Impact of Machine Learning and Artificial Intelligence in Business Based on Intuitionistic Fuzzy Soft WASPAS Method

Abstract

Artificial intelligence (AI) is a well-known and reliable technology that enables a machine to simulate human behavior. While the major theme of AI is to make a smart computer system that thinks like a human to solve awkward problems, machine learning allows a machine to automatically learn from past information without the need for explicit programming. In this analysis, we aim to derive the idea of Aczel–Alsina aggregation operators based on an intuitionistic fuzzy soft set. The initial stage was the discovery of the primary and critical Aczel–Alsina operational laws for intuitionistic fuzzy soft sets. Subsequently, we pioneer a range of applicable theories (set out below) and identify their essential characteristics and key results: intuitionistic fuzzy soft Aczel–Alsina weighted averaging; intuitionistic fuzzy soft Aczel–Alsina ordered weighted averaging; intuitionistic fuzzy soft Aczel–Alsina weighted geometric operators; and intuitionistic fuzzy soft Aczel–Alsina ordered weighted geometric operators. Additionally, by utilizing certain key information, including intuitionistic fuzzy soft Aczel–Alsina weighted averaging and intuitionistic fuzzy soft Aczel–Alsina weighted geometric operators, we also introduce the theory of the weighted aggregates sum product assessment method for intuitionistic fuzzy soft information. This paper also introduces a multi-attribute decision-making method, which is based on derived operators for intuitionistic fuzzy soft numbers and seeks to assess specific industrial problems using artificial intelligence or machine learning. Finally, to underline the value and reasonableness of the information described herein, we compare our obtained results with some pre-existing information in the field. This comparison is supported by a range of numerical examples to demonstrate the practicality of the invented theory.

1. Introduction

Machine learning is a field of inquiry that seeks to comprehend and construct methods that “learn”, that is, methods that manipulate information to improve the execution of tasks. A leading field and one of considerable value, it is considered to be an element within the broader area of artificial intelligence in which machine learning techniques develop a novel model based on simple information known as “training data” and use this to generate predictions or opinions without being explicitly programmed to do so. Machine learning techniques are employed in a wide range of applications in different fields, such as agriculture, email filtering, computer vision, and speech recognition. Machine learning may involve clustering or cluster analysis, a technique of substantial value and utility in the classification of objects into collections according to the extent of similarity between their variable characteristics. A wide range of valuable methods has been established by scholars of classical set theory, including pattern recognition, medical diagnosis, artificial intelligence, machine learning, data mining, and decision making. However significant problems may occur in the application of classical information, notwithstanding the expertise with which it is deployed. Substantial amounts of valuable and knowledgeable information may be lost during the decision-making procedure due to the availability of only two possibilities, “zero” and “one”. Experts identified the need for ideas that allow decision makers greater freedom of choice beyond merely “zero” and “one”. Therefore, Zadeh [1] evaluated the novel theory of fuzzy set (FS). Consequently, a novel fuzzy set (FS) theory was advanced by Zadeh [1] who observed that FS incorporates only a single grade, entitled “membership”, with the mathematical form ${\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}}(\widetilde{{\mathcal{O}}_E})\in \left[\mathrm{0,1}\right]$. Numerous other well-regarded scholars have also undertaken studies of FS. These include Mahmood and Ali [2] who derived fuzzy superior mandelbrot sets; the exploration of fuzzy set consensus analysis by de Souza Oliveira et al. [3]; the analysis of cryptocurrency prices in the light of fuzzy information by Maciel et al. [4]; the analysis, by Cao et al. [5] of multiple neural networks in the presence of fuzzy rough information; the investigation by Chinnasamy et al. [6] of the utilization of the ELECTRE method according to fuzzy set; the introduction of clustering analysis and node attribute in complex networks based on fuzzy information by Naderipour et al. [7]; and finally, the derivation by Liu et al. [8] of the theory of distance and cross-entropy measures based on complex fuzzy information. The use of FS alone is insufficient to replace classical information in decision making as it can handle “yes” but overlooks “no”, even though, in the numerous real-world problems that require both “yes” and “no”, the “no” element is amongst the most critical. A concept was therefore required by experts working in the field which would incorporate both “yes” and “no” information. Therefore, in early 1986 Atanassov [9] developed a novel theory known as intuitionistic FS (IFS). This deployed important new types of information, such as “yes” and “no”. The mathematical shape for yes or membership grades was ${\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}}(\widetilde{{\mathcal{O}}_E})\in \left[\mathrm{0,1}\right],$ and for no or non-membership grades, it was ${\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}(\widetilde{{\mathcal{O}}_E})\in \left[\mathrm{0,1}\right]$. The primary feature of IFS was represented by the mathematical structure $0\le {\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}}(\widetilde{{\mathcal{O}}_E})+{\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}(\widetilde{{\mathcal{O}}_E})\le 1$. A substantial body of work now exists regarding IFS, produced by renowned scholars, such as Garg and Rani [10], who established new distance measures for IFS, while a new concept of bipolar fuzzy information was developed by Mahmood [11]. Jia and Wang [12] introduced the Choquet arithmetic aggregation operators for IFS, while Ecer [13] presented the MAIRCA technique based on IFS. Jebadass and Balasubramaniam [14] developed modifications of low light analysis for color pictures and outlined historical equalization, while Atanassov [15] established a novel class of topological operators based on intuitionistic fuzzy information. Yu et al. [16] investigated and identified evolutionary procedures in light of intuitionistic fuzzy information. Thao and Chou [17] defined novel similarity and entropy measures for IFS and applied them for purposes of enhancing software, and finally, Rahman et al. [18] defined a new algorithm to utilize aggregation operators and similarity measures in the context of IFS.

The definition of FS is based upon the universal set of the unit interval, which performs a highly significant role in the context of decision-making problems. Further elaborating the theory of FS, Molodtsov [19] established the theory of fuzzy information, while recognizing it contains a fundamental problem in that the magnitude of “yes” information for any given component of a universal set is reliant upon the knowledge of the scholar providing the information and therefore is inclined to include variations. Maji et al. [20] extended the theory of fuzzy soft sets by combining the theory of FS and soft sets and additionally originated the theory of intuitionistic fuzzy soft set (IFSS) [21]. A substantial body of work was subsequently produced by scholars in the course of utilizing IFSS. Arora and Garg are key proponents of this work. They invented robust aggregation operators for IFSS (Arora and Garg [22]); developed averaging and geometric aggregation operators for IFSS (Garg and Arora [23]); introduced Bonferroni mean operators for IFSS (Garg and Arora [24]); identified the requirement to prioritize averaging\geometric aggregation operators in the context of IFSS [25]; and finally, provided an evaluation of the generalized Maclaurin symmetric mean aggregation operators for IFSS (Garg and Arora [26]).

The WSM and WPM are distinct methods, both operating similarly to aggregation operators, which are useful to experts in evaluating the optimum information from a collection. In 2012, Zavadskas et al. [27] combined WSM and WPM to create the primary theory for the WASPAS “weighted aggregates sum product assessment” method. This is a sub-element of decision making utilized to investigate the optimum selection from a range of available decisions. Some studies subsequently adjusted the theory underlying the WASPAS method to establish the extended WASPAS technique, for example, Zavadskas et al. [28]. The WASPAS method was deployed by Turskis et al. [29] based on FS, while Stanuikis and Karabasevic [30] utilized the WASPAS method based on IFS. T-norm and t-conorm were developed by Aczel and Alsina [31] as valuable and effective substitutes for algebraic norms given their propensity to support the more precise derivation of any type of aggregation operator. Additionally, Senapati et al. [32] developed the theory of Aczel–Alsina aggregation operators for IFS and considered their application in decision-making processes.

