TOPSIS Method Based on the Correlation Coefficient of Interval-Valued Intuitionistic Fuzzy Soft Sets and Aggregation Operators with Their Application in Decision-Making

The correlation coeﬃcient betweenthe two parametersplays a signiﬁcant part in statistics. Furthermore,the exactness ofthe assessmentof correlation depends upon information from the set of discourses. The data collected for various statistical studies are full of ambiguities. The idea of interval-valued intuitionistic fuzzy soft sets is an extension of intuitionistic fuzzy soft sets that is used to express insuﬃcient evaluation, uncertainty, and anxiety in decision-making. Intuitionistic fuzzy soft sets consider two diﬀerent types of information, such as membership degree and nonmembership degree. In this paper, the concepts and properties of the correlation coeﬃcient and the weighted correlation coeﬃcient of interval-valued intuitionistic fuzzy soft sets are proposed. A prioritization technique for order preference by similarity to the ideal solution based on interval-valued intuitionistic fuzzy soft sets of correlation coeﬃcients and the weighted correlation coeﬃcient is introduced. We also proposed interval-valued intuitionistic fuzzy soft weighted average and interval-valued intuitionistic fuzzy soft weighted geometric operators and developed decision-making techniques based on the proposed operators. By using the developed techniques, a method for solving decision-making problems is proposed. To ensure the applicability of the proposed methods, an illustrative example is given. Finally, we present a comparison of some existing methods with our proposed techniques.


