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

Authors: Rana Muhammad Zulqarnain; Xiao Long Xin; Muhammad Saqlain; Waseem Asghar Khan
Journal: Journal of Mathematics
 (2021)

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## Abstract

The correlation coefficient between the two parameters plays a significant part in statistics. Furthermore, the exactness of the assessment of 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.


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## Body

## 1. 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. Therefore, 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. Therefore, 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 interval-valued 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]. The 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. Çağman and Enginoğlu [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. They 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). The 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 [19–28]. 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. They also extended the Maclaurin symmetric mean (MSM) operators to IFSS based on Archimedean T-conorm and T-norm [31]. The 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. Thao [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, interval-valued 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. The 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. The 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.The rest of the article is organized as follows. In Section2, 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. The 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.


## 2. 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 PU 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(1)ℱ:A⟶PU.
It is also defined as(2)ℱ,A=ℱe∈PU:e∈ℰ,ℱe=∅,ife∉A.Definition 2 (see [11]).
LetFU 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⟶FU.Definition 3 (see [15]).
A mappingℱ:A⟶FU is known as an IVFSS and defined as ℱUie=ui,σAui|ui∈U, where σAui is the interval-valued fuzzy membership value of ui against parameter e∈ℰ, FU is a collection of interval-valued fuzzy subsets of U, and σAui=σAℓui,σAUui in which σAℓui and σAUui represent the lower and upper limits of the interval.Definition 4 (see [3]).
An IFS is an object of the formA=ui,σAui,τAui|ui∈U on a universe U, where σA and τA:⟶0,1 represents the degree of membership and nonmembership, respectively, of any element ui∈U to set A with the following condition 0≤σAui+τAui≤1.Definition 5.
(see [12]). A mapping ℱ:A⟶FU is known as an IFSS and defined as ℱuie=ui,σAui,τAui|ui∈U, where σAui and τAui are the degrees of acceptance and rejection, respectively, for all ui∈U and 0≤σAui,τAui,σAui+τAui≤1.Definition 6.
(see [13]). An IVIFS is an object of the form A=ui,σAui,τAui|ui∈U on a universe U, where σA and τA:U⟶Int0,1. Int0,1 represents all closed subintervals of 0,1 which satisfy the following condition ∀ui∈U, supσAui+supτAui≤1. IVIFS U represents the class of all IVIFS over U.Definition 7 (see [13]).
Letℱ,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 ℱe⊆Ge∀ui∈U and e∈A, that is,σAℓui≤τBℓui,σAUui≤τBUui,σAℓui≥τBℓui, and τAUui≥σBUui(2)
Letℱ,A=ui,σAui,τAui|ui∈U be an IVIFS, and then its complement is defined as follows: ℱ,Ac=ui,τAui,σAui|ui∈U(3)
ℱ,A=g,B, if ℱ,A⊆g,B and g,B⊆ℱ,ADefinition 8 (see [16]).
LetU 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⟶IVIFSU.
IVIFSS is a parameterized family of IVIFSs ofU, and consequently its universe is IVIFSU. There exists a mapping from parameters to IVIFSU, so we can say that IVIFSS is also a special case of SS.
ℱe states the interval intuitionistic fuzzy value set of the parameter for any e∈A, and it is an IVIFS of U where ui∈U and e∈A can be expressed in the mathematical form such as ℱe=ui,σAui,τAui|ui∈U. Here, σAui=⌈σAℓui,σAUui⌉ and σAu=σAℓui,σAUui 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 as ℱ=σℓ,σU,τℓ,τU, where 0≤σU+τU≤1.Definition 9 (see [16]).
Letℱ,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 ∀ui∈U and e∈A, we haveℱe⊆ge, that is, σAℓui≤σBℓui,σAUui≤σBUui,τAℓui≥τBℓui, and τAUui≥σBUui. It can be represented as ℱ,A⊆g,B.(2)
Letℱ,A=ui,σAui,τAui|ui∈U be an IVIFSS, and then its complement is defined as follows: ℱ,Ac=ui,τAui,σAui|ui∈U.(3)
ℱ,A=g,B, if ℱ,A⊆g,B and g,B⊆ℱ,A.(4)
IfσAui=0,0, and τAui=1,1∀ui∈U and e∈A, then ℱ,A are called null IVIFSS over U. It can be denoted as Φ.(5)
IfσAui=1,1, and τAui=0,0∀ui∈U and e∈A, then ℱ,A are called absolute IVIFSS over U. It can be denoted as Ω.


