Aggregation Operators Based on Algebraic t-Norm and t-Conorm for Complex Linguistic Fuzzy Sets and Their Applications in Strategic Decision Making

Abstract

Aggregation operators perform a valuable and significant role in various decision-making processes. Averaging and geometric aggregation operators are both used for capturing the interrelationships of the aggregated preferences, even if the preferences are independent. The purpose of this paper is to analyze the theory of complex linguistic fuzzy (CLF) sets and their important laws, such as algebraic laws, score values, and accuracy values, and to describe the relationship between the score and accuracy values with the help of their properties. Additionally, based on the proposed CLF information, we introduce the theory of CLF weighted averaging (CLFWA), CLF ordered weighted averaging (CLFOWA), CLF hybrid averaging (CLFHA), CLF weighted geometric (CLFWG), CLF ordered weighted geometric (CLFOWG), and CLF hybrid geometric (CLFHG) operators. The fundamental properties and some valuable results of these operators are evaluated, and their particular cases are described. Based on the presented operators, a technique for evaluating the “multi-attribute decision-making” (MADM) problems in the consideration of CLF sets is derived. The superiority of the derived technique is illustrated via a practical example, a set of experiments, and significant and qualitative comparisons. The illustration results indicate that the derived technique can be feasible and superior in evaluating CLF information. Further, it can be used for determining the interrelationships of attributes and the criteria of experts. Moreover, it is valuable and capable of evaluating the MADM problems using CLF numbers.

1. Introduction

Various people have developed many types of apps for evaluating complicated and awkward situations. Initially, engineers saw most of these apps as being only suitable “for fun”; however, their development attracted a great deal of attention, and people have begun using them to evaluate serious problems. Some of the most beneficial engineering apps for Android and IOS are noted below: Auto CAD 360 is useful: Auto CAD is highly valuable for engineers because it enables users to easily sketch their ideas. This app is commonly used in project management, electrical management, and architecture. Further, snip by MathPix solves equations in front of the user’s eye; the mechanical engineering implementation is highly useful. Finger CAD is another CAD app for engineers, and the Graphing calculator X84 is a robust engineering app for prediction. Droid Tesla and iCricuit are also highly useful apps for electrical engineers.

To find the best of these under the consideration of classical information, certain applications have been tested by various scholars. However, in the presence of classical information, experts face a large number of errors due to vagueness and uncertainty. Handling the awkward and unreliable problem discussed above, Zadeh [1] evaluated fuzzy set (FS) theory by extending the codomain of the characteristic function on the classical set from “{0, 1}” to “[0, 1]”. This is because, in the set {0, 1}, only two possibilities in the shape of “0” or “1” exist, whereas a full range of possibilities exists within [0, 1]. FS theory has been much more modified than classical set theory due to its effectiveness. As a result, various scholars have employed it in the circumstance of distinct areas [2,3,4]. For instance, Brown [5] introduced a note on FSs, Dubois and Prade [6] evaluated the three semantics of FS theory, Akram et al. [7] presented fuzzy N-soft sets, and Fatimah and Alcantud [8] diagnosed multi-fuzzy N-soft sets. Additionally, the theory of intuitionistic FSs [9], interval-valued FSs [10,11], and type-2 fuzzy sets [12] also played a very important role in the environment of fuzzy set theory.

Based on this brief literature review, the operators computed based on FS theory contain a great deal of ambiguity and uncertainty, as FS information manages only membership grade ${\mathfrak{k}}_{\mathfrak{m}}\left(𝓏\right)=\left({\mathfrak{k}}_{\mathfrak{m}}^R\left(𝓏\right)\right)$, which is computed in the shape of a one-term element. It thus fails to evaluate those types of complicated problem that depend on two terms of information ${\mathfrak{k}}_{\mathfrak{m}}\left(𝓏\right)=\left({\mathfrak{k}}_{\mathfrak{m}}^R\left(𝓏\right),{\mathfrak{k}}_{\mathfrak{m}}^I\left(𝓏\right)\right)$. An example where this would be important is where company A seeks to launch a new kind of car in a market, and, for each potential model, it provides two types of information: the name and price of each car. In terms of handling even this minor complication, the theory of FS will fail. To evaluate the above dilemma perfectly, Ramot et al. [13] introduced the main theory of complex FS (CFS) by extending the main range of the FS “[0, 1]” into “$\left[0, 1]\times [0, 1\right]$”. Further, Liu et al. [14] introduced the cross-entropy measures for CFS, Ramot et al. [15] diagnosed the complex fuzzy logic, Chen et al. [16] evaluated the neuro-fuzzy information using CFSs, and Dick [17] formulated the complex fuzzy logic.

