right\}\\ & = & \left\{\sum\limits_{j = 1}^{m} \sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{J}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{J}_{j}}^{2}(u_{i})\right)\right\}\times \left\{\sum\limits_{j = 1}^{m} \sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)\right\}\\ & = & E_{IFSS}\big((\mathbf{J}, \mathbf{E})\big). E_{IFSS}\big((\mathbf{K}, \mathbf{E})\big) \end{eqnarray*}$

Therefore, $\left(C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big)\right)^{2}\leq E_{IFSS}\big((\mathbf{J}, \mathbf{E})\big). E_{IFSS}\big((\mathbf{K}, \mathbf{E})\big).$ Thus, from Eq. (3.4), we get $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1$ and hence $0\leq \rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1.$

(P2) It is obvious so omitted here.

(P3) As we have $\zeta_{\mathbf{J}_{j}}(u_{i}) = \zeta_{\mathbf{K}_{j}}(u_{i})$ and $\vartheta_{\mathbf{J}_{j}}(u_{i}) = \vartheta_{\mathbf{K}_{j}}(u_{i}), \forall i, j,$ then from Eq. (3.4), we get

$\begin{eqnarray*} \rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) & = & \frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left( \zeta_{\mathbf{K}_{j}}(u_{i})\zeta_{\mathbf{K}_{j}}(u_{i})+ \vartheta_{\mathbf{K}_{j}}(u_{i})\vartheta_{\mathbf{K}_{j}}(u_{i})\right)}{\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)}\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)}}\\ & = &\frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)}{\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)}\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)}}\\ & = & 1 \end{eqnarray*}$

Thus, $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) = 1$ and hence the result.

Definition 3.3. For two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E}),$ the correlation coefficient between them is described as:

$\begin{eqnarray} \rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)& = &\frac{C_{IFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big)}{\max\left\{E_{\text{IFSS}}(\mathbf{J}, \mathbf{E}), E_{\text{IFSS}}(\mathbf{K}, \mathbf{E})\right\}}\\ & = & \frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left( \zeta_{\mathbf{J}_{j}}(u_{i})\zeta_{\mathbf{K}_{j}}(u_{i})+ \vartheta_{\mathbf{J}_{j}}(u_{i})\vartheta_{\mathbf{K}_{j}}(u_{i})\right)}{\max\left\{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{J}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{J}_{j}}^{2}(u_{i})\right), \sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n} \left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)\right\}} \end{eqnarray}$

(3.5)

Theorem 3.2. The correlation coefficient between two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ denoted as $\rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ stated in Eq. (3.5), fulfill the properties as follows:

(P1) $0\leq \rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1;$

(P2) $\rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) = \rho_{2}\left((\mathbf{K}, \mathbf{E}), (\mathbf{J}, \mathbf{E})\right);$

(P3) If $(\mathbf{J}, \mathbf{E}) = (\mathbf{K}, \mathbf{E}),$ i.e. if $\forall i, j, \zeta_{\mathbf{J}_{j}}(u_{i}) = \zeta_{\mathbf{K}_{j}}(u_{i})$ and $\vartheta_{\mathbf{J}_{j}}(u_{i}) = \vartheta_{\mathbf{K}_{j}}(u_{i}),$ then $\rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) = 1.$

Proof. (P1) Since $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ are the IFSSs therefore, the inequality $\rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)\geq 0$ is obvious. The only need is to prove $\rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1,$ which we can easily prove by applying Cauchy-Schwarz inequality:

$\begin{eqnarray} \sum\limits_{i = 1}^{n}x_{i}y_{i}\leq \sqrt{\left(\sum\limits_{i = 1}^{n}x_{i}^{2}\right).\left(\sum\limits_{i = 1}^{n}y_{i}^{2}\right)} \end{eqnarray}$

(3.6)

with equality if and only if the two vectors $x = (x_1, x_2, \ldots, x_n)$ and $y = (y_1, y_2, \ldots, y_n)$ are linearly dependent. Also, from Eq. (3.6), we have

$\begin{eqnarray*} \sum\limits_{i = 1}^{n}x_{i}y_{i}&\leq& \sqrt{\left(\sum\limits_{i = 1}^{n}x_{i}^{2}\right).\left(\sum\limits_{i = 1}^{n}y_{i}^{2}\right)} \leq \sqrt{\left(\max\left\{\sum\limits_{i = 1}^{n}x_{i}^{2}, \sum\limits_{i = 1}^{n}y_{i}^{2}\right\}\right)^{2}} = \max\left\{\sum\limits_{i = 1}^{n}x_{i}^{2}, \sum\limits_{i = 1}^{n}y_{i}^{2}\right\} \end{eqnarray*}$

and hence, by Eq. (3.5), we have $\rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1.$ The proof of remaining parts are similar as in Theorem 3.1.

However, in numerous viable circumstances, the distinctive set can take diverse weights and in this way, weight $\eta_{i}$ for the expert $u_{i}$ and $\xi_{j}$ for the parameter $e_{j},$ ought to be taken under consideration. Let $\eta = (\eta_{1}, \eta_{2}, \ldots, \eta_{n})$ be the weight vector for the experts $u_{i}, (i = 1, 2, \ldots, n)$ such that $\eta_{i} > 0,$ $\sum_{i = 1}^{n}\eta_{i} = 1$ and let the weight vector of the parameters $e_{j}, (j = 1, 2, \ldots, m)$ is $\xi = (\xi_{1}, \xi_{2}, \ldots, \xi_{m})$ with $\xi > 0, \sum_{j = 1}^{m}\xi_{j} = 1.$ Further, the above-defined correlation coefficient $\rho_{1},$ $\rho_{2}$ are extended to the weighted correlation coefficients between the IFSSs which are as follows:

Definition 3.4. For two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E}),$ the weighted correlation coefficient is described as:

$\begin{eqnarray} \rho_{3}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)& = &\frac{C_{WIFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big)}{\sqrt{E_{\text{WIFSS}}(\mathbf{J}, \mathbf{E}).E_{\text{WIFSS}}(\mathbf{K}, \mathbf{E})}}\\ & = & \frac{\sum\limits_{j = 1}^{m}\xi_{j}\left(\sum\limits_{i = 1}^{n} \eta_{i}\left( \zeta_{\mathbf{J}_{j}}(u_{i})\zeta_{\mathbf{K}_{j}}(u_{i})+ \vartheta_{\mathbf{J}_{j}}(u_{i})\vartheta_{\mathbf{K}_{j}}(u_{i})\right)\right)}{\sqrt{\sum\limits_{j = 1}^{m}\xi_{j}\left(\sum\limits_{i = 1}^{n} \eta_{i} \left(\zeta_{\mathbf{J}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{J}_{j}}^{2}(u_{i})\right)\right)}\sqrt{\sum\limits_{j = 1}^{m}\xi_{j}\left(\sum\limits_{i = 1}^{n} \eta_{i}\left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)\right)}} \end{eqnarray}$

(3.7)

Remark 3.1. If we take $\eta = \left(\frac{1}{n}, \frac{1}{n}, \ldots, \frac{1}{n}\right)$ and $\xi = \left(\frac{1}{m}, \frac{1}{m}, \ldots, \frac{1}{m}\right),$ then the correlation coefficient $\rho_{3}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ reduces to $\rho_{1}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ define in Eq. (3.4).

Theorem 3.3. For two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E}),$ the correlation coefficient $\rho_{3}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ fulfill the properties as follows:

(P1) $0\leq \rho_{3}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) \leq 1;$

(P2) $\rho_{3}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) = \rho_{3}\left((\mathbf{K}, \mathbf{E}), (\mathbf{J}, \mathbf{E})\right);$

(P3) If $(\mathbf{J}, \mathbf{E}) = (\mathbf{K}, \mathbf{E}),$ i.e. if $\forall i, j, \zeta_{\mathbf{J}_{j}}(u_{i}) = \zeta_{\mathbf{K}_{j}}(u_{i})$ and $\vartheta_{\mathbf{J}_{j}}(u_{i}) = \vartheta_{\mathbf{K}_{j}}(u_{i}),$ then $\rho_{3}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right) = 1.$

Proof. Proof is same as Theorem 3.1.

Definition 3.5. For two IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E}),$ the weighted correlation coefficient is described as:

$\begin{eqnarray} \rho_{4}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)& = &\frac{C_{WIFSS}\big((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\big)}{\max\left\{E_{\text{WIFSS}}(\mathbf{J}, \mathbf{E}).E_{\text{WIFSS}}(\mathbf{K}, \mathbf{E})\right\}}\\ & = & \frac{\sum\limits_{j = 1}^{m}\xi_{j}\left(\sum\limits_{i = 1}^{n} \eta_{i}\left( \zeta_{\mathbf{J}_{j}}(u_{i})\zeta_{\mathbf{K}_{j}}(u_{i})+ \vartheta_{\mathbf{J}_{j}}(u_{i})\vartheta_{\mathbf{K}_{j}}(u_{i})\right)\right)}{\max\left\{\sum\limits_{j = 1}^{m}\xi_{j}\left(\sum\limits_{i = 1}^{n} \eta_{i} \left(\zeta_{\mathbf{J}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{J}_{j}}^{2}(u_{i})\right)\right), \sum\limits_{j = 1}^{m}\xi_{j}\left(\sum\limits_{i = 1}^{n} \eta_{i}\left(\zeta_{\mathbf{K}_{j}}^{2}(u_{i})+\vartheta_{\mathbf{K}_{j}}^{2}(u_{i})\right)\right)\right\}} \qquad \end{eqnarray}$

(3.8)

Remark 3.2. If we take $\eta = \left(\frac{1}{n}, \frac{1}{n}, \ldots, \frac{1}{n}\right)$ and $\xi = \left(\frac{1}{m}, \frac{1}{m}, \ldots, \frac{1}{m}\right),$ then the correlation coefficient $\rho_{4}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ reduces to $\rho_{2}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ define in Eq. (3.5).

Theorem 3.4. The correlation coefficient $\rho_{4}\left((\mathbf{J}, \mathbf{E}), (\mathbf{K}, \mathbf{E})\right)$ defined in Eq. (3.8) fulfills same properties as those in Theorem 3.3.

Proof. Proof is same as Theorem 3.2.

From above defined correlation coefficients, we can easily examine that the correlation coefficient given in Eqs. (3.4) and (3.7), utilize the geometric mean between the information energies of the IFSSs $(\mathbf{J}, \mathbf{E})$ and $(\mathbf{K}, \mathbf{E})$ . On the other hand, the correlation coefficients defined in Eqs. (3.5) and (3.8), consider the greatest between them. For the idealistic evaluators, they tend to utilize the correlation coefficients characterized by Eqs. (3.4) and (3.7). As opposed to the positive thinking evaluators, the cynical evaluators have a tendency to apply the correlation coefficients characterized by Eqs. (3.5) and (3.8).

4. TOPSIS approach for solving DM problems based on correlation coefficient

This section presents a technique for solving DM problems based on the proposed correlation coefficient by extending the TOPSIS method for the IFSS environment.