## 3. 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.Definition 10.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs defined over a set of attributes e1,e2,e3,…,em; here σAkℓui=σAkℓui,σAkひui, τAkℓui=τAkℓui,τAkひui, σBkℓui=σBkℓui,σBkひui, and τBkℓui=τBkℓui,τBkひui. Then, the informational intuitionistic energies of ℱ,A and g,B can be described as follows:(3)ςIVIFSSℱ,A=∑k=1m∑i=1nσAkℓui2+σAkひui2+τAkℓui2+τAkひui2,(4)ςIVIFSSg,B=∑k=1m∑i=1nσBkℓui2+σBkひui2+τBkℓui2+τBkひui2.Definition 11.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then the correlation between them is defined as follows:(5)ςIVIFSSℱ,A,g,B=∑k=1m∑i=1nσAkℓui∗σBkℓui+σAkUui∗σBkUui+τAkℓui∗τBkℓui+τAkUui∗τBkUui.Proposition 1.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs and CIVIFSSℱ,A,g,B be a correlation between them, and then the following properties hold:(1)
CIVIFSSℱ,A,ℱ,A=ςIVIFSSℱ,A(2)
CIVIFSSg,B,g,B=ςIVIFSSg,BProof.
The proof is trivial.Definition 12.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then their correlation coefficient is given as δIVIFSSℱ,A,g,B and expressed as follows:(6)δIVIFSSℱ,A,g,B=CIVIFSSℱ,A,g,BςIVIFSSℱ,AςIVIFSSg,B.Example 1.
Letℱ,A and g,B be two IVIFSSs over U=u1,u2 and ℰ=e be a set of attributes which consist of only one parameter given as follows: ℱ,A=e,u1,.2,.4,.4,.5,u2,.3,.5,.1,.4 and g,B=e,u1,.3,.6,.1,.4,u2,.3,.5,.2,.3. Then, by using Definitions 1–3, we have ςIVIFSSℱ,A=1.12, ςIVIFSSg,B=1.09, and CIVIFSSℱ,A,g,B=1.02. Now, by utilizing equation (6), we get(7)δIVIFSSℱ,A,g,B=0.92316.Proposition 2.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then CC between them satisfies the following properties:(1)
0≤δIVIFSSℱ,A,g,B≤1(2)
δIVIFSSℱ,A,g,B=δIVIFSSg,B,ℱ,A(3)
Ifℱ,A=g,B, that is, ∀i,k,σAkℓui=σBkℓui, σAkひui=σBkひui, τAkℓui=τBkℓui, and τAkひui=τBkひui, then δIVIFSSℱ,A,g,B=1Proof 1.
δIVIFSSℱ,A,g,B≥0 is trivial, and here we only need to prove that δIVIFSSℱ,A,g,B≤1.
From equation (5), we have(8)CIVIFSSℱ,A,g,B=∑k=1m∑i=1nσAkℓui∗σBkℓui+σAkひui∗σBkひui+τAkℓui∗τBkℓui+τAkひui∗τBkひui=∑k=1mσAkℓu1∗σBkℓu1+σAkひu1∗σBkひu1+τAkℓu1∗τBkℓu1+τAkひu1∗τBkひu1+∑k=1mσAkℓu2∗σBkℓu2+σAkひu2∗σBkひu2+τAkℓu2∗τBkℓu2+τAkひu2∗τBkひu2+⋮+∑k=1mσAkℓun∗σBkℓun+σAkひun∗σBkひun+τAkℓun∗τBkℓun+τAkひun∗τBkひun,(9)CIVIFSSℱ,A,g,B=σA1ℓu1∗σB1ℓu1+σA1ひu1∗σB1ひu1+τA1ℓu1∗τB1ℓu1+τA1ひu1∗τB1ひu1+σA2ℓu1∗σB2ℓu1+σA2ひu1∗σB2ひu1+τA2ℓu1∗τB2ℓu1+τA2ひu1∗τB2ひu1⋮σAmℓu1∗σBmℓu1+σAmひu1∗σBmひu1+τAmℓu1∗τBmℓu1+τAmひu1∗τBmひu1+σA1ℓu2∗σB1ℓu2+σA1ひu2∗σB1ひu2+τA1ℓu2∗τB1ℓu2+τA1ひu2∗τB1ひu2+σA2ℓu2∗σB2ℓu2+σA2ひu2∗σB2ひu2+τA2ℓu2∗τB2ℓu2+τA2ひu2∗τB2ひu2⋮σAmℓu2∗σBmℓu2+σAmひu2∗σBmひu2+τAmℓu2∗τBmℓu2+τAmひu2∗τBmひu2+⋮+σA1ℓun∗σB1ℓun+σA1ひun∗σB1ひun+τA1ℓun∗τB1ℓun+τA1ひun∗τB1ひun+σA2ℓun∗σB2ℓun+σA2ひun∗σB2ひun+τA2ℓun∗τB2ℓun+τA2ひun∗τB2ひun⋮σAmℓun∗σBmℓun+σAmひun∗σBmひun+τAmℓun∗τBmℓun+τAmひun∗τBmひun=∑k=1mσAkℓu1∗σBkℓu1+σAkひu1∗σBkひu1+σAkℓu2∗σBkℓu2+σAkひu2∗σBkひu2+…+σAkℓun∗σBkℓun+σAkひun∗σBkひun+∑k=1mτAkℓu1∗τBkℓu1+τAkひu1∗τBkひu1+τAkℓu2∗τBkℓu2+τAkひu2∗τBkひu2+…+τAkℓun∗τBkℓun+τAkひun∗τBkひun.
By using Cauchy–Schwarz inequality,(10)CIVIFSSℱ,A,g,B2≤∑k=1mσAkℓu12+σAkひu12+σAkℓu22+σAkひu22+⋯+σAkℓun2+σAkひun2+τAkℓu12+τAkひu12+τAkℓu22+τAkひu22+⋯+τAkℓun2+τAkひun2×∑k=1mσBkℓu12+σBkひu12+σBkℓu22+σBkひu22+⋯+σBkℓun2+σBkひun2+τBkℓu12+τBkひu12+τBkℓu22+τBkひu22+⋯+τBkℓun2+τBkひun2,ςIVIFSSℱ,A,g,B2=∑k=1m∑i=1nσAkℓui2+σAkひui2+τAkℓui2+τAkひui2×∑k=1m∑i=1nσBkℓui2+σBkひui2+τBkℓui2+τBkひui2,CIVIFSSℱ,A,g,B2≤ςIVIFSSℱ,A×ςIVIFSSg,B.
Therefore,CIVIFSSℱ,A,g,B2≤ςIVIFSSℱ,A×ςIVIFSSg,B. Hence, by using Definition 12, we get ςIVIFSSℱ,A,g,B≤1, so 0≤ςIVIFSSℱ,A,g,B≤1.Proof 2.
The proof is obvious.Proof 3.
From equation (6), we have(11)δIVIFSSℱ,A,g,B=∑k=1m∑i=1nσAkℓui∗σBkℓui+σAkUui∗σBkUui+τAkℓui∗τBkℓui+τAkUui∗τBkUui∑k=1m∑i=1nσAkℓui2+σAkUui2+τAkℓui2+τAkUui2∑k=1m∑i=1nσBkℓui2+σBkUui2+τBkℓui2+τBkUui2.
As we know thatσAkℓui=σBkℓui, σAkひui=σBkひui, τAkℓui=τBkℓui, and τAkひui=τBkひui∀i,k, we get(12)δIVIFSSℱ,A,g,B=∑k=1m∑i=1nσBkℓui2+σBkひui2+τBkℓui2+τBkひui2∑k=1m∑i=1nσAkℓui2+σAkひui2+τAkℓui2+τAkひui2∑k=1m∑i=1nσBkℓui2+σBkひui2+τBkℓui2+τBkひui2,δIVIFSSℱ,A,g,B=1.
Thus, proving the required result.Definition 13.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then their correlation coefficient is given as δIVIFSS1ℱ,A,g,B and expressed as follows:(13)δIVIFSS1ℱ,A,g,B=CIVIFSSℱ,A,g,BmaxςIVIFSSℱ,A,ςIVIFSSg,B.Proposition 3.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then CC between them satisfies the following properties:(1)
0≤δIVIFSS1ℱ,A,g,B≤1(2)
δIVIFSS1ℱ,A,g,B=δIVIFSS1g,B,ℱ,A(3)
Ifℱ,A=g,B, that is, ∀i,k, σAkℓui=σBkℓui, σAkひui=σBkひui, τAkℓui=τBkℓui, and τAkひui=τBkひui, then δIVIFSSℱ,A,g,B=1Proof.
The proof is similar to Proposition2.
Nowadays, considering the weight of IVIFSS is essential for practical applications. The 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. Letω˙k=ω˙1,ω˙2,ω˙3,…,ω˙m be a weight vector for experts such as ω˙k>0,∑k=1mω˙k=1, and γ=γ1,γ2,γ3,…,γn be a weight vector for parameters such as γi>0,∑i=1nγi=1. In the following, we develop the WCC IVIFSS by extending Definitions 12 and 13.Definition 14.
Letℱ,A=ui,σAkui,τAkui|ui∈U andg,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then their weighted correlation coefficient is given as δWIVIFSSℱ,A,g,B and expressed as follows:(14)δWIVIFSSℱ,A,g,B=CIVIFSSℱ,A,g,BmaxςIVIFSSℱ,A,ςIVIFSSg,B.Definition 15.
Letℱ,A=ui,σAkui,τAkui|ui∈U and g,B=ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then their weighted correlation coefficient is also given as δWIVIFSS1ℱ,A,g,B and expressed as follows:(15)δWIVIFSS1ℱ,A,g,B=CIVIFSSℱ,A,g,BmaxςIVIFSSℱ,A,ςIVIFSSg,B,δWIVIFSS1ℱ,A,g,B=∑k=1mω˙k∑i=1nγiσAkℓui∗σBkℓui+σAkひui∗σBkひui+τAkℓui∗τBkℓui+τAkひui∗τBkひuimax∑k=1mω˙kσAkℓui2+σAkひui2+τAkℓui2+τAkひui2,∑k=1nω˙k∑i=1nγ1σBkℓui2+σBkひui2+τBkℓui2+τBkひui2.
If we considerω˙ = {1/m, 1/m, …, 1/m} and γ = {1/n, 1/n, …, 1/n}, then δWIVIFSS (ℱA⌣, GB⌣) and δWIVIFSS1 (ℱA⌣, GB⌣) are reduced to δIVIFSS (ℱA⌣, GB⌣) and δIVIFSS1 (ℱA⌣, GB⌣), respectively, defined in Definitions 12 and 13.Proposition 4.
Letℱ,A = ui,σAkui,τAkui|ui∈U and G,B = ui,σBkui,τBkui|ui∈U be two IVIFSSs, and then WCC between them satisfies the following properties:(1)
0≤ δWIVIFSSℱ,A,G,B ≤ 1(2)
δWIVIFSSℱ,A,G,B = δWIVIFSSG,B,ℱ,A(3)
Ifℱ,A = G,B, that is, ∀i, k, σAkℓui=σBkℓui, σAkひui=σBkひui, τAkℓui=τBkℓui, and τAkひui=τBkひui, then δWIVIFSSℱ,A,G,B = 1Proof.
Similar to Proposition2.


## 4. TOPSIS Approach on IVIFSS for Solving DM Problems Based on the Correlation Coefficient

In this section, we develop a technique to solve decision-making 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. The 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.