In the above literature review, we observed that different scholars have proposed certain types of operators, including averaging, geometric, and Aczel–Alsina aggregation operators for fuzzy sets, soft sets, fuzzy soft sets, and intuitionistic fuzzy sets. The theory of IFSS can be considered very reliable because it is the modified version of all existing ideas. In the presence of the IFSS, it is very challenging for researchers to define the theory of Aczel–Alsina operational laws and their aggregation operators. For this reason, it has not previously been evaluated where the proposed aggregation operators are more massively modified than the existing operators. Although the field has benefitted from the development of theories based on IFS, for example, the Aczel–Alsina aggregation operators and the WASPAS technique, the challenge of successfully utilizing or combining theories (including intuitionistic fuzzy sets, soft sets, aggregation operators, Aczel–Alsina t-norm and t-conorm, and WAPSAS) to derive a theory of the WASPAS method and Aczel–Alsina aggregation operators based on intuitionistic fuzzy soft set theory remains. The theory of averaging, geometric operators, Aczel–Alsina aggregation operators, and the WASPAS technique based on fuzzy sets, soft sets, fuzzy soft sets, and intuitionistic fuzzy sets are the special cases of the Aczel–Alsina aggregation operators and the WASPAS method for IFSS. Inspired by the above analysis, this paper pursues the development of such a theory, and the key themes of this analysis are set out below:

• To identify the primary, significant Aczel–Alsina operational laws for IFSSs.
• To develop and introduce the theory of IFSAAWA, IFSAAOWA, IFSAAWG, and IFSAAOWG operators, identifying their essential characteristics and key outcomes.
• To present a theory of the WASPAS method for IFS information, for example, IFSAAWA and IFSAAWG operators.
• To describe a novel MADM method based on derived operators for IFS numbers.
• To reinforce the validity and reasonableness of the information presented in this paper, we compare the results obtained from their analysis with pre-existing information. The major flowchart of the proposed work is shown in Figure 1.

This paper is structured as follows: firstly, Section 2 considers the concepts of the WASPAS method, IFSS, and their operational laws. In Section 3, the discovery of the key, critical Aczel–Alsina operational laws for IFSSs is discussed. This section also proceeds to introduce the theories of IFSAAWA, IFSAAOWA, IFSAAWG, and IFSAAOWG operators and to analyze their essential properties and key results. Utilizing significant information regarding, for example, the IFSAAWA and IFSAAWG operators, Section 4 introduces the theory of the WASPAS method for IFS information, validating the theory concerning numerical examples. Section 5 introduces a MADM method founded upon derived operators for IFS numbers. Section 6 provides a comparison of the results obtained by the authors with information available within the existing literature. Finally, Section 7 completes the work by offering concluding remarks.

2. Preliminaries

Zyadskas et al. [27] developed the theory underlying the WASPAS technique, which became the prevalent—and invaluable—procedure for presenting difficult and unpredictable information relating to real-life problems. The WASPAS technique is entirely reliant on weighted sum assessment (WSA) and the weighted product assessment (WPA) models. Its primary procedure concerning classical information is set out below:

Point 1: Compute a decision information matrix. This combines two categories of information, such as benefit and cost types.

Point 2: Normalize the decision information matrix if it includes cost types of information such as:

$${\widetilde{{\mathcal{O}}_{𝒷\mathfrak{p}}}}^{\prime }=\left\{\begin{array}{cc}\stackrel{~}{\frac{{\mathcal{O}}_{𝒷\mathfrak{p}}}{\underset{𝒷}{\max }\widetilde{{\mathcal{O}}_{𝒷\mathfrak{p}}}}}& if \mathfrak{p}\in B\\ \frac{{\mathcal{O}}_{𝒷\mathfrak{p}}}{\underset{𝒷}{\min }\widetilde{{\mathcal{O}}_{𝒷\mathfrak{p}}}}& if \mathfrak{p}\in C\end{array}\right.$$

(1)

where $B$ and $C$ are used for benefit and cost types, if the information in the decision matrix is benefit types, then it does not need to be normalized.

Point 3: Compute the WSA and WPA with the help of the below procedure, such as:

$$Q_{𝒷}^{WSA}=\sum _{\mathfrak{p}=1}^m{w_{\mathfrak{p}}\widetilde{{\mathcal{O}}_{𝒷\mathfrak{p}}}}^{\prime }$$

(2)

$$Q_{𝒷}^{WPA}=\prod _{\mathfrak{p}=1}^m{\left({\widetilde{{\mathcal{O}}_{𝒷\mathfrak{p}}}}^{\prime }\right)}^{w_{\mathfrak{p}}}$$

(3)

Point 4: Calculate the aggregated measure based on a convex formula of the WASPAS method, such as:

$$Q_{𝒷}=\partial Q_{𝒷}^{WSA}+\left(1-\partial \right)Q_{𝒷}^{WPA}$$

(4)

where $\partial \in \left[\mathrm{0,1}\right]$. If $\partial =1$, then the information in Equation (4) will be changed to Equation (2), and if $\partial =0$, then the information in Equation (4) will be changed to Equation (3).

Point 5: Evaluate the ranking results and derive the optimal benefit.

Here, we aim to recall the Aczel–Alsina theory of t-norm and t-conorm which are very beneficial for derived work.

Definition 1

([31]). The mathematical structure of the Aczel–Alsina t-norm ${\left({\Delta }_A^{\Sigma }\right)}_{\Sigma \in \left[0,\infty \right]}$ is pioneered by:

$${\Delta }_A^{\Sigma }\left(\widetilde{{\mathcal{O}}_E},{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)=\left\{\begin{array}{cc}\Delta \left(\widetilde{{\mathcal{O}}_E},{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)& if \Sigma =0\\ min\left(\widetilde{{\mathcal{O}}_E},{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)& if \Sigma =\infty \\ e^{-{\left({\left(-log\widetilde{{\mathcal{O}}_E}\right)}^{\Sigma }+{\left(-log{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}& otherwise.\end{array}\right.$$

(5)

The mathematical structure of the Aczel–Alsina t-conorm  ${\left({\nabla }_A^{\Sigma }\right)}_{\Sigma \in \left[0,\infty \right]}$ is discovered by:

$${\nabla }_A^{{\Sigma }^t}\left(\widetilde{{\mathcal{O}}_E},{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)=\left\{\begin{array}{cc}\nabla \left(\widetilde{{\mathcal{O}}_E},{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)& if \Sigma =0\\ max\left(\widetilde{{\mathcal{O}}_E},{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)& if \Sigma =\infty \\ 1-e^{-{\left({\left(-\log \left(1-\widetilde{{\mathcal{O}}_E}\right)\right)}^{\Sigma }+{\left(-\log \left(1-{\widetilde{{\mathcal{O}}_E}}^{\prime }\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}& otherwise.\end{array}\right.$$

(6)

Furthermore, we revise the theory of the IFS set and its operational laws.

Definition 2

([21]). The mathematical and theoretical shape of IFS information $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C-e_{𝒷}}}$ is discovered by:

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})=\left\{\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E}),{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E}(\right):\widetilde{{\mathcal{O}}_E}\in \widetilde{{\mathbb{X}}_U}\right\}$$

(7)

Notice that the terms ${\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})$ and ${\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}\left(\widetilde{{\mathcal{O}}_E}\right)$ represented or stated the grade of truth and the grade of falsity with a characteristic: $0\le {\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})+{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})\le 1$. Furthermore, the complicated structure $R=1-\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})+{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})\right)$ stated the refusal or neutral grade, and the simple form of IFS number (IFSNs) is stated by: $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$.