The TOPSIS method is firstly introduced by Hwang and Yoon ^[21] and utilized to affirm the order of the assessment objects concerning the positive and negative ideal solutions of DM issues. This method depends on the fact that the best choice should have the minimum distance from the positive ideal solution and maximum distance from the negative ideal solution. A TOPSIS technique is displayed in which the correlation measure is used to perceive the positive and negative ideals along with the ranking of the choices. In most of the previous TOPSIS techniques, distinctive type of distance and comparability measures are used to discover the closeness coefficient. If close things are connected, at that point far off things, albeit less related, will be connected as well and in distinctive ways reflecting their integration versus isolation within the information investigation process. The advantage of taking correlation coefficient instead of taking distance or similarity measure in TOPSIS method is that correlation measure preserves the linear relationship among the factors under consideration. The algorithm to choose the most excellent option by utilizing TOPSIS method based on proposed correlation coefficient is described as follows:

4.1. Description of DM problem

Consider a DM problem in which evaluation is done to select the best choice from a set of distinct choices $B = \{B^{(1)}, B^{(2)}, \ldots, B^{(b)}\}.$ The ratings of the alternatives $B^{(z)}(z = 1, 2, \ldots, b)$ are given by a team of experts $\{u_1, u_2, \ldots, u_n\}$ by taking certain parameters $\{e_1, e_2, \ldots, e_m\}$ under consideration. The preferences of each alternative $B^{(z)}(z = 1, 2, \ldots, b)$ are described as IFSNs which is stated as $B_{ij}^{(z)} = \left (u_i, \zeta_{B_{j}}^{(z)}(u_i), \vartheta_{B_{j}}^{(z)}(u_i)\right)$ where $\zeta_{B_{j}}^{(z)}(u_i)$ and $\vartheta_{B_{j}}^{(z)}(u_i)$ indicate the degree of acceptance and rejection respectively, given by $u_i$ over parameter $e_j$ for an alternative $B^{(z)}.$ For simplicity, we can write $B_{ij}^{(z)}$ as $\beta_{ij}^{(z)} = \left(\zeta_{{ij}}^{(z)}, \vartheta_{{ij}}^{(z)}\right), (i = 1, 2, \ldots, n; j = 1, 2, \ldots, m).$ Therefore, a DM problem with IFSS information can be written in the form of decision matrix as $B^{(z)} = \left(u_i, \zeta_{{ij}}^{(z)}(u_i), \vartheta_{{ij}}^{(z)}(u_i)\right)_{n\times m}$ .

$\left( {{B^{(z)}}, {\mathbf{E}}} \right) = \begin{array}{*{20}{c}} {{u_1}}\\ {{u_2}}\\ \vdots \\ {{u_n}} \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{e_1}}&{{e_2}}& \cdots &{{e_m}} \end{array}}\\ {\left( {\begin{array}{*{20}{c}} {\left( {\zeta _{11}^{(z)}, \vartheta _{11}^{(z)}} \right)}&{\left( {\zeta _{12}^{(z)}, \vartheta _{12}^{(z)}} \right)}& \ldots &{\left( {\zeta _{1m}^{(z)}, \vartheta _{1m}^{(z)}} \right)}\\ {\left( {\zeta _{21}^{(z)}, \vartheta _{21}^{(z)}} \right)}&{\left( {\zeta _{22}^{(z)}, \vartheta _{22}^{(z)}} \right)}& \ldots &{\left( {\zeta _{2m}^{(z)}, \vartheta _{2m}^{(z)}} \right)}\\ \vdots & \vdots & \ddots & \vdots \\ {\left( {\zeta _{n1}^{(z)}, \vartheta _{n1}^{(z)}} \right)}&{\left( {\zeta _{n2}^{(z)}, \vartheta _{n2}^{(z)}} \right)}& \ldots &{\left( {\zeta _{nm}^{(z)}, \vartheta _{nm}^{(z)}} \right)} \end{array}} \right)} \end{array}$

4.2. Determine the PIA and NIA for IFSSs

As the data is given in the form of IFSNs so the positive ideal alternative (PIA) and negative ideal alternative (NIA) can be taken as $1$ and $0,$ respectively. Thus, the rating values of PIA and NIA can be formatted as $r^{+} = \left(1, 0\right)_{n\times m}$ and $r^{-} = \left(0, 1\right)_{n\times m},$ respectively. Of these, it can be observed that $r^{+}$ and $r^{-}$ are complementary to one another. However, in case, we take these fixed reference points, at that point any change in alternatives does not effect on the ranking of the alternatives. Therefore, these points can easily be amplified to the circumstances in which these reference points are not settled a priori. To address this, a decision-maker can characterize them as

$\begin{eqnarray} \beta^{+}& = & \left(\zeta^{+}, \vartheta^{+}\right)_{n\times m} = \left(\zeta_{{ij}}^{(h)}, \vartheta_{{ij}}^{(h)}\right)_{n\times m} \end{eqnarray}$

(4.1)

$\begin{eqnarray} \beta^{-}& = & \left(\zeta^{-}, \vartheta^{-}\right)_{n\times m} = \left(\zeta_{{ij}}^{(g)}, \vartheta_{{ij}}^{(g)}\right)_{n\times m} \end{eqnarray}$

(4.2)

where $h$ and $g$ , respectively, are the indices at which correlation coefficient with positive ideal $r^{+}$ is maximum and minimum with $r^-$ for all $i, j$ .

4.3. Compute the correlation coefficient

The correlation coefficient between each alternative $B^{(z)}, (z = 1, 2, \ldots, b)$ and PIA can be computed as follows:

$\begin{eqnarray} \rho\left(\beta_{ij}^{(z)}, \beta^{+}\right) = \frac{C_{IFSS}\left(\beta_{ij}^{(z)}, \beta^{+}\right)}{\sqrt{E_{IFSS}\left(\beta_{ij}^{(z)}\right)\cdot E_{IFSS}\left(\beta^{+}\right)}} \end{eqnarray}$

(4.3)

Similarly, the correlation coefficient between each alternative $B^{(z)}, (z = 1, 2, \ldots, b)$ and NIA can be calculated as follows:

$\begin{eqnarray} \rho\left(\beta_{ij}^{(z)}, \beta^{-}\right) = \frac{C_{IFSS}\left(\beta_{ij}^{(z)}, \beta^{-}\right)}{\sqrt{E_{IFSS}\left(\beta_{ij}^{(z)}\right) \cdot E_{IFSS}\left(\beta^{-}\right)}} \end{eqnarray}$

(4.4)

4.4. Calculate the relative closeness coefficient

The closeness coefficient corresponding to each alternative can be obtained by utilizing the values of correlation coefficient between each alternative and PIA as well as NIA. For each alternative $B^{(z)}, (z = 1, 2, \ldots, b)$ , the relative closeness coefficient is defined as:

$\begin{eqnarray} R^{(z)} = \frac{K\left(\beta_{ij}^{(z)}, \beta^{-}\right)}{K\left(\beta_{ij}^{(z)}, \beta^{+}\right) + K\left(\beta_{ij}^{(z)}, \beta^{-}\right)} \end{eqnarray}$

(4.5)

where $K\left(\beta_{ij}^{(z)}, \beta^{-}\right) = 1-\rho\left(\beta_{ij}^{(z)}, \beta^{-}\right)$ and $K\left(\beta_{ij}^{(z)}, \beta^{+}\right) = 1-\rho\left(\beta_{ij}^{(z)}, \beta^{+}\right)$ for all $i, j.$

To analyze how linearly the alternatives are correlated with the ideal alternatives with respect to the closeness coefficient, we need a computational property, which is $K(\beta_{ij}^{(z)}, \beta^{-}) = 1-\rho(\beta_{ij}^{(z)}, \beta^{-})$ and $K(\beta_{ij}^{(z)}, \beta^{+}) = 1-\rho(\beta_{ij}^{(z)}, \beta^{+})$ . From this property, it can be easily seen that if the correlation coefficient between any alternative and positive ideal is $1$ i.e. if the alternative is highly correlated with PIA then their corresponding closeness coefficient value is also $1,$ which is the maximum. It shows that closeness coefficient of an alternative is maximum when it has maximum distant form NIA. Also, we have $0\leq K(\beta_{ij}^{(z)}, \beta^{-})\leq K(\beta_{ij}^{(z)}, \beta^{+})+K(\beta_{ij}^{(z)}, \beta^{-})$ and hence, for all $z$ , $0\leq R^{(z)}\leq 1.$

5. Proposed approach and its application

In this section, a methodology is exhibited to solve the of DM issues by using proposed TOPSIS technique based on correlation coefficients.

5.1. Proposed approach

A set of $b$ alternatives $B = \{B^{(1)}, B^{(2)}$ , $\ldots$ , $B^{(b)}\}$ is considered for evaluation under the experts $\mathbf{U} = \{u_1, u_2, \ldots, u_n\}$ with weights $\eta = (\eta_1, \eta_2, \ldots, \eta_n)^T$ such that $\eta_i > 0$ and $\sum\limits_{i = 1}^n \eta_i = 1$ . The judgement is given on the set of parameters $\mathbf{E} = \{e_1, e_2, \ldots, e_m\}$ having weights $\xi = (\xi_1, \xi_2, \ldots, \xi_m)^T$ such that $\xi_j > 0$ and $\sum\limits_{j = 1}^m \xi_j = 1$ . The evaluation of alternative $B^{(z)}, (z = 1, 2, \ldots, b)$ by the experts $u_{i}, (i = 1, 2, \ldots, n)$ on the parameters $e_j$ , $(j = 1, 2, \ldots, m)$ are given as IFSNs denoted by $\beta_{ij}^{(z)} = \left(\zeta_{ij}^{(z)}, \vartheta_{ij}^{(z)}\right)$ such that $0\leq \zeta_{ij}^{(z)}, \vartheta_{ij}^{(z)} \leq 1$ and $\zeta_{ij}^{(z)} + \vartheta_{ij}^{(z)} \leq 1.$ Then, the procedure for choosing the finest alternative(s) by utilizing the proposed operators are summarized in the following steps:

Step 1: Collect the data related to each alternative $B^{(z)}; z = 1, 2, \ldots, b$ and summarized in a matrix as follows:

Step 2: Construct the weighted decision matrix $\overline{B}^{(z)} = \left(\overline{\beta}_{ij}^{(z)}\right)$ , where

$\begin{eqnarray} \overline{\beta}_{ij}^{(z)} & = &\xi_{j}\eta_{i}\beta_{ij}^{(z)}\\ & = &\left( 1-\left(\left(1-\zeta_{ij}^{(z)}\right)^{\eta_{i}}\right)^{\xi_{j}}, \left(\left(\vartheta_{ij}^{(z)}\right)^{\eta_{i}}\right)^{\xi_{j}}\right)\\ & = & \left( \overline{\zeta}_{ij}^{(z)}, \overline{\vartheta}_{ij}^{(z)}\right) \end{eqnarray}$

(5.1)

where $\eta_{i}$ is the weight vector for $i$ th expert, $\xi_{j}$ is the weight vector for $j$ th parameter and $\overline{B}^{(z)} = \left(\overline{\beta}_{ij}^{(z)}\right)_{n\times m}.$ The matrices so obtained are stated as weighted decision matrices.

