Identification and Classification of Multi-Attribute Decision-Making Based on Complex Intuitionistic Fuzzy Frank Aggregation Operators

Abstract

Invented by Frank in 1979, Frank’s t-norm and t-conorm operations possess improved modifications and can be applied more generally than the existing algebraic t-norm and t-conorm. The major objective of this article is to determine Frank’s operational laws based on complex intuitionistic fuzzy (CIF) information. Moreover, we examine the Frank aggregation operators (averaging and geometric) based on CIF set theory and Frank operational laws, such as the CIF Frank weighted averaging (CIFFWA) operator, CIF Frank ordered weighted averaging (CIFFOWA) operator, CIF Frank hybrid averaging (CIFFHA) operator, CIF Frank weighted geometric (CIFFWG) operator, CIF Frank ordered weighted geometric (CIFFOWG) operator and CIF Frank hybrid geometric (CIFFHG) operator. Some dominant and feasible properties of the invented techniques are also stated. Additionally, to evaluate the problem of osteoporosis in human bodies based on their causes and risk factors, we illustrate an application of the multi-attribute decision-making (MADM) technique with consideration of the invented methods to show the supremacy and validity of the derived techniques. Finally, we aim to compare the proposed scenarios with some valid existing or prevailing techniques to increase the value of the presented approaches.

1. Introduction

Multi-attribute decision-making (MADM) [1,2] is a type of procedure used for determining the most optimal solution with the help of a decision-making procedure from a finite collection of alternatives. In the last few years, investigations of companies have been conducted by many scholars. In a genuine decision-making tool [3], a common dilemma is how to represent the criteria terms feasibly and effectively. Further, due to uncertainty in real life, for every expert, it can be challenging to determine the best solution considering the availability of classical information [4,5]. For this, to determine the best optimal solution based on a fuzzy set instead of classical information, in 1965, Zadeh [6] evaluated the main theory of fuzzy information (FI). FI depends on the value of the supporting function, and the final or resultant value is included in [0, 1]. Further, to improve the worth of FI, certain theories are available regarding the shape: a process based on the MADM technique for FI was pioneered by Jain [7]; Akram et al. [8,9] diagnosed a mathematical structure that is different from FI, called fuzzy N-soft sets and hesitant fuzzy N-soft sets; Ohlan and Ohlan [10] evaluated the problem of bibliometrics under the consideration of FI; Abdullah et al. [11] mixed the theory of fractions and FI; and recently, Mahmood [12] evaluated the main concept of bipolar soft sets. But certain scholars have demonstrated that the theory of FI is incomplete. To remove these deficiencies, Atanassov [13,14] diagnosed intuitionistic FI (IFI), where IFI contained two supporting functions, called positive and negative grades. The condition of IFI stated that the sum of both values of the IFI is necessarily contained in the unit interval. Moreover, an extended MAIRCA technique based on IFI was diagnosed by Ecer [15]; a technique to analyze the quality of software in different fields based on IFI was developed by Thao and Chou [16]; a pattern recognition and decision-making evaluation based on IFI was presented by Gohain et al. [17]; an analysis of three-way decision-making using IFI was evaluated by Yang et al. [18]; and distance measures based on IFI were diagnosed by Garg and Rani [19].

A brief explanation of existing FI and IFI is available, and it is clear that the prevailing information has received a lot of attention from different individuals, but it is also clear that the prevailing FI can handle only one dimension of information at a time. When conducting research, experts in most fields typically work with two-dimensional information; however, when we consider the example of medical research, biometrics and facial recognition can dynamically change data over time. Thus, to address information challenges in various fields of research, the theory of complex FI (CFI) was pioneered by Ramot et al. [20] which fixes the imaginary term, or one more extra term, in the supporting grade and constructs the terms in the order of the polar coordinates. Based on the following work, we identified that the theory of CFI has a lot of applications in different fields, for instance, entropy measures [21], neighborhood operators [22], complex fuzzy logic [23,24], operation properties [25], complex fuzzy neuro structure [26], distance measures and continuity [27], complex fuzzy N-soft sets [28], and complex fuzzy soft sets [29]. Further, it was observed that the theory of CFI ignores the grade of anti-support, which results in a loss of information during the decision-making procedure. To remove these deficiencies, Alkouri and Salleh [30] diagnosed a complex IFI (CIFI), where CIFI contained two supporting functions, called positive and negative grades, in the order of complex values. The conditions of CIFI state that the sum of both values of the CIFI is necessarily contained in the unit interval (for both real and imaginary parts). Moreover, Ali et al. [31] proposed another form of complex intuitionistic fuzzy soft set. Garg and Rani [32,33] evaluated the information measures and correlation coefficient measures for CIFI. Jan et al. [34] investigated cyber-security and cybercrime with consideration of CIF relations. After a long assessment, we noticed that every expert faced the following queries:

• How can we collect data for osteoporosis problems?

• How can we aggregate the arranged information?

• How can we find the most optimal solution?

The proposed techniques are very suitable and reliable for addressing the above three problems. In many real-world situations, it is challenging to detect the most severe type of osteoporosis in human beings. Therefore, this paper aims to collect information on five different types of osteoporosis and evaluates them with the help of the proposed theory in the presence of some hypothetical or artificial information. The main theory of Frank’s t-norm and t-conorm was evaluated by Frank [35]. Further, Seikh and Mandel [36] utilized Frank aggregation operators based on IFI. But to date, no studies have applied the theory of Frank aggregation operators to CIFS. The main hypotheses of this work are stated below:

• The theory of averaging, geometry, Frank averaging, and Frank geometric aggregation operators for FS are special cases of the proposed theory.

• The theory of averaging, geometry, Frank averaging, and Frank geometric aggregation operators for IFS are special cases of the proposed theory.

• The theory of averaging, geometry, Frank averaging, and Frank geometric aggregation operators for CFS are special cases of the proposed theory.

• The theory of averaging, geometry, Frank averaging, and Frank geometric aggregation operators for CIFS are special cases of the proposed theory.

Considering the advantages of the proposed structure, it is clear that Frank aggregation operators are more reliable and more dominant compared with other notable aggregation operators because they possess improved modifications and can be applied more generally. Further, in this study, we explain the importance of complex-valued functions. For this, we provide an example to show the importance of the phase word. For example, consider a case where XYZ Company wishes to install biometric-based attendance devices (BBADs) at every one of its workplaces spread out across the nation. To achieve this, the corporation contacts an expert who provides information on (i) BBAD models and (ii) BBAD production dates. The business wants to choose the best BBAD model and manufacturing date at the same time. Since the issue at hand is two-dimensional, conventional IFS theory cannot be used to represent both dimensions at once. Utilizing CIFS theory is the most effective technique to express all of the knowledge offered by the expert. The phase terms and amplitude terms in CIFS may be used to inform the company’s decision on the BBAD model. The major influences of this theory are listed below:

• We derive Frank’s operational laws based on CIF information.

• We examine the CIFFWA operator, CIFFOWA operator, CIFFHA operator, CIFFWG operator, CIFFOWG operator, and CIFFHG operator. Some dominant and feasible properties of the invented techniques are also stated.

• To evaluate the problem of osteoporosis in human bodies based on their causes and risk factors, we illustrate an application of the MADM technique with consideration of the invented methods to show the supremacy and validity of the derived techniques.

• We compare the proposed scenarios with some valid existing or prevailing techniques to increase the worth of the presented approaches.

This manuscript is organized as follows: In Section 2, information on the CIF set and its algebraic laws with Frank’s t-norm and t-conorm is revised. In Section 3, we diagnose certain Frank’s operational laws under the availability of CIF information. Furthermore, using the evaluated operational laws, we pioneer the theory of CIFFWA, CIFFOWA, CIFFHA, CIFFWG, CIFFOWG, and CIFFHG operators, and describe their beneficial and valuable results with certain useful properties. In Section 4, we aim to evaluate the most severe type of osteoporosis based on its symptoms, causes, and risk factors using diagnostic approaches. In Section 5, to enhance the worth of the evaluated operators, we compare the diagnostic result with certain prevailing results and illustrate their geometrical representation with the help of MATLAB. Our final remarks and concluding information are provided in Section 6.