Introduction
Correlation performs a vital part in statistics and engineering; through correlation analysis, the joint relationship of two variables can be used to evaluate the interdependence of two variables. Although probabilistic methods have been applied to various practical engineering problems, there are some obstacles to probabilistic strategies. For example, the probability of a process depends on the large amount of data collected, which is random. However, large-scale complex systems have many fuzzy uncertainties, so it is difficult to obtain accurate probability events. erefore, the results based on probability theory do not always provide useful information for experts due to the limited quantitative information. Besides, in practical applications, sometimes there is not enough data to properly process standard statistical data. Due to the abovementioned obstacles, the results based on probability theory are not always available to experts. erefore, probabilistic methods are usually not enough to resolve such inherent uncertainties in the data. A lot of researchers in the world proposed and recommended different approaches to solve those problems that contain uncertainty. First of all, Zadeh developed the notion of fuzzy sets (FSs) [1] to solve those problems which contain uncertainty and vagueness. It is observed that in some cases, circumstances cannot be handled by FSs, and to overcome such types of situations, Turksen [2] gave the idea of intervalvalued fuzzy sets (IVFSs). In some cases, we must deliberate membership unbiased as a nonmembership value for the suitable representation of an object in uncertain and indeterminate conditions that could not be handled by FSs nor by IVFSs. To overcome these difficulties, Atanassov presented the idea of intuitionistic fuzzy sets (IFSs) [3]. e theory which was presented by Atanassov only deals with insufficient data because of both the membership and nonmembership values, but the IFSs cannot handle incompatible and imprecise information.
A general mathematical tool was proposed by Molodtsov [4] to deal with indeterminate, fuzzy, and not clearly defined substances known as a soft set (SS). Maji et al. [5] extended the work on SS and developed some operations with their properties. In [6], they also used the SS theory for decision-making. Ali et al. [7] revised the Maji approach to SS and developed some new operations with their properties. De Morgan's law on SS theory was proved [8] by using different operators. Çagman and Enginoglu [9] developed the concept of soft matrices with operations and discussed their properties, and they also introduced a decision-making method to resolve those problems which contain uncertainty. In [10], they revised the operations proposed by Molodtsov's SS. Maji et al. [11] developed the notion of the fuzzy soft set (FSS) by combining the FS and SS. ey also proposed the intuitionistic fuzzy soft sets (IFSSs) with basic operations and properties [12]. Atanassov and Gargov [13] extended the IFS theory and established a new notion which is known as interval-valued intuitionistic fuzzy sets (IFSs). e authors in [14] established a novel technique to solve multiattribute decision-making (MADM) problems by using set pair analysis (SPA) under the IVIFS environment. Yang et al. [15] developed the concept of the interval-valued fuzzy soft set (IVFSS) with operations and proved some important results by combining the IVFS and SS, and they also used the developed notion for decision-making. Jiang et al. [16] proposed the concept of interval-valued intuitionistic fuzzy soft sets (IVIFSSs) by extending the IVIFS, and they also proposed the necessity and possibility of operations on IVIFSS with their properties. In [17], the authors constructed an algorithm based on IVIFSS and used the developed algorithm for decision-making.
Hwang and Yoon [18] developed the TOPSIS method to solve decision-making problems. By using the TOPSIS method, the minimum distance from a positive ideal solution which supports to elect the finest alternative is easily obtained. After the invention of the TOPSIS method, many researchers used the TOPSIS method for decision-making and extended this approach to the fuzzy and intuitionistic fuzzy environment [19][20][21][22][23][24][25][26][27][28]. Garg and Kumar [29] developed the idea of linguistic interval-valued Atanassov intuitionistic fuzzy sets and presented basic operational laws, score, and accuracy functions with their properties. Garg and Arora [30] developed a generalized version of the intuitionistic fuzzy soft set (IFSS) with weighted averaging and geometric aggregation operators and constructed a decision-making technique to solve problems under an intuitionistic fuzzy environment. ey also extended the Maclaurin symmetric mean (MSM) operators to IFSS based on Archimedean T-conorm and T-norm [31]. e idea of entropy measure and TOPSIS based on the correlation coefficient (CC) has been developed by using complex Q-rung orthopair fuzzy information and used the established techniques for decision-making [32].
In [33], the authors proposed the functional measuring of the interrelation of IFSs, nowadays, known as correlation, and developed its coefficient properties. To measure the interrelation of fuzzy numbers, Yu [34] established the CC of fuzzy numbers. Evaluating the CC for fuzzy data has been developed by Chiang and Lin [35]. Hung and Wu [36] proposed the centroid method to calculate the CC of IFSs and extended the proposed method to IVIFS. Bustince and Burillo [37] introduced the correlation and CC of IVIFS and proved the decomposition theorem on the correlation of IVIFS. Hong [38] and Mitchell [39] also established the CC for IFSs and IVIFSs, respectively. Garg and Arora introduced the correlation measures on IFSS and constructed the TOPSIS technique based on developed correlation measures [40]. Huang and Guo [41] gave an improved CC on IFS with their properties, and they also established the coefficient of IVIFS. Singh et al. [42] developed the one-and two-parametric generalization of CC on IFS and used the proposed technique in multiattribute group decision-making problems. ao [43] introduced the variance and covariance to establish the novel CC among IFS. Garg and Arora [44] developed the aggregate operators by using dual hesitant fuzzy soft numbers and utilized the proposed operators to solve MCDM problems. Jana et al. [45] developed various aggregation operators under Q-rung orthopair fuzzy environment.
In this research, the TOPSIS technique extends to IVIFSS information, where the mechanisms are assumed in terms of IVIFSNs. To measure the degree of dependency of IVIFSSs, we propose a new CC on IVIFSSs and examine some properties of developed CC. To achieve the goal accurately, the TOPSIS technique may be extended to solve multiattribute decision-making (MADM) problems. In the present research, our main objective is to introduce a new CC under IVIFSS information and develop the TOPSIS method for IVIFSS based on the proposed CC, intervalvalued intuitionistic fuzzy soft weighted average (IVIFSWA), and interval-valued intuitionistic fuzzy soft weighted geometric (IVIFSWG) operators. To solve MADM problems based on the extended TOPSIS approach, an algorithm is developed and the validity of the proposed technique is checked with a numerical illustration. e correlation measures are given that IVIFSS has been considered for the pairs of IVIFSSs, which will be used to compute the interrelation as well as the scope of dependence between the elements. Since the prevailing IFS and IFSS are special cases of IVIFSSs, therefore, the developed measure is more generalized than the prevailing measures. e CC conserves the linear relationship between the underconsidered elements. To find the closeness coefficient, generally, researchers used the basic TOPSIS method, similarity measures, and distance. Meanwhile, in our developed method, the closeness coefficient can be computed by utilizing the CC. e rest of the article is organized as follows. In Section 2, we remember some basic definitions such as SS, FSS, IVFSS, IFS, and IVIFSS and IVIFSS with some operations. In Section 3, we propose the correlation and informational energies for IVIFSS and develop the CC and WCC with their properties by using the correlation and informational energies. An extended TOPSIS technique is presented based on CC, and an algorithm is developed based on the proposed TOPSIS method to solve the MADM problem with a numerical illustration in Section 4. e IVIFSWA and IVIFSWG operators with decision-making techniques are presented in Section 5. Section 6 provides a comparative analysis of some existing techniques to developed methods. Finally, a comprehensive conclusion and future directions are given in Section 7.