Step 3: The correlation coefficient of each value of $\overline{\beta}_{ij}^{(z)}$ with the perfect positive ideal $r^{+} = \left(1, 0\right)$ is obtained as:

$\begin{eqnarray} \rho_{1}\left(\overline{\beta}_{ij}^{(z)}, r^{+}\right) = \frac{C_{\text{IFSS}}\left(\overline{\beta}_{ij}^{(z)}, r^{+}\right)}{\sqrt{E_{IFSS}(\overline{\beta}_{ij}^{(z)}) \cdot E_{IFSS}(r^{+})}} \end{eqnarray}$

(5.2)

$\begin{eqnarray} \rho_{2}\left(\overline{\beta}_{ij}^{(z)}, r^{+}\right) = \frac{C_{\text{IFSS}}\left(\overline{\beta}_{ij}^{(z)}, r^{+}\right)}{\max\left\{E_{IFSS}(\overline{\beta}_{ij}^{(z)}), E_{IFSS}(r^{+})\right\}} \end{eqnarray}$

(5.3)

By applying these, we get a decision matrix corresponding to each alternative and this matrix is called as correlation coefficient matrix and is denoted by $\phi^{(z)} = \left(\phi_{ij}^{(z)}\right)_{n \times m}, (z = 1, 2, \ldots, b).$ Here $\phi_{ij}^{(z)}$ is the correlation coefficient between the value $\overline{\beta}_{ij}^{(z)}$ and the positive ideal $r^{+}.$

Step 4: Find the indices $h_{ij}$ and $g_{ij}$ for each expert $u_i$ and parameter $e_j$ from the correlation coefficient matrices $\phi^{(z)}, (z = 1, 2, \ldots, b)$ such that $h_{ij} = \arg\max_z\{\phi_{ij}^{(z)}\}$ and $g_{ij} = \arg\min_z\{\phi_{ij}^{(z)}\}$ . Based on these indices, determine the PIA $\beta^{+}$ and NIA $\beta^{-}$ as

$\begin{eqnarray} \beta^+ & = & \left(\zeta^{+}, \vartheta^{+}\right)_{n\times m} = \left(\overline{\zeta}_{ij}^{(h_{ij})}, \overline{\vartheta}_{ij}^{(h_{ij})}\right) \end{eqnarray}$

(5.4)

$\begin{eqnarray} \text{and }\; \; \beta^- & = & \left(\zeta^{-}, \vartheta^{-}\right)_{n\times m} = \left(\overline{\zeta}_{ij}^{(g_{ij})}, \overline{\vartheta}_{ij}^{(g_{ij})}\right) \end{eqnarray}$

(5.5)

Step 5: Compute the correlation coefficient between weighted decision matrices $\overline{B}^{(z)}, (z = 1, 2, \ldots, b)$ of each alternative $B^{(z)}, (z = 1, 2, \ldots, b)$ and the PIA $\beta^{+}$ by utilizing either $\rho_1$ or $\rho_2$ proposed correlation coefficients as:

$\begin{eqnarray} p^{(z)} & = & \rho_{1}\left(\overline{B}^{(z)}, \beta^{+}\right)\\ & = & \frac{C_{\text{IFSS}}\left(\overline{B}^{(z)}, \beta^{+}\right)}{\sqrt{E_{\text{IFSS}}(\overline{B}^{(z)})\cdot E_{\text{IFSS}}(\beta^{+})}} \\ & = &\frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\overline{\zeta}_{ij}^{(z)}\zeta^{+}+ \overline{\vartheta}_{ij}^{(z)}\vartheta^{+}\right)} {\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\left(\overline{\zeta}_{ij}^{(z)}\right)^{2}+ \left(\overline{\vartheta}_{ij}^{(z)}\right)^{2}\right)} \sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\Big(\zeta^{+}\Big)^{2}+ \Big(\vartheta^{+}\Big)^{2}\right)}} \end{eqnarray}$

(5.6)

$\begin{eqnarray} \text{or } p^{(z)} & = &\rho_{2}\left(\overline{B}^{(z)}, \beta^{+}\right)\\ & = &\frac{ C_{\text{IFSS}}\left(\overline{B}^{(z)}, \beta^{+}\right)}{\max\left\{E_{\text{IFSS}}(\overline{B}^{(z)}), E_{\text{IFSS}}(\beta^{+})\right\}}\\ & = &\frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\overline{\zeta}_{ij}^{(z)}\zeta^{+} +\overline{\vartheta}_{ij}^{(z)}\vartheta^{+}\right)} {\max\left\{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\left(\overline{\zeta}_{ij}^{(z)}\right)^{2} +\left(\overline{\vartheta}_{ij}^{(z)}\right)^{2}\right), \sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\Big(\zeta^{+}\Big)^{2} +\Big(\vartheta^{+}\Big)^{2}\right)\right\}} \end{eqnarray}$

(5.7)

Step 6: Compute the correlation coefficient between weighted decision matrices $\overline{B}^{(z)}, (z = 1, 2, \ldots, b)$ and the NIA $\beta^{-}$ as:

$\begin{eqnarray} n^{(z)} & = &\rho_{1}\left(\overline{B}^{(z)}, \beta^{-}\right)\\ & = &\frac{ C_{\text{IFSS}}\left(\overline{B}^{(z)}, \beta^{-}\right)}{\sqrt{E_{\text{IFSS}}(\overline{B}^{(z)})\cdot E_{\text{IFSS}}(\beta^{-})}}\\ & = &\frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\overline{\zeta}_{ij}^{(z)}\zeta^{-} +\overline{\vartheta}_{ij}^{(z)}\vartheta^{-}\right)} {\sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\left(\overline{\zeta}_{ij}^{(z)}\right)^{2} +\left(\overline{\vartheta}_{ij}^{(z)}\right)^{2}\right)} \sqrt{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\Big(\zeta^{-}\Big)^{2} +\Big(\vartheta^{-}\Big)^{2}\right)}} \end{eqnarray}$

(5.8)

$\begin{eqnarray} \text{or } n^{(z)}& = &\rho_{2}\left(\overline{B}^{(z)}, \beta^{-}\right)\\ & = &\frac{ C_{\text{IFSS}}\left(\overline{B}^{(z)}, \beta^{-}\right)}{\max\left\{E_{\text{IFSS}}(\overline{B}^{(z)}), E_{\text{IFSS}}(\beta^{-})\right\}}\\ & = &\frac{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\overline{\zeta}_{ij}^{(z)}\zeta^{-}+ \overline{\vartheta}_{ij}^{(z)}\vartheta^{-}\right)} {\max\left\{\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\left(\overline{\zeta}_{ij}^{(z)}\right)^{2} +\left(\overline{\vartheta}_{ij}^{(z)}\right)^{2}\right), \sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left(\Big(\zeta^{-}\Big)^{2}+\Big(\vartheta^{-}\Big)^{2}\right)\right\}} \end{eqnarray}$

(5.9)

Step 7: Obtain the closeness coefficient for each alternative $B^{(z)}, (z = 1, 2, \ldots, b)$ as:

$\begin{eqnarray} R^{(z)} = \frac{K\left(\overline{B}^{(z)}, \beta^{-}\right)}{K\left(\overline{B}^{(z)}, \beta^{+}\right) + K\left(\overline{B}^{(z)}, \beta^{-}\right)} \end{eqnarray}$

(5.10)

where $K\left(\overline{B}^{(z)}, \beta^{-}\right) = 1-n^{(z)}$ and $K\left(\overline{B}^{(z)}, \beta^{+}\right) = 1-p^{(z)}.$

Step 8: By utilizing descending value of $R^{(z)}$ give ranking to the alternatives $B^{(z)} (z = 1, 2, \ldots, b)$ and then find the desired one(s).

5.2. Illustrative example

In this section, an illustrative example for solving DM problem based upon outsourcing supplier selection problem is presented to describe the application of proposed approaches.

In the following, we discuss the IT Outsourcing Provider Selection problem. Millennium Semiconductors(MS), established in October $1995$ , is an ISO $9001-2015$ association with a circulation of electronic parts as its center aptitude. This driving wholesaler of electronic segments in India is synonymous with advancement and today, it is a standout amongst the most rumored name in the market. MS has set up in nearly every locale of India with catering more than $1500$ clients in all sections from the recent two decades. It engages in investigation and improvement, production and promoting of items, for example, full shading ultra high shine LED epitaxial items, chips, compound sun oriented cells and high control concentrating sunlight based items. The branch offices of MS is situated in Delhi, Bangalore, Hyderabad, Ahmedabad, Chennai and Mumbai in India and Overseas workplaces in Singapore and Shenzhen(China).

MS contributes to the extraordinary larger part of labor and financial resources to its center competition rather than IT. It's the outsourcing of IT is a better choice for MS as of its absence of the ability to do it efficiently. Therefore, MS selects the following outsourcing providers(alternatives): Tata Constancy services $(B^{(1)})$ , Infosys $(B^{(2)})$ , Wipro $(B^{(3)})$ , HCL $(B^{(4)})$ . Now, to find the more suitable or best outsourcing provider among the above choices, MS hires a team of five experts $\{u_1, u_2, u_3, u_4, u_5\}$ having weight vectors $(0.2, 0.3, 0.1, 0.3, 0.1)$ . These experts evaluate each provider against the five parameters: Design development $(e_1),$ Quality of product $(e_2),$ Delivery Time $(e_3),$ Risk factor $(e_4)$ and Cost $(e_5)$ having weights $(0.15, 0.25, 0.10, 0.30, 0.20)$ . Each expert assesses its rating values for each provider over the parameters in terms of IFSNs. The procedure for selecting the best alternative by applying the proposed approach is listed as:

Step 1: The ratings given by the experts for each provider are summarized in Table 1.

Table 1. Decision matrices for each alternative in terms of IFSNs.

	For alternative $B^{(1)}$					For alternative $B^{(2)}$
	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$
$u_1$	$(0.7, 0.2)$	$(0.6, 0.3)$	$(0.8, 0.1)$	$(0.6, 0.2)$	$(0.7, 0.2)$	$(0.9, 0.1)$	$(0.7, 0.3)$	$(0.6, 0.1)$	$(0.4, 0.3)$	$(0.9, 0.1)$
$u_2$	$(0.5, 0.1)$	$(0.7, 0.1)$	$(0.9, 0.1)$	$(0.5, 0.3)$	$(0.6, 0.3)$	$(0.7, 0.2)$	$(0.8, 0.2)$	$(0.7, 0.1)$	$(0.5, 0.2)$	$(0.8, 0.2)$
$u_3$	$(0.6, 0.2)$	$(0.5, 0.4)$	$(0.7, 0.2)$	$(0.5, 0.4)$	$(0.7, 0.1)$	$(0.8, 0.1)$	$(0.6, 0.3)$	$(0.5, 0.3)$	$(0.6, 0.1)$	$(0.8, 0.1)$
$u_4$	$(0.7, 0.3)$	$(0.7, 0.2)$	$(0.7, 0.3)$	$(0.4, 0.3)$	$(0.6, 0.1)$	$(0.7, 0.3)$	$(0.9, 0.1)$	$(0.7, 0.2)$	$(0.5, 0.3)$	$(0.7, 0.1)$
$u_5$	$(0.5, 0.4)$	$(0.5, 0.5)$	$(0.6, 0.1)$	$(0.5, 0.1)$	$(0.8, 0.1)$	$(0.6, 0.1)$	$(0.7, 0.1)$	$(0.6, 0.3)$	$(0.4, 0.5)$	$(0.6, 0.2)$
	For alternative $B^{(3)}$					For alternative $B^{(4)}$
	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$
$u_1$	$(0.8, 0.2)$	$(0.7, 0.3)$	$(0.9, 0.1)$	$(0.7, 0.2)$	$(0.8, 0.1)$	$(0.9, 0.1)$	$(0.4, 0.3)$	$(0.5, 0.2)$	$(0.6, 0.2)$	$(0.4, 0.3)$
$u_2$	$(0.7, 0.1)$	$(0.6, 0.2)$	$(0.6, 0.1)$	$(0.6, 0.3)$	$(0.7, 0.2)$	$(0.7, 0.2)$	$(0.6, 0.2)$	$(0.6, 0.4)$	$(0.7, 0.2)$	$(0.5, 0.4)$
$u_3$	$(0.6, 0.2)$	$(0.8, 0.1)$	$(0.5, 0.5)$	$(0.5, 0.4)$	$(0.6, 0.3)$	$(0.6, 0.1)$	$(0.7, 0.1)$	$(0.7, 0.3)$	$(0.7, 0.1)$	$(0.6, 0.3)$
$u_4$	$(0.5, 0.4)$	$(0.6, 0.3)$	$(0.8, 0.1)$	$(0.6, 0.2)$	$(0.9, 0.1)$	$(0.8, 0.2)$	$(0.7, 0.2)$	$(0.4, 0.6)$	$(0.5, 0.4)$	$(0.5, 0.2)$
$u_5$	$(0.8, 0.1)$	$(0.8, 0.2)$	$(0.7, 0.3)$	$(0.7, 0.1)$	$(0.8, 0.2)$	$(0.7, 0.1)$	$(0.6, 0.3)$	$(0.5, 0.4)$	$(0.6, 0.3)$	$(0.6, 0.2)$

| Show Table

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Step 2: The weighted decision matrices $\overline{B}^{(z)}, (z = 1, 2, 3, 4)$ are computed by utilizing Eq. (5.1). The results of it are summarized in Table 2.