2. Preliminary Information

This section discusses the CIF set and its algebraic laws with Frank’s t-norm and t-conorm, where the universal set is shown by $\mathcal{X},$ and the terms $m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)}$ and $n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}$ represent the truth and falsity grades with real parts $m_{{\mathfrak{l}}_R}\left(\mathcal{x}\right), n_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)$ and imaginary parts $m_{{\mathfrak{l}}_I}\left(\mathcal{x}\right), n_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)$.

Definition 1

([30]). Assume $\mathcal{X}$ as a universal set, and then, the shape of ${\mathfrak{P}}_{𝓈}$, as shown below:

(1)${\mathfrak{l}}_{𝓈}=\left\{\left(m_{{\mathfrak{l}}_{𝓈}}\left(\mathcal{X}\right),n_{{\mathfrak{l}}_{𝓈}}\left(\mathcal{X}\right)\right):\mathcal{x}\in \mathcal{X}\right\}$

With a well-known condition, $0\le m_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)+n_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)\le 1$ and $0\le m_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)+n_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)\le 1$ represent the CIF information. The mathematical shape ${ℛ}_{{\mathfrak{l}}_{𝓈}}\left(\mathcal{x}\right)={ℛ}_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)e^{i2\pi \left({ℛ}_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)\right)}=\left(1-\left(m_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)+n_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)\right)\right)e^{i2\pi \left(1-\left(m_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)+n_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)\right)\right)}$ is shown as refusal information. Furthermore, the theory of the CIF number (CIFN) is represented by: ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$.

Definition 2

([30]). We recall some algebraic laws based on CIFNs ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2$, such that

(2)${\mathfrak{l}}_1\oplus {\mathfrak{l}}_2=\left(\left(m_{{\mathfrak{l}}_{R-1}}+m_{{\mathfrak{l}}_{R-2}}-m_{{\mathfrak{l}}_{R-1}}{m}_{{\mathfrak{l}}_{R-2}}\right)e^{i2\pi \left(m_{{\mathfrak{l}}_{I-1}}+m_{{\mathfrak{l}}_{I-2}}-m_{{\mathfrak{l}}_{I-1}}{m}_{{\mathfrak{l}}_{I-2}}\right)},\left(n_{{\mathfrak{l}}_{R-1}}{n}_{{\mathfrak{l}}_{R-2}}\right)e^{i2\pi \left(n_{{\mathfrak{l}}_{I-1}}{n}_{{\mathfrak{l}}_{I-2}}\right)}\right)$

(3)${\mathfrak{l}}_1\otimes {\mathfrak{l}}_2=\left(\left(m_{{\mathfrak{l}}_{R-1}}{m}_{{\mathfrak{l}}_{R-2}}\right)e^{i2\pi \left(m_{{\mathfrak{l}}_{I-1}}{m}_{{\mathfrak{l}}_{I-2}}\right)},\left(n_{{\mathfrak{l}}_{R-1}}+n_{{\mathfrak{l}}_{R-2}}-n_{{\mathfrak{l}}_{R-1}}{n}_{{\mathfrak{l}}_{R-2}}\right)e^{i2\pi \left(n_{{\mathfrak{l}}_{I-1}}+n_{{\mathfrak{l}}_{I-2}}-n_{{\mathfrak{l}}_{I-1}}{n}_{{\mathfrak{l}}_{I-2}}\right)}\right)$

(4)$\check{\rho }{\mathfrak{l}}_{𝓈}=\left(\left(1-{\left(1-m_{{\mathfrak{l}}_{R-𝓈}}\right)}^{\check{\rho }}\right)e^{i2\pi \left(1-{\left(1-m_{{\mathfrak{l}}_{I-𝓈}}\right)}^{\check{\rho }}\right)},n_{{\mathfrak{l}}_{R-𝓈}}^{\check{\rho }}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}^{\check{\rho }}\right)}\right)$

(5)${\mathfrak{l}}_{𝓈}^{\check{\rho }}=\left(m_{{\mathfrak{l}}_{R-𝓈}}^{\check{\rho }}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}^{\check{\rho }}\right)},\left(1-{\left(1-n_{{\mathfrak{l}}_{R-𝓈}}\right)}^{\check{\rho }}\right)e^{i2\pi \left(1-{\left(1-n_{{\mathfrak{l}}_{I-𝓈}}\right)}^{\check{\rho }}\right)}\right)$

where the notations $\oplus$ and $\otimes$ are used for algebraic (also Frank) addition and algebraic (also Frank) multiplications.

Definition 3

([30]). We recall some important ideas based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that

(6)${\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_{𝓈}\right)=\frac{1}{2}\left(m_{{\mathfrak{l}}_{R-𝓈}}+m_{{\mathfrak{l}}_{I-𝓈}}-n_{{\mathfrak{l}}_{R-𝓈}}-n_{{\mathfrak{l}}_{I-𝓈}}\right),{\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_{𝓈}\right)\in \left[-1,1\right]$

(7)${\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_{𝓈}\right)=\frac{1}{2}\left(m_{{\mathfrak{l}}_{R-𝓈}}+m_{{\mathfrak{l}}_{I-𝓈}}+n_{{\mathfrak{l}}_{R-𝓈}}+n_{{\mathfrak{l}}_{I-𝓈}}\right),{\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_{𝓈}\right)\in \left[0,1\right]$

where, ${\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_{𝓈}\right)$ and ${\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_{𝓈}\right)$ are represented by the score value and accuracy value.

Definition 4

([30]). We recall some important rules based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that if ${\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_1\right)>{\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_2\right)$, then ${\mathfrak{l}}_1>{\mathfrak{l}}_2$, if ${\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_1\right)={\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_2\right)$; if ${\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_1\right)>{\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_2\right)$, then ${\mathfrak{l}}_1>{\mathfrak{l}}_2$; and if ${\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_1\right)={\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_2\right)$, then ${\mathfrak{l}}_1={\mathfrak{l}}_2$.

Definition 5

([35]). Examples of Frank’s t-norm and t-conorm are illustrated below:

(8)$T\left(x,y\right)={log}_{\Xi }\left(1+\frac{\left({\Xi }^x-1\right)\left({\Xi }^y-1\right)}{\Xi -1}\right)$

(9)$S\left(x,y\right)=1-{log}_{\Xi }\left(1+\frac{\left({\Xi }^{1-x}-1\right)\left({\Xi }^{1-y}-1\right)}{\Xi -1}\right)$

where $x,y\in \left[0,1\right]$ and $\mathsf{\Xi }\in \left(1,+\infty \right)$.

3. Frank Aggregation Operators for CIF Information

The objective of this section is to determine Frank’s operational laws based on CIF information. Moreover, we examine the Frank aggregation operators (averaging and geometric) based on CIF set theory and Frank’s operational laws, such as the CIFFWA operator, CIFFOWA operator, CIFFHA operator, CIFFWG operator, CIFFOWG operator, and CIFFHG operator. Some dominant and feasible properties of the invented techniques are also stated.

Definition 6.

We diagnose some Frank laws based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2$, such that

(10)${\mathfrak{l}}_1\oplus {\mathfrak{l}}_2=\left(\begin{array}{c}1-{log}_{\Xi }\left(1+\frac{\left({\Xi }^{1-m_{{\mathfrak{l}}_{R-1}}}-1\right)\left({\Xi }^{1-m_{{\mathfrak{l}}_{R-2}}}-1\right)}{\Xi -1}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{\left({\Xi }^{1-m_{{\mathfrak{l}}_{I-1}}}-1\right)\left({\Xi }^{1-m_{{\mathfrak{l}}_{I-2}}}-1\right)}{\Xi -1}\right)\right)},\\ {log}_{\Xi }\left(1+\frac{\left({\Xi }^{n_{{\mathfrak{l}}_{R-1}}}-1\right)\left({\Xi }^{n_{{\mathfrak{l}}_{R-2}}}-1\right)}{\Xi -1}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{\left({\Xi }^{n_{{\mathfrak{l}}_{I-1}}}-1\right)\left({\Xi }^{n_{{\mathfrak{l}}_{I-2}}}-1\right)}{\Xi -1}\right)\right)}\end{array}\right)$

