An Approach for the Assessment of Multi-National Companies Using a Multi-Attribute Decision Making Process Based on Interval Valued Spherical Fuzzy Maclaurin Symmetric Mean Operators

Abstract

Many fuzzy concepts have been researched and described with uncertain information. Collecting data under uncertain information is a difficult task, especially when there is a difference between the opinions of experts. To deal with such situations, different types of operators have been introduced. This paper aims to develop the Maclaurin symmetric mean (MSM) operator for the information in the shape of the interval-valued spherical fuzzy set (IVSFS). In this article, a family of aggregation operators (AOs) is proposed which consists of interval valued spherical fuzzy Maclaurin symmetric mean operator (IVSFMSM), interval valued spherical fuzzy weighted Maclaurin symmetric mean (IVSFWMSM), interval valued spherical fuzzy dual Maclaurin symmetric mean (IVSFDMSM), and interval valued spherical fuzzy dual weighted Maclaurin symmetric mean (IVSFDWMSM) operators. In this paper, we studied an elucidative example to discuss the evaluation of multi-national companies for the application of the proposed operator. Then the obtained results from the proposed operators are compared. The results obtained are graphed and tabulated for a better understanding.

1. Introduction

A fuzzy set (FS) was introduced by Zadeh [1] in 1965 to deal with an uncertain situation. In FS, there is only one side we can read about how much a phenomenon relates to a set. To deal with the two-sided opinions of experts Atanassov [2] gave the idea of an intuitionistic fuzzy set (IFS), in which we use (MD) and (NMD), and an (RD) shows the accuracy of IFS. IFS also describes the disconnection of phenomena, and the sum is existing in $\left[0,1\right]$. Atanassov [3] extended IFS into the interval-valued intuitionistic fuzzy set (IVIFS). To increase the accuracy of IFS, Yager [4] proposed the idea of the Pythagorean fuzzy set (PyFS). In which, we take $0\le {\left(f\left(\nu \right)\right)}^2+{\left(w\left(\nu \right)\right)}^2\le 1$. Peng and Yager extended the concept of PyFS into an interval-valued Pythagorean fuzzy set (IVPyFS). Coung [5] introduced the concept of a picture fuzzy set (PFS), in which we use an abstain degree (AD). A picture-fuzzy set eliminates the loss of information due to four possibilities (MD, NMD, AD, and RD). Coung [6] also worked on the extended form of PFS and introduced the concept of an interval-valued picture fuzzy set (IVPFS). The concept of a spherical fuzzy set (SFS) was introduced by Ullah et al. [7]. The notion of q-Rung Orthopair Fuzzy Set (q-ROFS) was given by Yager [8], in which we use cubic power for the MD, NMD, AD, and RD. A (SFS) increases the range of accuracy by taking the square of all the degrees. FS theory plays an important role in different mathematical fields, such as work by Ghaznavi et al. [9] on the parametric fuzzy equation, and Jafri et al. [10] work on the fuzzy differential equation. Ullah et al. [11] also worked on the Interval Valued Spherical Fuzzy Set (IVSFS). Some more work on the SFS and TSFS can be found in [12,13,14].

Aggregation operators (AOs) [15] are important tools for gathering information under uncertain information. Over the last decade, a lot of AOs have been developed to aggregate the results of fuzzy concepts in undefined situations. AOs are very important due to the use of these operators in different fields of fuzzy theory. If we talk about AOs, the Bonferroni mean operator (BMO) was developed by Bonferroni [16], and Liang et al. extended it into the Weighted Pythagorean Fuzzy Geometric Bonferroni mean operator (WFGBM) [17]. Sykora had developed the Heronian Mean operator (HMO) [18], and Yu developed the concept of an intuitionistic fuzzy geometric weighted Heronian Mean operator (IFGHMO) [19]. The generalized Heronian mean operator based on q rung Orthopair (q-ROPFGHMO) [20] was developed by Wei et al. Over time, many mathematicians made extensions to (BMO), such as the partitioned Heronian mean (PHM) operator based on linguistic fuzzy numbers [21]. To solve (MADM) problems with more accuracy, Xing YP et al. [22] combined (HMO) with interactional operational law. Dombi t norm (t-conorm) is an elastic operator; the (DMO) based on (IFS) was developed by Liu et al. [23]. In fuzzy theory, Chen and Ye developed the concept of generalized Dombi operations (GDO) based on the neutrosophic cubic fuzzy set [24]. To solve (MADM) problem, Shi and Ye developed the idea of (WNSCFS) based on the t norm (t-conorm) [25]. Yang and Pang worked on more extensions to BMO and Dombi t norm (t-conorm) [26]. The related literature can be found in [13,27,28,29].