Table 2. Weighted decision matrices

$\overline{B}$ for each alternative.

	For alternative $B^{(1)}$					For alternative $B^{(2)}$
	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$
$u_1$	$(0.0355, 0.9529)$	$(0.0448, 0.9416)$	$(0.0317, 0.9550)$	$(0.0535, 0.9079)$	$(0.0470, 0.9377)$	$(0.0667, 0.9333)$	$(0.0584, 0.9416)$	$(0.0182, 0.9550)$	$(0.0302, 0.9303)$	$(0.0880, 0.9120)$
$u_2$	$(0.0307, 0.9016)$	$(0.0863, 0.8414)$	$(0.0667, 0.9333)$	$(0.0605, 0.8973)$	$(0.0535, 0.9303)$	$(0.0527, 0.9301)$	$(0.1137, 0.8863)$	$(0.0355, 0.9333)$	$(0.0605, 0.8652)$	$(0.0921, 0.9079)$
$u_3$	$(0.0137, 0.9761)$	$(0.0172, 0.9774)$	$(0.0120, 0.9840)$	$(0.0206, 0.9729)$	$(0.0238, 0.9550)$	$(0.0239, 0.9661)$	$(0.0226, 0.9703)$	$(0.0069, 0.9880)$	$(0.0271, 0.9333)$	$(0.0317, 0.9550)$
$u_4$	$(0.0527, 0.9473)$	$(0.0863, 0.8863)$	$(0.0355, 0.9645)$	$(0.0449, 0.8973)$	$(0.0535, 0.8710)$	$(0.0527, 0.9473)$	$(0.1586, 0.8414)$	$(0.0355, 0.9529)$	$(0.0605, 0.8973)$	$(0.0697, 0.8710)$
$u_5$	$(0.0103, 0.9863)$	$(0.0172, 0.9828)$	$(0.0091, 0.9772)$	$(0.0206, 0.9333)$	$(0.0317, 0.9550)$	$(0.0137, 0.9661)$	$(0.0297, 0.9441)$	$(0.0091, 0.9880)$	$(0.0152, 0.9794)$	$(0.0182, 0.9683)$
	For alternative $B^{(3)}$					For alternative $B^{(4)}$
	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$
$u_1$	$(0.0471, 0.9529)$	$(0.0584, 0.9416)$	$(0.0450, 0.9550)$	$(0.0697, 0.9079)$	$(0.0623, 0.9120)$	$(0.0667, 0.9333)$	$(0.0252, 0.9416)$	$(0.0138, 0.9683)$	$(0.0535, 0.9079)$	$(0.0202, 0.9530)$
$u_2$	$(0.0137, 0.9016)$	$(0.0664, 0.8863)$	$(0.0271, 0.9333)$	$(0.0792, 0.8973)$	$(0.0697, 0.9079)$	$(0.0527, 0.9301)$	$(0.0664, 0.8863)$	$(0.0271, 0.9729)$	$(0.1027, 0.8652)$	$(0.0407, 0.9465)$
$u_3$	$(0.0137, 0.9761)$	$(0.0394, 0.9441)$	$(0.0069, 0.9931)$	$(0.0206, 0.9729)$	$(0.0182, 0.9762)$	$(0.0137, 0.9661)$	$(0.0297, 0.9441)$	$(0.0120, 0.9880)$	$(0.0355, 0.9333)$	$(0.0182, 0.9762)$
$u_4$	$(0.0307, 0.9596)$	$(0.0664, 0.9137)$	$(0.0471, 0.9333)$	$(0.0792, 0.8652)$	$(0.1290, 0.8710)$	$(0.0699, 0.9301)$	$(0.0863, 0.8863)$	$(0.0152, 0.9848)$	$(0.0605, 0.9208)$	$(0.0407, 0.9079)$
$u_5$	$(0.0239, 0.9661)$	$(0.0394, 0.9606)$	$(0.0120, 0.9880)$	$(0.0355, 0.9333)$	$(0.0317, 0.9683)$	$(0.0179, 0.9661)$	$(0.0226, 0.9703)$	$(0.0069, 0.9909)$	$(0.0271, 0.9645)$	$(0.0182, 0.9683)$

| Show Table

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Step 3: By utilizing Eq. (5.2) to determine the correlation coefficients between each alternative $B^{(z)}, (z = 1, 2, 3, 4)$ and the positive ideal $r^{+}$ , we get

$\phi^{(1)} = \left[ {\begin{array}{*{20}{c}} &0.0372 & 0.0475 & 0.0332& 0.0588 & 0.0501\\ &0.0340 & 0.1021 & 0.0713 & 0.0672 & 0.0574\\ &0.0140 & 0.0176 & 0.0122 & 0.0211 & 0.0249\\ &0.0556 & 0.0970 & 0.0368 & 0.0500 & 0.0613\\ &0.0105 & 0.0175 & 0.0093 & 0.0220 & 0.0332\\ \end{array}} \right] ; \phi^{(2)} = \left[ {\begin{array}{*{20}{c}} &0.0713 & 0.0619 & 0.0190 & 0.0324 & 0.0960\\ &0.0566 & 0.1273 & 0.0380 & 0.0697 & 0.1009\\ &0.0247 & 0.0233 & 0.0070 & 0.0290 & 0.0332\\ &0.0556 & 0.1852 & 0.0372 & 0.0672 & 0.0798\\ &0.0141 & 0.0314 & 0.0092 & 0.0155 & 0.0187\\ \end{array}} \right] \\ \phi^{(3)} = \left[ {\begin{array}{*{20}{c}} &0.0494 & 0.0619 & 0.0471 & 0.0765 & 0.0682\\ &0.0584 & 0.0747 & 0.0290 & 0.0879 & 0.0765\\ &0.0140 & 0.0417 & 0.0070 & 0.0211 & 0.0186\\ &0.0320 & 0.0725 & 0.0504 & 0.0911 & 0.1466\\ &0.0247 & 0.0410 & 0.0121 & 0.0380 & 0.0327\\ \end{array}} \right] ; \phi^{(4)} = \left[ {\begin{array}{*{20}{c}} &0.0713 & 0.0268 & 0.0142 & 0.0588 & 0.0212\\ &0.0566 & 0.0747 & 0.0279 & 0.1179 & 0.0430\\ &0.0141 & 0.0314 & 0.0121 & 0.0380 & 0.0186\\ &0.0749 & 0.0970 & 0.0154 & 0.0655 & 0.0448\\ &0.0185 & 0.0233 & 0.0070 & 0.0281 & 0.0187\\ \end{array}} \right]$

Step 4: By utilizing Eqs. (5.4) and (5.5), the PIA and NIA are computed as:

$\beta^{+} = \left[ {\begin{array}{*{20}{c}} & (0.0667, 0.9333) & (0.0584, 0.9416) & (0.0450, 0.9550) & (0.0697, 0.9079) & (0.0880, 0.9120) \\ & (0.0527, 0.9016) & (0.1137, 0.8863) & (0.0667, 0.9333) & (0.1027, 0.8652) & (0.0921, 0.9079) \\ & (0.0239, 0.9661) & (0.0394, 0.9441) & (0.0120, 0.9840) & (0.0355, 0.9333) & (0.0317, 0.9550) \\ & (0.0699, 0.9301) & (0.1586, 0.8414) & (0.0471, 0.9333) & (0.0792, 0.8652) & (0.1290, 0.8710) \\ & (0.0239, 0.9661) & (0.0394, 0.9606) & (0.0120, 0.9880) & (0.0355, 0.9333) & (0.0317, 0.9550) \\ \end{array}} \right] \\ \text{and }\; \; \beta^{-} = \left[ {\begin{array}{*{20}{c}} & (0.0355, 0.9529) & (0.0252, 0.9416) & (0.0138, 0.9683) & (0.0302 0.9303) & (0.0202, 0.9530) \\ & (0.0307, 0.9016) & (0.0664, 0.8863) & (0.0271, 0.9729) & (0.0605, 0.8973) & (0.0407, 0.9465) \\ & (0.0137, 0.9761) & (0.0172, 0.9774) & (0.0069, 0.9931) & (0.0206, 0.9729) & (0.0182, 0.9762) \\ & (0.0307, 0.9596) & (0.0664, 0.9137) & (0.0152, 0.9848) & (0.0449, 0.8973) & (0.0407, 0.9079) \\ & (0.0103, 0.9863) & (0.0172, 0.9828) & (0.0069, 0.9909) & (0.0152 0.9794) &, (0.0182, 0.9683) \\ \end{array}} \right]$

Step 5: By using Eq. (5.6), the correlation coefficients between each alternative and the PIA $\beta^{+}$ are computed and get $p^{(1)} = 0.99926$ , $p^{(2)} = 0.99964$ , $p^{(3)} = 0.99944$ and $p^{(4)} = 0.99911$ .

Step 6: By using Eq. (5.8), the correlation coefficients between each alternative and the NIA $\beta^{-}$ are computed and get $n^{(1)} = 0.99973$ , $n^{(2)} = 0.99921$ , $n^{(3)} = 0.99937$ and $n^{(4)} = 0.99970$ .

Step 7: The closeness coefficient of alternatives $B^{(z)}, (z = 1, 2, 3, 4)$ are computed by using Eq. (5.10) and the results are obtained as:

$\begin{eqnarray*} R^{(1)} = 0.26298; \quad R^{(2)} = 0.68716;\quad R^{(3)} = 0.527702; \quad R^{(4)} = 0.25051. \end{eqnarray*}$

Step 8: Since $R^{(2)} > R^{(3)} > R^{(1)} > R^{(4)}$ and hence the given alternatives are ranked as $B^{(2)}\succ B^{(3)} \succ B^{(1)}\succ B^{(4)}$ . Therefore, the best alternative is obtained as $B^{(2)}$ .