(11)${\mathfrak{l}}_1\otimes {\mathfrak{l}}_2=\left(\begin{array}{c}{log}_{\Xi }\left(1+\frac{\left({\Xi }^{m_{{\mathfrak{l}}_{R-1}}}-1\right)\left({\Xi }^{m_{{\mathfrak{l}}_{R-2}}}-1\right)}{\Xi -1}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{\left({\Xi }^{m_{{\mathfrak{l}}_{I-1}}}-1\right)\left({\Xi }^{m_{{\mathfrak{l}}_{I-2}}}-1\right)}{\Xi -1}\right)\right)},\\ 1-{log}_{\Xi }\left(1+\frac{\left({\Xi }^{1-n_{{\mathfrak{l}}_{R-1}}}-1\right)\left({\Xi }^{1-n_{{\mathfrak{l}}_{R-2}}}-1\right)}{\Xi -1}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{\left({\Xi }^{1-n_{{\mathfrak{l}}_{I-1}}}-1\right)\left({\Xi }^{1-n_{{\mathfrak{l}}_{I-2}}}-1\right)}{\Xi -1}\right)\right)}\end{array}\right)$

(12)$\check{\rho }{\mathfrak{l}}_1=\left(\begin{array}{c}1-{log}_{\Xi }\left(1+\frac{{\left({\Xi }^{1-m_{{\mathfrak{l}}_{R-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{\left({\Xi }^{1-m_{{\mathfrak{l}}_{I-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)\right)},\\ {log}_{\Xi }\left(1+\frac{{\left({\Xi }^{n_{{\mathfrak{l}}_{R-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{\left({\Xi }^{n_{{\mathfrak{l}}_{I-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)\right)}\end{array}\right)$

(13)${\mathfrak{l}}_{𝓈}^{\check{\rho }}=\left(\begin{array}{c}{log}_{\Xi }\left(1+\frac{{\left({\Xi }^{m_{{\mathfrak{l}}_{R-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{\left({\Xi }^{m_{{\mathfrak{l}}_{I-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)\right)},\\ 1-{log}_{\Xi }\left(1+\frac{{\left({\Xi }^{1-n_{{\mathfrak{l}}_{R-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{\left({\Xi }^{1-n_{{\mathfrak{l}}_{I-1}}}-1\right)}^{\check{\rho }}}{{\left(\Xi -1\right)}^{\check{\rho }-1}}\right)\right)}\end{array}\right)$

Definition 7.

We diagnose the idea of a CIFFWA operator based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that

(14)$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\mu }_{w-1}{\mathfrak{l}}_1\oplus {\mu }_{w-2}{\mathfrak{l}}_2\oplus \dots \oplus {\mu }_{w-t}{\mathfrak{l}}_t={\oplus }_{𝓈=1}^t{\mu }_{w-𝓈}{\mathfrak{l}}_{𝓈}$

where ${\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}=1,{\mu }_{w-𝓈}\in \left[0,1\right]$ represents the weight vector. In particular, if we use the value of ${\mu }_{w-𝓈}=\left(\frac{1}{t},\frac{1}{t},\dots ,\frac{1}{t}\right)$ in Equation (14), then we obtain

(15)$CIFFA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\frac{1}{t}\left({\mathfrak{l}}_1\oplus {\mathfrak{l}}_2\oplus \dots \oplus {\mathfrak{l}}_t\right)=\frac{1}{t}{\oplus }_{𝓈=1}^t{\mathfrak{l}}_{𝓈}$

which is called the CIF Frank averaging (CIFFA) operator.

Theorem 1.

Verify that the resultant value of Equation (14) is again in the shape of CIFNs, such that

(16) $CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ {log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$

Proof of Theorem 1.

The proof of Theorem 1 is given in Appendix A. □

Property 1.

(Idempotency.) When ${\mathfrak{l}}_{𝓈}=\mathfrak{l}=\left(m_{{\mathfrak{l}}_R}{e}^{i2\pi \left(m_{{\mathfrak{l}}_I}\right)},n_{{\mathfrak{l}}_R}{e}^{i2\pi \left(n_{{\mathfrak{l}}_I}\right)}\right),𝓈=1,2,\dots ,t$, then

(17)$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_k\right)=\mathfrak{l}$

Proof of Property 1.

The proof of Property 1 is given in Appendix B. □

Property 2.

(Monotonicity.) When ${\mathfrak{l}}_{𝓈}\le {\mathfrak{l}}_{𝓈}^{*}$ where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)$ and ${\mathfrak{l}}_{𝓈}^{*}=\left(m_{{\mathfrak{l}}_{R-𝓈}}^{*}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}^{*}\right)},n_{{\mathfrak{l}}_{R-𝓈}}^{*}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}^{*}\right)}\right),𝓈=1,2,\dots ,t$, then

(18)$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\le CIFFWA\left({\mathfrak{l}}_1^{*},{\mathfrak{l}}_2^{*},\dots ,{\mathfrak{l}}_t^{*}\right)$

Proof of Property 2.

The proof of Property 2 is given in Appendix C. □

Property 3.

(Boundedness.) When ${\mathfrak{l}}_{𝓈}^{+}=\left(\underset{𝓈}{max} m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{max} m_{{\mathfrak{l}}_{I-𝓈}}\right)},\underset{𝓈}{min} n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{min} n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)$ and ${\mathfrak{l}}_{𝓈}^{-}=\left(\underset{𝓈}{min} m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{min} m_{{\mathfrak{l}}_{I-𝓈}}\right)},\underset{𝓈}{max} n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{max} n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)$, then

(19)${\mathfrak{l}}_{𝓈}^{-}\le CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\le {\mathfrak{l}}_{𝓈}^{+}$

Proof of Property 3.

The proof of Property 3 is given in Appendix D. □

Definition 8.

We diagnose the idea of a CIFFOWA operator based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that

(20)$CIFFOWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\mu }_{w-1}{\mathfrak{l}}_{o\left(1\right)}\oplus {\mu }_{w-2}{\mathfrak{l}}_{o\left(2\right)}\oplus \dots \oplus {\mu }_{w-t}{\mathfrak{l}}_{o\left(t\right)}={\oplus }_{𝓈=1}^t{\mu }_{w-𝓈}{\mathfrak{l}}_{o\left(𝓈\right)}$

where ${\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}=1,{\mu }_{w-𝓈}\in \left[0,1\right]$ represents the weight vector with ${\mathfrak{l}}_{o\left(𝓈\right)}\le {\mathfrak{l}}_{o\left(𝓈-1\right)}$. In particular, if we use the value of ${\mu }_{w-𝓈}=\left(\frac{1}{t},\frac{1}{t},\dots ,\frac{1}{t}\right)$ in Equation (20), then we obtain:

(21)$CIFFOA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\frac{1}{t}\left({\mathfrak{l}}_{o\left(1\right)}\oplus {\mathfrak{l}}_{o\left(2\right)}\oplus \dots \oplus {\mathfrak{l}}_{o\left(t\right)}\right)=\frac{1}{t}{\oplus }_{𝓈=1}^t{\mathfrak{l}}_{o\left(𝓈\right)}$

which is called the CIF Frank ordered averaging (CIFFOA) operator.

Theorem 2.

Verify that the resultant value of Equation (20) is again in the shape of CIFNs, such that

(22) $CIFFOWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-m_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-m_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ {log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{n_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{n_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$

Proof. 

The proof of Theorem 2 is similar to the proof of Theorem 1. □

Definition 9.

We diagnose the idea of a CIFFHA operator based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that

(23)$CIFFHA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\mu }_{w-1}{\mathfrak{l}}_{o\left(1\right)}^{*}\oplus {\mu }_{w-2}{\mathfrak{l}}_{o\left(2\right)}^{*}\oplus \dots \oplus {\mu }_{w-t}{\mathfrak{l}}_{o\left(t\right)}^{*}={\oplus }_{𝓈=1}^t{\mu }_{w-𝓈}{\mathfrak{l}}_{o\left(𝓈\right)}^{*}$

where ${\mathfrak{l}}_{o\left(𝓈\right)}^{*}=t{\mu }_{w-𝓈}^{*}{\mathfrak{l}}_{𝓈}$ and ${\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}^{*}=1,{\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}=1,{\mu }_{w-𝓈}^{*},{\mu }_{w-𝓈}\in \left[0,1\right]$ represent the weight vector with ${\mathfrak{l}}_{o\left(𝓈\right)}^{*}\le {\mathfrak{l}}_{o\left(𝓈-1\right)}^{*}$. In particular, if we use the value of ${\mu }_{w-𝓈}=\left(\frac{1}{t},\frac{1}{t},\dots ,\frac{1}{t}\right)$ in Equation (23), then we obtain

(24)$CIFFHA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\frac{1}{t}\left({\mathfrak{l}}_{o\left(1\right)}^{*}\oplus {\mathfrak{l}}_{o\left(2\right)}^{*}\oplus \dots \oplus {\mathfrak{l}}_{o\left(t\right)}^{*}\right)=\frac{1}{t}{\oplus }_{𝓈=1}^t{\mathfrak{l}}_{o\left(𝓈\right)}^{*}$

which is called the CIF Frank hybrid averaging (CIFFHA) operator.