Maclaurin [30] gave the concept of the MSM operator, which is a high-significance form of AOs. MSM operators are very important due to their correlation with four (MD, NMD, AD, and RD) input arguments, like (BMO) and (HMO). As we know, all the existing operators correlate with two input arguments, but MSM eliminates the loss of information. Liu [31] proposed an extended form of the IFMSM operator, and Liu et al. [32] did extensive work on IVIFMSM. Wei and Lu [33] developed the PyFMSM operator, which was later expanded by Wei et al. [34]. The idea of PFMSM was given by Ullah et al. [35], and extended work on IVPMSM operators was done by Ashraf et al. [36]. The idea of q-ROPFMSM was developed by Liu [37], in which we take the cubic powers of MD, NMD, AD, and RD. MSM operators are unique due to their correlation with more than two input arguments. As we know, PFMSM [31] operators increased the range of accuracy, so if we take a square of the four possible degrees, the range of accuracy increases. More work on the MSM operator can be found in [38,39,40,41].

The IVSFS is the framework that covers information with the least amount of data loss from real-life scenarios. Furthermore, the MSM operator is an interesting AO that aggregates the information by preserving the relationship of the components of the information. The major contribution of this article is to develop a family of AOs for IVSFS based on the MSM operator. In Section 2, we define the background of FS theory and the importance of aggregation operators (AOs). We proposed IVSFMSM and IVSFWMSM in Section 3. In Section 4, we developed the concept of IVSFDMSM and IVSFDWMSM operators. We analyze some special cases of the developed AOs in Section 5. In Section 6, we applied the developed AOs to the MADM problem. In Section 7, we analyze the comparative study of developed AOs with traditional operators. Conclusive remarks are in Section 8.

2. Preliminaries

In this section, we define the SFS, IVSFS, MSM, and score function of the IVSFS. We also described the basic operations of the IVSFS and MSM operators.

Definition 1:

[11] For a universal set Z, a SFS is defined as $X=\left\{\left(v,\left(f\left(\nu \right),g\left(\nu \right),w\left(\nu \right)\right)\right):\nu \in Z\right\}$ where$f, g,$ and w are mapped from Z to$\left[0,1\right]$, with the condition that $0\le {\left(f\left(\nu \right)\right)}^2+{\left(g\left(\nu \right)\right)}^2+{\left(w\left(\nu \right)\right)}^2\le 1$. A refusal degree can be defined as $A\left(\nu \right)={\left(1-\left({\left(f\left(\nu \right)\right)}^2+{\left(g\left(\nu \right)\right)}^2+{\left(w\left(\nu \right)\right)}^2\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ and a triplet $\left(f\left(\nu \right),g\left(\nu \right),w\left(\nu \right)\right)$ is known as SFV. Further, $f\left(\nu \right)$ shows MD $g\left(\nu \right)$ is AD, and $w\left(\nu \right)$ is ND.