5.3. Comparative Studies

To justify the predominance of our methodology, this section comprises the comparative investigation of the proposed approach with the prevailing methodologies ^{[44,45,50,51]}. The computational procedure and the obtained results so described as follows:

1) Comparison with the score function of Arora and Garg ^[44]:

For any IFSN $(\mathbf{F}, \mathbf{E}) = (\zeta, \vartheta)$ , Arora and Garg ^[44] defined the score function as

$\begin{eqnarray*} S(\mathbf{F}, \mathbf{E}) = \zeta-\vartheta \end{eqnarray*}$

To implement the TOPSIS approach based on this score function, we utilize the information as presented in . Based on this table, we compute the score matrix between each alternative $B^{(z)}$ and the positive ideal $r^+$ and hence the PIA $\beta^+$ and NIA $\beta^-$ are computed as

$\beta^{+} = \left[ {\begin{array}{*{20}{c}} & (0.0667, 0.9333) & (0.0584, 0.9416) & (0.0450, 0.9550) & (0.0697, 0.9079) & (0.0880, 0.9120) \\ & (0.0527, 0.9016) & (0.0863, 0.8414) & (0.0667, 0.9333) & (0.1027, 0.8652) & (0.0921, 0.9079) \\ & (0.0239, 0.9661) & (0.0394, 0.9441) & (0.0120, 0.9840) & (0.0355, 0.9333) & (0.0317, 0.9550) \\ & (0.0699, 0.9301) & (0.1586, 0.8414) & (0.0471, 0.9333) & (0.0792, 0.8652) & (0.1290, 0.8710) \\ & (0.0239, 0.9661) & (0.0297, 0.9441) & (0.0091, 0.9772) & (0.0355, 0.9333) & (0.0317, 0.9550) \\ \end{array}} \right] \\ \text{and }\; \; \beta^{-} = \left[ {\begin{array}{*{20}{c}} & (0.0355, 0.9529) & (0.0252, 0.9416) & (0.0138, 0.9683) & (0.0302 0.9303) & (0.0202, 0.9530) \\ & (0.0527, 0.9301) & (0.0664, 0.8863) & (0.0271, 0.9729) & (0.0605, 0.8973) & (0.0407, 0.9465) \\ & (0.0137, 0.9761) & (0.0172, 0.9774) & (0.0069, 0.9931) & (0.0206, 0.9729) & (0.0182, 0.9762) \\ & (0.0307, 0.9596) & (0.0664, 0.9137) & (0.0152, 0.9848) & (0.0605, 0.9208) & (0.0407, 0.9079) \\ & (0.0103, 0.9863) & (0.0172, 0.9828) & (0.0069, 0.9909) & (0.0152, 0.9794) &, (0.0182, 0.9683) \\ \end{array}} \right]$

Now, the correlation coefficient of alternatives with PIA are computed as $p^{(1)} = 0.99935$ , $p^{(2)} = 0.99961$ , $p^{(3)} = 0.99945$ and $p^{(4)} = 0.99914$ , while correlation coefficients with NIA are $n^{(1)} = 0.99972$ , $n^{(2)} = 0.99927$ , $n^{(3)} = 0.99937$ and $n^{(4)} = 0.99976$ . Finally, the relative closeness coefficient of the given alternatives is obtained as

$\begin{eqnarray*} R^{(1)} = 0.29818; \quad R^{(2)} = 0.65022; \quad R^{(3)} = 0.53415; \quad R^{(4)} = 0.21257. \end{eqnarray*}$

Thus, the ranking order is $B^{(2)}\succ B^{(3)} \succ B^{(1)}\succ B^{(4)}$ and hence the best alternative is $B^{(2)},$ which is same as that of proposed approach.

2) Comparison with the score function of Arora and Garg ^[45]:

For any IFSN $(\mathbf{F}, \mathbf{E}) = (\zeta, \vartheta)$ , Arora and Garg ^[45] defined the score function as

$\begin{eqnarray*} S(\mathbf{F}, \mathbf{E}) = \frac{1+\zeta-\vartheta}{2} \end{eqnarray*}$

To compare the decision making process with this score function, we compute the score matrix between each alternative $B^{(z)}$ and the positive ideal $r^+$ and hence the PIA $\beta^+$ and NIA $\beta^-$ are computed as

By using PIA and NIA, the closeness coefficients are obtained as

$\begin{eqnarray*} R^{(1)} = 0.29818; \quad R^{(2)} = 0.65022; \quad R^{(3)} = 0.53415; \quad R^{(4)} = 0.21257. \end{eqnarray*}$

Hence, the ranking order of the alternative is $B^{(2)}\succ B^{(3)} \succ B^{(1)}\succ B^{(4)}$ and the best alternative is $B^{(2)}.$

3) Comparison with similarity measure proposed by Sarala and Suganya ^[50]:

For IFSSs, Sarala and Suganya ^[50] defined the similarity measure as

$\begin{eqnarray*} S((\mathbf{F}, \mathbf{E}), (\mathbf{K}, \mathbf{E})) = 1 - \frac{1}{2mn} \sum\limits_{j = 1}^m \sum\limits_{i = 1}^n \left[ \mid\zeta_{\mathbf{F}_j}(u_i) - \zeta_{\mathbf{K}_j}(u_i) \mid + \mid \vartheta_{\mathbf{F}_j}(u_i) - \vartheta_{\mathbf{K}_j}(u_i)\mid \right] \end{eqnarray*}$

To implement the TOPSIS approach based on this similarity measure, we have taken the weighted decision matrix as given in . Now, based on this table, we compute the similarity matrices between each alternative and the positive ideal $r^+$ and get

$\phi^{(1)} = \left[ {\begin{array}{*{20}{c}} & 0.0413 & 0.0516 & 0.0383 & 0.0728 & 0.0547 \\ & 0.0646 & 0.1225 & 0.0667 & 0.0816 & 0.0616 \\ & 0.0188 & 0.0199 & 0.0140 & 0.0238 & 0.0344 \\ & 0.0527 & 0.1000 & 0.0355 & 0.0738 & 0.0913 \\ & 0.0120 & 0.0172 & 0.0159 & 0.0437 & 0.0383 \\ \end{array}} \right] ; \phi^{(2)} = \left[ {\begin{array}{*{20}{c}} & 0.0667 & 0.0584 & 0.0316 & 0.0499 & 0.0880 \\ & 0.0613 & 0.1137 & 0.0511 & 0.0977 & 0.0921 \\ & 0.0289 & 0.0261 & 0.0094 & 0.0469 & 0.0383 \\ & 0.0527 & 0.1586 & 0.0413 & 0.0816 & 0.0994 \\ & 0.0238 & 0.0428 & 0.0105 & 0.0179 & 0.0249 \\ \end{array}} \right] \\ \phi^{(3)} = \left[ {\begin{array}{*{20}{c}} & 0.0471 & 0.0584 & 0.0450 & 0.0809 & 0.0752 \\ & 0.0756 & 0.0901 & 0.0469 & 0.0909 & 0.0809 \\ & 0.0188 & 0.0477 & 0.0069 & 0.0238 & 0.0210 \\ & 0.0356 & 0.0764 & 0.0569 & 0.1070 & 0.1290 \\ & 0.0289 & 0.0394 & 0.0120 & 0.0511 & 0.0317 \\ \end{array}} \right] ; \phi^{(4)} = \left[ {\begin{array}{*{20}{c}} & 0.0667 & 0.0418 & 0.0227 & 0.0728 & 0.0336 \\ & 0.0613 & 0.0901 & 0.0271 & 0.1188 & 0.0471 \\ & 0.0238 & 0.0428 & 0.0120 & 0.0511 & 0.0210 \\ & 0.0699 & 0.1000 & 0.0152 & 0.0698 & 0.0664 \\ & 0.0259 & 0.0261 & 0.0080 & 0.0313 & 0.0249 \\ \end{array}} \right]$

Based on this ratings, we compute the PIA and NIA and hence the correlation coefficients between the given alternatives and PIA are computed as $p^{(1)} = 0.55378$ , $p^{(2)} = 0.69324$ , $p^{(3)} = 0.67131$ and $p^{(4)} = 0.47709$ , while correlation coefficients with NIA are $n^{(1)} = 0.64577$ , $n^{(2)} = 0.51651$ , $n^{(3)} = 0.55318$ and $n^{(4)} = 0.76003$ . The relative closeness coefficients of each alternative are computed as

$\begin{eqnarray*} R^{(1)} = 0.44253; \quad R^{(2)} = 0.61182; \quad R^{(3)} = 0.57616; \quad R^{(4)} = 0.31455. \end{eqnarray*}$

Hence the ranking is $B^{(2)}\succ B^{(3)} \succ B^{(1)}\succ B^{(4)}$ and the best alternative is $B^{(2)}.$

4) Comparison with similarity measure defined by Muthukumar and Krishnan ^[51]:

Muthukumar and Krishnan ^[51] defined the similarity measure for a pair of IFSS as

$\begin{eqnarray*} S((\mathbf{F}, \mathbf{E}), (\mathbf{K}, \mathbf{E})) = \frac{\sum \left[\zeta_{\mathbf{F}}(e_i)\cdot \zeta_\mathbf{K}(e_i) + \vartheta_{\mathbf{F}}(e_i)\cdot \vartheta_{\mathbf{K}}(e_i)\right]}{\sum\left[\max(\zeta_{\mathbf{F}}^2(e_i), \zeta_{\mathbf{K}}^2(e_i)) + \max(\vartheta_{\mathbf{F}}^2(e_i), \vartheta_{\mathbf{K}}^2(e_i))\right]} \end{eqnarray*}$

By using this similarity measure, the matrices between each alternative and the positive ideal $r^+$ and get

$\phi^{(1)} = \left[ {\begin{array}{*{20}{c}} & 0.0186 & 0.0237 & 0.0166 & 0.0293 & 0.0250 \\ & 0.0169 & 0.0506 & 0.0357 & 0.0335 & 0.0287 \\ & 0.0070 & 0.0088 & 0.0061 & 0.0106 & 0.0124 \\ & 0.0278 & 0.0484 & 0.0184 & 0.0249 & 0.0304 \\ & 0.0052 & 0.0087 & 0.0047 & 0.0110 & 0.0166 \\ \end{array}} \right] ; \phi^{(2)} = \left[ {\begin{array}{*{20}{c}} & 0.0357 & 0.0310 & 0.0095 & 0.0162 & 0.0480 \\ & 0.0283 & 0.0637 & 0.0190 & 0.0346 & 0.0505 \\ & 0.0123 & 0.0117 & 0.0035 & 0.0145 & 0.0166 \\ & 0.0278 & 0.0929 & 0.0186 & 0.0335 & 0.0396 \\ & 0.0071 & 0.0157 & 0.0046 & 0.0078 & 0.0094 \\ \end{array}} \right] \\ \phi^{(3)} = \left[ {\begin{array}{*{20}{c}} & 0.0247 & 0.0310 & 0.0235 & 0.0382 & 0.0340 \\ & 0.0291 & 0.0372 & 0.0145 & 0.0439 & 0.0382 \\ & 0.0070 & 0.0209 & 0.0035 & 0.0106 & 0.0093 \\ & 0.0160 & 0.0362 & 0.0252 & 0.0453 & 0.0734 \\ & 0.0123 & 0.0205 & 0.0061 & 0.0190 & 0.0163 \\ \end{array}} \right] ; \phi^{(4)} = \left[ {\begin{array}{*{20}{c}} & 0.0357 & 0.0134 & 0.0071 & 0.0293 & 0.0106 \\ & 0.0283 & 0.0372 & 0.0139 & 0.0587 & 0.0215 \\ & 0.0071 & 0.0157 & 0.0061 & 0.0190 & 0.0093 \\ & 0.0375 & 0.0484 & 0.0077 & 0.0327 & 0.0223 \\ & 0.0093 & 0.0117 & 0.0035 & 0.0140 & 0.0094 \\ \end{array}} \right]$

and hence computed the PIA $\beta^+$ and NIA $\beta^-$ . By using this information, the similarity between each alternative and PIA as well as NIA are obtained as, $p^{(1)} = 0.51418$ , $p^{(2)} = 0.72021$ , $p^{(3)} = 0.70004$ , $p^{(4)} = 0.49318$ and $n^{(1)} = 0.69060$ , $n^{(2)} = 0.49521$ , $n^{(3)} = 0.56044$ , $n^{(4)} = 0.71519$ . Hence, the relative closeness coefficient degrees of the considered alternatives are computed as

$\begin{eqnarray*} R^{(1)} = 0.38907; \quad R^{(2)} = 0.64338; \quad R^{(3)} = 0.59438; \quad R^{(4)} = 0.35977. \end{eqnarray*}$

and therefore, the ranking order is $B^{(2)}\succ B^{(3)} \succ B^{(1)}\succ B^{(4)}$ which implies that the best alternative is $B^{(2)}$ and coincides with the proposed approach ones.