Theorem 3.

Verify that the resultant value of Equation (20) is again in the shape of CIFNs, such that

(25) $CIFFHA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-m_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-m_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ {log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{n_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{n_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$

Proof. 

The proof of Theorem 3 is similar to the proof of Theorem 1. □

Definition 10.

We diagnose the idea of a CIFFWG operator based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that

(26)$CIFFWG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\mathfrak{l}}_1^{{\mu }_{w-1}}\otimes {\mathfrak{l}}_2^{{\mu }_{w-2}}\otimes \dots \otimes {\mathfrak{l}}_t^{{\mu }_{w-t}}={\otimes }_{𝓈=1}^t{\mathfrak{l}}_{𝓈}^{{\mu }_{w-𝓈}}$

where ${\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}=1,{\mu }_{w-𝓈}\in \left[0,1\right]$ represents the weight vector. In particular, if we use the value of ${\mu }_{w-𝓈}=\left(\frac{1}{t},\frac{1}{t},\dots ,\frac{1}{t}\right)$ in Equation (26), then we obtain

(27)$CIFFA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\left({\mathfrak{l}}_1\otimes {\mathfrak{l}}_2\otimes \dots \otimes {\mathfrak{l}}_t\right)}^{\frac{1}{t}}={\left({\otimes }_{𝓈=1}^t{\mathfrak{l}}_{𝓈}\right)}^{\frac{1}{t}}$

which is called the CIF Frank geometric (CIFFG) operator.

Theorem 4.

Verify that the resultant value of Equation (26) is again in the shape of CIFNs, such that

(28) $CIFFWG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ 1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$

Proof. 

The proof of Theorem 4 is similar to the proof of Theorem 1. □

Definition 11.

We diagnose the idea of a CIFFOWG operator based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that

(29)$CIFFOWG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\mathfrak{l}}_{o\left(1\right)}^{{\mu }_{w-1}}\otimes {\mathfrak{l}}_{o\left(2\right)}^{{\mu }_{w-2}}\otimes \dots \otimes {\mathfrak{l}}_{o\left(t\right)}^{{\mu }_{w-t}}={\otimes }_{𝓈=1}^t{\mathfrak{l}}_{o\left(𝓈\right)}^{{\mu }_{w-𝓈}}$

where ${\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}=1,{\mu }_{w-𝓈}\in \left[0,1\right]$ represents the weight vector with ${\mathfrak{l}}_{o\left(𝓈\right)}\le {\mathfrak{l}}_{o\left(𝓈-1\right)}$. In particular, if we use the value of ${\mu }_{w-𝓈}=\left(\frac{1}{t},\frac{1}{t},\dots ,\frac{1}{t}\right)$ in Equation (29), then we obtain

(30)$CIFFG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\left({\mathfrak{l}}_{o\left(1\right)}\otimes {\mathfrak{l}}_{o\left(2\right)}\otimes \dots \otimes {\mathfrak{l}}_{o\left(t\right)}\right)}^{\frac{1}{t}}={\left({\otimes }_{𝓈=1}^t{\mathfrak{l}}_{o\left(𝓈\right)}\right)}^{\frac{1}{t}}$

which is called the CIF Frank ordered geometric (CIFFOG) operator.

Theorem 5.

Verify that the resultant value of Equation (29) is again in the shape of CIFNs, such that

(31) $CIFFOWG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{m_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{m_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ 1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-n_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-n_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$

Proof: 

The proof of Theorem 5 is similar to the proof of Theorem 1. □

Definition 12.

We diagnose the idea of a CIFHG operator based on CIFNs, where ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, such that

(32)$CIFFHG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\mathfrak{l}}^{*}{}_{o\left(1\right)}^{{\mu }_{w-1}}\otimes {\mathfrak{l}}^{*}{}_{o\left(2\right)}^{{\mu }_{w-2}}\otimes \dots \otimes {\mathfrak{l}}^{*}{}_{o\left(t\right)}^{{\mu }_{w-t}}={\otimes }_{𝓈=1}^t{\mathfrak{l}}^{*}{}_{o\left(𝓈\right)}^{{\mu }_{w-𝓈}}$

where ${\mathfrak{l}}_{o\left(𝓈\right)}^{*}=t{\mu }_{w-𝓈}^{*}{\mathfrak{l}}_{𝓈}$ and ${\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}^{*}=1,{\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}=1,{\mu }_{w-𝓈}^{*},{\mu }_{w-𝓈}\in \left[0,1\right]$ represent the weight vector with ${\mathfrak{l}}_{o\left(𝓈\right)}^{*}\le {\mathfrak{l}}_{o\left(𝓈-1\right)}^{*}$. In particular, if we use the value of ${\mu }_{w-𝓈}=\left(\frac{1}{t},\frac{1}{t},\dots ,\frac{1}{t}\right)$ in Equation (32), then we obtain

(33)$CIFFHG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)={\left({\mathfrak{l}}^{*}{}_{o\left(1\right)}\otimes {\mathfrak{l}}^{*}{}_{o\left(2\right)}\otimes \dots \otimes {\mathfrak{l}}^{*}{}_{o\left(t\right)}\right)}^{\frac{1}{t}}={\left({\otimes }_{𝓈=1}^t{\mathfrak{l}}^{*}{}_{o\left(𝓈\right)}\right)}^{\frac{1}{t}}$

which is called the CIF Frank hybrid geometric (CIFFHG) operator.

Theorem 6.

Verify that the resultant value of Equation (32) is again in the shape of CIFNs, such that

(34) $CIFFHG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{m_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{m_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ 1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-n_{{\mathfrak{l}}_{R-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\Xi }^{1-n_{{\mathfrak{l}}_{I-o\left(𝓈\right)}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\Xi -1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$

Proof: 

The proof of Theorem 6 is similar to the proof of Theorem 1. □

4. MADM Technique Based on Proposed Methods

In this section, we aim to evaluate the problem of bone diseases with consideration of the invented operators.

For this, we use the collection of alternatives ${\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_m$ and their criteria ${\mathfrak{l}}_{AT-1},{\mathfrak{l}}_{AT-2},\dots ,{\mathfrak{l}}_{AT-n}$ with the weight vector ${\displaystyle \sum }_{𝓈=1}^t{\mu }_{w-𝓈}=1,{\mu }_{w-𝓈}\in \left[0,1\right]$. To obtain this information, we need to construct a matrix by combining the value of CIF information, ${\mathfrak{l}}_{𝓈}=\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right),𝓈=1,2,\dots ,t$, with well-known conditions: $0\le m_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)+n_{{\mathfrak{l}}_R}\left(\mathcal{x}\right)\le 1$ and $0\le m_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)+n_{{\mathfrak{l}}_I}\left(\mathcal{x}\right)\le 1$, which represent the CIF information. To formulate the above dilemmas, we organized the stages of the planning procedure in the following order:

• Stage 1. Usually, we address two types of criteria, benefits and cost types. Therefore, the information or computed matrix is normalized using the following equation:

  $N=\{\begin{array}{cc}\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)& {\mathfrak{l}}_{𝓈}\in B\\ \left(n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)},m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)& {\mathfrak{l}}_{𝓈}\in C\end{array}$

  where $B$ is used to represent the benefit and $C$ is used to represent the cost type.

• Stage 2. After normalization, we use the theory of CIFFWA and CIFFWG operators to aggregate the information in the decision matrix.