Definition 2:

[11] In a universal set Z an IVSFS is defined as $X=\left\{\left(v,\left(f\left(\nu \right),g\left(\nu \right),w\left(\nu \right)\right)\right):\nu \in Z\right\}$ where $f, g$ and $w$ are mapped from Z to $\left[0,1\right]$ such that $f\left(\nu \right)=\left[f^{inf}\left(\nu \right),f^{sup}\left(\nu \right)\right]$,$g\left(\nu \right)=\left[g^{inf}\left(\nu \right),g^{sup}\left(\nu \right)\right]$ and $w\left(\nu \right)=\left[\left(w^{inf}\right)\left(\nu \right),w^{sup}\left(\nu \right)\right]$ with the condition that $0\le {\left(f{\left(\nu \right)}^{sup}\right)}^2+{\left(g{\left(\nu \right)}^{sup}\right)}^2+{\left(w{\left(\nu \right)}^{sup}\right)}^2\le 1$. A refusal degree can be derived as $A\left(\nu \right)=\left[A^{inf}\left(\nu \right),A^{sup}\left(\nu \right)\right]$

$$=\left[\begin{array}{c}{\left(1-\left({\left(f^{inf}\right)}^2\left(\nu \right)+{\left(g^{inf}\right)}^2\left(\nu \right)+{\left(w^{inf}\right)}^2\left(\nu \right)\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ \left({\left(1-\left({\left(f^{sup}\right)}^2\left(\nu \right)+{\left(g^{sup}\right)}^2\left(\nu \right)+{\left(w^{sup}\right)}^2\left(\nu \right)\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\end{array}\right]$$

and a triplet $\left(f\left(\nu \right),g\left(\nu \right),w\left(\nu \right)\right)=\left(\left[f^{inf}\left(\nu \right),f^{sup}\left(\nu \right)\right],\left[g^{inf}\left(\nu \right),g^{sup}\left(\nu \right)\right],\left[w^{inf}\left(\nu \right),w^{sup}\left(\nu \right)\right]\right)$ is known as IVSFV here $f\left(\nu \right)$ shows MD $g\left(\nu \right)$ are AD and $w\left(\nu \right)$ is ND.

The score function for IVSFVs is given below.

Definition 3:

Let $A_i=\left(\left[f^{inf}\left(\nu \right),f^{sup}\left(\nu \right)\right],\left[g^{inf}\left(\nu \right),g^{sup}\left(\nu \right)\right],\left[w^{inf}\left(\nu \right),w^{sup}\left(\nu \right)\right]\right)$be values of IVSFS, then score function is defined as

$$SC\left(\nu \right)=\frac{\begin{array}{c}{\left(f^{inf}\left(v\right)\right)}^2\left(1-{\left(g^{inf}\left(v\right)\right)}^2-{\left(w^{inf}\left(v\right)\right)}^2\right)+\\ {\left(f^{sup}\left(v\right)\right)}^2\left(1-{\left(g^{sup}\left(v\right)\right)}^2-{\left(w^{sup}\left(v\right)\right)}^2\right)\end{array}}{3}$$

(1)

the accuracy function is defined as

$$H\left(\nu \right)=\frac{\begin{array}{c}{\left(f^{inf}\left(v\right)\right)}^2\left(1+{\left(g^{inf}\left(v\right)\right)}^2+{\left(w^{inf}\left(v\right)\right)}^2\right)+\\ {\left(f^{sup}\left(v\right)\right)}^2\left(1+{\left(g^{sup}\left(v\right)\right)}^2+{\left(w^{sup}\left(v\right)\right)}^2\right)\end{array}}{3}$$

(2)

Definition 4:

[11] Let$L_1=\left(\left[f_1^{inf}\left(v\right),f_1^{sup}\left(v\right)\right],\left[g_1^{inf}\left(v\right),g_1^{sup}\left(v\right)\right],\left[w_1^{inf}\left(v\right),w_1^{sup}\left(v\right)\right]\right)$ and $L_2=\left(\left[f_2^{inf}\left(v\right),f_2^{sup}\left(v\right)\right],\left[g_2^{inf}\left(v\right),g_2^{sup}\left(v\right)\right],\left[w_2^{inf}\left(v\right),w_2^{sup}\left(v\right)\right]\right)$ be two IVSFVs, then we can define the following operations. Note that $\otimes$ and $\oplus$ denote the multiplication and addition of two IVSFVs.