From these studies and the comparative analysis, we conclude that the results computed through the presented approach coincide with the existing studies, and hence it is conservative. However, concerning the decision-making process, the main advantage of the proposed approach over the existing decision-making approaches is that they contain much more information to handle the uncertainties in the data. In it, information related to the object is expressed more precisely and objectively and hence is a useful tool for handling the imprecise and ambiguous information during the decision-making process. Furthermore, it is noted from the study that the computational procedure of the proposed approach is different from the existing approaches. This is due to the determination of the PIA and NIA. In the former approaches, these ideals, PIA and NIA, are computed based on the given decision matrices or by taking the extreme values only. Thus, the influence of rating values corresponding to each parameter doesn't impact other parameters and hence there is a certain loss of information during the process. On the other hand, in the proposed approach, these ideals are computed based on the impact of the maximum correlation coefficient on the given alternative ratings, and hence there is certainly less loss of information during the process. Finally, the strength of the correlation measures for each of these ideals is taken from the rating observation and hence compute the correlation between them to find the optimal solution. The advantage of the stated TOPSIS method with the stated correlation measures over the existing ones is that it not only taken into the account the degree of discrimination but also takes the degree of similarity between the observation, to avoid the decision only based on the negative reasons.

6. Conclusion

The key contribution of the work can be summarized below.

1) The examined study employs the IFSS to handle the inadequate, vague and conflicting data by considering membership degree, non-membership degree over the set of the parameters.

2) This paper offers new correlation measures to compute the degree of reliance between the two or more IFSSs. The various properties of the measures are also investigated in detail. The various existing correlation measures under the intuitionistic fuzzy set environment can be treated as a special case of the proposed measures by setting only one parameter in the analysis.

3) An extended TOPSIS method based on the proposed correlation measures has been introduced by considering the set of parameters and the experts. The advantages of the stated method are that it not only taken into account the degree of discrimination but also takes the degree of similarity between the observation, to avoid any short decision. Also, in the proposed approach, the ideal alternatives i.e., PIA ( $\beta^+$ ) and NIA ( $\beta^-$ ) are computed based on the index of the maximum correlation coefficient matrix of the given alternative ratings from its positive $r^+ = (1, 0)$ . By using these PIA and NIA, correlation indices and hence define the relative closeness coefficient degree to rank the alternatives.

4) To demonstrate the performance of the stated algorithm, a numerical example is given and compare their results with the existing studies ^{[44,45,50,51]}. It is concluded from this study that the proposed work gives more reasonable ways to handle the fuzzy information to solve the practical problems.

In the future, we shall lengthen the application of the proposed measures to the diverse fuzzy environment as well as different fields of application such as emerging decision problems, brain hemorrhage, risk evaluation, etc ^[58,59,60].

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	M. Randi $\text {c̀ }$ , Generalized molecular descriptors. J. Math. Chem., 7 (1991), 155–168. https://doi.org/10.1007/BF01200821 doi: 10.1007/BF01200821
[2]	S. Nikolić, N. Trinajstić, The Wiener index: Development and applications, Croat. Chem. Acta, 68 (1995), 105–129.
[3]	I. Gutman, B. Furtula, V. Katanić, Randić index and information, AKCE Int. J. Graphs Comb., 15 (2018), 307–312. https://doi.org/10.1016/j.akcej.2017.09.006 doi: 10.1016/j.akcej.2017.09.006
[4]	M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math., 157 (2009), 804–811. https://doi.org/10.1016/j.dam.2008.06.015 doi: 10.1016/j.dam.2008.06.015
[5]	G. H. Shirdel, H. Rezapour, A. M. Sayadi, The hyper-Zagreb index of graph operations, Iran. J. Math. Chem., 4 (2013), 213–220.
[6]	H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[7]	A. A. Khabyah, S. Zaman, A. N. A. Koam, A. Ahmad, A. Ullah, Minimum Zagreb eccentricity indices of two-mode network with applications in boiling point and benzenoid hydrocarbons, Mathematics, 10 (2022), 1–18. https://doi.org/10.3390/math10091393 doi: 10.3390/math10091393
[8]	X. J. Wang, M. F. Hanif, H. Mahmood, S. Manzoor, M. K. Siddiqui, M. Cancan, On computation of entropy measures and their statistical analysis for complex benzene systems, Polycycl. Aromat. Comp., 43 (2023), 7754–7768. https://doi.org/10.1080/10406638.2022.2139734 doi: 10.1080/10406638.2022.2139734
[9]	F. Arjmand, F. Shafiei, Prediction of the normal boiling points and enthalpy of vaporizations of alcohols and phenols using topological indices, J. Struct. Chem., 59 (2018), 748–754. https://doi.org/10.1134/S0022476618030393 doi: 10.1134/S0022476618030393
[10]	I. Redžepović, B. Furtula, Predictive potential of eigenvalue-based topological molecular descriptors, J. Comput. Aided Mol. Des., 34 (2020), 975–982. https://doi.org/10.1007/s10822-020-00320-2 doi: 10.1007/s10822-020-00320-2
[11]	A. Rauf, M. Naeem, S. U. Bukhari, Quantitative structure-property relationship of Ev-degree and Ve-degree based topological indices: Physico-chemical properties of benzene derivatives, Int. J. Quantum Chem., 122 (2022), e26851. https://doi.org/10.1002/qua.26851 doi: 10.1002/qua.26851
[12]	Y. L. Shang, Sombor index and degree-related properties of simplicial networks, Appl. Math. Comput., 419 (2022), 126881. https://doi.org/10.1016/j.amc.2021.126881 doi: 10.1016/j.amc.2021.126881
[13]	M. Rizwan, A. A. Bhatti, M. Javaid, Y. L. Shang, Conjugated tricyclic graphs with maximum variable sum exdeg index, Heliyon, 9 (2023), E15706. https://doi.org/10.1016/j.heliyon.2023.e15706 doi: 10.1016/j.heliyon.2023.e15706
[14]	B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem., 53 (2015), 1184–1190. https://doi.org/10.1007/s10910-015-0480-z doi: 10.1007/s10910-015-0480-z
[15]	I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total $\varphi$ -electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 (1972), 535–538. https://doi.org/10.1016/0009-2614(72)85099-1 doi: 10.1016/0009-2614(72)85099-1
[16]	E. D. Molina, J. M. Rodriguez, J. L. Sanchez, J. M. Sigarreta, Applications of the inverse degree index to molecular structures, J. Math. Chem., 2023. https://doi.org/10.1007/s10910-023-01526-z
[17]	G. F. Su, S. Wang, J. F. Du, M. J. Gao, K. C. Das, Y. L. Shang, Sufficient conditions for a graph to be $\ell$ -connected, $\ell$ -deficient, $\ell$ -Hamiltonian and $\ell$ -independent in terms of the forgotten topological index, Mathematics, 10 (2022), 1–11. https://doi.org/10.3390/math10111802 doi: 10.3390/math10111802
[18]	H. Ahmed, M. R. Salestina, A. Alwardi, N. D. Soner, Forgotten domination, hyper domination and modified forgotten domination indices of graphs, J. Discrete Math. Sci. Cryptogr., 24 (2021), 353–368. https://doi.org/10.1080/09720529.2021.1885805 doi: 10.1080/09720529.2021.1885805
[19]	N. De, S. M. A. Nayeem, A. Pal, F-index of some graph operations, Discrete Math. Algorithms Appl., 8 (2016), 1650025. https://doi.org/10.1142/S1793830916500257 doi: 10.1142/S1793830916500257
[20]	H. S. Ramane, R. B. Jummannaver, Note on forgotten topological index of chemical structure in drugs, Appl. Math. Nonlinear Sci., 1 (2016), 369–374. https://doi.org/10.21042/AMNS.2016.2.00032 doi: 10.21042/AMNS.2016.2.00032
[21]	S. Mondal, N. De, A. Pal, Onsome new neighbourhood degree based indices, Acta Chem. Iasi, 27 (2019), 31–46.
[22]	F. Harary, Graph theory, New Delhi: Narosa Publishing House, 2001.
[23]	H. Ahmed, A. Saleh, R. Ismail, M. R. Salestina, A. Alameri, Computational analysis for eccentric neighborhood Zagreb indices and their significance, Heliyon, 9 (2023), E17998. https://doi.org/10.1016/j.heliyon.2023.e17998 doi: 10.1016/j.heliyon.2023.e17998
[24]	S. Wazzan, A. Saleh, New versions of locating indices and their significance in predicting the physicochemical properties of benzenoid hydrocarbons, Symmetry, 14 (2022), 1–18. https://doi.org/10.3390/sym14051022 doi: 10.3390/sym14051022
[25]	S. Wazzan, H. Ahmed, Symmetry-adapted domination indices: The enhanced domination sigma index and its applications in QSPR studies of octane and its isomers, Symmetry, 15 (2023), 1–32. https://doi.org/10.3390/sym15061202 doi: 10.3390/sym15061202
[26]	S. Wazzan, N. U. Ozalan, Exploring the symmetry of curvilinear regression models for enhancing the analysis of fibrates drug activity through molecular descriptors, Symmetry, 15 (2023), 1–22. https://doi.org/10.3390/sym15061160 doi: 10.3390/sym15061160
[27]	K. C. Das, S. Mondal, On neighborhood inverse sum indeg index of molecular graphs with chemical significance, Inform. Sci., 623 (2023), 112–131. https://doi.org/10.1016/j.ins.2022.12.016 doi: 10.1016/j.ins.2022.12.016
[28]	M. Demirci, S. Delen, A. S. Cevik, I. N. Cangul, Omega index of line and total graphs, J. Math., 2021 (2021), 1–6. https://doi.org/10.1155/2021/5552202 doi: 10.1155/2021/5552202
[29]	N. U. Özalan, Some indices over a new algebraic graph, J. Math., 2021 (2021), 1–8. https://doi.org/10.1155/2021/5510384 doi: 10.1155/2021/5510384
[30]	B. H. Xing, N. U. Ozalan, J. B. Liu, The degree sequence on tensor and cartesian products of graphs and their omega index, AIMS Math., 8 (2023), 16618–16632. https://doi.org/10.3934/math.2023850 doi: 10.3934/math.2023850
[31]	T. Doslic, M. Saheli, Eccentric connectivity index of composite graphs, Util. Math., 95 (2014), 3–22.
[32]	W. C. Chen, H. Lu, Y. N. Yeh, Operations of interlaced trees and graceful trees, Southeast Asian Bull. Math., 21 (1997), 337–348.
[33]	GaussView 6. Available from: https://gaussian.com/gaussview6/.
[34]	M. J. Frisch, A. B Nielsen, H. P. Hratchian, Gaussian 09 programmer's reference, 2009.
[35]	B. Mennucci, J. Tomasi, R. Cammi, J. R. Cheeseman, M. J. Frisch, F. J. Devlin, et al., Polarizable continuum model (PCM) calculations of solvent effects on optical rotations of chiral molecules, J. Phys. Chem. A, 106 (2002), 6102–6113. https://doi.org/10.1021/jp020124t doi: 10.1021/jp020124t
[36]	P. Y. Chan, C. M. Tong, M. C. Durrant, Estimation of boiling points using density functional theory with polarized continuum model solvent corrections, J. Mol. Graph. Model., 30 (2011), 120–128. https://doi.org/10.1016/j.jmgm.2011.06.010 doi: 10.1016/j.jmgm.2011.06.010
[37]	A. P. Scott, L. Radom, Harmonic vibrational frequencies: an evaluation of Hartree-Fock, Møller-Plesset, quadratic configuration interaction, density functional theory, and semiempirical scale factors, J. Phys. Chem., 100 (1996), 16502–16513. https://doi.org/10.1021/jp960976r doi: 10.1021/jp960976r
[38]	Z. Raza, S. Akhter, Y. L. Shang, Expected value of first Zagreb connection index in random cyclooctatetraene chain, random polyphenyls chain, and random chain network, Front. Chem., 10 (2023), 1067874. https://doi.org/10.3389/fchem.2022.1067874 doi: 10.3389/fchem.2022.1067874
[39]	S. Nikolić, A. Miličević, N. Trinajstić, A. Jurić, On use of the variable Zagreb vM2 index in QSPR: Boiling points of benzenoid hydrocarbons, Molecules, 9 (2004), 1208–1221. https://doi.org/10.3390/91201208 doi: 10.3390/91201208
[40]	Ö. C. Havare, Quantitative structure analysis of some molecules in drugs used in the treatment of COVID-19 with topological indices, Polycycl. Aromat. Comp., 42 (2022), 5249–5260. https://doi.org/10.1080/10406638.2021.1934045 doi: 10.1080/10406638.2021.1934045
[41]	G. V. Rajasekharaiah, U. P. Murthy, Hyper-Zagreb indices of graphs and its applications, J. Algebra Combin. Discrete Struct. Appl., 8 (2021), 9–22. https://doi.org/10.13069/jacodesmath.867532 doi: 10.13069/jacodesmath.867532
[42]	F. C. Manso, H. S. Júnior, R. E. Bruns, A. F. Rubira, E. C. Muniz, Development of a new topological index for the prediction of normal boiling point temperatures of hydrocarbons: The $Fi$ index, J. Mol. Liq., 165 (2012), 125–132. https://doi.org/10.1016/j.molliq.2011.10.019 doi: 10.1016/j.molliq.2011.10.019
[43]	S. A. K. Kirmani, P. Ali, J. Ahmad, Topological coindices and quantitative structure-property analysis of antiviral drugs investigated in the treatment of COVID-19, J. Chem., 2022 (2022), 1–15. https://doi.org/10.1155/2022/3036655 doi: 10.1155/2022/3036655
[44]	H. C. Liu, H. L. Chen, Q. Q. Xiao, X. N. Fang, Z. K. Tang, More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons, Int. J. Quantum Chem., 121 (2021), e26689. https://doi.org/10.1002/qua.26689 doi: 10.1002/qua.26689