Definition 3

([21]). In the presence of any two IFSNs $\stackrel{{\displaystyle ̿}}{{\mathfrak{C}\mathfrak{d}}_{C_{1\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}\right),\mathfrak{p}=\mathrm{1,2}$, we have

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\oplus \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}+{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)$$

(8)

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\otimes \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}+{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)$$

(9)

$$\Lambda \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}=\left(1-{\left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)}^{\Lambda },{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}^{\Lambda }\right)$$

(10)

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{\Lambda }=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}^{\Lambda },1-{\left(1-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)}^{\Lambda }\right)$$

(11)

Definition 4

([21]). In the presence of any two IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{1\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}\right),\mathfrak{p}=\mathrm{1,2}$, we have

$${\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})={\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}},{\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})\in \left[-\mathrm{1,1}\right]$$

(12)

$${\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})={\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}+{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}},{\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})\in \left[\mathrm{0,1}\right]$$

(13)

Stating the score and accuracy information with characteristics: when ${\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})>{\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}})\Rightarrow \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}>\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}$; if ${\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})<{\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}})\Rightarrow \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}<\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}$; if ${\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})={\psi }_{SV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}})\Rightarrow$; if ${\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})>{\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}})\Rightarrow \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}>\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}$; if ${\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})<{\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}})\Rightarrow \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}<\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}$; if ${\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}})={\psi }_{AV}(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}})\Rightarrow \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}$.

3. Aczel–Alsina Aggregation Operators for IFSSs

Here, we aim to discover the theory of Aczel–Alsina operational laws and their related results. Furthermore, with the help of derived operational laws, we aim to pioneer the theory of IFSAAWA, IFSAAOWA, IFSAAWG, and IFSAAOWG operators and also utilize their valuable properties with some important results.

Definition 5.

In the presence of any two IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{1\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}\right),\mathfrak{p}=\mathrm{1,2}$, we have

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\oplus \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}=\left(\left(1-e^{-{\left({\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)\right)}^{\Sigma }+{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right),\left(e^{-{\left({\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)\right)}^{\Sigma }+{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)\right)$$

(14)

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\otimes \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}=\left(\left(e^{-{\left({\left(-\log \left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)\right)}^{\Sigma }+{\left(-\log \left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right),\left(1-e^{-{\left({\left(-\log \left(1-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)\right)}^{\Sigma }+{\left(-\log \left(1-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)\right)$$

(15)

$$\stackrel{=}{{\theta }_S}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}=\left(\left(1-e^{-{\left(\stackrel{=}{{\theta }_S}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right),\left(e^{-{\left(\stackrel{=}{{\theta }_S}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)\right)$$

(16)

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{\stackrel{=}{{\theta }_S}}=\left(\left(e^{-{\left(\stackrel{=}{{\theta }_S}{\left(-\log \left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{11}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right),\left(1-e^{-{\left(\stackrel{=}{{\theta }_S}{\left(-\log \left(1-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{12}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)\right)$$

(17)

Theorem 1.

In the presence of any two IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{1\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}\right),\mathfrak{p}=\mathrm{1,2}$, we have

• $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\oplus \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}\oplus \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}$;
• $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\otimes \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}\otimes \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}$;
• $\stackrel{=}{{\theta }_S}\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\oplus \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}\right)=\stackrel{=}{{\theta }_S}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\oplus \stackrel{=}{{\theta }_S}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}$;
• $\left(\stackrel{=}{{\theta }_{S_1}}+\stackrel{=}{{\theta }_{S_1}}\right)\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}=\stackrel{=}{{\theta }_{S_1}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\oplus \stackrel{=}{{\theta }_{S_2}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}$;
• ${\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}\otimes \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}\right)}^{\stackrel{=}{{\theta }_S}}={\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{\stackrel{=}{{\theta }_S}}\otimes {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{\stackrel{=}{{\theta }_S}}$;
• ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{\stackrel{=}{{\theta }_{S_1}}}\otimes {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{\stackrel{=}{{\theta }_{S_1}}}={\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{\stackrel{=}{{\theta }_{S_1}}+\stackrel{=}{{\theta }_{S_2}}}$.

Proof. 

Straightforward. □

Definition 6.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then

$$IFSAAWA(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}})={\oplus }_{𝒷=1}^n({\xi }_{𝒷}^{\prime }{\oplus }_{\mathfrak{p}=1}^m(\stackrel{̿}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}))$$

(18)

Stating the IFSAAWA operator with weight vectors $\stackrel{=}{\xi }={({\stackrel{̿}{\xi }}_1,{\stackrel{̿}{\xi }}_2,\dots ,{\stackrel{̿}{\xi }}_n)}^T$ and ${\xi }^{\prime }={\left({\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_m^{\prime }\right)}^T$, such as: $\sum _{i=1}^m{\xi }_i^{\prime }=1$ and $\sum _{𝒷=1}^n{\stackrel{=}{\xi }}_{𝒷}=1$.

Theorem 2.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then we diagnose that the final or concluding value of Equation (18) is again an IFSN, such as:

$$IFSAAWA(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}})=\left(1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

(19)

Proof. 

To discover the theory in Equation (19), we aim to use mathematical induction and try to evaluate the information in Equation (19), such as:

For $m=1$ and $\stackrel{=}{{\xi }_{\mathfrak{p}}}=1$, then

$$IFSAAWA(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{n1}}})={\oplus }_{𝒷=1}^n({\xi }_{𝒷}^{\prime }\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷1}}})$$

$$=\left(1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷1}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷1}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)$$

$$=\left(1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^1\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^1\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

For $n=1$ and ${\xi }_{𝒷}^{\prime }=1$, then

$$IFSAAWA(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{1m}}})={\oplus }_{\mathfrak{p}=1}^m(\stackrel{=}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{1\mathfrak{p}}}})$$

$$=\left(1-e^{-{\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{1\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)$$

$$=\left(1-e^{-{\left(\sum _{𝒷=1}^1{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^1{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

The information in Equation (19) is holding successfully. Further, we derive that the information in Equation (19) is also valid $m=d_2,n=d_1+1$ and $m=d_2+1,n=d_1$, such as:

$$IFSAAWA(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}})={\oplus }_{𝒷=1}^{d_1+1}\left({\xi }_{𝒷}^{\prime }{\oplus }_{\mathfrak{p}=1}^{d_2}\left(\stackrel{̿}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\right)\right)$$

$$=\left(1-e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

$${\oplus }_{𝒷=1}^{d_1}\left({\xi }_{𝒷}^{\prime }{\oplus }_{\mathfrak{p}=1}^{d_2+1}\left(\stackrel{̿}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\right)\right)$$

$$=\left(1-e^{-{\left(\sum _{𝒷=1}^{d_1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2+1}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^{d_1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2+1}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

Further, we derive for $n=d_1+1$ and $m=d_2+1$; then

$${\oplus }_{𝒷=1}^{d_1+1}\left({\xi }_{𝒷}^{\prime }{\oplus }_{\mathfrak{p}=1}^{d_2+1}\left(\stackrel{̿}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\right)\right)={\oplus }_{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left({\oplus }_{\mathfrak{p}=1}^{d_2}\left(\stackrel{̿}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\right)\oplus \left(\stackrel{=}{{\xi }_{𝒷+1}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{\left(d_2+1\right)𝒷}}}\right)\right)$$

$$=\left({\oplus }_{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left({\oplus }_{\mathfrak{p}=1}^{d_2}\left(\stackrel{=}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\right)\right)\right)\oplus \left({\oplus }_{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left(\left(\stackrel{=}{{\xi }_{𝒷+1}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{\left(d_2+1\right)𝒷}}}\right)\right)\right)$$

$$=\left(1-e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)\oplus \left(1-e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷+!}^{\prime }{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{\left(d_2+1\right)𝒷}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷+1}^{\prime }{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{\left(d_2+1\right)𝒷}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)$$

$$=\left(1-e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2+1}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^{d_1+1}{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^{d_2+1}\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

Hence, our main point has been proved for all possible values of $m$ and $n$. Moreover, under the consideration of information in Equation (19), we derive their fundamental properties, such as idempotency, monotonicity, and boundedness, by using the collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$. □

Property 1.