  $CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}1-{log}_{\Xi }\left(1+{\displaystyle {\displaystyle \prod }_{𝓈=1}^t}{\left({\Xi }^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+{\displaystyle \prod }_{𝓈=1}^t{\left({\Xi }^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)\right)},\\ {log}_{\Xi }\left(1+{\displaystyle {\displaystyle \prod }_{𝓈=1}^t}{\left({\Xi }^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+{\displaystyle \prod }_{𝓈=1}^t{\left({\Xi }^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)\right)}\end{array}\right)$

  $CIFFWG\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}{log}_{\Xi }\left(1+{\displaystyle {\displaystyle \prod }_{𝓈=1}^t}{\left({\Xi }^{m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)e^{i2\pi \left({log}_{\Xi }\left(1+{\displaystyle \prod }_{𝓈=1}^t{\left({\Xi }^{m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)\right)},\\ 1-{log}_{\Xi }\left(1+{\displaystyle {\displaystyle \prod }_{𝓈=1}^t}{\left({\Xi }^{1-n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)e^{i2\pi \left(1-{log}_{\Xi }\left(1+{\displaystyle \prod }_{𝓈=1}^t{\left({\Xi }^{1-n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}\right)\right)}\end{array}\right)$

• Stage 3. We determine the score value of the obtained accumulated information.

• Stage 4. Finally, we determine the ordering of the alternatives based on their score values.

To justify the above problem with the help of an example, we use an issue that is faced by old and new generations called osteoporosis disease. We aim to evaluate the most severe type of osteoporosis based on its symptoms, causes, and risk factors using diagnostic approaches.

Analysis of Osteoporosis-Based Causes and Risk Factors

The information in this example is taken from different references [1,2,3,4,5]. There are many different kinds of osteoporosis, each with different causes and characteristics. The main aim of this study is to identify the most severe kind of osteoporosis among the five most serious types based on their features, symptoms, or characteristics, where the five major types of osteoporosis are represented by different families, including:

• Primary Osteoporosis (${\mathfrak{l}}_1$): This kind of osteoporosis is the most prevalent and often results from aging naturally. It is also separated into two subtypes:

  • (i). Postmenopausal osteoporosis.

  • (ii). Age-related osteoporosis.

• Secondary Osteoporosis (${\mathfrak{l}}_2$): This particular form of osteoporosis is brought on by a few underlying illnesses or drugs. Secondary osteoporosis may have several causes, such as:

  • (i). Hormonal problems.

  • (ii). Pharmaceuticals.

  • (iii). Medical conditions.

• Idiopathic Juvenile Osteoporosis (${\mathfrak{l}}_3$): There is no known underlying cause for this uncommon kind of osteoporosis, which affects children and teenagers. Significant bone loss and fractures may result from it.

• Secondary Osteoporosis due to Endocrine Disorders (${\mathfrak{l}}_4$): This type of osteoporosis is characterized by decreased bone density and an increased risk of fractures, which can be caused by certain endocrine conditions such as hyperparathyroidism or diabetes.

• Osteoporosis Imperfecta (${\mathfrak{l}}_5$): Collagen production is impacted by this genetic condition, which is necessary for strong bones. It causes bones to become very brittle and highly fracture-prone.

To identify the most severe of the above-described osteoporosis types, we use the values of criteria, represented by the shapes of the symptoms of osteoporosis: ${\mathfrak{l}}_{AT-1}$: deformity and mobility ${\mathfrak{l}}_{AT-2}$: pain; ${\mathfrak{l}}_{AT-3}$: anxiety and sleeping complications, and ${\mathfrak{l}}_{AT-4}$: chronic illness. To evaluate the problem discussed above, we use weight vectors of 0.4, 0.3, 0.2, and 0.1. Then, to formulate the above dilemmas, we organize the planning procedure stages into the following order:

• Stage 1. Usually, we address two types of criteria, benefits and cost types. Therefore, the information or computed matrix is normalized using the following equation:

  $N=\{\begin{array}{cc}\left(m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)},n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)& {\mathfrak{l}}_{𝓈}\in B\\ \left(n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(n_{{\mathfrak{l}}_{I-𝓈}}\right)},m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(m_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)& {\mathfrak{l}}_{𝓈}\in C\end{array}$

  where $B$ is used to represent the benefit and $C$ is used to represent the cost type. But the information given in Table 1 is based on benefit types, so we cannot normalize it. Table 1

  The expressed CIF matrix.

  	${\mathfrak{l}}_{\mathit{A}\mathit{T}-\mathbf{1}}$	${\mathfrak{l}}_{\mathit{A}\mathit{T}-\mathbf{2}}$
  ${\mathfrak{l}}_{\mathbf{1}}$	$\left(0.7e^{i2\pi \left(0.6\right)},0.1e^{i2\pi \left(0.2\right)}\right)$	$\left(0.71e^{i2\pi \left(0.61\right)},0.11e^{i2\pi \left(0.21\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{2}}$	$\left(0.3e^{i2\pi \left(0.4\right)},0.2e^{i2\pi \left(0.3\right)}\right)$	$\left(0.31e^{i2\pi \left(0.41\right)},0.21e^{i2\pi \left(0.31\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{3}}$	$\left(0.4e^{i2\pi \left(0.5\right)},0.2e^{i2\pi \left(0.1\right)}\right)$	$\left(0.41e^{i2\pi \left(0.51\right)},0.21e^{i2\pi \left(0.11\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{4}}$	$\left(0.6e^{i2\pi \left(0.5\right)},0.3e^{i2\pi \left(0.3\right)}\right)$	$\left(0.61e^{i2\pi \left(0.51\right)},0.31e^{i2\pi \left(0.31\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{5}}$	$\left(0.8e^{i2\pi \left(0.7\right)},0.1e^{i2\pi \left(0.1\right)}\right)$	$\left(0.81e^{i2\pi \left(0.71\right)},0.11e^{i2\pi \left(0.11\right)}\right)$
  	${\mathfrak{l}}_{\mathit{A}\mathit{T}-\mathbf{3}}$	${\mathfrak{l}}_{\mathit{A}\mathit{T}-\mathbf{4}}$
  ${\mathfrak{l}}_{\mathbf{1}}$	$\left(0.7e^{i2\pi \left(0.6\right)},0.1e^{i2\pi \left(0.2\right)}\right)$	$\left(0.7e^{i2\pi \left(0.6\right)},0.1e^{i2\pi \left(0.2\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{2}}$	$\left(0.32e^{i2\pi \left(0.42\right)},0.22e^{i2\pi \left(0.32\right)}\right)$	$\left(0.33e^{i2\pi \left(0.43\right)},0.23e^{i2\pi \left(0.33\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{3}}$	$\left(0.42e^{i2\pi \left(0.52\right)},0.22e^{i2\pi \left(0.12\right)}\right)$	$\left(0.43e^{i2\pi \left(0.53\right)},0.23e^{i2\pi \left(0.13\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{4}}$	$\left(0.62e^{i2\pi \left(0.52\right)},0.32e^{i2\pi \left(0.32\right)}\right)$	$\left(0.63e^{i2\pi \left(0.53\right)},0.33e^{i2\pi \left(0.33\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{5}}$	$\left(0.82e^{i2\pi \left(0.72\right)},0.12e^{i2\pi \left(0.12\right)}\right)$	$\left(0.83e^{i2\pi \left(0.73\right)},0.13e^{i2\pi \left(0.13\right)}\right)$

• Stage 2. After normalization, we used the theory of CIFFWA and CIFFWG operators to aggregate the information in the decision matrix, as shown in Table 2. Table 2

  The expressed aggregated values.

  	CIFFWA Operator	CIFFWG Operator
  ${\mathfrak{l}}_{\mathbf{1}}$	$\left(0.7101e^{i2\pi \left(0.6101\right)},0.1095e^{i2\pi \left(0.2097\right)}\right)$	$\left(0.7099e^{i2\pi \left(0.6099\right)},0.11004e^{i2\pi \left(0.21005\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{2}}$	$\left(0.31006e^{i2\pi \left(0.41007\right)},0.2097e^{i2\pi \left(0.2097\right)}\right)$	$\left(0.3098e^{i2\pi \left(0.4099\right)},0.21005e^{i2\pi \left(0.21005\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{3}}$	$\left(0.41007e^{i2\pi \left(0.51009\right)},0.2097e^{i2\pi \left(0.1095\right)}\right)$	$\left(0.4099e^{i2\pi \left(0.5099\right)},0.21005e^{i2\pi \left(0.11004\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{4}}$	$\left(0.61011e^{i2\pi \left(0.51009\right)},0.30986e^{i2\pi \left(0.30986\right)}\right)$	$\left(0.60993e^{i2\pi \left(0.50992\right)},0.31006e^{i2\pi \left(0.31006\right)}\right)$
  ${\mathfrak{l}}_{\mathbf{5}}$	$\left(0.81025e^{i2\pi \left(0.71016\right)},0.10958e^{i2\pi \left(0.10958\right)}\right)$	$\left(0.80995e^{i2\pi \left(0.70995\right)},0.11004e^{i2\pi \left(0.11004\right)}\right)$

• Stage 3. Moreover, we determine the score value of the obtained accumulated information, as shown in Table 3. Table 3

  The expressed score values.