### 4.1. TOPSIS Method Based on CC for IVIFSS to Solve MADM Problem

Assume a set of “s” alternatives such as β = β1,β2,β3,…,βs for assessment under the team of experts such as U = u1,u2,u3,…,un with weights Ω = Ω1,Ω1,…,ΩnT such that Ωi > 0, ∑i=1nΩi = 1. Let E = e1,e2,…,em be a set of attributes with weights and γ = γ1,γ2,γ3,…,γmT be a weight vector for parameters such as γj > 0, ∑j=1mγj = 1. The team of experts {ui: i = 1, 2, …, n} evaluates the alternatives {βz: z = 1, 2, …, s} under the considered parameters {ej: j = 1, 2, …, m} given in the form of IVIFSNs such as ℒijz = σijz,τijz, where σijz = σijℓ,σijひ and τijz = τijℓ,τijひ, in which 0 ≤ σijℓ,σijひ, τijℓ,τijひ ≤ 1 and supσijz + supτijz ≤ 1. So, ℒijz = σijℓ,σijひ,τijℓ,τijひ 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 1. Construct a matrix for each alternative {βz: z = 1, 2, …, s} as follows:(16)e1e2⋯emβz,εn×m=u1u2⋮unℒ11zℒ12z⋯ℒ2mzℒ21zℒ22z⋯ℒ2mz⋮⋮⋮⋮ℒn1zℒn2z⋯ℒnmz.Step 2. Develop the weighted decision matrix β¯z = ℒ¯ijzn×m, where(17)ℒ¯ijz=γjΩiℒijz=1−1−σijℓ,σijひΩiγj,τijℓ,τijひΩiγj=σ¯ijz,τ¯ijz,in whichΩi and γj are the weights for the ith expert and the jth parameter, respectively.Step 3. To find the CC of every element of ℒ¯ijz with the perfect positive ideal ℐ+ = 1,1,0,0 as follows:(18)δIVIFSSℒ¯ijz,ℐ+=CIVIFSSℒ¯ijz,ℐ+ςIVIFSSℒ¯ijz∗ςIVIFSSℐ+.We obtained a correlation coefficient matrix which can be represented asθz = θijzn×m, where θijz is the CC between every element of ℒ¯ijz and ℐ+.Step 4. For each expert ui and parameter from CC matrices ej, we find the indices such as hij = arg maxzθijz and gij = arg minzθijz and determine the PIA and NIA based on indices as follows:(19)ℒ+=σ+,τ+n×m=σ¯ijhij,τ¯ijhij,(20)ℒ−=σ−,τ−n×m=σ¯ijgij,τ¯ijgij.Step 5. To find the CC between each alternative of weighted decision matrices β¯z and PIA ℒ+ as follows:(21)pz=δIVIFSSβ¯z,ℒ+=CIVIFSSβ¯z,ℒ+ςIVIFSSβ¯z∗ςIVIFSSℒ+=∑j=1m∑i=1nσ¯ijz∗σ++τ¯ijz∗τ+∑j=1m∑i=1nσ¯ijz2+τ¯ijz2∑j=1m∑i=1nσ+2+τ+2.Step 6. To find the CC between each alternative of weighted decision matrix β¯z and NIA ℒ− as follows:(22)qz=δIVIFSSβ¯z,ℒ−=CIVIFSSβ¯z,ℒ−ςIVIFSSβ¯z∗ςIVIFSSℒ−=∑j=1m∑i=1nσ¯ijz∗σ−+τ¯ijz∗τ−∑j=1m∑i=1nσ¯ijz2+τ¯ijz2∑j=1m∑i=1nσ−2+τ−2.Step 7. To find the closeness coefficient for each alternative:(23)ℛz=Kβ¯z,ℒ−Kβ¯z,ℒ++Kβ¯z,ℒ−,whereKβ¯z,ℒ− = 1 − qz and Kβ¯z,ℒ+ = 1 − pz.Step 8. Ranking the alternatives and choosing the best alternative.Figure 1
Algorithm of the TOPSIS method based on CC for IVIFSS.Table 1
Decision matrix for alternativeβ1.
β1e1e2e3e4u1.3,.5,.2,.4.2,.6,.2,.3.5,.7,.1,.3.5,.6,.1,.3u2.5,.7,.1,.2.4,.6,.2,.4.3,.4,.2,.5.6,.8,.1,.2u3.4,.6,.2,.3.1,.4,.2,.5.2,.3,.1,.6.3,.4,.2,.5u4.2,.4,.3,.5.3,.6,.2,.3.2,.4,.4,.5.4,.6,.1,.3Table 2
Decision matrix for alternativeβ2.
β2e1e2e3e4u1.5,.6,.2,.3.3,.4,.4,.6.4,.5,.3,.5.3,.5,.4,.5u2.3,.5,.4,.5.1,.3,.2,.4.7,.8,.1,.2.1,.2,.7,.8u3.6,.7,.2,.3.3,.4,.4,.5.5,.8,.1,.2.1,.2,.5,.8u4.5,.7,.1,.2.2,.4,.5,.6.4,.6,.2,.3.2,.3,.4,.6Table 3
Decision matrix for alternativeβ3.
β3e1e2e3e4u1.4,.5,.2,.4.3,.5,.4,.5.4,.6,.3,.4.3,.4,.4,.6u2.3,.4,.4,.6.1,.3,.3,.7.6,.8,.1,.2.1,.2,.6,.8u3.6,.7,.1,.2.3,.4,.4,.5.7,.8,.1,.2.1,.2,.7,.8u4.5,.6,.1,.3.2,.3,.6,.7.4,.6,.3,.4.3,.4,.4,.6Table 4
Decision matrix for alternativeβ4.
β4e1e2e3e4u1.4,.7,.1,.2.4,.5,.2,.4.2,.4,.3,.4.3,.4,.2,.4u2.3,.5,.3,.4.2,.4,.4,.5.6,.8,.1,.2.1,.2,.6,.8u3.6,.7,.1,.2.4,.5,.3,.4.5,.7,.1,.3.1,.3,.5,.7u4.5,.6,.1,.3.1,.2,.7,.8.5,.7,.2,.3.2,.3,.5,.7Table 5
Weighted decision matrix forβ¯1.
β¯1e1e2e3e4u1.0307,.0588,.6687,.9229.0138,.0557,.9043,.9275.0172,.0297,.9441,.9703.0507,.0664,.8414,.9137u2.0474,.0808,.8511,.8935.0252,.0448,.9227,.9552.0071,.0102,.9683,.9862.0535,.0921,.8709,.9079u3.0265.0469,.9189,.9387.0039,.0189,.9414,.9743.0033,.0053,.9661,.9924.0159,.0227,.9301,.9693u4.0308,.0690,.8449,.9075.0350,.0876,.8513,.8866.0089,.0202,.9640,.9727.0595,.1041,.7586,.8655Table 6
Weighted decision matrix forβ¯2.
β¯2e1e2e3e4u1.0588,.0770,.8686,.9000.0220,.0314,.9443,.9686.0127,.0172,.9703,.9828.0264,.0507,.9336,.9493u2.0247,.0474,.9379,.9526.0053,.0177,.9227,.9552.0238,.0317,.9549,.9683.0063,.0133,.9788,.9867u3.0469.0613,.9189,.9387.0133,.0189,.9414,.9559.0103,.0239,.9661,.9761.0047,.0099,.9693,.9900u4.0925,.1551,.7244,.7983.0221,.0498,.9330,.9502.0202,.0359,.9377,.9529.0264,.0419,.8959,.9405Table 7
Weighted decision Matrix forβ¯3.
β¯3e1e2e3e4u1.0437,.0588,.8686,.9229.0220,.0424,.9443,.9576.0127,.0226,.9703,.9774.0264,.0376,.9336,.9624u2.0247,.0351,.9379,.9649.0053,.0177,.9416,.9823.0182,.0317,.9549,.9683.0063,.0133,.9698,.9867u3.0469.0613,.8861,.9189.0133,.0189,.9662,.9743.0179,.0239,.9661,.9761.0047,.0099,.9841,.9900u4.0925,.1204,.7244,.8449.0221,.0350,.9309,.9649.0202,.0359,.9529,.0.9640.0264,.0419,.8959,.9405Table 8
Weighted decision Matrix forβ¯4.
β¯4e1e2e3e4u1.0437,.0999,.8175,.8686.0314,.0424,.9043,.9443.0056,.0127,.9703,.9774.0264,.0376,.8863,.9336u2.0247,.0474,.9192,.9379.0111,.0252,.9552,.9659.0182,.0317,.9549,.9683.0063,.0133,.9698,.9867u3.0469.0613,.8861,.9189.0189,.0257,.9559,.9662.0103,.0179,.9661,.9821.0047,.0159,.9693,.9841u4.0925,.1204,.7244,.8449.0105,.0221,.9649,.9779.0273,.0470,.9377,.0.9529.0264,.0419,.9202,.9581