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24.	Muhammad Riaz, Dragan Pamucar, Anam Habib, Mishal Riaz, Ali Ahmadian, A New TOPSIS Approach Using Cosine Similarity Measures and Cubic Bipolar Fuzzy Information for Sustainable Plastic Recycling Process, 2021, 2021, 1563-5147, 1, 10.1155/2021/4309544
25.	Zhanhong Shi, Dinghai Zhang, Attribute significance based on inclusion degree for incomplete information systems and their applications to MADM problems, 2021, 41, 10641246, 3635, 10.3233/JIFS-211046
26.	Rana Muhammad Zulqarnain, Xiao Long Xin, Harish Garg, Rifaqat Ali, Interaction aggregation operators to solve multi criteria decision making problem under pythagorean fuzzy soft environment, 2021, 41, 10641246, 1151, 10.3233/JIFS-210098
27.	Rana Muhammad Zulqarnain, Imran Siddique, Rifaqat Ali, Fahd Jarad, Aiyared Iampan, Ewa Rak, Multicriteria Decision-Making Approach for Pythagorean Fuzzy Hypersoft Sets’ Interaction Aggregation Operators, 2021, 2021, 1563-5147, 1, 10.1155/2021/9964492
28.	Muhammad Riaz, Maryam Saba, Muhammad Abdullah Khokhar, Muhammad Aslam, Medical diagnosis of nephrotic syndrome using m-polar spherical fuzzy sets, 2022, 15, 1793-5245, 10.1142/S1793524521500947
29.	Sisi Xia, Lin Chen, Haoran Yang, A soft set based approach for the decision-making problem with heterogeneous information, 2022, 7, 2473-6988, 20420, 10.3934/math.20221119
30.	Paul Augustine Ejegwa, Shiping Wen, Yuming Feng, Wei Zhang, Jinkui Liu, A three-way Pythagorean fuzzy correlation coefficient approach and its applications in deciding some real-life problems, 2023, 53, 0924-669X, 226, 10.1007/s10489-022-03415-5
31.	Imran Siddique, Rana Muhammad Zulqarnain, Rifaqat Ali, Fahd Jarad, Aiyared Iampan, Ahmed Mostafa Khalil, Multicriteria Decision-Making Approach for Aggregation Operators of Pythagorean Fuzzy Hypersoft Sets, 2021, 2021, 1687-5273, 1, 10.1155/2021/2036506
32.	P. A. Ejegwa, C. Jana, 2021, Chapter 2, 978-981-16-1988-5, 39, 10.1007/978-981-16-1989-2_2
33.	Muhammad Riaz, Maryam Saba, Muhammad Abdullah Khokhar, Muhammad Aslam, Novel concepts of $m$ -polar spherical fuzzy sets and new correlation measures with application to pattern recognition and medical diagnosis, 2021, 6, 2473-6988, 11346, 10.3934/math.2021659
34.	Rana Muhammad Zulqarnain, Imran Siddique, Fahd Jarad, Aiyared Iampan, Thomas Hanne, Aggregation Operators for Interval-Valued Intuitionistic Fuzzy Hypersoft Set with Their Application in Material Selection, 2022, 2022, 1563-5147, 1, 10.1155/2022/8321964
35.	Gang Sun, Mingxin Wang, Xiaoping Li, Wei Huang, Distance measure and intuitionistic fuzzy TOPSIS method based on the centroid coordinate representation*, 2023, 44, 10641246, 555, 10.3233/JIFS-221732
36.	Nimra Jamil, Muhammad Riaz, Bipolar disorder diagnosis with cubic bipolar fuzzy information using TOPSIS and ELECTRE-I, 2022, 15, 1793-5245, 10.1142/S1793524522500309
37.	Rana Muhammad Zulqarnain, Wen-Xiu Ma, Imran Siddique, Alhanouf Alburaikan, Hamiden Abd El-Wahed Khalifa, Agaeb Mahal Alanzi, Prioritization of Thermal Energy Storage Techniques Using TOPSIS Method Based on Correlation Coefficient for Interval-Valued Intuitionistic Fuzzy Hypersoft Set, 2023, 15, 2073-8994, 615, 10.3390/sym15030615
38.	Vikas Arya, Satish Kumar, A Novel VIKOR-TODIM Approach Based on Havrda–Charvat–Tsallis Entropy of Intuitionistic Fuzzy Sets to Evaluate Management Information System, 2019, 11, 1616-8658, 357, 10.1080/16168658.2020.1840317
39.	Rana Muhammad Zulqarnain, Imran Siddique, Rifaqat Ali, Dragan Pamucar, Dragan Marinkovic, Darko Bozanic, Robust Aggregation Operators for Intuitionistic Fuzzy Hypersoft Set with Their Application to Solve MCDM Problem, 2021, 23, 1099-4300, 688, 10.3390/e23060688
40.	Shanshan Qiu, Dan Fu, Xiaofang Deng, Wei Quan, A Multicriteria Selection Framework for Wireless Communication Infrastructure with Interval-Valued Pythagorean Fuzzy Assessment, 2021, 2021, 1530-8677, 1, 10.1155/2021/9913737
41.	Chittaranjan Shit, Ganesh Ghorai, Multiple attribute decision-making based on different types of Dombi aggregation operators under Fermatean fuzzy information, 2021, 25, 1432-7643, 13869, 10.1007/s00500-021-06252-9
42.	Paul Augustine Ejegwa, Idoko Charles Onyeke, A Novel Intuitionistic Fuzzy Correlation Algorithm and Its Applications in Pattern Recognition and Student Admission Process, 2021, 11, 2156-177X, 1, 10.4018/IJFSA.285984
43.	Pranjal Bansal, Himanshu Dhumras, Rakesh K Bajaj, 2022, On T-Spherical Fuzzy Hypersoft Sets and Their Aggregation Operators with Application in Soft Computing, 978-1-6654-7647-8, 1, 10.1109/IMPACT55510.2022.10029247
44.	Paul Augustine Ejegwa, Shiping Wen, Yuming Feng, Wei Zhang, Ning Tang, Novel Pythagorean Fuzzy Correlation Measures Via Pythagorean Fuzzy Deviation, Variance, and Covariance With Applications to Pattern Recognition and Career Placement, 2022, 30, 1063-6706, 1660, 10.1109/TFUZZ.2021.3063794
45.	P. A. Ejegwa, I. C. Onyeke, B. T. Terhemen, M. P. Onoja, A. Ogiji, C. U. Opeh, Modified Szmidt and Kacprzyk’s Intuitionistic Fuzzy Distances and their Applications in Decision-making, 2022, 2714-4704, 174, 10.46481/jnsps.2022.530
46.	Muhammad Zeeshan Hanif, Naveed Yaqoob, Muhammad Riaz, Muhammad Aslam, Linear Diophantine fuzzy graphs with new decision-making approach, 2022, 7, 2473-6988, 14532, 10.3934/math.2022801
47.	Rana Muhammad Zulqarnain, Imran Siddique, Rifaqat Ali, Fahd Jarad, Abdul Samad, Thabet Abdeljawad, Ahmed Mostafa Khalil, Neutrosophic Hypersoft Matrices with Application to Solve Multiattributive Decision-Making Problems, 2021, 2021, 1099-0526, 1, 10.1155/2021/5589874
48.	Abdul Samad, Rana Muhammad Zulqarnain, Emre Sermutlu, Rifaqat Ali, Imran Siddique, Fahd Jarad, Thabet Abdeljawad, Ahmed Mostafa Khalil, Selection of an Effective Hand Sanitizer to Reduce COVID-19 Effects and Extension of TOPSIS Technique Based on Correlation Coefficient under Neutrosophic Hypersoft Set, 2021, 2021, 1099-0526, 1, 10.1155/2021/5531830
49.	Pongsakorn Sunthrayuth, Fahd Jarad, Jihen Majdoubi, Rana Muhammad Zulqarnain, Aiyared Iampan, Imran Siddique, Lazim Abdullah, A Novel Multicriteria Decision-Making Approach for Einstein Weighted Average Operator under Pythagorean Fuzzy Hypersoft Environment, 2022, 2022, 2314-4785, 1, 10.1155/2022/1951389
50.	Muhammad Tahir Hamid, Muhammad Riaz, Khalid Naeem, A study on weighted aggregation operators for q‐ rung orthopair m‐ polar fuzzy set with utility to multistage decision analysis , 2022, 37, 0884-8173, 6354, 10.1002/int.22847
51.	S. Memiş, S. Enginoğlu, U. Erkan, Fuzzy parameterized fuzzy soft k-nearest neighbor classifier, 2022, 500, 09252312, 351, 10.1016/j.neucom.2022.05.041
52.	Rana Muhammad Zulqarnain, Xiao Long Xin, Muhammad Saeed, Extension of TOPSIS method under intuitionistic fuzzy hypersoft environment based on correlation coefficient and aggregation operators to solve decision making problem, 2021, 6, 2473-6988, 2732, 10.3934/math.2021167
53.	Muhammad Riaz, Sania Batool, Yahya Almalki, Daud Ahmad, Topological Data Analysis with Cubic Hesitant Fuzzy TOPSIS Approach, 2022, 14, 2073-8994, 865, 10.3390/sym14050865
54.	Harish Garg, Fathima Perveen P A, Sunil Jacob John, Luis Perez-Dominguez, Adiel T. de Almeida-Filho, Spherical Fuzzy Soft Topology and Its Application in Group Decision-Making Problems, 2022, 2022, 1563-5147, 1, 10.1155/2022/1007133
55.	Muhammad Akram, Faiza Wasim, Faruk Karaaslan, MCGDM with complex Pythagorean fuzzy ‐soft model , 2021, 38, 0266-4720, 10.1111/exsy.12783
56.	T. Umamakeswari, Evaluating the quality factors of leaf plates by fuzzy TOPSIS method, 2023, 22147853, 10.1016/j.matpr.2023.01.390
57.	Muhammad Riaz, Shaista Tanveer, Dragan Pamucar, Dong-Sheng Qin, Topological Data Analysis with Spherical Fuzzy Soft AHP-TOPSIS for Environmental Mitigation System, 2022, 10, 2227-7390, 1826, 10.3390/math10111826
58.	Shahid Hussain Gurmani, Huayou Chen, Yuhang Bai, Extension of TOPSIS Method Under q-Rung Orthopair Fuzzy Hypersoft Environment Based on Correlation Coefficients and Its Applications to Multi-Attribute Group Decision-Making, 2022, 1562-2479, 10.1007/s40815-022-01386-w
59.	Paul Augustine Ejegwa, Yuming Feng, Shiping Wen, Wei Zhang, 2021, Determination of Pattern Recognition Problems based on a Pythagorean Fuzzy Correlation Measure from Statistical Viewpoint, 978-1-6654-1254-4, 132, 10.1109/ICACI52617.2021.9435895