Preliminaries
In this section, we recollect some basic definitions which will be used in the following sequel, such as SS, FSS, IVFSS, IFS, and IVIFSS.
Definition 1 (see [4]). Let U be the universal set and E be the set of attributes concerning U. Let P(U) be the power set of U and A⊆E. A pair (F, A) is called an SS over U, and its mapping is given as It is also defined as Definition 2 (see [11]). Let F(U) be a collection of all fuzzy subsets over U and E be a set of attributes. Let A⊆E, and then a pair (F, A) is called FSS over U, where F is a mapping such as F: A ⟶ F(U).
Definition 3 (see [15]). A mapping F: A ⟶ F(U) is known as an IVFSS and defined as F U i (e) � (u i , σ A (u i ))|u i ∈ U , where σ A (u i ) is the interval-valued fuzzy membership value of u i against parameter e ∈ E, F(U) is a collection of interval-valued fuzzy subsets of U, and in which σ ℓ A (u i ) and σ U A (u i ) represent the lower and upper limits of the interval.
Definition 4 (see [3]). An IFS is an object of the form A � (u i , σ A (u i ), τ A (u i ))|u i ∈ U on a universe U, where σ A and τ A : ⟶ [0, 1] represents the degree of membership and nonmembership, respectively, of any element u i ∈ U to set A with the following condition 0 Definition 5 (see [12]). A mapping F: A ⟶ F(U) is known as an IFSS and defined as are the degrees of acceptance and rejection, respectively, for all u i ∈ U and 0 ≤ σ Definition 6 (see [13]). An IVIFS is an object of the form A � (u i , σ A (u i ), τ A (u i ))|u i ∈ U on a universe U, where σ A and τ A : U ⟶ Int([0, 1]). Int([0, 1]) represents all closed subintervals of [0, 1] which satisfy the following condition ∀u i ∈ U, supσ A (u i ) + supτ A (u i ) ≤ 1. IVIFS (U) represents the class of all IVIFS over U.
Definition 7 (see [13]). Let (F, A) and (g, B) be two IVIFSs over U, and then (1) (F, A) is an interval-valued intuitionistic fuzzy subset of (g, B); if A⊆B, then F(e)⊆G(e) ∀u i ∈ U and e ∈ A, that is, IVIFS, and then its complement is defined as follows: Definition 8 (see [16]). Let U and E be the initial universe and set of parameters, respectively, and IVIFS (U) be the set of all IVIFS of U. Let A⊆E, and then a pair (F, A) is called IVIFSS over U, where F is a mapping such that F: A ⟶ IVIFS(U). IVIFSS is a parameterized family of IVIFSs of U, and consequently its universe is IVIFS (U).
ere exists a mapping from parameters to IVIFS (U), so we can say that IVIFSS is also a special case of SS.
F(e) states the interval intuitionistic fuzzy value set of the parameter for any e ∈ A, and it is an IVIFS of U where u i ∈ U and e ∈ A can be expressed in the mathematical form such as represent that the interval-valued fuzzy membership degree of U is held or not on parameter. Simply, an interval-valued intuitionistic fuzzy soft number (IVIFSN) can be expressed Definition 9 (see [16]). Let (F, A) and (g, B) be two IVIFSSs over U, and then (1) (F, A) is an interval-valued intuitionistic fuzzy soft subset of (g, B); if A⊆B, and ∀u i ∈ U and e ∈ A, we have F(e)⊆g(e), that is, IVIFSS, and then its complement is defined as follows: e ∈ A, then (F, A) are called null IVIFSS over U. It can be denoted as Φ.
e ∈ A, then (F, A) are called absolute IVIFSS over U. It can be denoted as Ω.