$$L_1\otimes L_2=\left\{\left(\begin{array}{c}\left[f_1^{inf}\left(\nu \right)f_2^{inf}\left(\nu \right),f_1^{sup}\left(\nu \right)f_2^{sup}\left(\nu \right)\right]\\ \left[\begin{array}{c}1-\left(1-g_1^{inf}{\left(\nu \right)}^2\right){\left(1-g_2^{inf}{\left(\nu \right)}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-\left(1-g_1^{sup}{\left(\nu \right)}^2\right)\left(1-g_2^{sup}{\left(\nu \right)}^2\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\\ \left[\begin{array}{c}{\left(1-\left(1-w_1^{inf}{\left(\nu \right)}^2\right)\left(1-w_2^{inf}{\left(\nu \right)}^2\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-\left(1-w_1^{sup}{\left(\nu \right)}^2\right)\left(1-w_2^{sup}{\left(\nu \right)}^2\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array},\right),\nu \in Z\right\}$$

$$L_1\oplus L_2=\left\{\left(\begin{array}{c}\left[\begin{array}{c}{\left(1-\left(1-f_1^{inf}{\left(\nu \right)}^2\right)\left(1-f_2^{sup}{\left(\nu \right)}^2\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-\left(1-f_1^{sup}\left(\nu \right)\right)\left(1-f_2^{sup}\left(\nu \right)\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\\ \left[g_1^{inf}\left(\nu \right)g_2^{inf}\left(\nu \right),g_1^{sup}\left(\nu \right)g_2^{sup}\left(\nu \right)\right],\\ \left[w_1^{inf}\left(\nu \right)w_2^{inf}\left(\nu \right),w_1^{sup}\left(\nu \right)w_2^{sup}\left(\nu \right)\right]\end{array},\right),\nu \in Z\right\}$$

$$\xi L_1=\left\{\left(\begin{array}{c}\left[{\left(1-{\left(1-f_1^{inf}\left(\nu \right)\right)}^{\xi }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},{\left(1-{\left(1-f_1^{sup}\left(\nu \right)\right)}^{\xi }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right],\\ \left[{\left(g_1^{inf}\left(\nu \right)\right)}^{\xi },{\left(g_1^{sup}\left(\nu \right)\right)}^{\xi }\right],\\ \left[{\left(w_1^{inf}\left(\nu \right)\right)}^{\xi },{\left(w_1^{sup}\left(\nu \right)\right)}^{\xi }\right]\end{array}\right), \nu \in Z\right\}$$

$$L_1{}^{\xi }=\left\{\left(\begin{array}{c}\left[{\left(f_1^{inf}\left(\nu \right)\right)}^{\xi },{\left(f_1^{sup}\left(\nu \right)\right)}^{\xi }\right],\\ \left[{\left(1-{\left(1-g_1^{inf}\left(\nu \right)\right)}^{\xi }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},{\left(1-{\left(1-g_1^{sup}\left(\nu \right)\right)}^{\xi }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right],\\ \left[{\left(1-{\left(1-w_1^{inf}\left(\nu \right)\right)}^{\xi }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},{\left(1-{\left(1-w_1^{sup}\left(\nu \right)\right)}^{\xi }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]\end{array}\right), \nu \in Z\right\}$$

Definition 5:

[35] Let $A_i=\left(i=1,2,3,\dots ,r\right)$ be a collection of positive real numbers. Then

$$MS{M}^{\left(\nu \right)}\left(c_1,c_2,\dots ,c_r\right)={\left(\frac{{{\displaystyle \sum }}_{1\le i_1\le ,\dots ,i_{{\rm Y}}\le r}{{\displaystyle \prod }}_{\tau =1}^{{\rm Y}}{A}_{i_{\tau }}}{C_r^{{\rm Y}}}\right)}^{\frac{1}{{\rm Y}}}$$

is called MSM. Where $\left(c_1,c_2,\dots ,c_{{\rm Y}}\right)$ convert all the l-tuple combinations of $\left(1,2,\dots ,r\right)$ and $C_r^{{\rm Y}}$is the binomial coefficient.