Assume that if $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}=\left({\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}},{\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$$IFSAAWA(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}})=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}$$

(20)

Proof. 

In the presence of the information in Equation (21), we assume that if $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}=\left({\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}},{\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$; then

$$IFSAAWA(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}})=\left(1-e^{-{(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{(-\log (1-{\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}}))}^{\Sigma }))}^{\frac{1}{\Sigma }}},e^{-{(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{(-\log ({\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}))}^{\Sigma }))}^{\frac{1}{\Sigma }}}\right)$$

$$=\left(1-e^{-{\left({\left(-\log \left(1-{\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}},e^{-{\left({\left(-\log \left({\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right),\sum _{i=1}^m{\xi }_i^{\prime }=1,\sum _{𝒷=1}^n{\stackrel{=}{\xi }}_{𝒷}=1$$

$$=\left(1-e^{-\left(-\log \left(1-{\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}}\right)\right)},e^{-\left(-\log \left({\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right)\right)}\right)$$

$$=\left(1-e^{\log \left(1-{\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}}\right)},e^{\log \left({\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right)}\right)=\left({\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}},{\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right)=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}.$$

□

Property 2.

Assume that when $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{*}$, then prove that

$$IFSAAWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le IFSAAWA\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{*},{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{*},\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}}^{*}\right)$$

(21)

Proof. 

Note that $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{\mathrm{*}}$. It means that ${\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\le {\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{\mathrm{*}}$ and ${\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\ge {\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{\mathrm{*}}$; then, we prove that the information in Equation (21) is correct. For this, we assume that

$${\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\le {\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\Rightarrow 1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\ge 1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\Rightarrow \log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\ge \log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)$$

$$\Rightarrow -\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\le -\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\Rightarrow {\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\le {\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }$$

$$\Rightarrow {\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}\le {\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}$$

$$\Rightarrow -{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}\ge -{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}$$

$$\Rightarrow e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\ge e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}$$

$$\Rightarrow -e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\le -e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}$$

$$\Rightarrow 1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\le 1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}$$

Moreover, we assume that ${\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\ge {\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{\mathrm{*}}$; then,

$${\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\ge {\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\Rightarrow \log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\ge \log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\Rightarrow -\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\le -\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)$$

$$\Rightarrow {\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}\le {\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}$$

$$\Rightarrow -{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}\ge -{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}$$

$$\Rightarrow e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\ge e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{̿}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}^{*}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}$$

Then, in the presence of Equations (12) and (13), we can easily derive the required information, such as:

$$IFSAAWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le IFSAAWA\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{*},{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{*},\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}}^{*}\right).$$

□

Property 3.

Assume that when ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}=\left(\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}=\left(\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}\le IFSAAWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}$$

(22)

Proof. 

Note that in the presence of ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}=\left(\underset{𝒷}{\min }\underset{\mathfrak{p}}{\min }{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\max }\underset{\mathfrak{p}}{\max }{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}=\left(\underset{𝒷}{\max }\underset{\mathfrak{p}}{\max }{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\min }\underset{\mathfrak{p}}{\min }{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m,$ and by using the theory in property 1 and property 2, we obtain

$$IFSAAWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le IFSAAWA\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{+},{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{+},\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}}^{+}\right)={\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}$$

$$IFSAAWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\ge IFSAAWA\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{-},{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{-},\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}}^{-}\right)={\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}$$

Thus,

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}\le IFSAAWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}.$$

□

Definition 7.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then

$$IFSAAOWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)={\oplus }_{𝒷=1}^n\left({\xi }_{𝒷}^{\prime }{\oplus }_{\mathfrak{p}=1}^m\left(\stackrel{̿}{{\xi }_{\mathfrak{p}}}\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0\left(𝒷\right)0\left(\mathfrak{p}\right)}}}\right)\right)$$

(23)

Stating the IFSAAOWA operator with weight vectors $\stackrel{=}{\xi }={\left({\stackrel{̿}{\xi }}_1,{\stackrel{̿}{\xi }}_2,\dots ,{\stackrel{̿}{\xi }}_n\right)}^T$ and ${\xi }^{\prime }={\left({\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_m^{\prime }\right)}^T$, such as: $\sum _{i=1}^m{\xi }_i^{\prime }=1$ and $\sum _{𝒷=1}^n{\stackrel{=}{\xi }}_{𝒷}=1$ with $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0\left(𝒷\right)\left(\mathfrak{p}\right)}}}\le \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0\left(𝒷-1\right)\left(\mathfrak{p}\right)}}}$ and $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{\left(𝒷\right)0\left(\mathfrak{p}\right)}}}\le \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{\left(𝒷\right)0\left(\mathfrak{p}-1\right)}}}$.

Theorem 3.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$ we diagnose that the final or concluding value of Equation (23) is again an IFSN, such as:

$$IFSAAOWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)=\left(1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{0\left(𝒷\right)0\left(\mathfrak{p}\right)}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{0\left(𝒷\right)0\left(\mathfrak{p}\right)}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

(24)

Proof. 

Straightforward. □

Moreover, under the consideration of information in Equation (24), we derive their fundamental properties, such as idempotency, monotonicity, and boundedness, by using the collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$.

Property 4.

Assume that if $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}=\left({\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}},{\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$$IFSAAOWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}$$

(25)

Property 5.

Assume that when $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{*}$, then prove that

$$IFSAAOWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le IFSAAOWA\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{*},{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{*},\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}}^{*}\right)$$

(26)

Property 6.

Assume that when ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}=\left(\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}=\left(\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}\le IFSAAOWA\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}$$

(27)

Definition 8.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then

$$IFSAAWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)={\otimes }_{𝒷=1}^n{\left({\otimes }_{\mathfrak{p}=1}^m\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{\stackrel{=}{{\xi }_{\mathfrak{p}}}}\right)\right)}^{{\xi }_{𝒷}^{\prime }}$$

(28)

Stating the IFSAAWG operator with weight vectors $\stackrel{̿}{\xi }={\left({\stackrel{̿}{\xi }}_1,{\stackrel{̿}{\xi }}_2,\dots ,{\stackrel{̿}{\xi }}_n\right)}^T$ and ${\xi }^{\prime }={\left({\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_m^{\prime }\right)}^T$, such as: $\sum _{i=1}^m{\xi }_i^{\prime }=1$ and $\sum _{𝒷=1}^n{\stackrel{̿}{\xi }}_{𝒷}=1$.

Theorem 4.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, we diagnose that the final or concluding value of Equation (28) is again an IFSN, such as:

$$IFSAAWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)=\left(e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

(29)

Proof. 