60.	Khalid Naeem, Muhammad Riaz, 2021, Chapter 16, 978-981-16-1988-5, 407, 10.1007/978-981-16-1989-2_16
61.	Tuğçe Aydın, Serdar Enginoğlu, Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft matrices and their application to performance-based value assignment to noise-removal filters, 2022, 41, 2238-3603, 10.1007/s40314-022-01893-4
62.	Jabbar Ahmmad, Tahir Mahmood, Picture Fuzzy Soft Prioritized Aggregation Operators and Their Applications in Medical Diagnosis, 2023, 15, 2073-8994, 861, 10.3390/sym15040861
63.	Muhammad Imran Harl, Muhammad Saeed, Muhammad Haris Saeed, Sanaa Ahmed Bajri, Alhanouf Alburaikan, Hamiden Abd El-Wahed Khalifa, Development of an Investment Sector Selector Using a TOPSIS Method Based on Novel Distances and Similarity Measures for Picture Fuzzy Hypersoft Sets, 2024, 12, 2169-3536, 45118, 10.1109/ACCESS.2024.3380025
64.	Rana Muhammad Zulqarnain, Imran Siddique, Muhammad Asif, Hijaz Ahmad, Sameh Askar, Shahid Hussain Gurmani, Muhammet Gul, Extension of correlation coefficient based TOPSIS technique for interval-valued Pythagorean fuzzy soft set: A case study in extract, transform, and load techniques, 2023, 18, 1932-6203, e0287032, 10.1371/journal.pone.0287032
65.	Paul Augustine Ejegwa, Nasreen Kausar, John Abah Agba, Francis Ugwuh, Emre Özbilge, Ebru Ozbilge, Determination of medical emergency via new intuitionistic fuzzy correlation measures based on Spearman's correlation coefficient, 2024, 9, 2473-6988, 15639, 10.3934/math.2024755
66.	Rana Muhammad Zulqarnain, Wen-Xiu Ma, Imran Siddique, Hijaz Ahmad, Sameh Askar, A fair bed allocation during COVID-19 pandemic using TOPSIS technique based on correlation coefficient for interval-valued pythagorean fuzzy hypersoft set, 2024, 14, 2045-2322, 10.1038/s41598-024-53923-2
67.	Aman Sharma, Rakesh Kumar Bajaj, On identifying suitable hydrogen power plant location under T-spherical fuzzy hypersoft matrix structures, 2024, 68, 03603199, 1057, 10.1016/j.ijhydene.2024.04.221
68.	Muhammad Ihsan, Muhammad Saeed, Atiqe Ur Rahman, An intuitionistic fuzzy hypersoft expert set-based robust decision-support framework for human resource management integrated with modified TOPSIS and correlation coefficient, 2024, 36, 0941-0643, 1123, 10.1007/s00521-023-09085-9
69.	Vikash Patel, Harendra Kumar, Ashu Redhu, Kamal Kumar, Multiattribute decision-making based on TOPSIS technique and novel correlation coefficient of q-rung orthopair fuzzy sets, 2024, 9, 2364-4966, 10.1007/s41066-024-00493-3
70.	Tahir Mahmood, Jabbar Ahmmad, Ubaid Ur Rehman, Muhammad Bilal Khan, Analysis and Prioritization of the Factors of the Robotic Industry With the Assistance of EDAS Technique Based on Intuitionistic Fuzzy Rough Yager Aggregation Operators, 2023, 11, 2169-3536, 50462, 10.1109/ACCESS.2023.3272388
71.	Palash Dutta, Abhilash Kangsha Banik, A novel generalized similarity measure under intuitionistic fuzzy environment and its applications to criminal investigation, 2024, 57, 1573-7462, 10.1007/s10462-023-10682-2
72.	Rukhsana Kausar, Muhammad Riaz, Vladimir Simic, Khadija Akmal, Muhammad Umar Farooq, Enhancing solid waste management sustainability with cubic m-polar fuzzy cosine similarity, 2023, 1432-7643, 10.1007/s00500-023-08801-w
73.	Rashmi Singh, Milar Sinamcha, Rakesh Kumar Phanden, 2024, Chapter 51, 978-981-97-3172-5, 731, 10.1007/978-981-97-3173-2_51
74.	Jin Zheng, Ming-Yang Li, Duo-Ning Yuan, Song Xie, Product ranking considering differences across online review platforms: a method based on intuitionistic fuzzy soft sets, 2024, 0160-5682, 1, 10.1080/01605682.2024.2417727
75.	Saima Mustafa, Sadia Mahmood, Zabidin Salleh, Abdullah Hussein Abdullah Alamoodi, An application of fuzzy bipolar weighted correlation coefficient in decision-making problem, 2023, 18, 1932-6203, e0283516, 10.1371/journal.pone.0283516
76.	Huiping Chen, Yan Liu, Group Decision-Making Method with Incomplete Intuitionistic Fuzzy Soft Information for Medical Diagnosis Model, 2024, 12, 2227-7390, 1823, 10.3390/math12121823
77.	Palash Dutta, Abhilash Kangsha Banik, Crime linkage and psychological profiling of offenders under intuitionistic fuzzy environment using a novel resemblance measure, 2023, 56, 0269-2821, 893, 10.1007/s10462-023-10538-9
78.	Rana Muhammad Zulqarnain, Hong-Liang Dai, Wen-Xiu Ma, Imran Siddique, Sameh Askar, Hamza Naveed, Supplier selection in green supply chain management using correlation-based TOPSIS in a q-rung orthopair fuzzy soft environment, 2024, 10, 24058440, e32145, 10.1016/j.heliyon.2024.e32145
79.	Jayakumar Vimala, P. Mahalakshmi, Atiqe Ur Rahman, Muhammad Saeed, A customized TOPSIS method to rank the best airlines to fly during COVID-19 pandemic with q-rung orthopair multi-fuzzy soft information, 2023, 27, 1432-7643, 14571, 10.1007/s00500-023-08976-2
80.	S. Anita Shanthi, N. Nishadevi, S. Sampathu, 2023, 2931, 0094-243X, 030015, 10.1063/5.0178701
81.	Rakesh K. Bajaj, Aman Sharma, On some new aggregation operators for T-spherical fuzzy hypersoft sets with application in renewable energy sources, 2023, 15, 2511-2104, 2457, 10.1007/s41870-023-01258-y
82.	Siti Norziahidayu Amzee Zamri, Muhammad Azeem, Muhammad Imran, Muhammad Kamran Jamil, Bandar Almohsen, A study of novel linear Diophantine fuzzy topological numbers and their application to communicable diseases, 2024, 47, 1292-8941, 10.1140/epje/s10189-024-00460-5
83.	Paul Augustine Ejegwa, Nasreen Kausar, Nezir Aydin, Muhammet Deveci, A novel Pythagorean fuzzy correlation coefficient based on Spearman’s technique of correlation coefficient with applications in supplier selection process, 2024, 2452414X, 100762, 10.1016/j.jii.2024.100762
84.	Rana Muhammad Zulqarnain, Imran Siddique, Sameh Askar, Ahmad M. Alshamrani, Dragan Pamucar, Vladimir Simic, An extended TOPSIS technique based on correlation coefficient for interval-valued q-rung orthopair fuzzy hypersoft set in multi-attribute group decision-making, 2025, 11, 2199-4536, 10.1007/s40747-025-01838-4
85.	Hua Tian, Chenyang Tian, Ruolin Zhang, Multi-Objective Optimization and Allocation of Water Resources in Hancheng City Based on NSGA Algorithm and TOPSIS-CCDM Decision-Making Model, 2025, 17, 2071-1050, 4616, 10.3390/su17104616
86.	Rakesh Kumar Bajaj, Aman Sharma, On Clustering and TOPSIS Decision-Making Technique with New Trigonometric Information Measures under T-spherical Fuzzy Hypersoft Structures, 2025, 09574174, 128356, 10.1016/j.eswa.2025.128356
87.	Aman Sharma, Rakesh Kumar Bajaj, 2025, Chapter 2, 978-981-96-3255-8, 15, 10.1007/978-981-96-3256-5_2
88.	Himanshu Dhumras, Varun Shukla, Rakesh Kumar Bajaj, 2025, Chapter 36, 978-981-96-3255-8, 525, 10.1007/978-981-96-3256-5_36

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AIMS Mathematics

Unveiling novel eccentric neighborhood forgotten indices for graphs and gaph operations: A comprehensive exploration of boiling point prediction

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Abstract

1. Introduction

2. Preliminaries

3. Correlation coefficients for IFSSs

4. TOPSIS approach for solving DM problems based on correlation coefficient

4.1. Description of DM problem

4.2. Determine the PIA and NIA for IFSSs

4.3. Compute the correlation coefficient

4.4. Calculate the relative closeness coefficient

5. Proposed approach and its application

5.1. Proposed approach

5.2. Illustrative example

5.3. Comparative Studies

6. Conclusion

Conflict of interest

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AIMS Mathematics

Unveiling novel eccentric neighborhood forgotten indices for graphs and gaph operations: A comprehensive exploration of boiling point prediction

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Correlation coefficients for IFSSs

4. TOPSIS approach for solving DM problems based on correlation coefficient

4.1. Description of DM problem

4.2. Determine the PIA and NIA for IFSSs

4.3. Compute the correlation coefficient

4.4. Calculate the relative closeness coefficient

5. Proposed approach and its application

5.1. Proposed approach

5.2. Illustrative example

5.3. Comparative Studies

6. Conclusion

Conflict of interest

References

This article has been cited by:

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