### 4.2. 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 asβ1,β2,β3,β4. The managing director of an electrical company hires a team of four decision makers u1,u2,u3,u4 having weight vectors 0.25,0.2,0.15,0.4T to conduct the interview. The team of decision makers evaluates the alternatives according to the following parameters: e1 = qualification, e2 = experience, e3 = leadership quality, and e4 = personality with weights 0.35,0.25,0.10,0.30T. Every decision maker evaluates the ratings for every alternative in IVIFSN form under the considered parameters. The 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¯z and perfect positive ideal ℐ+ = 1,1,0,0 and construct a matrix for each alternative which is known as the CC matrix as follows:(24)θ1=0.05540.03790.02450.00650.07330.06820.00890.08150.03950.02760.00440.02030.05680.07030.01500.0999,θ2=0.07650.02790.01530.04090.03810.01220.02880.00990.05810.01690.01760.00750.16040.03810.02970.0372,θ3=0.05710.03380.01810.03370.03140.01190.02590.01000.05980.01660.02150.00740.13400.03010.02930.0372,θ4=0.08480.03990.00940.03520.03880.01890.02590.01000.05980.02320.01440.01050.13400.01680.03930.0363.Step 4. Find the PIA and NIA by using equations (19) and (20):(25)ℒ+=.0588,.0770,.8686,.9000.0314,.0424,.9043,.9443.0172,.0297,.9441,.9703.0507,.0664,.8414,.9137.0247,.0351,.9379,.9649.0053,.0177,.9416,.9823.0071,.0102,.9683,.9862.0063,.0133,.9698,.9867.0469.0613,.8861,.9189.0039,.0189,.9414,.9743.0033,.0053,.9661,.9924.0159,.0227,.9301,.9693.0925,.1204,.7244,.8449.0221,.0350,.9309,.9649.0202,.0359,.9529,.0.9640.0595,.1041,.7586,.8655,ℒ−=.0437,.0999,.8175,.8686.0220,.0424,.9443,.9576.0056,.0127,.9703,.9774.0264,.0507,.9336,.9493.0247,.0474,.9192,.9379.0111,.0252,.9552,.9659.0182,.0317,.9549,.9683.0063,.0133,.9788,.9867.0265.0469,.9189,.9387.0133,.0189,.9662,.9743.0103,.0239,.9661,.9761.0047,.0099,.9841,.9900.0925,.1551,.7244,.7983.0105,.0221,.9649,.9779.0202,.0359,.9377,.9529.0264,.0419,.9202,.9581.Step 5. Compute the CC between each alternative of the weighted decision matrix β¯z and PIA ℒ+.Correlation coefficient betweenβ¯z and ℒ+ is computed by using equation (21), and p1 = 0.99701, p2 = 0.99822, p3 = 0.99986, and p4 = 0.99759 are obtained.Step 6. Find the CC between each alternative of the weighted decision matrix β¯z and NIA ℒ−.Correlation coefficient betweenβ¯z and ℒ− is computed by using equation (22), and q1 = 0.99776, q2 = 0.99811, q3 = 0.99988, and q4 = 0.99237 are obtained.Step 7. By using equation (23), we get the closeness coefficient of alternatives given as follows:ℛ1 = 0.42839, ℛ2 = 0.51499, ℛ3 = 0.46154, and ℛ4 = 0.98198.Step 8. We conclude from the above calculations that ℛ4 > ℛ2>ℛ3 > ℛ1, so the ranking of the alternatives is described as β4 > β2>β3 > β1. Therefore, β4 is the best alternative for the electrical company.