The existing information discussed briefly in the above paragraph is very valuable and certain scholars have employed it in the environments of different fields. Frequently, however, in genuine dilemmas, various decision-making issues arise due to the limited available qualitative information based on ways of expressing complicated and ambiguous information; for instance, if regarding the intelligence of an expert the decision maker mostly used linguistic terms such as very low, low, medium, high, very high, and perfect. In this kind of situation, an expert may prefer linguistics, as diagnosed by Zadeh [18,19,20]. Further, in 1983, Zadeh [21] again evaluated some real-life dilemmas using linguistic information. Moreover, linguistic information based on FSs was evaluated by Tong and Bonissone [22]. Boran et al. [23] diagnosed linguistic summarization with FSs. One of the most important questions is what happens if linguistic information is provided in the shape of complex values. This type of information is only handled by CLF information because CLF information contains linguistic variables in the shape of complex numbers. The major purposes of this paper are described below:

• To analyze the theory of CLF sets and their important laws, such as algebraic laws, score values, and accuracy values, and describe the relationship between the score and accuracy values with the help of their properties.
• To pioneer the theory of CLFWA, CLFOWA, CLFHA, CLFWG, CLFOWG, and CLFHG operators. The fundamental properties and some valuable results of these operators are evaluated, and their particular cases are described.
• To evaluate some real-life problems based on the presented operators, a technique for evaluating the MADM problems in the consideration of CLF sets is derived.
• We assess the superiority of the derived technique via a practical example, a set of experiments, and significant and qualitative comparisons. The illustration results indicate that the derived technique has the ability to be feasible and superior in evaluating CLF information. Further, it can be used for determining the interrelationships of attributes and the criteria of experts. Moreover, it is valuable and capable of evaluating the MADM problems using CLF numbers.

The main construction of this theory is presented as follows: In Section 2, we recall the main theory of CF set theory and the fundamental information, alongside the idea of linguistic variables. In Section 3, we analyze the theory of CLF sets and their important laws, such as algebraic laws, score values, and accuracy values, and describe the relationship between the score and accuracy values with the help of their properties. In Section 4, we introduce the theory of CLFWA, CLFOWA, CLFHA, CLFWG, CLFOWG, and CLFHG operators. The fundamental properties and some valuable results of these operators are evaluated, and their particular cases are described. Based on the presented operators, a technique for evaluating the MADM problems in the consideration of CLF sets is derived. The superiority of the derived technique is illustrated via a practical example, a set of experiments, and significant and qualitative comparisons. The illustration results indicate that the derived technique has the ability to be feasible and superior in evaluating CLF information. Further, it can be used for determining the interrelationships of attributes and the criteria of experts. Moreover, it is valuable and capable of evaluating the MADM problems using CLF numbers, as discussed in Section 5. In Section 6, we present some concluding remarks.

2. Preliminaries

This section recalls the basics of CF set theory and the linguistic term set (TLS).

Definition 1.

Ref. ([13]) assume that the symbol${\hat{\mathcal{Q}}}_{𝓈}$, represents a CF set computed in the presence of a fixed set${\mathfrak{T}}_u$, such that:

$${\hat{\mathcal{Q}}}_{𝓈}=\left\{\left({\mathfrak{k}}_{\mathfrak{m}}\left(𝓏\right)\right):𝓏\in {\mathfrak{T}}_u\right\}$$

(1)

where${\mathfrak{k}}_{\mathfrak{m}}\left(𝓏\right)=\left({\mathfrak{k}}_{\mathfrak{m}}^R\left(𝓏\right),{\mathfrak{k}}_{\mathfrak{m}}^I\left(𝓏\right)\right)$represents the supported grade with$0\le {\mathfrak{k}}_{\mathfrak{m}}^R\left(𝓏\right)\le 1,0\le {\mathfrak{k}}_{\mathfrak{m}}^I\left(𝓏\right)\le 1$. Finally, the main idea of the CF number is represented by:${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\mathfrak{k}}_{{\mathfrak{m}}_j}^R,{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right),j=1,2,\dots ,u$. Further, to find the addition, multiplication, scalar multiplication, and exponential multiplication, we diagnose some operational laws under the consideration of algebraic t-norm and t-conorm.