  	CIFFWA Operator	CIFFWG Operator
  ${\mathfrak{l}}_{\mathbf{1}}$	$0.50046$	$0.4999$
  ${\mathfrak{l}}_{\mathbf{2}}$	$0.15028$	$0.14983$
  ${\mathfrak{l}}_{\mathbf{3}}$	$0.3004$	$0.29986$
  ${\mathfrak{l}}_{\mathbf{4}}$	$0.25024$	$0.24987$
  ${\mathfrak{l}}_{\mathbf{5}}$	$0.65063$	$0.64991$

• Stage 4. Finally, we find the ordering of the alternatives based on their score values, as shown in Table 4. Table 4

  The expressed ranking values.

  Methods	Ranking Values
  CIFFWA Operator	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$
  CIFFWG Operator	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$

We consider most optimal value to be ${\mathfrak{l}}_5$, which represents osteoporosis imperfecta. We notice that the diagnostic operators can easily evaluate the CIF types of data. But if we used the theory of intuitionistic fuzzy information instead of CIF information, what would happen? For this, we consider the information in Table 1 without imaginary parts. Then, using imaginary parts, the evaluated information is reviewed. For this, we determine the score value of the obtained accumulated information, as shown in Table 5 (using the information in Table 1 without imaginary parts).

Finally, we identify the ordering of the alternatives based on their score values, as shown in Table 6.

We consider the most optimal value to be l₅, which represents osteoporosis imperfecta. We notice that the diagnostic operators evaluated the intuitionistic and CIF types of data very easily.

5. Discussion

Many scholars have proposed different types of operator based on fuzzy sets and their extensions. Upon reviewing the existing techniques, it was noted that the scholars compared their methods with existing techniques to demonstrate the supremacy and validity of their proposed techniques. Therefore, motivated by this information, we compared our diagnostic result with certain prevailing results. To compare the diagnostic operators with some existing operators, we considered the following: the use of aggregation operators (AOs) with consideration of IFI was pioneered by Xu [37]; the use of geometric AOs under the availability of IFI was evaluated by Xu and Yager [38], novel AOs for CIFI were presented by Garg and Rani [39], and generalized geometric AOs for CIFI were explored by Garg and Rani [40]. We also considered the evaluated operators in this manuscript, called CIFFWA and CIFFWG operators. Then, we used the information in Table 2, and the comparative analysis of the diagnostic and existing operators is presented in Table 7.

From the information in Table 7, we determined that the operators defined in Refs. [39,40] are easily evaluated using the information given in Table 2, but noticed that the theory of Garg and Rani [39,40] is special cases in the proposed work because the proposed techniques are more general and reliable than the existing operators of Garg and Rani [39,40]. Furthermore, the theory of Garg and Rani [39,40] was invented based on algebraic norms, which is different from the proposed work, in which values are computed based on Frank norms, where the algebraic norms are special cases of the Frank norms. Additionally, the operators defined based on IFI in Refs. [37,38] failed to evaluate the theory given in Table 2. AOs with consideration of IFI were pioneered by Xu [37], and geometric AOs under the availability of IFI were evaluated by Xu and Yager [38] and have a lot of limitations. However, we try to demonstrate some supremacy of the existing information. For this, we used the information in Table 2 (without imaginary parts), and the comparative analysis of the diagnostic and existing operators is presented in Table 8.

From the information in Table 8, we determined that the operators defined in Refs. [39,40] are easily evaluated using the information given in Table 2 (without imaginary parts), and the operators defined based on IFI in Refs. [37,38] can also be evaluated based on the theory given in Table 2 (without imaginary parts). AOs with consideration of IFI were pioneered by Xu [37], and geometric AOs under the availability of IFI were evaluated by Xu and Yager [38]; we resolved the above information based on their limitations. This means that the diagnostic operators are beneficial and more dominant compared with the existing operators [37,38,39,40].

6. Conclusions

This manuscript aims to combine the major theory of aggregation operators in the presence of Frank’s t-norm and t-conorm with the CIF set theory. Our main study contributions are listed below:

• We derived Frank’s operational laws based on CIF information.

• We examined the CIFFWA operator, CIFFOWA operator, CIFFHA operator, CIFFWG operator, CIFFOWG operator, and CIFFHG operator. Some dominant and feasible properties of the invented techniques are also stated.

• We evaluated the problem of osteoporosis in human bodies based on its causes and risk factors and illustrated an application of the MADM technique with consideration of the invented methods to show the supremacy and validity of the derived techniques.

• We compared the proposed scenarios with some valid existing or prevailing techniques to increase the value of the presented approaches.

6.1. Limitations

Complex intuitionistic fuzzy set theory is very beneficial and reliable for evaluating awkward and complicated information, but in some real-life situations, CIFS fails when experts are faced with the problem of data sets in triplicate, such as data divided into yes, no, and abstinence categories. In such cases, we require the theory of complex picture fuzzy sets and their extensions.

6.2. Future Directions

In the future, we will review information on the Maclaurin symmetric mean, Dombi operators, the TOPSIS technique, decision-making, and aggregation operators, and try to improve the limitations and barriers of these theories. We aim to apply our results to different fields, such as fuzzy sets and their extensions.

Author Contributions

Conceptualization, K.H.; Methodology, X.Y. and Z.A.; Formal analysis, T.M.; Investigation, Z.A. and K.H.; Resources, X.Y. and T.M.; Writing—original draft, Z.A.; Writing—review & editing, X.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The employed data of this study is artificial and imaginary and anyone can employ this data by merely citing this paper before prior permission.

Acknowledgments

We would like to express our appreciation to the editor and the anonymous reviewers for their valuable comments, which have been very helpful in improving the paper.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Tables

Table 5

The expressed score values (without imaginary parts).

	CIFFWA Operator	CIFFWG Operator
${\mathfrak{l}}_{\mathbf{1}}$	$0.60058$	$0.5999$
${\mathfrak{l}}_{\mathbf{2}}$	$0.1003$	$0.0998$
${\mathfrak{l}}_{\mathbf{3}}$	$0.2003$	$0.1998$
${\mathfrak{l}}_{\mathbf{4}}$	$0.3003$	$0.2999$
${\mathfrak{l}}_{\mathbf{5}}$	$0.7007$	$0.6999$

Table 6

The expressed ranking values (without imaginary parts).

Methods	Ranking Values
CIFFWA Operator	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_2$
CIFFWG Operator	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_2$

Table 7

The represented ranking values using the information in Table 2.

Methods	Score Values	Ranking Values
Xu [37]	$\mathrm{Not}\mathrm{evaluated}\mathrm{due}\mathrm{to}\mathrm{some}\mathrm{complications}$	$\mathrm{Not}\mathrm{evaluated}\mathrm{due}\mathrm{to}\mathrm{some}\mathrm{complications}$
Xu and Yager [38]	$\mathrm{Not}\mathrm{evaluated}\mathrm{due}\mathrm{to}\mathrm{some}\mathrm{complications}$	$\mathrm{Not}\mathrm{evaluated}\mathrm{due}\mathrm{to}\mathrm{some}\mathrm{complications}$
Garg and Rani [39]	$0.7001,0.3006,0.6009,0.5006,0.8013$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$
Garg and Rani [40]	$0.9997,0.2996,0.5997,0.4997,0.9998$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$
CIFFWA Operator	$0.50046,0.15028,0.3004,0.25024,0.65063$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$
CIFFWG Operator	$0.4999,0.14983,0.29986,0.24987,0.64991$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$

Table 8

The represented ranking values using the information in Table 2 (without imaginary parts).