## 4.1. TOPSIS Method Based on CC for IVIFSS to Solve MADM Problem

Assume a set of “s” alternatives such as β = β1,β2,β3,…,βs for assessment under the team of experts such as U = u1,u2,u3,…,un with weights Ω = Ω1,Ω1,…,ΩnT such that Ωi > 0, ∑i=1nΩi = 1. Let E = e1,e2,…,em be a set of attributes with weights and γ = γ1,γ2,γ3,…,γmT be a weight vector for parameters such as γj > 0, ∑j=1mγj = 1. The team of experts {ui: i = 1, 2, …, n} evaluates the alternatives {βz: z = 1, 2, …, s} under the considered parameters {ej: j = 1, 2, …, m} given in the form of IVIFSNs such as ℒijz = σijz,τijz, where σijz = σijℓ,σijひ and τijz = τijℓ,τijひ, in which 0 ≤ σijℓ,σijひ, τijℓ,τijひ ≤ 1 and supσijz + supτijz ≤ 1. So, ℒijz = σijℓ,σijひ,τijℓ,τijひ 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 1. Construct a matrix for each alternative {βz: z = 1, 2, …, s} as follows:(16)e1e2⋯emβz,εn×m=u1u2⋮unℒ11zℒ12z⋯ℒ2mzℒ21zℒ22z⋯ℒ2mz⋮⋮⋮⋮ℒn1zℒn2z⋯ℒnmz.Step 2. Develop the weighted decision matrix β¯z = ℒ¯ijzn×m, where(17)ℒ¯ijz=γjΩiℒijz=1−1−σijℓ,σijひΩiγj,τijℓ,τijひΩiγj=σ¯ijz,τ¯ijz,in whichΩi and γj are the weights for the ith expert and the jth parameter, respectively.Step 3. To find the CC of every element of ℒ¯ijz with the perfect positive ideal ℐ+ = 1,1,0,0 as follows:(18)δIVIFSSℒ¯ijz,ℐ+=CIVIFSSℒ¯ijz,ℐ+ςIVIFSSℒ¯ijz∗ςIVIFSSℐ+.We obtained a correlation coefficient matrix which can be represented asθz = θijzn×m, where θijz is the CC between every element of ℒ¯ijz and ℐ+.Step 4. For each expert ui and parameter from CC matrices ej, we find the indices such as hij = arg maxzθijz and gij = arg minzθijz and determine the PIA and NIA based on indices as follows:(19)ℒ+=σ+,τ+n×m=σ¯ijhij,τ¯ijhij,(20)ℒ−=σ−,τ−n×m=σ¯ijgij,τ¯ijgij.Step 5. To find the CC between each alternative of weighted decision matrices β¯z and PIA ℒ+ as follows:(21)pz=δIVIFSSβ¯z,ℒ+=CIVIFSSβ¯z,ℒ+ςIVIFSSβ¯z∗ςIVIFSSℒ+=∑j=1m∑i=1nσ¯ijz∗σ++τ¯ijz∗τ+∑j=1m∑i=1nσ¯ijz2+τ¯ijz2∑j=1m∑i=1nσ+2+τ+2.Step 6. To find the CC between each alternative of weighted decision matrix β¯z and NIA ℒ− as follows:(22)qz=δIVIFSSβ¯z,ℒ−=CIVIFSSβ¯z,ℒ−ςIVIFSSβ¯z∗ςIVIFSSℒ−=∑j=1m∑i=1nσ¯ijz∗σ−+τ¯ijz∗τ−∑j=1m∑i=1nσ¯ijz2+τ¯ijz2∑j=1m∑i=1nσ−2+τ−2.Step 7. To find the closeness coefficient for each alternative:(23)ℛz=Kβ¯z,ℒ−Kβ¯z,ℒ++Kβ¯z,ℒ−,whereKβ¯z,ℒ− = 1 − qz and Kβ¯z,ℒ+ = 1 − pz.Step 8. Ranking the alternatives and choosing the best alternative.Figure 1
Algorithm of the TOPSIS method based on CC for IVIFSS.Table 1
Decision matrix for alternativeβ1.
β1e1e2e3e4u1.3,.5,.2,.4.2,.6,.2,.3.5,.7,.1,.3.5,.6,.1,.3u2.5,.7,.1,.2.4,.6,.2,.4.3,.4,.2,.5.6,.8,.1,.2u3.4,.6,.2,.3.1,.4,.2,.5.2,.3,.1,.6.3,.4,.2,.5u4.2,.4,.3,.5.3,.6,.2,.3.2,.4,.4,.5.4,.6,.1,.3Table 2
Decision matrix for alternativeβ2.
β2e1e2e3e4u1.5,.6,.2,.3.3,.4,.4,.6.4,.5,.3,.5.3,.5,.4,.5u2.3,.5,.4,.5.1,.3,.2,.4.7,.8,.1,.2.1,.2,.7,.8u3.6,.7,.2,.3.3,.4,.4,.5.5,.8,.1,.2.1,.2,.5,.8u4.5,.7,.1,.2.2,.4,.5,.6.4,.6,.2,.3.2,.3,.4,.6Table 3
Decision matrix for alternativeβ3.
β3e1e2e3e4u1.4,.5,.2,.4.3,.5,.4,.5.4,.6,.3,.4.3,.4,.4,.6u2.3,.4,.4,.6.1,.3,.3,.7.6,.8,.1,.2.1,.2,.6,.8u3.6,.7,.1,.2.3,.4,.4,.5.7,.8,.1,.2.1,.2,.7,.8u4.5,.6,.1,.3.2,.3,.6,.7.4,.6,.3,.4.3,.4,.4,.6Table 4
Decision matrix for alternativeβ4.
β4e1e2e3e4u1.4,.7,.1,.2.4,.5,.2,.4.2,.4,.3,.4.3,.4,.2,.4u2.3,.5,.3,.4.2,.4,.4,.5.6,.8,.1,.2.1,.2,.6,.8u3.6,.7,.1,.2.4,.5,.3,.4.5,.7,.1,.3.1,.3,.5,.7u4.5,.6,.1,.3.1,.2,.7,.8.5,.7,.2,.3.2,.3,.5,.7Table 5
Weighted decision matrix forβ¯1.
β¯1e1e2e3e4u1.0307,.0588,.6687,.9229.0138,.0557,.9043,.9275.0172,.0297,.9441,.9703.0507,.0664,.8414,.9137u2.0474,.0808,.8511,.8935.0252,.0448,.9227,.9552.0071,.0102,.9683,.9862.0535,.0921,.8709,.9079u3.0265.0469,.9189,.9387.0039,.0189,.9414,.9743.0033,.0053,.9661,.9924.0159,.0227,.9301,.9693u4.0308,.0690,.8449,.9075.0350,.0876,.8513,.8866.0089,.0202,.9640,.9727.0595,.1041,.7586,.8655Table 6
Weighted decision matrix forβ¯2.
β¯2e1e2e3e4u1.0588,.0770,.8686,.9000.0220,.0314,.9443,.9686.0127,.0172,.9703,.9828.0264,.0507,.9336,.9493u2.0247,.0474,.9379,.9526.0053,.0177,.9227,.9552.0238,.0317,.9549,.9683.0063,.0133,.9788,.9867u3.0469.0613,.9189,.9387.0133,.0189,.9414,.9559.0103,.0239,.9661,.9761.0047,.0099,.9693,.9900u4.0925,.1551,.7244,.7983.0221,.0498,.9330,.9502.0202,.0359,.9377,.9529.0264,.0419,.8959,.9405Table 7
Weighted decision Matrix forβ¯3.
β¯3e1e2e3e4u1.0437,.0588,.8686,.9229.0220,.0424,.9443,.9576.0127,.0226,.9703,.9774.0264,.0376,.9336,.9624u2.0247,.0351,.9379,.9649.0053,.0177,.9416,.9823.0182,.0317,.9549,.9683.0063,.0133,.9698,.9867u3.0469.0613,.8861,.9189.0133,.0189,.9662,.9743.0179,.0239,.9661,.9761.0047,.0099,.9841,.9900u4.0925,.1204,.7244,.8449.0221,.0350,.9309,.9649.0202,.0359,.9529,.0.9640.0264,.0419,.8959,.9405Table 8
Weighted decision Matrix forβ¯4.
β¯4e1e2e3e4u1.0437,.0999,.8175,.8686.0314,.0424,.9043,.9443.0056,.0127,.9703,.9774.0264,.0376,.8863,.9336u2.0247,.0474,.9192,.9379.0111,.0252,.9552,.9659.0182,.0317,.9549,.9683.0063,.0133,.9698,.9867u3.0469.0613,.8861,.9189.0189,.0257,.9559,.9662.0103,.0179,.9661,.9821.0047,.0159,.9693,.9841u4.0925,.1204,.7244,.8449.0105,.0221,.9649,.9779.0273,.0470,.9377,.0.9529.0264,.0419,.9202,.9581


## 4.2. 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 asβ1,β2,β3,β4. The managing director of an electrical company hires a team of four decision makers u1,u2,u3,u4 having weight vectors 0.25,0.2,0.15,0.4T to conduct the interview. The team of decision makers evaluates the alternatives according to the following parameters: e1 = qualification, e2 = experience, e3 = leadership quality, and e4 = personality with weights 0.35,0.25,0.10,0.30T. Every decision maker evaluates the ratings for every alternative in IVIFSN form under the considered parameters. The 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¯z and perfect positive ideal ℐ+ = 1,1,0,0 and construct a matrix for each alternative which is known as the CC matrix as follows:(24)θ1=0.05540.03790.02450.00650.07330.06820.00890.08150.03950.02760.00440.02030.05680.07030.01500.0999,θ2=0.07650.02790.01530.04090.03810.01220.02880.00990.05810.01690.01760.00750.16040.03810.02970.0372,θ3=0.05710.03380.01810.03370.03140.01190.02590.01000.05980.01660.02150.00740.13400.03010.02930.0372,θ4=0.08480.03990.00940.03520.03880.01890.02590.01000.05980.02320.01440.01050.13400.01680.03930.0363.Step 4. Find the PIA and NIA by using equations (19) and (20):(25)ℒ+=.0588,.0770,.8686,.9000.0314,.0424,.9043,.9443.0172,.0297,.9441,.9703.0507,.0664,.8414,.9137.0247,.0351,.9379,.9649.0053,.0177,.9416,.9823.0071,.0102,.9683,.9862.0063,.0133,.9698,.9867.0469.0613,.8861,.9189.0039,.0189,.9414,.9743.0033,.0053,.9661,.9924.0159,.0227,.9301,.9693.0925,.1204,.7244,.8449.0221,.0350,.9309,.9649.0202,.0359,.9529,.0.9640.0595,.1041,.7586,.8655,ℒ−=.0437,.0999,.8175,.8686.0220,.0424,.9443,.9576.0056,.0127,.9703,.9774.0264,.0507,.9336,.9493.0247,.0474,.9192,.9379.0111,.0252,.9552,.9659.0182,.0317,.9549,.9683.0063,.0133,.9788,.9867.0265.0469,.9189,.9387.0133,.0189,.9662,.9743.0103,.0239,.9661,.9761.0047,.0099,.9841,.9900.0925,.1551,.7244,.7983.0105,.0221,.9649,.9779.0202,.0359,.9377,.9529.0264,.0419,.9202,.9581.Step 5. Compute the CC between each alternative of the weighted decision matrix β¯z and PIA ℒ+.Correlation coefficient betweenβ¯z and ℒ+ is computed by using equation (21), and p1 = 0.99701, p2 = 0.99822, p3 = 0.99986, and p4 = 0.99759 are obtained.Step 6. Find the CC between each alternative of the weighted decision matrix β¯z and NIA ℒ−.Correlation coefficient betweenβ¯z and ℒ− is computed by using equation (22), and q1 = 0.99776, q2 = 0.99811, q3 = 0.99988, and q4 = 0.99237 are obtained.Step 7. By using equation (23), we get the closeness coefficient of alternatives given as follows:ℛ1 = 0.42839, ℛ2 = 0.51499, ℛ3 = 0.46154, and ℛ4 = 0.98198.Step 8. We conclude from the above calculations that ℛ4 > ℛ2>ℛ3 > ℛ1, so the ranking of the alternatives is described as β4 > β2>β3 > β1. Therefore, β4 is the best alternative for the electrical company.