Definition 2.

Ref. ([13]) let${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\mathfrak{k}}_{{\mathfrak{m}}_j}^R,{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right),j=1,2,\dots ,u$, represent the family of CF numbers. Then:

$${\hat{\mathcal{Q}}}_{{𝓈}_1}\oplus {\hat{\mathcal{Q}}}_{{𝓈}_2}=\left({\mathfrak{k}}_{{\mathfrak{m}}_1}^R+{\mathfrak{k}}_{{\mathfrak{m}}_2}^R-{\mathfrak{k}}_{{\mathfrak{m}}_1}^R{\mathfrak{k}}_{{\mathfrak{m}}_2}^R,{\mathfrak{k}}_{{\mathfrak{m}}_1}^I+{\mathfrak{k}}_{{\mathfrak{m}}_2}^I-{\mathfrak{k}}_{{\mathfrak{m}}_1}^I{\mathfrak{k}}_{{\mathfrak{m}}_2}^I\right)$$

(2)

$${\hat{\mathcal{Q}}}_{{𝓈}_1}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_2}=\left({\mathfrak{k}}_{{\mathfrak{m}}_1}^R{\mathfrak{k}}_{{\mathfrak{m}}_2}^R,{\mathfrak{k}}_{{\mathfrak{m}}_1}^I{\mathfrak{k}}_{{\mathfrak{m}}_2}^I\right)$$

(3)

$${\mathsf{\Psi }}_s{\hat{\mathcal{Q}}}_{{𝓈}_j}=\left(\left(1-{\left(1-{\mathfrak{k}}_{{\mathfrak{m}}_j}^R\right)}^{{\mathsf{\Psi }}_s}\right),\left(1-{\left(1-{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right)}^{{\mathsf{\Psi }}_s}\right)\right)$$

(4)

$${\hat{\mathcal{Q}}}_{{𝓈}_j}^{{\mathsf{\Psi }}_s}=\left({\left({\mathfrak{k}}_{{\mathfrak{m}}_j}^R\right)}^{{\mathsf{\Psi }}_s},{\left({\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right)}^{{\mathsf{\Psi }}_s}\right)$$

(5)

To evaluate the order between any two CF numbers, we recall the theory of score and accuracy function.

Definition 3.

Ref. ([13]) let${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left(\left({\mathfrak{k}}_{{\mathfrak{m}}_j}^R,{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right)\right),j=1,2,\dots ,u$, represent the family of CF numbers. Then:

$$\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)=\frac{1}{2}\left|{\mathfrak{k}}_{{\mathfrak{m}}_j}^R-{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right|,\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)\in \left[0,1\right]$$

(6)

$$\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)=\frac{1}{2}\left|{\mathfrak{k}}_{{\mathfrak{m}}_j}^R+{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right|, \mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)\in \left[0,1\right]$$

(7)

Equations (6) and (7) represent the score and accuracy value with a condition, such that:

1. 

Confirmed${\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if$\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)>\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

2. 

Confirmed${\hat{\mathcal{Q}}}_{{𝓈}_1}<{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if$\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)<\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

3. 

But if$\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)=\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$, then

(i) 

Confirmed${\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if$\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)>\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

(ii) 

Confirmed${\hat{\mathcal{Q}}}_{{𝓈}_1}<{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if$\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)\mathscr{H}\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

To find the intelligence of an expert, decision makers mostly use linguistic terms such as very low, low, medium, high, very high, and perfect. In this kind of situation, an expert can prefer linguistics, as diagnosed by Zadeh [18,19,20], and their mathematical shape is described below.