Now we discuss the main work regarding this article.

3. Interval Valued Spherical Fuzzy Maclaurin Symmetric Mean (IVSFMSM) Operator

In this section, we developed the concept of the IVSFMSM and IVSFWMSM operators.

Definition 6:

Let $A_i=\left(\left[f_i^{inf}\left(v\right), f_i^{sup}\left(v\right)\right] \left[g_i^{inf}\left(v\right), g_i^{sup}\left(v\right)\right] \left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$and be two IVSFVs. Then, the IVSFMSM operator is given by

$$IVSFMSM\left(A_1,A_{2,\dots ,}{A}_r\right)={\left(\frac{\underset{1\le i_1\le ,\dots ,i_{{\rm Y}}\le r}{\oplus }\left(\stackrel{{\rm Y}}{\underset{\tau =1}{\otimes }}{A}_{i_{\tau }}\right)}{C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}.$$

(3)

Theorem 1:

Let $A_i=\left(\left[f_i^{inf}\left(v\right),f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right),g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right),w_i^{sup}\left(v\right)\right]\right)$be a collection of IVSFVs. Then, using IVSFMSM operator, we get

$$\begin{gathered} IVSFMSM\left(A_1,A_2,\cdots ,A_r\right) \\ =\left(\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/{\rm Y}}\right)}^{1/2}\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/{\rm Y}}\right)}^{1/2}\end{array}\right],\\ \left[\begin{array}{c}1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}},\\ 1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{\dot{{\rm Y}}}}\end{array}\right],\\ \end{array}\\ \left[\begin{array}{c}{\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array}\right) \end{gathered}$$

Proof: 

By using Definition 6, we have

$$\stackrel{{\rm Y}}{\underset{\tau =1}{⨂}}{A}_{i_{\tau }}=\left(\begin{array}{c}\left[{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2,{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right],\\ \left[1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right),1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right],\\ \left[1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right),1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right]\end{array}\right),$$

$$\underset{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}{⨁}\left(\stackrel{{\rm Y}}{\underset{\tau =1}{⨂}}{A}_{i_{\tau }}\right)=\left(\begin{array}{c}\left[\begin{array}{c}{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/2},\\ {\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/2}\end{array}\right],\\ \left[\begin{array}{c}{\left(\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/2}\\ ,{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/2}\end{array}\right],\\ \left[\begin{array}{c}{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/2},\\ {\left({\displaystyle \prod _{1\le {{\rm Y}}_1\le ,\dots ,<i_{{\rm Y}}\le r}^{{\rm Y}}}1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/2}\end{array}\right]\\ \end{array}\right)$$

$$\frac{1}{C_r^{{\rm Y}}}{\displaystyle \underset{1\le {{\rm Y}}_1\le ,\dots ,<{{\rm Y}}_{{\rm Y}}\le r}{⨂}}\left(\stackrel{{\rm Y}}{\underset{\tau =1}{⨂}}{A}_{i_{\tau }}\right)=\left(\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left(1-{\left({\displaystyle \prod _{1\le {{\rm Y}}_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/2}\end{array}\right]\end{array}\\ \left[\begin{array}{c}1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}},\\ 1-{\left({\displaystyle \prod _{1\le {{\rm Y}}_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\end{array}\right]\end{array}\right)$$

therefore, we get

$$\begin{gathered} IVSFMSM\left(A_1,A_2,\dots ,A_r\right) \\ =\left(\begin{array}{c}\begin{array}{c}\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/{\rm Y}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array},\\ \left[\begin{array}{c}{\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{\dot{{\rm Y}}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array}\right) \end{gathered}$$

as we know, an aggregation operator fulfills the criteria of three properties (boundedness, idempotency, and monotonicity). So, IVSFMSM operators satisfied these properties as given below