Methods	Score Values	Ranking Values
Xu [37]	$0.6006,0.1003,0.2003,0.3003,0.7007$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_2$
Xu and Yager [38]	$0.5999,0.0998,0.1998,0.2499,0.6999$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_2$
Garg and Rani [39]	$0.7001,0.3006,0.6009,0.5006,0.8013$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$
Garg and Rani [40]	$0.9997,0.2996,0.5997,0.4997,0.9998$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_2$
CIFFWA Operator	$0.60058,0.1003,0.2003,0.3003,0.7007$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_2$
CIFFWG Operator	$0.5999,0.0998,0.1998,0.2999,0.6999$	${\mathfrak{l}}_5\ge {\mathfrak{l}}_1\ge {\mathfrak{l}}_4\ge {\mathfrak{l}}_3\ge {\mathfrak{l}}_2$

Appendix A

Proof of Theorem 1.

Mathematical induction is used in the proof of Equation (16), so, for this, we consider $t=2$, and then, $${\mu }_{w-1}{\mathfrak{l}}_1=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)\right)}\end{array}\right)$$ $${\mu }_{w-2}{\mathfrak{l}}_2=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)\right)}\end{array}\right)$$

Then$$\begin{gathered} {\mu }_{w-1}{\mathfrak{l}}_1\oplus {\mu }_{w-2}{\mathfrak{l}}_2=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-1}}}-1\right)}^{{\mu }_{w-1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-1}-1}}\right)\right)}\end{array}\right) \\ \oplus \left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-2}}}-1\right)}^{{\mu }_{w-2}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-2}-1}}\right)\right)}\end{array}\right) \\ =\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^2{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^2{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^2{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^2{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^2{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^2{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^2{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^2{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right) \end{gathered}$$

Hence, Equation (16) fulfills the criteria for the value of $t=2$; further, if we assume that $t=k$, then $$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_k\right)=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$$

Then, we prove that Equation (16) also fulfills the criteria for the value of $t=k+1$, such that $$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_{k+1}\right)={\mu }_{w-1}{\mathfrak{l}}_1\oplus {\mu }_{w-2}{\mathfrak{l}}_2\oplus \dots \oplus {\mu }_{w-k}{\mathfrak{l}}_k\oplus {\mu }_{w-k+1}{\mathfrak{l}}_{k+1}={\oplus }_{𝓈=1}^k{\mu }_{w-𝓈}{\mathfrak{l}}_{𝓈}\oplus {\mu }_{w-k+1}{\mathfrak{l}}_{k+1}$$ $$=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)\oplus {\mu }_{w-k+1}{\mathfrak{l}}_{k+1}$$ $$\begin{gathered} =\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^k{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^k{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right) \\ \oplus \left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-k+1}}}-1\right)}^{{\mu }_{w-k+1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-k+1}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-k+1}}}-1\right)}^{{\mu }_{w-k+1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-k+1}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-k+1}}}-1\right)}^{{\mu }_{w-k+1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-k+1}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-k+1}}}-1\right)}^{{\mu }_{w-k+1}}}{{\left(\mathsf{\Xi }-1\right)}^{{\mu }_{w-k+1}-1}}\right)\right)}\end{array}\right) \end{gathered}$$ $$=(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^{k+1}{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^{k+1}{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^{k+!}{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^{k+1}{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^{k+1}{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^{k+1}{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^{k+1}{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^{k+!}{\mu }_{w-𝓈}-1}}\right)\right)}\end{array})$$

Hence, Equation (16) holds for all possible values of $t$. □

Appendix B

Proof of Property 1.

Given that ${\mathfrak{l}}_{𝓈}=\mathfrak{l}=\left(m_{{\mathfrak{l}}_R}{e}^{i2\pi \left(m_{{\mathfrak{l}}_I}\right)},n_{{\mathfrak{l}}_R}{e}^{i2\pi \left(n_{{\mathfrak{l}}_I}\right)}\right)$, $$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$$ $$=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_R}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_I}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_R}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_I}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$$ $$=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_R}}-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_I}}-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_R}}-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_I}}-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\right)}\end{array}\right)$$ $$=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_R}}-1\right)}^1}{{\left(\mathsf{\Xi }-1\right)}^{1-1}}\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_I}}-1\right)}^1}{{\left(\mathsf{\Xi }-1\right)}^{1-1}}\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_R}}-1\right)}^1}{{\left(\mathsf{\Xi }-1\right)}^{1-1}}\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+\frac{{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_I}}-1\right)}^1}{{\left(\mathsf{\Xi }-1\right)}^{1-1}}\right)\right)}\end{array}\right)$$ $$=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}\left(1+{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_R}}-1\right)}^1\right)e^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}\left(1+{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_I}}-1\right)}^1\right)\right)},\\ {\log }_{\mathsf{\Xi }}\left(1+\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_R}}-1\right)\right)e^{i2\pi \left({\log }_{\mathsf{\Xi }}\left(1+{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_I}}-1\right)}^1\right)\right)}\end{array}\right)$$ $$=\left(\begin{array}{c}1-{\log }_{\mathsf{\Xi }}{\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_R}}{e}^{i2\pi \left(1-{\log }_{\mathsf{\Xi }}{\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_I}}\right)},\\ {\log }_{\mathsf{\Xi }}{\mathsf{\Xi }}^{n_{{\mathfrak{l}}_R}}{e}^{i2\pi \left({\log }_{\mathsf{\Xi }}{\mathsf{\Xi }}^{n_{{\mathfrak{l}}_I}}\right)}\end{array}\right)=\left(m_{{\mathfrak{l}}_R}{e}^{i2\pi \left(m_{{\mathfrak{l}}_I}\right)},n_{{\mathfrak{l}}_R}{e}^{i2\pi \left(n_{{\mathfrak{l}}_I}\right)}\right)=\mathfrak{l}.$$□

Appendix C

Proof of Property 2.

Considering that ${\mathfrak{l}}_{𝓈}\le {\mathfrak{l}}_{𝓈}^{*}$, this means that $m_{{\mathfrak{l}}_{R-𝓈}}\le m_{{\mathfrak{l}}_{R-𝓈}}^{*},m_{{\mathfrak{l}}_{I-𝓈}}\le m_{{\mathfrak{l}}_{I-𝓈}}^{*}$ and $n_{{\mathfrak{l}}_{R-𝓈}}\ge n_{{\mathfrak{l}}_{R-𝓈}}^{*},n_{{\mathfrak{l}}_{I-𝓈}}\ge n_{{\mathfrak{l}}_{I-𝓈}}^{*}$, and then, $$1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le 1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)$$ $$1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le 1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)$$

And$${\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\ge {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)$$ $${\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\ge {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)$$

Then, by using Equation (6), we obtain:$$\begin{array}{l}{\mathcal{C}}_{s𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)\\ =\frac{1}{2}(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)+1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\\ -{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right))\\ \le \frac{1}{2}(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)+1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\\ -{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right))\\ ={\mathcal{C}}_{s𝓈}\left(CIFFWA\left({\mathfrak{l}}_1^{*},{\mathfrak{l}}_2^{*},\dots ,{\mathfrak{l}}_t^{*}\right)\right)\end{array}$$

Hence, $$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\le CIFFWA\left({\mathfrak{l}}_1^{*},{\mathfrak{l}}_2^{*},\dots ,{\mathfrak{l}}_t^{*}\right).$$

If ${\mathcal{C}}_{s𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)={\mathcal{C}}_{s𝓈}\left(CIFFWA\left({\mathfrak{l}}_1^{*},{\mathfrak{l}}_2^{*},\dots ,{\mathfrak{l}}_t^{*}\right)\right)$, then $$\begin{array}{l}{\mathcal{A}}_{a𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)\\ =\frac{1}{2}(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)+1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\\ +{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)+{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right))\\ \le \frac{1}{2}(1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)+1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\\ +{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)+{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}^{*}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right))\\ ={\mathcal{A}}_{a𝓈}\left(CIFFWA\left({\mathfrak{l}}_1^{*},{\mathfrak{l}}_2^{*},\dots ,{\mathfrak{l}}_t^{*}\right)\right)\end{array}$$

Hence, $$CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\le CIFFWA\left({\mathfrak{l}}_1^{*},{\mathfrak{l}}_2^{*},\dots ,{\mathfrak{l}}_t^{*}\right).$$□

Appendix D

Proof of Property 3.