## 5. Aggregation Operators for Interval-Valued Intuitionistic Fuzzy Soft Numbers (IVIFSNs)

In this section, interval-valued intuitionistic fuzzy soft weighted average and interval-valued intuitionistic fuzzy soft weighted geometric operators for IVIFSNs are presented.


### 5.1. Operational Laws for IVIFSNs

Definition 16.
Letℵe=σℓ,σひ,τℓ,τひ, ℵe11=σ11ℓ,σ11ひ,τ11ℓ,τ11ひ, and ℵe12=σ12ℓ,σ12ひ,τ12ℓ,τ12ひ be three IVIFSNs and α be a positive real number, and by algebraic norms, we have(1)
ℵe11⊕ℵe12=σ11ℓ+σ12ℓ−σ11ℓσ12ℓ,σ11ひ+σ12ひ−σ11ひσ12ひ,τ11ℓτ12ℓ,τ11ひτ12ひ(2)
ℵe11⊗ℵe12=σ11ℓσ12ℓ,σ11ひσ12ひ,τ11ℓ+τ12ℓ−τ11ℓτ12ℓ,τ11ひ+τ12ひ−τ11ひτ12ひ(3)
αℵe=1−1−σℓ,σひα,τℓ,τひα(4)
ℵeα=σℓ,σひα,1−1−σℓ,σひα
Some averaging and geometric aggregation operators for IVIFSNs have been defined based on the above laws for the collection of IVIFSNΔ.Definition 17.
Letℵeij=σijℓ,σijひ,τijℓ,τijひ be a collection of IVIFSNs, and Ωi and γj are the weight vector for experts and parameters, respectively, with given conditions Ωi > 0, ∑i=1nΩi = 1; γj > 0, ∑j=1mγj = 1, where i=1,2,…,nandj=1,2,…,m. Then, the IVIFSWA operator is defined as IVIFSWA: Δn ⟶ Δ defined as follows:(26)IVIFSWAℵe11,ℵe12,…,ℵenm=⊕j=1mγj⊕i=1nΩiℵeij.Theorem 1.
Letℵeij=σijℓ,σijひ,τijℓ,τijひ be an IVIFSN, where i=1,2,…,nandj=1,2,…,m, and the aggregated value is also an IVIFSN, such as(27)IVIFSWAℵe11,ℵe12,…,ℵenm=1−∏j=1m∏i=1n1−σijℓ,σijひΩiγj,∏j=1m∏i=1nτijℓ,τijひΩiγj.Proof.
Similar to Theorem 3.1 [46].Example 2.
LetU = {u1, u2, u3, u4} be a set of decision makers, who are going to describe the beauty of a city under the defined set of attributes E = e1,e2,e3. Assume Ωi = 0.35,0.25,0.10,0.30T and γj = 0.3,0.5,0.2T be the weight vectors for experts and attributes, respectively. The assumed rating values of the experts for each attribute in the form of IVIFSNs ℵ,E = σijℓ,σijひ,τijℓ,τijひ4×3 are given as follows:(28)ℵ,E=.3,.5,.2,.4.2,.6,.2,.3.5,.7,.1,.3.5,.7,.1,.2.4,.6,.2,.4.3,.4,.2,.5.4.6,.2,.3.1,.4,.2,.5.2,.3,.1,.6.2,.4,.3,.5.3,.6,.2,.3.2,.4,.4,.5.
By using equation (27),(29)IVIFSWAℵe11,ℵe12,…,ℵe43=1−∏j=13∏i=141−σijℓ,σijひΩiγj,∏j=13∏i=14τijℓ,τijひΩiγj=1−.5,.70.35.3,.50.25.4,.60.1.6,.80.3.3.4,.80.35.4,.60.25.6,.90.1.4,.70.3.5.3,.50.35.6,.70.25.7,.80.1.6,.80.3.2.2,.40.35.1,.20.25.2,.30.1.3,.50.3.3.2,.30.35.2,.40.25.2,.50.1.2,.30.3.5.1,.30.35.2,.50.25.1,.60.1.4,.50.3.2=0.3098,0.5554,0.1929,0.3582.Definition 18.
Letℵeij=σijℓ,σijひ,τijℓ,τijひ be an IVIFSN, and the experts ui’s and parameters ej’s have standardized weight vectors Ωi and γj, respectively; then the IVIFSWG operator is defined as(30)IVIFSWGℵe11,ℵe12,…,ℵenm=⊗j=1m⊗i=1nℵenmΩiγj.Theorem 2.
By using the IVIFSWG operator, the obtained value is also an IVIFSN and given as follows:(31)IVIFSWAℵe11,ℵe12,…,ℵenm=∏j=1m∏i=1nσijℓ,σijひΩiγj,1−∏j=1m∏i=1n1−τijℓ,τijひΩiγj.Proof.
Similar to Theorem 3.2 [46].Example 3.
LetU = {u1, u2, u3, u4} be a set of experts with weight vector Ωi = 0.35,0.25,0.10,0.30T, and all the experts give their suggestion for the beauty of a city under considered attributes E = e1,e2,e3 with weight vector γj = 0.3,0.5,0.2T. Suggested values of the experts in the form of IVIFSNs are given ℵ,E in the form of a decision matrix as follows:(32)ℵ,E=.3,.5,.2,.4.2,.6,.2,.3.5,.7,.1,.3.5,.7,.1,.2.4,.6,.2,.4.3,.4,.2,.5.46,.2,.3.1,.4,.2,.5.2,.3,.1,.6.2,.4,.3,.5.3,.6,.2,.3.2,.4,.4,.5.
By using equation (31), we get(33)IVIFSWGℵe11,ℵe12,…,ℵe43=.2779,.5364,.2079,.3797.


### 5.2. Decision-Making Approach with the Proposed Operators

An MADM approach is presented here based on the proposed operators and describes numerical examples for showing their effectivity.


#### 5.2.1. Proposed Approach

Assume a set of “s” alternatives such as β = β1,β2,β3,…,βs for assessment under the team of experts such as U = u1,u2,u3,…,un with weights Ω = Ω1,Ω1,…,ΩnT such that Ωi > 0, ∑i=1nΩi = 1. Let E = e1,e2,…,em be a set of attributes with weights andγ = γ1,γ2,γ3,…,γmT be a weight vector for parameters such as γj > 0, ∑j=1mγj = 1. The team of experts {ui: i = 1, 2, …, n} evaluates the alternatives {βz: z = 1, 2, …, s} under the considered parameters {ej: j = 1, 2, …, m} given in the form of IVIFSNs such as ℒijz = σijz,τijz, where σijz = σijℓ,σijひ and τijz = τijℓ,τijひ, in which 0 ≤ σijℓ,σijひ, τijℓ,τijひ ≤ 1 and supσijz + supτijz ≤ 1. So, ℒijz = σijℓ,σijひ,τijℓ,τijひ for all i, j. Experts give their preferences for each alternative in terms of IVIFSNs such as ℵeij = 〈σijℓ,σijひ,τijℓ,τijひ〉 and are given in Tables 1–4. Develop the aggregated IVIFSNs Δk based on the expert’s preference values for each alternative by using IVIFSWG and IVIFSWA operators. Finally, utilizing the alternative, the scoring values [47] of the aggregated IVIFSNs are obtained for the ranking of the alternatives. The algorithm is presented in Figure 2.Figure 2
Decision-making approaches with the proposed operators.


#### 5.2.2. 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 Δk by using the IVIFSWA or IVIFSWG operator.Step 3. Find the scoring values Δk for each alternative [47]:(34)SΔk=σijℓ+σijひ+τijℓ+τijひ4.Step 4. Analyze the ranking.


#### 5.2.3. 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 {β1, β2, β3, β4}. The manager of the company hires a group of four decision makers {u1, u2, u3, u4} whose weight vector is 0.25,0.2,0.15,0.4T. The decision makers give their opinion for each alternative under considered parameters {e1, e2, e3, e4} with weights 0.35,0.25,0.10,0.30T. Listed below are the steps to get the best alternative by utilizing the developed method.