Property 1:

(Idempotency Property) Let ${\alpha }_i=\left(\left[f_i^{inf}\left(v\right) ,f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right) ,g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$ and ${\alpha }_j=\left(\left[f_j^{inf}\left(v\right) ,f_j^{sup}\left(v\right)\right],\left[g_j^{inf}\left(v\right) ,g_j^{sup}\left(v\right)\right],\left[w_j^{inf}\left(v\right), w_j^{sup}\left(v\right)\right]\right)$ be two collections of IVSFVs. If $A_i=A$ then (all are identical)

$$IVSFMSM\left(A_1,A_2,A_3,\dots ,A_r\right)=A$$

Property 2:

(Monotonicity Property) Let $A_{i\tau }$ and${\check{A}}_{i\tau }$ be two IVSFVs. If$f_i^{inf}\left(v\right)\le {\check{f}}_i^{inf}\left(v\right)$,$f_i^{sup}\left(v\right)\le {\check{f}}_i^{sup}\left(v\right)$,$g_i^{inf}\left(v\right)\ge {\check{g}}_i^{inf}\left(v\right)$,$g_i^{sup}\left(v\right)\ge {\check{g}}_i^{sup}\left(v\right)$, and $w_i^{inf}\left(v\right)\ge {\check{w}}_i^{inf}\left(v\right)$, $w_i^{sup}\left(v\right)\ge {\check{w}}_i^{sup}\left(v\right)$ then

$$IVSFMSM\left(A_1,A_2,\dots ,A_r\right)\le IVSFMSM\left({\check{A}}_1,{\check{A}}_2,\dots ,{\check{A}}_r\right)$$

Property 3:

(Boundedness Property) Let $A_i=\left(\left[f_i^{inf}\left(v\right) ,f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right) ,g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$ be a collection of IVSFVs and let$A^{inf}$and$A^{sup}$ denote the smallest and the greatest IVSFVs respectively. Then

$$A^{inf}\le IVSFMSM\left(A_1,A_2,\dots ,A_r\right)\le A^{sup}$$

Definition 7:

Let $A_i=\left(\left[f_i^{inf}\left(v\right) ,f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right) ,g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$be a collection of IVSFVs and${\omega }_i$be the weight vector of$A_i$such that ${{\displaystyle \sum }}_{i=1}^n{\omega }_i=1$

. Then, the IVSFWMSM operator is defined by

$$IVSFWMSM\left(A_1,A_2,\dots ,A_r\right)=\left[{\left(\frac{\underset{1\le i_1\le ,\dots ,<i_{{\rm Y}}}{\otimes }{\left(\stackrel{{\rm Y}}{\underset{\tau =1}{\otimes \left(A_{i_{\tau }}\right)}}\right)}^{{\omega }_{i\tau }}}{C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right].$$

(4)

Theorem 2:

Let $A_i=\left(\left[f_i^{inf}\left(v\right) ,f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right) ,g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$be a collection of IVSFVs. Then, using IVSFWMSM operator, we get

$$\begin{gathered} IVSFWMSM\left(A_1,A_2,\dots ..,A_r\right) \\ =\left(\begin{array}{c}\left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\left({\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{\dot{{\rm Y}}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\left({\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left(1-{\left(1-{\left({\displaystyle \prod _{1\le {{\rm Y}}_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{\dot{{\rm Y}}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\left({\displaystyle \prod _{1\le {{\rm Y}}_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{\dot{{\rm Y}}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left(1-{\left(1-{\left({\displaystyle \prod _{1\le {{\rm Y}}_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{\dot{{\rm Y}}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\left({\displaystyle \prod _{1\le {{\rm Y}}_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{\dot{{\rm Y}}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array}\right). \end{gathered}$$

Proof: 

Proof is skipped.

4. Interval-Valued IVSFDMSM Operator

The main purpose of this part of the paper is to develop the ideas of IVSFDMSM and IVSFDWMSM by using MD, NMD, and AD.