Straightforward. □

Moreover, under the consideration of information in Equation (29), we derive their fundamental properties, such as idempotency, monotonicity, and boundedness, by using the collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$.

Property 7.

Assume that if $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}=\left({\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}},{\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$$IFSAAWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}$$

(30)

Property 8.

Assume that when $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{*}$, then prove that

$$IFSAAWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le IFSAAWG\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{*},{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{*},\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}}^{*}\right)$$

(31)

Property 9.

Assume that when ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}=\left(\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}=\left(\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}\le IFSAAWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}$$

(32)

Definition 9.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then

$$IFSAAOWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)={\otimes }_{𝒷=1}^n{\left({\otimes }_{\mathfrak{p}=1}^m\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0\left(𝒷\right)0\left(\mathfrak{p}\right)}}}}^{\stackrel{=}{{\xi }_{\mathfrak{p}}}}\right)\right)}^{{\xi }_{𝒷}^{\prime }}$$

(33)

Stating the IFSAAOWG operator with weight vectors $\stackrel{=}{\xi }={\left({\stackrel{̿}{\xi }}_1,{\stackrel{̿}{\xi }}_2,\dots ,{\stackrel{̿}{\xi }}_n\right)}^T$ and ${\xi }^{\prime }={\left({\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_m^{\prime }\right)}^T$, such as: $\sum _{i=1}^m{\xi }_i^{\prime }=1$ and $\sum _{𝒷=1}^n{\stackrel{=}{\xi }}_{𝒷}=1$ with $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0\left(𝒷\right)\left(\mathfrak{p}\right)}}}\le \stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0\left(𝒷-1\right)\left(\mathfrak{p}\right)}}}$ and $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{\left(𝒷\right)0\left(\mathfrak{p}\right)}}}\le \stackrel{̿}{{\mathfrak{C}\mathfrak{d}}_{C_{\left(𝒷\right)0\left(\mathfrak{p}-1\right)}}}$.

Theorem 5.

In the presence of any collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, we diagnose that the final or concluding value of Equation (33) is again an IFSN, such as:

$$IFSAAOWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)=\left(e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{0\left(𝒷\right)0\left(\mathfrak{p}\right)}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}},1-e^{-{\left(\sum _{𝒷=1}^n{\xi }_{𝒷}^{\prime }\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{0\left(𝒷\right)0\left(\mathfrak{p}\right)}}}\right)\right)}^{\Sigma }\right)\right)}^{\frac{1}{\Sigma }}}\right)$$

(34)

Proof. 

Straightforward. □

Moreover, under the consideration of information in Equation (34), we derive their fundamental properties, such as idempotency, monotonicity, and boundedness, by using the collection of IFSNs $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$.

Property 10.

Assume that if $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}=\left({\mathfrak{T}}_{\mathfrak{C}\mathfrak{d}},{\mathfrak{K}}_{\mathfrak{C}\mathfrak{d}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$$IFSAAOWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)=\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}$$

(35)

Property 11.

Assume that when $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{*}$, then prove that

$$IFSAAOWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le IFSAAOWG\left({\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}}}^{*},{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}}}^{*},\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}}^{*}\right)$$

(36)

Property 12.

Assume that when ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}=\left(\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}=\left(\underset{𝒷}{\mathit{max}}\underset{\mathfrak{p}}{\mathit{max}}{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathit{min}}\underset{\mathfrak{p}}{\mathit{min}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$, then prove that

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{-}\le IFSAAOWG\left(\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{11}}},\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{12}}},\dots ,\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{nm}}}\right)\le {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{+}$$

(37)

All theories which are derived in this section are massively modified as Aczel–Alsina aggregation operators for FSs, SSs, FSSs, and IFSs.

4. WASPAS Method for IFS Information

In this section, we seek to develop the novel theory of WASPAS for IFS information. The key objective of this technique is the identification of the optimal decision, supported by both the operators—IFSAAWA and IFSAAWG—to demonstrate the dependability and effectiveness of the resulting information. The central procedure of the WASPAS technique for intuitionistic fuzzy soft information is therefore explained below:

Point 1: Compute a decision information matrix, which is the combination of two types of information, such as benefit and cost types.

Point 2: Normalize the decision information matrix if the matrix contains cost types of information, such as:

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0,\mathfrak{p}}}}=\left(\underset{𝒷}{\max }{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\min }{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$$

(38)

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0,\mathfrak{p}}}}=\left(\underset{𝒷}{\min }{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\max }{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$$

(39)

For the benefit type of data, the $\left(\underset{𝒷}{\max }{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\min }{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)$ is used in Equation (38); for the cost type, the $\left(\underset{𝒷}{\min }{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\mathrm{ax}}{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)$ is used in Equation (39). Then, by using the information in Equation (40), that data can be normalized as follows:

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{/}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)=\left\{\begin{array}{cc}0& otherwise\\ \frac{{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}}{1+{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{0,\mathfrak{p}}}}}& if{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\le {\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{0,\mathfrak{p}}}}\\ \frac{{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}}{1+{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{0,\mathfrak{p}}}}}& fornon-membershipgrade\end{array}\right.$$

(40)

Point 3: Compute the IFSAAWA operator and IFSAAWG operator with the help of the below procedure, such as:

$$Q_{𝒷}^{WSA}=\left(1-e^{-{\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}},e^{-{\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)$$

(41)

$$Q_{𝒷}^{WPA}=\left(e^{-{\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}},1-e^{-{\left(\sum _{\mathfrak{p}=1}^m\stackrel{=}{{\xi }_{\mathfrak{p}}}{\left(-\log \left(1-{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)\right)}^{\Sigma }\right)}^{\frac{1}{\Sigma }}}\right)$$

(42)

Point 4: Calculate the aggregated measure based on a convex formula of the WASPAS method, such as:

$$Q_{𝒷}=\partial Q_{𝒷}^{WSA}+\left(1-\partial \right)Q_{𝒷}^{WPA}$$

(43)

where $\partial \in \left[\mathrm{0,1}\right]$. If $\partial =1$, then the information in Equation (43) will be changed to Equation (41), and if $\partial =0$, then the information in Equation (43) will be changed to Equation (43).

Point 5: Evaluate the ranking results and derive the optimal benefit.

Example 1.

Consider four different types of machine learning that are used as an alternative, such as:

$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_1}}$: Supervised learning.

 $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_2}}$: Unsupervised learning.

 $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_3}}$: Semi-Supervised learning.

 $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_4}}$: Reinforced learning.