Correlation Coefficient of Interval-Valued Intuitionistic Fuzzy Soft Set
In this section, the concept of the correlation coefficient and the weighted correlation coefficient of IVIFSS has been proposed with some basic properties.
two IVIFSSs defined over a set of attributes e 1 , e 2 , e 3 , . . . , e m ; here . en, the informational intuitionistic energies of (F, A) and (g, B) can be described as follows: (4) two IVIFSSs, and then the correlation between them is defined as follows: ) be a correlation between them, and then the following properties hold: Proof.
e proof is trivial.
be two IVIFSSs, and then their correlation coefficient is given as δ IVIFSS ( (F, A), (g, B) ) and expressed as follows: be two IVIFSSs, and then CC between them satisfies the following properties: ) ≥ 0 is trivial, and here we only need to prove that δ IVIFSS ((F, A), (g, B)) ≤ 1.
From equation (5), we have Journal of Mathematics 5 By using Cauchy-Schwarz inequality, erefore, g, B). Hence, by using Definition 12, we get e proof is obvious.
□ Proof 3. From equation (6), we have As we know that σ ℓ us, proving the required result.
two IVIFSSs, and then their correlation coefficient is given as δ 1 IVIFSS ( (F, A), (g, B) ) and expressed as follows: )|u i ∈ U be two IVIFSSs, and then CC between them satisfies the following properties: Proof.
e proof is similar to Proposition 2. Nowadays, considering the weight of IVIFSS is essential for practical applications. e result of a decision may be varying, whenever decision makers adjust the different weights to every element in the universe of discourse. Consequently, it is particularly significant to plan the weight before decision-making. and c �  c 1 , c 2 , c 3 , . . . , c n be a weight vector for parameters such as c i > 0, n i�1 c i � 1. In the following, we develop the WCC IVIFSS by extending Definitions 12 and 13.

Proposition 4. Let
be two IVIFSSs, and then WCC between them satisfies the following properties:

TOPSIS Approach on IVIFSS for Solving DM Problems Based on the Correlation Coefficient
In this section, we develop a technique to solve decisionmaking problems by extending the TOPSIS method for IVIFSS information on the base of the correlation coefficient. Hwang and Yoon [18] developed the TOPSIS method and utilized them to encourage the order of the assessment substances regarding the positive and negative ideal solutions for decision-making matters. By using the TOPSIS method, we can find the best alternative from different alternatives having minimum and maximum distance from PIS and NIS, respectively. e TOPSIS technique demonstrates that the correlation measure is used to distinguish the positive and negative ideals in the choice ranking. Most researchers used the TOPSIS method to discover the closeness coefficient with a different type of distance and comparability measure. TOPSIS technique with a correlation coefficient is more appropriate to find the closeness coefficient instead of distance and similarity measure, since the correlation measure preserves the linear relationship among those factors which are under consideration. By using the developed CC, an algorithm based on the TOPSIS method will be introduced to select the most appropriate option.

TOPSIS Method Based on CC for IVIFSS to Solve MADM
Problem. Assume a set of "s" alternatives such as β = β 1 , β 2 , β 3 , . . . , β s for assessment under the team of experts such as U = u 1 , u 2 , u 3 , . . . , u n with weights ) for all i, j. A flowchart for the method is presented in Figure 1, and stepwise calculated results are presented in Tables 1-8.
Step 2. Develop the weighted decision matrix in which Ω i and c j are the weights for the i th expert and the j th parameter, respectively.
Step 3. To find the CC of every element of L We obtained a correlation coefficient matrix which can be represented as θ (z) = (θ (z) ij ) n×m , where θ (z) ij is the CC between every element of L (z) ij and I + .
Step 4. For each expert u i and parameter from CC matrices e j , we find the indices such as h ij = arg max z θ (z) ij and g ij = arg min z θ (z) ij and determine the PIA and NIA based on indices as follows: (20) Step 5. To find the CC between each alternative of weighted decision matrices β (z) and PIA L + as follows:   Step 6. To find the CC between each alternative of weighted decision matrix β (z) and NIA L − as follows: Step 7. To find the closeness coefficient for each alternative: where Step 8. Ranking the alternatives and choosing the best alternative.

Application of Proposed TOPSIS Technique for
Decision-Making. An electrical company calls for the appointment of an electrical engineer, and after initial scrutiny, four candidates (alternatives) remained for further assessment such as β (1) , β (2) , β (3) , β (4) . e managing director of an electrical company hires a team of four decision makers u 1 , u 2 , u 3 , u 4 having weight vectors (0.25, 0.2, 0.15, 0.4) T to conduct the interview. e team of decision makers evaluates the alternatives according to the following parameters: e 1 = qualification, e 2 = experience, e 3 = leadership quality, and e 4 = personality with weights (0.35, 0.25, 0.10, 0.30) T . Every decision maker evaluates the ratings for every alternative in IVIFSN form under the considered parameters. e developed method to find the best alternative for the electric company is given as follows: Step 1. Develop the decision matrices for each alternative under defined parameters according to the ratings of each decision maker in terms of IVIFSNs.
Step 2. Develop the weighted decision matrices for each alternative by using equation (17).
Step 3. Compute the CC between each alternative β Step 4. Find the PIA and NIA by using equations (19) and (20):

Proposed Approach.
Assume a set of "s" alternatives such as β = β 1 , β 2 , β 3 , . . . , β s for assessment under the team of experts such as U = u 1 , u 2 , u 3 , . . . , u n with weights   [47] of the aggregated IVIFSNs are obtained for the ranking of the alternatives. e algorithm is presented in Figure 2.