#### 5.2.4. By Using the IVIFSWA Operator

Step 1. These 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. The opinions of the experts for each alternative are aggregated by using equation (27), and hence we getΔ1=.3547.5695,.1689,.3457, Δ2=.3303.4884,.3018,.4429, Δ3=.4296.5670,.2988,.4815, and Δ4=.3391.4956,.2473,.7105.Step 3. Scoring values by using equation (34):SΔ1 = 0.3597, SΔ2 = 0.3909, SΔ3 = 0.4442, and SΔ4 = 0.4631.Step 4. Therefore, the ranking of the alternatives is as follows SΔ4 > SΔ3 > SΔ2 > SΔ1. So, β4 > β3>β2 > β1, and hence the alternative β4 is the most suitable alternative for the company.


#### 5.2.5. By Using the IVIFSWG Operator

Step 1. The 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). Therefore, we getΔ1 = .3144.5379,.1819,.3711, Δ2 = .2815.4420,.3546,.5037, Δ3 = .2904.4223,.3761,.5547, and Δ4 = .2713.5445.3530,.5200.Step 3. Scoring values by using equation (34):SΔ1 = 0.3533, SΔ2 = 0.3955, SΔ3 = 0.4109, and SΔ4 = 0.4222.Step 4. Therefore, the ranking of the alternatives is as follows SΔ4 > SΔ3 > SΔ2 > SΔ1. So, β4 > β3>β2 > β1, and hence the alternative β4 is the most suitable alternative for the company.


## 5.1. Operational Laws for IVIFSNs

Definition 16.
Letℵe=σℓ,σひ,τℓ,τひ, ℵe11=σ11ℓ,σ11ひ,τ11ℓ,τ11ひ, and ℵe12=σ12ℓ,σ12ひ,τ12ℓ,τ12ひ be three IVIFSNs and α be a positive real number, and by algebraic norms, we have(1)
ℵe11⊕ℵe12=σ11ℓ+σ12ℓ−σ11ℓσ12ℓ,σ11ひ+σ12ひ−σ11ひσ12ひ,τ11ℓτ12ℓ,τ11ひτ12ひ(2)
ℵe11⊗ℵe12=σ11ℓσ12ℓ,σ11ひσ12ひ,τ11ℓ+τ12ℓ−τ11ℓτ12ℓ,τ11ひ+τ12ひ−τ11ひτ12ひ(3)
αℵe=1−1−σℓ,σひα,τℓ,τひα(4)
ℵeα=σℓ,σひα,1−1−σℓ,σひα
Some averaging and geometric aggregation operators for IVIFSNs have been defined based on the above laws for the collection of IVIFSNΔ.Definition 17.
Letℵeij=σijℓ,σijひ,τijℓ,τijひ be a collection of IVIFSNs, and Ωi and γj are the weight vector for experts and parameters, respectively, with given conditions Ωi > 0, ∑i=1nΩi = 1; γj > 0, ∑j=1mγj = 1, where i=1,2,…,nandj=1,2,…,m. Then, the IVIFSWA operator is defined as IVIFSWA: Δn ⟶ Δ defined as follows:(26)IVIFSWAℵe11,ℵe12,…,ℵenm=⊕j=1mγj⊕i=1nΩiℵeij.Theorem 1.
Letℵeij=σijℓ,σijひ,τijℓ,τijひ be an IVIFSN, where i=1,2,…,nandj=1,2,…,m, and the aggregated value is also an IVIFSN, such as(27)IVIFSWAℵe11,ℵe12,…,ℵenm=1−∏j=1m∏i=1n1−σijℓ,σijひΩiγj,∏j=1m∏i=1nτijℓ,τijひΩiγj.Proof.
Similar to Theorem 3.1 [46].Example 2.
LetU = {u1, u2, u3, u4} be a set of decision makers, who are going to describe the beauty of a city under the defined set of attributes E = e1,e2,e3. Assume Ωi = 0.35,0.25,0.10,0.30T and γj = 0.3,0.5,0.2T be the weight vectors for experts and attributes, respectively. The assumed rating values of the experts for each attribute in the form of IVIFSNs ℵ,E = σijℓ,σijひ,τijℓ,τijひ4×3 are given as follows:(28)ℵ,E=.3,.5,.2,.4.2,.6,.2,.3.5,.7,.1,.3.5,.7,.1,.2.4,.6,.2,.4.3,.4,.2,.5.4.6,.2,.3.1,.4,.2,.5.2,.3,.1,.6.2,.4,.3,.5.3,.6,.2,.3.2,.4,.4,.5.
By using equation (27),(29)IVIFSWAℵe11,ℵe12,…,ℵe43=1−∏j=13∏i=141−σijℓ,σijひΩiγj,∏j=13∏i=14τijℓ,τijひΩiγj=1−.5,.70.35.3,.50.25.4,.60.1.6,.80.3.3.4,.80.35.4,.60.25.6,.90.1.4,.70.3.5.3,.50.35.6,.70.25.7,.80.1.6,.80.3.2.2,.40.35.1,.20.25.2,.30.1.3,.50.3.3.2,.30.35.2,.40.25.2,.50.1.2,.30.3.5.1,.30.35.2,.50.25.1,.60.1.4,.50.3.2=0.3098,0.5554,0.1929,0.3582.Definition 18.
Letℵeij=σijℓ,σijひ,τijℓ,τijひ be an IVIFSN, and the experts ui’s and parameters ej’s have standardized weight vectors Ωi and γj, respectively; then the IVIFSWG operator is defined as(30)IVIFSWGℵe11,ℵe12,…,ℵenm=⊗j=1m⊗i=1nℵenmΩiγj.Theorem 2.
By using the IVIFSWG operator, the obtained value is also an IVIFSN and given as follows:(31)IVIFSWAℵe11,ℵe12,…,ℵenm=∏j=1m∏i=1nσijℓ,σijひΩiγj,1−∏j=1m∏i=1n1−τijℓ,τijひΩiγj.Proof.
Similar to Theorem 3.2 [46].Example 3.
LetU = {u1, u2, u3, u4} be a set of experts with weight vector Ωi = 0.35,0.25,0.10,0.30T, and all the experts give their suggestion for the beauty of a city under considered attributes E = e1,e2,e3 with weight vector γj = 0.3,0.5,0.2T. Suggested values of the experts in the form of IVIFSNs are given ℵ,E in the form of a decision matrix as follows:(32)ℵ,E=.3,.5,.2,.4.2,.6,.2,.3.5,.7,.1,.3.5,.7,.1,.2.4,.6,.2,.4.3,.4,.2,.5.46,.2,.3.1,.4,.2,.5.2,.3,.1,.6.2,.4,.3,.5.3,.6,.2,.3.2,.4,.4,.5.
By using equation (31), we get(33)IVIFSWGℵe11,ℵe12,…,ℵe43=.2779,.5364,.2079,.3797.


## 5.2. Decision-Making Approach with the Proposed Operators

An MADM approach is presented here based on the proposed operators and describes numerical examples for showing their effectivity.


### 5.2.1. Proposed Approach

Assume a set of “s” alternatives such as β = β1,β2,β3,…,βs for assessment under the team of experts such as U = u1,u2,u3,…,un with weights Ω = Ω1,Ω1,…,ΩnT such that Ωi > 0, ∑i=1nΩi = 1. Let E = e1,e2,…,em be a set of attributes with weights andγ = γ1,γ2,γ3,…,γmT be a weight vector for parameters such as γj > 0, ∑j=1mγj = 1. The team of experts {ui: i = 1, 2, …, n} evaluates the alternatives {βz: z = 1, 2, …, s} under the considered parameters {ej: j = 1, 2, …, m} given in the form of IVIFSNs such as ℒijz = σijz,τijz, where σijz = σijℓ,σijひ and τijz = τijℓ,τijひ, in which 0 ≤ σijℓ,σijひ, τijℓ,τijひ ≤ 1 and supσijz + supτijz ≤ 1. So, ℒijz = σijℓ,σijひ,τijℓ,τijひ for all i, j. Experts give their preferences for each alternative in terms of IVIFSNs such as ℵeij = 〈σijℓ,σijひ,τijℓ,τijひ〉 and are given in Tables 1–4. Develop the aggregated IVIFSNs Δk based on the expert’s preference values for each alternative by using IVIFSWG and IVIFSWA operators. Finally, utilizing the alternative, the scoring values [47] of the aggregated IVIFSNs are obtained for the ranking of the alternatives. The algorithm is presented in Figure 2.Figure 2
Decision-making approaches with the proposed operators.