Under the presence of the four features below, we aim to identify and examine the best and worst from the following features: copyright checking, image classification and segmentation, language translation, online fraud detection, and plagiarism. The resulting information can be verified by computing a decision matrix, every term of which is in the shape of IFSNs. The main procedure of the WASPAS technique for intuitionistic fuzzy soft information can be described as follows:

Point 1: Compute a decision information matrix that is the combination of two types of information, such as benefit and cost types, where the information in matrix $D$ is arranged in the form of benefit types. Then, by using the theory in Equations (38) and (40), the data can be normalized, as follows:

$$D=\left[\begin{array}{c}\left(\mathrm{0.7,0.2}\right)\\ \left(\mathrm{0.6,0.4}\right)\\ \begin{array}{c}\left(\mathrm{0.5,0.3}\right)\\ \left(\mathrm{0.8,0.1}\right)\end{array}\end{array}\begin{array}{c}\left(\mathrm{0.9,0.1}\right)\\ \left(\mathrm{0.2,0.4}\right)\\ \begin{array}{c}\left(\mathrm{0.4,0.3}\right)\\ \left(\mathrm{0.7,0.2}\right)\end{array}\end{array}\begin{array}{c}\left(\mathrm{0.5,0.3}\right)\\ \left(\mathrm{0.6,0.2}\right)\\ \begin{array}{c}\left(\mathrm{0.3,0.3}\right)\\ \left(\mathrm{0.4,0.4}\right)\end{array}\end{array}\begin{array}{c}\left(\mathrm{0.4,0.2}\right)\\ \left(\mathrm{0.3,0.2}\right)\\ \begin{array}{c}\left(\mathrm{0.7,0.3}\right)\\ \left(\mathrm{0.9,0.1}\right)\end{array}\end{array}\right]$$

Point 2: It is clear that these are benefit types of data. By using the following equation $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0,\mathfrak{p}}}}=\left(\underset{𝒷}{\max }{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},\underset{𝒷}{\min }{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)$, we can normalize the decision information matrix as appears below

$$\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{0,\mathfrak{p}}}}=\left\{\left(\mathrm{0.8,0.1}\right),\left(\mathrm{0.9,0.1}\right),\left(\mathrm{0.6,0.2}\right),\left(\mathrm{0.9,0.1}\right)\right\}$$

Determine the value of normalized information by using the below theory:

$${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}}^{/}=\left[\begin{array}{c}\left(\mathrm{0.3888,0.1818}\right)\\ \left(\mathrm{0.3333,0.3636}\right)\\ \begin{array}{c}\left(\mathrm{0.2777,0.2727}\right)\\ \left(\mathrm{0.4444,0.0909}\right)\end{array}\end{array}\begin{array}{c}\left(\mathrm{0.4736,0.0909}\right)\\ \left(\mathrm{0.1052,0.3636}\right)\\ \begin{array}{c}\left(\mathrm{0.2105,0.2727}\right)\\ \left(\mathrm{0.3684,0.1818}\right)\end{array}\end{array}\begin{array}{c}\left(\mathrm{0.3125,0.25}\right)\\ \left(\mathrm{0.375,0.1666}\right)\\ \begin{array}{c}\left(\mathrm{0.1875,0.25}\right)\\ \left(\mathrm{0.25,0.3333}\right)\end{array}\end{array}\begin{array}{c}\left(\mathrm{0.2105,0.1818}\right)\\ \left(\mathrm{0.1578,0.1818}\right)\\ \begin{array}{c}\left(\mathrm{0.3684,0.2727}\right)\\ \left(\mathrm{0.4736,0.0909}\right)\end{array}\end{array}\right]$$

Point 3: Compute the IFSAAWA operator and IFSAAWG operator under the consideration of weight vectors 0.4, 0.3, 0.2, and 0.1 with the help of the below procedure, such as:

$$Q_{𝒷}^{WSA}=\left\{\left(\mathrm{0.001,0.8681}\right),\left(\mathrm{0.0002,0.8681}\right),\left(\mathrm{0.0002,0.9052}\right),\left(\mathrm{0.0003,0.7880}\right)\right\}$$

$$Q_{𝒷}^{WPA}=\left\{\left(\mathrm{0.9720,0.000001}\right),\left(\mathrm{0.9334,0.00007}\right),\left(\mathrm{0.9327,0.0001}\right),\left(\mathrm{0.9451,0.0003}\right)\right\}$$

Further, the score values of the above information are stated below:

$$Q_{𝒷}^{WSA}=\left\{-0.8672,-0.8679,-0.9049,-0.7877\right\}$$

$$Q_{𝒷}^{WPA}=\left\{\mathrm{0.9719,0.9333,0.9325,0.9451}\right\}$$

Point 4: Calculate the aggregated measure based on a convex formula of the WASPAS method, such as:

$$Q_1=0.4202,Q_2=0.3929,Q_3=0.3812,Q_4=0.4252$$

where $\partial =0.3$.

Point 5: Evaluate the ranking results and derive the optimal benefit.

$$Q_4\ge Q_1\ge Q_2\ge Q_3$$

The best decision is $Q_4$, which represents reinforced learning.

5. Multi-Attribute Decision-Making Methods

Here, we outline a procedure based on Aczel–Alsina aggregation operators— the IFSAAWA operator and IFSAAWG—in the presence of intuitionistic fuzzy soft information to demonstrate the validity and superiority of the derived operators.

For this, we have a collection of finite alternatives $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}=\left\{{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2,\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_m\right\}$ with a finite number of attributes or criteria, such as: $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}=\left\{{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}}_1,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}}_2,\dots ,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}}_n\right\}$. The number of attributes and alternatives need to be equal. Further, we have the weight vectors $\stackrel{̿}{\xi }={\left({\stackrel{̿}{\xi }}_1,{\stackrel{̿}{\xi }}_2,\dots ,{\stackrel{̿}{\xi }}_n\right)}^T$ and ${\xi }^{\prime }={\left({\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_{m^{\prime }}^{\prime }\right)}^T$, such as $\sum _{i=1}^{m^{\prime }}{\xi }_i^{\prime }=1$ and $\sum _{𝒷=1}^n{\stackrel{̿}{\xi }}_{𝒷}=1$, and the order of the attributes and weight vector will be necessarily equal. Further, a decision matrix will be computed based on alternatives and their attributes whose very values will be written in the shape of intuitionistic fuzzy soft numbers. For example, it is observed that the term ${\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})$ and ${\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})$ represent or state the grade of truth and the grade of falsity with a characteristic: $0\le {\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})+{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})\le 1$. Furthermore, the complicated structure $R=1-\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}\left(\widetilde{{\mathcal{O}}_E}\right)+{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷}}}(\widetilde{{\mathcal{O}}_E})\right)$ states the refusal or neutral grade, and the simple form of IFSNs is stated by $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{C_{𝒷\mathfrak{p}}}}=\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right),𝒷,\mathfrak{p}=\mathrm{1,2},\dots ,n,m$. To proceed with our primary problem with the assistance of a valuable and useful algorithm, it was necessary to compute a valuable and important procedure, the key stages of which are set out below:

Step 1: Compute a decision information matrix. This combines two categories of information, for example, benefit and cost types.

Step 2: Normalize the decision information matrix if it includes cost types of information, and proceed to identify the value of the normalized information, utilizing the theory set out below:

$$D=\left\{\begin{array}{cc}\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)& for benefit\\ \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)& for cost\end{array}\right.$$

When the information in the decision matrix is computed in the shape of benefit type, then it does not need to be evaluated.

Step 3: Compute the intuitionistic fuzzy soft number from the collection of information in the decision matrix with the help of the IFSAAWA operator and IFSAAWG operator.

Step 4: Calculate the score value based on the information in Equation (12). (If Equation (12) failed, then use the information in Equation (13).)

Point 5: Evaluate the ranking results and derive the optimal benefit.

Our next target is to justify our developed procedure with the help of some practical applications which are stated below.

Illustrated Example (Evaluation of the Problems in the Finance Market)

The primary focus of this section is the utilization of a practical example to illustrate the validity and superiority of the derived approaches. The illustration draws upon the field of business and concerns the application of artificial intelligence and machine learning in the financial markets.

A financial market is defined as a place where financial derivatives and securities may be exchanged at little to no cost. Examples of such securities, commonly known in the financial markets as commodities, include stocks, bonds, raw resources, and precious metals.

Finally, Figure 2 below explains the geometrical representation of the proposed application.