Definition 4. 

Ref. ([18]) assume that the mathematical shape$\hat{\Xi }=\left\{{\Xi }_{\mathrm{j}}:\mathrm{j}=1,2,\dots ,\mathrm{u}\right\}$represents the family of finite LTS with odd cardinality, where${\Xi }_{\mathrm{j}}$represents the possible linguistic variables, for instance,$\hat{\Xi }=\left\{{\Xi }_0,{\Xi }_1,{\Xi }_2,{\Xi }_3,{\Xi }_4,{\Xi }_5,{\Xi }_6,{\Xi }_7\right\}=\left\{none, very low, low, medium, high, very high, perfect\right\}$, which necessarily hold the basic conditions:

• ${\Xi }_j>{\Xi }_k\iff j>k$
• $Neg\left({\Xi }_j\right)={\Xi }_k, where k=u-j$
• $Max\left({\Xi }_j,{\Xi }_k\right)={\Xi }_j\iff j\ge k;$
• $Min\left({\Xi }_j,{\Xi }_k\right)={\Xi }_j\iff j\le k;$

Moreover, Xu [24] modified the theory of discrete LTS to continuous LTS $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$.

3. Complex Linguistic Fuzzy Sets

This section recalls the basics of CF set theory and their important laws, such as algebraic laws, score values, and accuracy values, and describes the relationship between the score and accuracy values with the help of their properties.

Definition 5. 

Assume that the symbol${\hat{\mathcal{Q}}}_{𝓈}$, represents a CLF set computed in the presence of a fixed set ${\mathfrak{T}}_u$and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS, such that:

$${\hat{\mathcal{Q}}}_{𝓈}=\left\{\left({\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^I}\left(𝓏\right)\right):𝓏\in {\mathfrak{T}}_u\right\}$$

(8)

where $\left({\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^I}\left(𝓏\right)\right)$represents the complex linguistic fuzzy number with $0\le {\mathfrak{k}}_{\mathfrak{m}}^R\left(𝓏\right)\le 1,0\le {\mathfrak{k}}_{\mathfrak{m}}^I\left(𝓏\right)\le 1$. Finally, the main idea of the CLF number is represented by: ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$. To evaluate the order between any two CF numbers, we recall the theory of score and accuracy function.

Definition 6. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers. Then:

$$\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)=\frac{1}{4}\left|z-{\mathfrak{k}}_{{\mathfrak{m}}_j}^R-{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right|,\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)\in \left[0,1\right]$$

(9)

$$\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)=\frac{1}{2}\left|{\mathfrak{k}}_{{\mathfrak{m}}_j}^R+{\mathfrak{k}}_{{\mathfrak{m}}_j}^I\right|, \mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_j}\right)\in \left[0,1\right]$$

(10)

Equations (9) and (10) represent the score and accuracy value with a condition, such that:

1. 

Confirmed ${\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if $\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)>\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

2. 

Confirmed ${\hat{\mathcal{Q}}}_{{𝓈}_1}<{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if $\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)<\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

3. 

But if $\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)=\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$, then

(i) 

Confirmed ${\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if $\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)>\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

(ii) 

Confirmed ${\hat{\mathcal{Q}}}_{{𝓈}_1}<{\hat{\mathcal{Q}}}_{{𝓈}_2}$, if $\mathscr{H}\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\right)\mathscr{H}\mathcal{S}\left({\hat{\mathcal{Q}}}_{{𝓈}_2}\right)$.

Further, we introduce the operations of addition, multiplication, scalar multiplication, and exponential multiplication through the following formulas. These operational laws can help us to compute some aggregation operators, because, without these operational laws, we cannot compute any hybrid structure, such as aggregation operators, similarity measures, and different kinds of methods.