Algorithm
Step 1. Develop the interval-valued intuitionistic fuzzy soft matrix for each alternative.
Step 2. Aggregate the IVIFSNs for each alternative into a collective decision matrix Δ k by using the IVIFSWA or IVIFSWG operator.
Step 3. Find the scoring values Δ k for each alternative [47]: Step 4. Analyze the ranking.

Numerical
Example. An electrical organization calls for the appointment of an electrical engineer. After the first preliminary review, there are still four candidates (substitutes) that need further evaluation, such as {β (1) , β (2) , β (3) , β (4) }. e manager of the company hires a group of four decision makers {u 1 , u 2 , u 3 , u 4 } whose weight vector is

By Using the IVIFSWA Operator
Step 1. ese experts will evaluate the condition in the case of IVIFSNs, and there are just four alternatives; parameters and a summary of their scores are given in Tables 1-4.
Step 2. e opinions of the experts for each alternative are aggregated by using equation (27) Step 3. Scoring values by using equation (34): Step 4. erefore, the ranking of the alternatives is as follows (1) , and hence the alternative β (4) is the most suitable alternative for the company.

By Using the IVIFSWG Operator
Step 1. e experts will evaluate the condition in the case of IVIFSNs, and there are just four alternatives; parameters and a summary of their scores are given in Tables 1-4. Step 2. Experts' opinions on each alternative are summarized by using equation (30  Step 4. erefore, the ranking of the alternatives is as follows (1) , and hence the alternative β (4) is the most suitable alternative for the company.

Result Comparison and Discussion
Since interval-valued soft sets are more precise than fuzzy soft sets when the case study consists of uncertain and inconsistent information, thus, its extension, an interval-valued intuitionistic fuzzy soft set, will play a vital role in uncertain and indeterminate problems. Existing Step 1 Step 2 Step 3 Step  methodologies of IFSSs have some limitations on membership and nonmembership grades, and they cannot deal with parameterizations. e proposed algorithms of IVFSSs enhance the existing methodologies, and the decision maker can choose the values from the interval with membership and nonmembership as its limitation. ere is a strong relationship between the proposed model and MADM problems. In this paper, we have proposed two types of algorithms. Firstly, the TOPSIS algorithm is proposed based on the correlation coefficients, and a numerical example is solved. Secondly, averaging aggregate and geometric aggregate operators is proposed. Moving on to the next, all the algorithms are applied to the real-world problem, i.e., the selection of electrical engineers. e graphical representation of all results is presented in Figure 3. e results show that the proposed algorithms are valid and practical. Finally, the rank of all the alternatives using the existing algorithms gives the same final decision that β (4) is selected for the post of the "electrical engineer." All rankings are also calculated by applying the existing approaches. e proposed methods are also compared with other existing methods by Imtiaz et al. [48], Wu and Su [49], and Mukherjee and Sarkar [50].
e listing in Table 9 shows the results of the comparison in the final ranking of the top 4 alternatives. It can be observed that the best selections made by the proposed methods are compared with the already established methods which are expressive in itself and approves the reliability and validity of the proposed method. e final score values of the proposed TOPSIS method, IVIFSWA, and IVIFSWG can be seen in Table 10.

Conclusion
e investigated study utilizes the IVIFSS to address the unsatisfactory, obscure, and inconsistent data by considering the membership degree and nonmembership degree over the set of parameters. e novel concept of the correlation coefficient and weighted correlation coefficient for IVIFSS with their properties is proposed in the present research. Based on the developed correlation measures, an extended TOPSIS method has been introduced by considering the set of attributes and decision makers. e proposed method not only debates the discrimination but also handles the degree of similarity to prevent short decisions during observation. We also develop the correlation matrices and find the correlation indices; the PIA and NIA are also developed by using the correlation indices. To find the ranking of the alternatives, we define the closeness coefficient for the developed method. Moreover, the IVIFSWA and IVIFSWG operators have also defined and presented the decisionmaking techniques based on developed operators. Finally, a numerical illustration has been described to solve the  Figure 3: Comparison of the alternative rank with the proposed algorithm. Table 9: Alternative rank comparison using the existing and proposed techniques.

Data Availability
No such type of data are used in this manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.