The term “market” is frequently utilized to refer to entities whose official purpose is to provide a locus for exchange. Examples include stock exchanges and commodity exchanges that support trading in financial securities. Such exchanges may take the form of a physical facility or an electronic system. Examples of the former include the New York Stock Exchange (NYSE), the London Stock Exchange (LSE), JSE Limited (JSE), and the Bombay Stock Exchange (BSE), while the NASDAQ illustrates an electronic system that is utilized for purposes of exchange. Substantial volumes of stock trading occur at an exchange, but corporate initiatives, such as mergers and spinoffs, are implemented outside the exchange by two or more businesses or individuals for a range of possible reasons. In this example, we will seek to determine which, according to their four key characteristics, of four different classes of financial market is superior or otherwise, employing consideration using derived operators.

For this purpose, the group of four specialists will be considered, whose mathematical representation is of the form: $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}=\left\{{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}}_1,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}}_2,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}}_3,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_C}}_4\right\}$ with weight vectors 0.1, 0.2, 0.4, and 0.3. The main advisory committee of experts will assign an individual valuation for each of the four different financial markets which are represented by alternatives, such as $\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}=\left\{{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\right\},$ for which brief explanations are provided below:

${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$: Money markets: These provide short-term debt financing and investment.

${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2$: Derivatives markets: These trade in instruments for managing financial risk.

${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$: Foreign exchange markets: These provide opportunities to trade in foreign exchange.

${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4$: Cryptocurrency markets: These support trading in digital assists and financial technologies.

Consider $\left\{{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_1,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_2,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_3,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_4\right\}$ to be a group of four experts with weight vectors ${\left(\mathrm{0.1,0.2,0.3,0.4}\right)}^T$. The panel of experts designates their valuation for four different financial technologies $\left\{{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\right\}$. The attributes information for all financial markets is given in the shape: $e_1:$ growth analysis; $e_2:$ social–political impact; $e_3:$ environmental impact; $e_4:$ current prices; and $e_5:$ customer expectations with components of weight vectors: 0.4, 0.1, 0.2, 0.2, and 0.1.

Proceeding to address our main problem assisted by a useful algorithm, it is necessary to compute a significant and valuable procedure, the key stages of which are set out below:

Step 1: Compute a decision information matrix that combines two types of information, such as benefit and cost. Examples of decision information matrices are provided in

Table 1. Intuitionistic fuzzy soft matrix for ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$.

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_1$	$\left(\mathrm{0.7,0.3}\right)$	$\left(\mathrm{0.4,0.3}\right)$	$\left(\mathrm{0.7,0.1}\right)$	$\left(\mathrm{0.3,0.2}\right)$	$\left(\mathrm{0.8,0.1}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_2$	$\left(\mathrm{0.6,0.1}\right)$	$\left(\mathrm{0.3,0.2}\right)$	$\left(\mathrm{0.6,0.2}\right)$	$\left(\mathrm{0.2,0.1}\right)$	$\left(\mathrm{0.7,0.2}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_3$	$\left(\mathrm{0.4,0.3}\right)$	$\left(\mathrm{0.4,0.1}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.1,0.1}\right)$	$\left(\mathrm{0.6,0.3}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_4$	$\left(\mathrm{0.2,0.1}\right)$	$\left(\mathrm{0.2,0.1}\right)$	$\left(\mathrm{0.4,0.4}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.5,0.4}\right)$

,

Table 2. Intuitionistic fuzzy soft matrix for ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2$ .

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_1$	$\left(\mathrm{0.4,0.3}\right)$	$\left(\mathrm{0.5,0.2}\right)$	$\left(\mathrm{0.7,0.3}\right)$	$\left(\mathrm{0.4,0.1}\right)$	$\left(\mathrm{0.8,0.1}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_2$	$\left(\mathrm{0.5,0.1}\right)$	$\left(\mathrm{0.6,0.3}\right)$	$\left(\mathrm{0.1,0.1}\right)$	$\left(\mathrm{0.6,0.3}\right)$	$\left(\mathrm{0.6,0.2}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_3$	$\left(\mathrm{0.6,0.2}\right)$	$\left(\mathrm{0.4,0.1}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.5,0.4}\right)$	$\left(\mathrm{0.5,0.3}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_4$	$\left(\mathrm{0.7,0.3}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.9,0.1}\right)$	$\left(\mathrm{0.2,0.1}\right)$	$\left(\mathrm{0.4,0.4}\right)$

,

Table 3. Intuitionistic fuzzy soft matrix for ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$ .

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_1$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.5,0.1}\right)$	$\left(\mathrm{0.6,0.3}\right)$	$\left(\mathrm{0.1,0.1}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_2$	$\left(\mathrm{0.7,0.2}\right)$	$\left(\mathrm{0.6,0.3}\right)$	$\left(\mathrm{0.1,0.1}\right)$	$\left(\mathrm{0.6,0.3}\right)$	$\left(\mathrm{0.5,0.3}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_3$	$\left(\mathrm{0.8,0.1}\right)$	$\left(\mathrm{0.4,0.1}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.5,0.4}\right)$	$\left(\mathrm{0.9,0.1}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_4$	$\left(\mathrm{0.9,0.1}\right)$	$\left(\mathrm{0.4,0.4}\right)$	$\left(\mathrm{0.1,0.1}\right)$	$\left(\mathrm{0.2,0.2}\right)$	$\left(\mathrm{0.3,0.3}\right)$

and

Table 4. Intuitionistic fuzzy soft matrix for ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4$ .

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_1$	$\left(\mathrm{0.8,0.1}\right)$	$\left(\mathrm{0.4,0.1}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.5,0.4}\right)$	$\left(\mathrm{0.9,0.1}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_2$	$\left(\mathrm{0.4,0.3}\right)$	$\left(\mathrm{0.4,0.1}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.1,0.1}\right)$	$\left(\mathrm{0.6,0.3}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_3$	$\left(\mathrm{0.7,0.3}\right)$	$\left(\mathrm{0.5,0.3}\right)$	$\left(\mathrm{0.9,0.1}\right)$	$\left(\mathrm{0.2,0.1}\right)$	$\left(\mathrm{0.4,0.4}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{C}}}}_4$	$\left(\mathrm{0.6,0.1}\right)$	$\left(\mathrm{0.3,0.2}\right)$	$\left(\mathrm{0.6,0.2}\right)$	$\left(\mathrm{0.2,0.1}\right)$	$\left(\mathrm{0.7,0.2}\right)$

.

Step 2: If the decision information matrix contains information on the cost type, it should be normalized, and the value of the normalized information should be established utilizing the theory below:

$$D=\left\{\begin{array}{cc}\left({\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)& for benefit\\ \left({\mathfrak{K}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}},{\mathfrak{T}}_{{\mathfrak{C}\mathfrak{d}}_{e_{𝒷\mathfrak{p}}}}\right)& for cost\end{array}\right.$$

(44)

It should be noted that where the decision matrix information is computed in the shape of benefit type, then it is not necessary to evaluate it as in the case of Table 1, Table 2, Table 3 and Table 4, where the information contained does not require evaluation (normalization).

Step 3: Compute the intuitionistic fuzzy soft number from the collection of information in the decision matrix with the help of the IFSAAWA operator and IFSAAWG operator, where $\Sigma =2$, see

Table 5. Aggregated values for weighted averaging/geometric operators.

Alternatives\Methods	IFSAAWA Operator	IFSAAWG Operator
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_1$	$\left(\mathrm{0.2512,0.464}\right)$	$\left(\mathrm{0.6054,0.1174}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_2$	$\left(\mathrm{0.3436,0.4839}\right)$	$\left(\mathrm{0.6932,0.1229}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_3$	$\left(\mathrm{0.4219,0.4474}\right)$	$\left(\mathrm{0.6475,0.1116}\right)$
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_4$	$\left(\mathrm{0.37,0.4531}\right)$	$\left(\mathrm{0.6652,0.1104}\right)$

.