### 5.2.2. 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 Δk by using the IVIFSWA or IVIFSWG operator.Step 3. Find the scoring values Δk for each alternative [47]:(34)SΔk=σijℓ+σijひ+τijℓ+τijひ4.Step 4. Analyze the ranking.


### 5.2.3. 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 {β1, β2, β3, β4}. The manager of the company hires a group of four decision makers {u1, u2, u3, u4} whose weight vector is 0.25,0.2,0.15,0.4T. The decision makers give their opinion for each alternative under considered parameters {e1, e2, e3, e4} with weights 0.35,0.25,0.10,0.30T. Listed below are the steps to get the best alternative by utilizing the developed method.


### 5.2.4. By Using the IVIFSWA Operator

Step 1. These 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. The opinions of the experts for each alternative are aggregated by using equation (27), and hence we getΔ1=.3547.5695,.1689,.3457, Δ2=.3303.4884,.3018,.4429, Δ3=.4296.5670,.2988,.4815, and Δ4=.3391.4956,.2473,.7105.Step 3. Scoring values by using equation (34):SΔ1 = 0.3597, SΔ2 = 0.3909, SΔ3 = 0.4442, and SΔ4 = 0.4631.Step 4. Therefore, the ranking of the alternatives is as follows SΔ4 > SΔ3 > SΔ2 > SΔ1. So, β4 > β3>β2 > β1, and hence the alternative β4 is the most suitable alternative for the company.


### 5.2.5. By Using the IVIFSWG Operator

Step 1. The 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). Therefore, we getΔ1 = .3144.5379,.1819,.3711, Δ2 = .2815.4420,.3546,.5037, Δ3 = .2904.4223,.3761,.5547, and Δ4 = .2713.5445.3530,.5200.Step 3. Scoring values by using equation (34):SΔ1 = 0.3533, SΔ2 = 0.3955, SΔ3 = 0.4109, and SΔ4 = 0.4222.Step 4. Therefore, the ranking of the alternatives is as follows SΔ4 > SΔ3 > SΔ2 > SΔ1. So, β4 > β3>β2 > β1, and hence the alternative β4 is the most suitable alternative for the company.


## 5.2.1. Proposed Approach

Assume a set of “s” alternatives such as β = β1,β2,β3,…,βs for assessment under the team of experts such as U = u1,u2,u3,…,un with weights Ω = Ω1,Ω1,…,ΩnT such that Ωi > 0, ∑i=1nΩi = 1. Let E = e1,e2,…,em be a set of attributes with weights andγ = γ1,γ2,γ3,…,γmT be a weight vector for parameters such as γj > 0, ∑j=1mγj = 1. The team of experts {ui: i = 1, 2, …, n} evaluates the alternatives {βz: z = 1, 2, …, s} under the considered parameters {ej: j = 1, 2, …, m} given in the form of IVIFSNs such as ℒijz = σijz,τijz, where σijz = σijℓ,σijひ and τijz = τijℓ,τijひ, in which 0 ≤ σijℓ,σijひ, τijℓ,τijひ ≤ 1 and supσijz + supτijz ≤ 1. So, ℒijz = σijℓ,σijひ,τijℓ,τijひ for all i, j. Experts give their preferences for each alternative in terms of IVIFSNs such as ℵeij = 〈σijℓ,σijひ,τijℓ,τijひ〉 and are given in Tables 1–4. Develop the aggregated IVIFSNs Δk based on the expert’s preference values for each alternative by using IVIFSWG and IVIFSWA operators. Finally, utilizing the alternative, the scoring values [47] of the aggregated IVIFSNs are obtained for the ranking of the alternatives. The algorithm is presented in Figure 2.Figure 2
Decision-making approaches with the proposed operators.


## 5.2.2. 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 Δk by using the IVIFSWA or IVIFSWG operator.Step 3. Find the scoring values Δk for each alternative [47]:(34)SΔk=σijℓ+σijひ+τijℓ+τijひ4.Step 4. Analyze the ranking.


## 5.2.3. 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 {β1, β2, β3, β4}. The manager of the company hires a group of four decision makers {u1, u2, u3, u4} whose weight vector is 0.25,0.2,0.15,0.4T. The decision makers give their opinion for each alternative under considered parameters {e1, e2, e3, e4} with weights 0.35,0.25,0.10,0.30T. Listed below are the steps to get the best alternative by utilizing the developed method.


## 5.2.4. By Using the IVIFSWA Operator

Step 1. These 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. The opinions of the experts for each alternative are aggregated by using equation (27), and hence we getΔ1=.3547.5695,.1689,.3457, Δ2=.3303.4884,.3018,.4429, Δ3=.4296.5670,.2988,.4815, and Δ4=.3391.4956,.2473,.7105.Step 3. Scoring values by using equation (34):SΔ1 = 0.3597, SΔ2 = 0.3909, SΔ3 = 0.4442, and SΔ4 = 0.4631.Step 4. Therefore, the ranking of the alternatives is as follows SΔ4 > SΔ3 > SΔ2 > SΔ1. So, β4 > β3>β2 > β1, and hence the alternative β4 is the most suitable alternative for the company.


## 5.2.5. By Using the IVIFSWG Operator

Step 1. The 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). Therefore, we getΔ1 = .3144.5379,.1819,.3711, Δ2 = .2815.4420,.3546,.5037, Δ3 = .2904.4223,.3761,.5547, and Δ4 = .2713.5445.3530,.5200.Step 3. Scoring values by using equation (34):SΔ1 = 0.3533, SΔ2 = 0.3955, SΔ3 = 0.4109, and SΔ4 = 0.4222.Step 4. Therefore, the ranking of the alternatives is as follows SΔ4 > SΔ3 > SΔ2 > SΔ1. So, β4 > β3>β2 > β1, and hence the alternative β4 is the most suitable alternative for the company.


## 6. 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 methodologies of IFSSs have some limitations on membership and nonmembership grades, and they cannot deal with parameterizations. The 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. There 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. The graphical representation of all results is presented in Figure3. The 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 β4 is selected for the post of the “electrical engineer.” All rankings are also calculated by applying the existing approaches. The proposed methods are also compared with other existing methods by Imtiaz et al. [48], Wu and Su [49], and Mukherjee and Sarkar [50]. The 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. The final score values of the proposed TOPSIS method, IVIFSWA, and IVIFSWG can be seen in Table 10.Figure 3
Comparison of the alternative rank with the proposed algorithm.Table 9
Alternative rank comparison using the existing and proposed techniques.
MethodAlternative final rankingOptimal choiceProposed algorithm using IVIFSWG operatorβ4>β3>β2>β1β4Proposed algorithm using IVIFSWA operatorβ4>β3>β2>β1β4Proposed TOPSIS algorithmβ4>β2>β3>β1β4Mukherjee and Sarkar [50]β4>β2>β3>β1β4Imtiaz et al. [48]β4>β3>β2>β1β4Wu and Su [49]β4>β3>β2>β1β4Table 10
Alternative final scores using the proposed techniques.
AlternativeProposed algorithm using IVIFSWG operatorProposed algorithm using IVIFSWA operatorProposed TOPSIS algorithmβ10.35330.35970.42839β20.39550.39090.51499β30.41090.44420.46154β40.42220.46310.98198


## 7. Conclusion

The 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. The 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. The 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 decision-making techniques based on developed operators. Finally, a numerical illustration has been described to solve the MADM problem by using the proposed TOPSIS method and other developed techniques. Furthermore, we have compared our proposed method with some already existing techniques. In the future, we plan to extend our work to (i) CC for multipolar interval-valued intuitionistic fuzzy soft sets, (ii) CC for interval-valued neutrosophic soft sets, and (iii) CC for the multipolar interval-valued neutrosophic soft sets.



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*Source: 6656858-2021-01-29.xml*