Definition 7. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then:

$${\hat{\mathcal{Q}}}_{{𝓈}_1}{{\displaystyle \cup }}^{}{\hat{\mathcal{Q}}}_{{𝓈}_2}=\left(\max \left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_1}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_2}^R}\left(𝓏\right)\right)+i\max \left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_1}^I}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_2}^I}\left(𝓏\right)\right)\right)$$

(11)

$${\hat{\mathcal{Q}}}_{{𝓈}_1}{{\displaystyle \cap }}^{}{\hat{\mathcal{Q}}}_{{𝓈}_2}=\left(\min \left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_1}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_2}^R}\left(𝓏\right)\right)+i\min \left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_1}^I}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_2}^I}\left(𝓏\right)\right)\right)$$

(12)

$${\hat{\mathcal{Q}}}_{{𝓈}_1}\oplus {\hat{\mathcal{Q}}}_{{𝓈}_2}=\left({\Xi }_{u\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_1}^R}{u}+\frac{{\mathfrak{k}}_{{\mathfrak{m}}_2}^R}{u}-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_1}^R}{u}\frac{{\mathfrak{k}}_{{\mathfrak{m}}_2}^R}{u}\right)}\left(𝓏\right)+i{\Xi }_{u\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_1}^I}{u}+\frac{{\mathfrak{k}}_{{\mathfrak{m}}_2}^I}{u}-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_1}^I}{u}\frac{{\mathfrak{k}}_{{\mathfrak{m}}_2}^I}{u}\right)}\left(𝓏\right)\right)$$

(13)

$${\hat{\mathcal{Q}}}_{{𝓈}_1}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_2}=\left({\Xi }_{u\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_1}^R}{u}\frac{{\mathfrak{k}}_{{\mathfrak{m}}_2}^R}{u}\right)}\left(𝓏\right)+i{\Xi }_{u\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_1}^I}{u}\frac{{\mathfrak{k}}_{{\mathfrak{m}}_2}^I}{u}\right)}\left(𝓏\right)\right)$$

(14)

$${\mathsf{\Psi }}_s{\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{u\left(1-{\left(1-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}{u}\right)}^{{\mathsf{\Psi }}_s}\right)}\left(𝓏\right)+i{\Xi }_{u\left(1-{\left(1-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}{u}\right)}^{{\mathsf{\Psi }}_s}\right)}\left(𝓏\right)\right)$$

(15)

$${\hat{\mathcal{Q}}}_{{𝓈}_j}^{{\mathsf{\Psi }}_s}=\left({\Xi }_{u{\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}{u}\right)}^{{\mathsf{\Psi }}_s}}\left(𝓏\right)+i{\Xi }_{u{\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}{u}\right)}^{{\mathsf{\Psi }}_s}}\left(𝓏\right)\right)$$

(16)

Lemma 1. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then:

• ${\hat{\mathcal{Q}}}_{{𝓈}_1}\oplus {\hat{\mathcal{Q}}}_{{𝓈}_2}={\hat{\mathcal{Q}}}_{{𝓈}_2}\oplus {\hat{\mathcal{Q}}}_{{𝓈}_1}$.
• ${\hat{\mathcal{Q}}}_{{𝓈}_1}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_2}={\hat{\mathcal{Q}}}_{{𝓈}_2}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_1}$.
• ${\mathsf{\Psi }}_s\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\oplus {\hat{\mathcal{Q}}}_{{𝓈}_2}\right)={\mathsf{\Psi }}_s{\hat{\mathcal{Q}}}_{{𝓈}_1}\oplus {\mathsf{\Psi }}_s{\hat{\mathcal{Q}}}_{{𝓈}_2}$.
• ${\left({\hat{\mathcal{Q}}}_{{𝓈}_1}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_2}\right)}^{{\mathsf{\Psi }}_s}={\hat{\mathcal{Q}}}_{{𝓈}_1}^{{\mathsf{\Psi }}_s}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_2}^{{\mathsf{\Psi }}_s}$.
• ${\hat{\mathcal{Q}}}_{{𝓈}_1}^{{\mathsf{\Psi }}_1}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_1}^{{\mathsf{\Psi }}_2}={\hat{\mathcal{Q}}}_{{𝓈}_1}^{{\mathsf{\Psi }}_1+{\mathsf{\Psi }}_2}$.

4. Aggregation Operators for CLFSs

The purpose of this section is to analyze the theory of CLFWA, CLFOWA, CLFHA, CLFWG, CLFOWG, and CLFHG operators. The fundamental properties and some valuable results of these operators are evaluated, and their particular cases are described.