Step 4: Calculate the score value based on the information in Equation (12), see

Table 6. Representation of score values.

Alternatives\Methods	IFSAAWA Operator	IFSAAWG Operator
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_1$	−0.213	0.488
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_2$	−0.14	0.5703
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_3$	−0.025	0.5359
${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_4$	−0.083	0.5548

. (If Equation (12) failed, then use the information in Equation (13).)

Point 5: Evaluate the ranking results and derive the optimal benefits which are stated in

Table 7. Mathematical representation of ranking results.

Methods	Ranking Values
IFSAAWA Operator	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
IFSAAWG Operator	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathrm{A}}}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathrm{A}}}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathrm{A}}}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathrm{A}}}}_1$

.

The best option is found to be ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$, based on implementing the theory of the IFSAAWA operator. According to the IFSAAWG operator, the best preference is ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2$. In Table 7, we acquire two different types of ranking results because of the value of parameter $\Sigma$. In other words, if we use the value of $\Sigma =1$, then we obtain the same ranking results, see

Table 8. Analysis of parameters for different values.

Parameters	Methods	Score Values	Ranking Results
$\mathbf{\Sigma }\mathbf{=}\mathbf{1}$	IFSAAWA	−0.259, −0.193, −0.101, −0.149	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
	IFSAAWG	0.5397, 0.6237, 0.6277, 0.6213	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
	WASPAS	0.3, 0.3786, 0.4092, 0.3902	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
$\mathbf{\Sigma }\mathbf{=}\mathbf{3}$	IFSAAWA	−0.174, −0.088, 0.0284, −0.028	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
	IFSAAWG	0.4441, 0.517, 0.4653, 0.5003	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
	WASPAS	0.2585, 0.3354, 0.3342, 0.3418	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
$\mathbf{\Sigma }\mathbf{=}\mathbf{5}$	IFSAAWA	−0.116, 0.0017, 0.0972, 0.0526	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
	IFSAAWG	0.3798, 0.4292, 0.3765, 0.4236	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$
	WASPAS	0.2309, 0.301, 0.2927, 0.3123	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
$\mathbf{\Sigma }\mathbf{=}\mathbf{7}$	IFSAAWA	−0.075, 0.0641, 0.1373, 0.1038	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
	IFSAAWG	0.3373, 0.3715, 0.3267, 0.3738	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$
	WASPAS	0.2136, 0.2792, 0.2699, 0.2928	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
$\mathbf{\Sigma }\mathbf{=}\mathbf{10}$	IFSAAWA	−0.031, 0.12, 0.1723, 0.1488	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
	IFSAAWG	0.2959, 0.3186, 0.284, 0.3248	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$
	WASPAS	0.1977, 0.259, 0.2505, 0.272	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$

. The supremacy and validity of the derived operators were subsequently checked by utilizing different values for parameter $\Sigma$, which are given in Table 8.

By using different values for the parameters, the best option produced is ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2,{\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3,$ and ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4$. However, of these, the best decision produced on the majority of occasions is ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$. To support additional modification, the key results will be compared with those produced using existing operators to demonstrate both the accuracy and the value of the derived operators.

6. Comparative Analysis

The operators described in this paper are superior to those currently in existence and are more capable of dealing with challenging and unpredictable behavior. In support of the validity and rationale of the information provided herein, the results obtained through the current analysis were compared with results available in the existing literature, supported by numerical examples to demonstrate the viability and usefulness of the method proposed. The comparison between the proposed and pre-existing operators is set out in

Table 9. Comparative analysis for IFS information.

Methods	Score Values	Ranking Values
Arora and Garg [22], WA	0.2537, 0.368, 0.4759, 0.4138	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
Arora and Garg [22], WG	0.1396, 0.2544, 0.2677, 0.2576	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
Arora and Garg [25]	0.2485, 0.3633, 0.3766, 0.3665	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
IFSAAWA operator	−0.213, −0.14, −0.025, −0.083	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
IFSAAWG operator	0.488, 0.5703, 0.5359, 0.5548	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_1$
WASPAS Method	0.3, 0.3786, 0.4092, 0.3902	${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_3\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_4\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_2\ge {\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_{\mathit{A}}}}_1$

and was based on a range of relevant information currently prevailing in the field. This included the robust aggregation operators for IFSS invented by Arora and Garg [22], the simple averaging and geometric aggregation operators for IFSS derived by Garg and Arora [23], and the prioritized averaging\geometric aggregation operators based upon IFSS as diagnosed by Arora and Garg [25].

If implementing the theory of the IFSAAWA operator and the theories presented in Arora and Garg [22], the WASPAS method, and Arora and Garg [25], the best option is ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_3$. However, according to the theory of the IFSAAWG operator, the best preference is ${\stackrel{=}{{\mathfrak{C}\mathfrak{d}}_A}}_2$. The presented work is modified significantly beyond the theories articulated in Arora and Garg [22] and Garg and Arora [25] because these operators are special cases of the derived operators. Furthermore, the simple averaging aggregation operators, simple geometric aggregation operators, and Aczel–Alsina aggregation operators based on fuzzy sets, intuitionistic fuzzy sets, soft sets, and fuzzy soft sets are the special cases of the derived operators.

Therefore, the novel WASPAS method and Aczel–Alsina aggregation operators based on intuitionistic fuzzy soft information are superior and of greater value in managing challenging and unpredictable information in comparison with existing operators [22,25].

7. Conclusions

The WASPAS method and the Aczel–Alsina aggregation operators are renowned for their ability to manage challenging and unreliable information because they are generalized to a significantly higher extent than many other aggregation operators. Listed below are the key features of the analysis is described in this paper:

• The key, significant Aczel–Alsina operational laws for IFSSs were identified.
• The theory introduced of IFSAAWA, IFSAAOWA, IFSAAWG, and IFSAAOWG operators and their essential characteristics and key results were identified.
• The paper presented the novel theory of the WASPAS method for IFS information, utilizing key information, such as IFSAAWA and IFSAAWG operators.
• An illustration was presented of a MADM method founded on derived operators for IFS numbers.
• The value and reasonableness of the information presented were supported in this paper utilizing a comparison between the results obtained in the current analysis and pre-existing results in the field. Numerical examples were cited to demonstrate the viability and effectiveness of the approaches proposed.

7.1. Limitations

IFS information theory has several restrictions in certain cases; for example, this paper found that during an election, experts face four types of information (truth, abstinence, falsity, neutral) that are complex and awkward to evaluate through the use of IFS set theory. In this case, it is necessary to use the novel approach of picture fuzzy soft sets theory and neutrosophic soft set theory and their expansions.

7.2. Future Works

In terms of the scope of future works, we will concentrate on developing novel ideas based on IFS sets, for instance, the Aczel–Alsina power aggregation operators, the Aczel–Alsina prioritized aggregation operators, the Aczel–Alsina Choquet-integral aggregation operators, fuzzy N-soft sets [33], hesitant fuzzy N-soft sets [34], TOPSIS methods [35], complex spherical fuzzy N-soft sets [36], bipolar fuzzy soft sets [37], and m-polar fuzzy soft sets [38]. The aim of this future work will be to improve the worth of the invented approaches and enhance the value and reliability of this research in the fields of AI, clustering analysis, game theory, machine learning, neural networks, and road signals.