Assume that ${\mathfrak{l}}_{𝓈}^{+}=\left(\underset{𝓈}{\max } m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{\max } m_{{\mathfrak{l}}_{I-𝓈}}\right)},\underset{𝓈}{\min } n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{\min } n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)$ and ${\mathfrak{l}}_{𝓈}^{-}=\left(\underset{𝓈}{\min } m_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{\min } m_{{\mathfrak{l}}_{I-𝓈}}\right)},\underset{𝓈}{\max } n_{{\mathfrak{l}}_{R-𝓈}}{e}^{i2\pi \left(\underset{𝓈}{\max } n_{{\mathfrak{l}}_{I-𝓈}}\right)}\right)$; then, $\underset{𝓈}{\min } m_{{\mathfrak{l}}_{R-𝓈}}\le m_{{\mathfrak{l}}_{R-𝓈}}\le \underset{𝓈}{\max } m_{{\mathfrak{l}}_{R-𝓈}},\underset{𝓈}{\min } m_{{\mathfrak{l}}_{I-𝓈}}\le m_{{\mathfrak{l}}_{I-𝓈}}\le \underset{𝓈}{\max } m_{{\mathfrak{l}}_{I-𝓈}},\underset{𝓈}{\min } n_{{\mathfrak{l}}_{R-𝓈}}\le n_{{\mathfrak{l}}_{R-𝓈}}\le \underset{𝓈}{\max } n_{{\mathfrak{l}}_{R-𝓈}},\underset{𝓈}{\min } n_{{\mathfrak{l}}_{I-𝓈}}\le n_{{\mathfrak{l}}_{I-𝓈}}\le \underset{𝓈}{\max } n_{{\mathfrak{l}}_{I-𝓈}}$; then, $$\begin{gathered} \underset{𝓈}{\min } m_{{\mathfrak{l}}_{R-𝓈}}=1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-\underset{𝓈}{\min } m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le 1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)=m_{{\mathfrak{l}}_{R-𝓈}} \\ \le 1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-\underset{𝓈}{\max } m_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le \underset{𝓈}{\max } m_{{\mathfrak{l}}_{R-𝓈}} \end{gathered}$$ $$\begin{gathered} \underset{𝓈}{\min } m_{{\mathfrak{l}}_{I-𝓈}}=1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-\underset{𝓈}{\min } m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le 1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)=m_{{\mathfrak{l}}_{R-𝓈}} \\ \le 1-{\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{1-\underset{𝓈}{\max } m_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le \underset{𝓈}{\max } m_{{\mathfrak{l}}_{I-𝓈}} \end{gathered}$$ $$\begin{gathered} \underset{𝓈}{\min } n_{{\mathfrak{l}}_{R-𝓈}}={\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{\underset{𝓈}{\min } n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)=n_{{\mathfrak{l}}_{R-𝓈}} \\ \le {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{\underset{𝓈}{\max } n_{{\mathfrak{l}}_{R-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le \underset{𝓈}{\max } n_{{\mathfrak{l}}_{R-𝓈}} \end{gathered}$$ $$\begin{gathered} \underset{𝓈}{\min } n_{{\mathfrak{l}}_{I-𝓈}}={\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{\underset{𝓈}{\min } n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)=n_{{\mathfrak{l}}_{I-𝓈}} \\ \le {\log }_{\mathsf{\Xi }}\left(1+\frac{{{\displaystyle \prod }}_{𝓈=1}^t{\left({\mathsf{\Xi }}^{\underset{𝓈}{\max } n_{{\mathfrak{l}}_{I-𝓈}}}-1\right)}^{{\mu }_{w-𝓈}}}{{\left(\mathsf{\Xi }-1\right)}^{{{\displaystyle \sum }}_{𝓈=1}^t{\mu }_{w-𝓈}-1}}\right)\le \underset{𝓈}{\max } n_{{\mathfrak{l}}_{I-𝓈}} \end{gathered}$$

Then by using Equations (6) and (7), we obtain:$$\begin{gathered} {\mathcal{C}}_{s𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)=\frac{1}{2}\left(m_{{\mathfrak{l}}_{R-𝓈}}+m_{{\mathfrak{l}}_{I-𝓈}}-n_{{\mathfrak{l}}_{R-𝓈}}-n_{{\mathfrak{l}}_{I-𝓈}}\right) \\ \ge \frac{1}{2}\left(\underset{𝓈}{\min } m_{{\mathfrak{l}}_{R-𝓈}}+\underset{𝓈}{\min } m_{{\mathfrak{l}}_{I-𝓈}}-\underset{𝓈}{\max } n_{{\mathfrak{l}}_{R-𝓈}}-\underset{𝓈}{\max } n_{{\mathfrak{l}}_{I-𝓈}}\right)={\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_{𝓈}^{-}\right) \end{gathered}$$ $$\begin{gathered} {\mathcal{C}}_{s𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)=\frac{1}{2}\left(m_{{\mathfrak{l}}_{R-𝓈}}+m_{{\mathfrak{l}}_{I-𝓈}}-n_{{\mathfrak{l}}_{R-𝓈}}-n_{{\mathfrak{l}}_{I-𝓈}}\right) \\ \le \frac{1}{2}\left(\underset{𝓈}{\max } m_{{\mathfrak{l}}_{R-𝓈}}+\underset{𝓈}{\max } m_{{\mathfrak{l}}_{I-𝓈}}-\underset{𝓈}{\min } n_{{\mathfrak{l}}_{R-𝓈}}-\underset{𝓈}{\min } n_{{\mathfrak{l}}_{I-𝓈}}\right)={\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_{𝓈}^{+}\right) \end{gathered}$$

In both situations, we obtain:$${\mathfrak{l}}_{𝓈}^{-}\le CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\le {\mathfrak{l}}_{𝓈}^{+}$$

But if ${\mathcal{C}}_{s𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)={\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_{𝓈}^{+}\right)={\mathcal{C}}_{s𝓈}\left({\mathfrak{l}}_{𝓈}^{-}\right)$, then $$\begin{gathered} {\mathcal{A}}_{a𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)=\frac{1}{2}\left(m_{{\mathfrak{l}}_{R-𝓈}}+m_{{\mathfrak{l}}_{I-𝓈}}-n_{{\mathfrak{l}}_{R-𝓈}}-n_{{\mathfrak{l}}_{I-𝓈}}\right) \\ \ge \frac{1}{2}\left(\underset{𝓈}{\min } m_{{\mathfrak{l}}_{R-𝓈}}+\underset{𝓈}{\min } m_{{\mathfrak{l}}_{I-𝓈}}-\underset{𝓈}{\max } n_{{\mathfrak{l}}_{R-𝓈}}-\underset{𝓈}{\max } n_{{\mathfrak{l}}_{I-𝓈}}\right)={\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_{𝓈}^{-}\right) \end{gathered}$$ $$\begin{gathered} {\mathcal{A}}_{a𝓈}\left(CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\right)=\frac{1}{2}\left(m_{{\mathfrak{l}}_{R-𝓈}}+m_{{\mathfrak{l}}_{I-𝓈}}-n_{{\mathfrak{l}}_{R-𝓈}}-n_{{\mathfrak{l}}_{I-𝓈}}\right) \\ \le \frac{1}{2}\left(\underset{𝓈}{\max } m_{{\mathfrak{l}}_{R-𝓈}}+\underset{𝓈}{\max } m_{{\mathfrak{l}}_{I-𝓈}}-\underset{𝓈}{\min } n_{{\mathfrak{l}}_{R-𝓈}}-\underset{𝓈}{\min } n_{{\mathfrak{l}}_{I-𝓈}}\right)={\mathcal{A}}_{a𝓈}\left({\mathfrak{l}}_{𝓈}^{+}\right) \end{gathered}$$

In both situations, we obtain:$${\mathfrak{l}}_{𝓈}^{-}\le CIFFWA\left({\mathfrak{l}}_1,{\mathfrak{l}}_2,\dots ,{\mathfrak{l}}_t\right)\le {\mathfrak{l}}_{𝓈}^{+}.$$□

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