Definition 8:

Let $A_i=\left(\left[f_i^{inf}\left(v\right) ,f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right) ,g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$be a collection of IVSFVs. Then, IVSFDMSM is defined as

$$IVSFDMSM\left(A_1,A_2,\dots ,A_r\right)=\left[\frac{1}{{\rm Y}}\left(\underset{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}{\oplus }{\left(\stackrel{{\rm Y}}{\underset{\tau =1}{\otimes }}{A}_{i_{\tau }}\right)}^{1/C_r^{{\rm Y}}}\right)\right]$$

Theorem 3:

Let $A_i=\left(\left[f_i^{inf}\left(v\right) ,f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right) ,g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$denote the collection of IVSFVs. Then, by using IVSFDMSM operators, we have

$$\begin{gathered} IVSFDMSM\left(A_1,A_2,\dots ,A_r\right) \\ =\left(\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\\ \end{array},\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array}\right). \end{gathered}$$

Proof: 

Using Definition 8, we have

$$\stackrel{{\rm Y}}{\underset{\tau =1}{⨂}}{A}_{i_{\tau }}=\left(\begin{array}{c}\left[{\left(1-{\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{\frac{1}{2}},{\left(1-{\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}\left(1-{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{\frac{1}{2}}\right],\\ \left[{\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}{g}_{i_{\tau }}^{inf}\left(v\right),{\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}{g}_{i_{\tau }}^{sup}\left(v\right)\right],\left[{\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}{w}_{i_{\tau }}^{inf}\left(v\right),{\displaystyle \prod _{\tau =1}^{\dot{{\rm Y}}}}{w}_{i_{\tau }}^{sup}\left(v\right)\right]\end{array}\right)$$

$${\left(\stackrel{{\rm Y}}{\underset{\tau =1}{⨂}}{A}_{i\tau }\right)}^{1/C_r^{{\rm Y}}}=\left(\begin{array}{c}\left[\begin{array}{c}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}},\\ {\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\end{array}\right],\\ \left[\begin{array}{c}1-{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}},\\ 1-{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\end{array}\right],\\ \left[\begin{array}{c}1-{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}},\\ 1-{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\end{array}\right]\end{array}\right)$$

$$\underset{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}{\dot{⨂}}{\left(\stackrel{{\rm Y}}{\underset{\tau =1}{⨁}}{A}_{i_{\tau }}\right)}^{1/C_r^{{\rm Y}}}=\left(\begin{array}{c}\left[\begin{array}{c}{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}},\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ {\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}},\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ {\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array}\right)$$

therefore, we get

$$\begin{gathered} IVSFDMSM\left(A_1,A_2,\dots ,A_r\right) \\ =\left(\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{\left(1-{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(1-{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({{\displaystyle \prod _{\tau =1}^{{\rm Y}}}\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\\ \end{array},\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\end{array}\right) \end{gathered}$$

IVSFDMSM operators also satisfied the aggregation properties (Boundedness, Idempotency, and Monotonicity).

Property 4:

(Idempotency property) Let $A_i=\left(\left[f_i^{inf}\left(v\right), f_i^{sup}\left(v\right)\right] \left[g_i^{inf}\left(v\right), g_i^{sup}\left(v\right)\right] \left[w_i^{inf}\left(v\right), w_i^{sup}\left(v\right)\right]\right)$ and if $A_i=A$ then (all are identical)

$$IVSFDMSM\left(A_1,A_2,A_3,\dots ,A_r\right)=A.$$

Property 5:

(Boundedness Property) Let$A_i=\left(\left[f_i^{inf}\left(v\right),f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right),g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right),w_i^{sup}\left(v\right)\right]\right)$ be a collection of IVSFVs and let$A^{inf}$ and$A^{sup}$ denote the smallest and the greatest IVSFVs, respectively. Then

$$A^{inf}\le IVSFFDMSM\left(A_1,A_2,\dots ,A_r\right)\le A^{sup}.$$

Definition 9:

Let$A_i=\left(\left[f_i^{inf}\left(v\right),f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right),g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right),w_i^{sup}\left(v\right)\right]\right)$be a collection of IVSFVs. Then, the IVSFDWMSM operator is given as

$$\begin{gathered} IVSFDWMSM\left(A_1,A_2,\dots ,A_r\right) \\ =\left(\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{\left(1-{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\\ \end{array},\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\end{array}\right]\end{array}\right). \end{gathered}$$

Theorem 4:

Let$A_i=\left(\left[f_i^{inf}\left(v\right),f_i^{sup}\left(v\right)\right],\left[g_i^{inf}\left(v\right),g_i^{sup}\left(v\right)\right],\left[w_i^{inf}\left(v\right),w_i^{sup}\left(v\right)\right]\right)$denote the collection of IVSPFNs. Then, by using IVSFDWMSM operators, we have

$$\begin{gathered} IVSFDWMSM\left(A_1,A_2,\dots ,A_r\right) \\ =\left(\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{\left(1-{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(f_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left(1-{\left(1-{\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}{\left(1-{\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left(1-{\left(f_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right],\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(g_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(g_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]\\ \end{array},\\ \left[\begin{array}{c}{\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(w_{i_{\tau }}^{inf}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ {\left({\left(1-{\left({\displaystyle \prod _{1\le i_1\le ,\dots ,<i_{{\rm Y}}\le r}}\left(1-\left({\displaystyle \prod _{\tau =1}^{{\rm Y}}}{\left({\left(w_{i_{\tau }}^{sup}\left(v\right)\right)}^2\right)}^{{\omega }_{i\tau }}\right)\right)\right)}^{1/C_r^{{\rm Y}}}\right)}^{1/\dot{{\rm Y}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\end{array}\right]\end{array}\right). \end{gathered}$$

Proof: 

The proof is the same as in Theorem 3 above.

We will discuss a numerical example to illustrate the calculation process.

We take the values $A_1=\left(\left[0.21,0.30\right],\left[0.18,0.25\right], \left[0.05,0.19\right]\right)$, $A_2=\left(\left[0.09,0.31\right],\left[0.15,0.20\right],\left[0.19,0.25\right]\right)$, $A_3=\left(\left[0.08,0.17\right],\left[0.14,0.20\right],\left[0.21,0.24\right]\right), A_4=\left(\left[0.17,0.31\right],\left[0.07,0.12\right],\left[0.20,0.25\right]\right)$ also $r=4$ and ${\rm Y}=1$. Now we take a weight vector such as ${\left(0.3, 0.2,0.1,0.4\right)}^T$.

For MD,

$$\begin{gathered} =\left(\left[\begin{array}{c}{\left(1-{\left(1-{\left(\begin{array}{c}1-{\left({\left({\left(1-0.21\right)}^2\right)}^{0.3}\times {\left(1-0.09\right)}^2\right)}^{0.2}\\ \times {\left({\left(1-0.08\right)}^2\right)}^{0.1}\times {\left({\left(1-0.17\right)}^2\right)}^{0.4}\end{array}\right)}^{\frac{1}{4}}\right)}^1\right)}^{\frac{1}{2}},\\ {\left(1-{\left(1-{\left(\begin{array}{c}1-{\left({\left({\left(1-0.30\right)}^2\right)}^{0.3}\times {\left(1-0.31\right)}^2\right)}^{0.2}\times \\ {\left({\left(1-0.17\right)}^2\right)}^{0.1}\times {\left({\left(1-0.31\right)}^2\right)}^{0.4}\end{array}\right)}^{1/4}\right)}^1\right)}^{1/2}\end{array}\right]\right), \\ =[0.11,0.12] \end{gathered}$$

for AD,

$$=\left[\begin{array}{c}{\left(1-{\left({\begin{array}{c}\left(1-{\left({\left(0.18\right)}^2\right)}^{0.3}\right)\times \end{array}}^{}\right)}^{}\right)}^{}\end{array}\right]$$