Definition 8. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, the CLFWA operator is introduced by:

$$CLFWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\mathsf{\Gamma }}_1{\hat{\mathcal{Q}}}_{{𝓈}_1}\oplus {\mathsf{\Gamma }}_2{\hat{\mathcal{Q}}}_{{𝓈}_2}\oplus {\mathsf{\Gamma }}_3{\hat{\mathcal{Q}}}_{{𝓈}_3}\oplus \dots \oplus {\mathsf{\Gamma }}_u{\hat{\mathcal{Q}}}_{{𝓈}_u}={\oplus }_{j=1}^u\left({\mathsf{\Gamma }}_j{\hat{\mathcal{Q}}}_{{𝓈}_j}\right)$$

(17)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$. The information in Equation (17) is superior to the information computed based on FS, CFS, and LTSs. This is because when we remove the linguistic terms, the information in Equation (17) is changed for CFSs, and similarly when we remove the linguistic term sets and imaginary parts, the information in Equation (17) is changed for FSs. Finally, when we remove only the imaginary part, the information in Equation (17) is changed for linguistic term sets, which are the particular cases of the introduced information.

Theorem 1. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, we evaluate Equation (17), and we obtain:

$$CLFWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)=\left({\Xi }_{u\left(1-{\displaystyle \prod }_{j=1}^u{\left(1-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)+i{\Xi }_{u\left(1-{\displaystyle \prod }_{j=1}^u{\left(1-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)\right)$$

(18)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$.

The Proof of Theorem 1 is given in Appendix A.

Property 1 (Idempotency). 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. When ${\hat{\mathcal{Q}}}_{{𝓈}_j}={\hat{\mathcal{Q}}}_{𝓈}=\left({\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^I}\left(𝓏\right)\right)$, then:

$$CLFWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\hat{\mathcal{Q}}}_{𝓈}$$

(19)

The Proof of Property 1 is given in Appendix B.

Property 2 (Monotonicity). 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. When ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)\le {\hat{\mathcal{Q}}}_{{𝓈}_j}^{\ast }=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}^{\ast }\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}^{\ast }\left(𝓏\right)\right)$, then:

$$CLFWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)\le CLFWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1}^{\ast },{\hat{\mathcal{Q}}}_{{𝓈}_2}^{\ast },\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}^{\ast }\right)$$

(20)

The Proof of Property 2 is given in Appendix C.

Property 3 (Boundedness). 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. When ${\hat{\mathcal{Q}}}_{{𝓈}_j}^{-}=\left(\underset{j}{\min }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i\left(\underset{j}{\min }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)\right)\right)$and ${\hat{\mathcal{Q}}}_{{𝓈}_j}^{+}=\left(\underset{j}{\max }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i\left(\underset{j}{\max }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)\right)\right)$, then:

$${\hat{\mathcal{Q}}}_{{𝓈}_j}^{-}\le CLFWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)\le {\hat{\mathcal{Q}}}_{{𝓈}_j}^{+}$$

(21)

The Proof of Property 3 is given in Appendix D.

Definition 9. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then the CLFOWA operator is introduced by:

$$CLFOWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\mathsf{\Gamma }}_1{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(1\right)}}\oplus {\mathsf{\Gamma }}_2{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(2\right)}}\oplus {\mathsf{\Gamma }}_3{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(3\right)}}\oplus \dots \oplus {\mathsf{\Gamma }}_u{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(u\right)}}={\oplus }_{j=1}^u\left({\mathsf{\Gamma }}_j{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(j\right)}}\right)$$

(22)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$, where $o\left(j\right)\le o\left(j-1\right)$. The information in Equation (22) is superior to the information computed based on FS, CFS, and LTSs, as by removing the linguistic terms, the information in Equation (22) is changed to be suitable for CFSs. Similarly, removing the linguistic term sets and imaginary parts causes the information in Equation (22) to be changed for use in FSs while removing only the imaginary part causes the information in Equation (22) to become suitable for linguistic term sets, which are particular cases of novel information.

Theorem 2. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, we evaluate Equation (22), and obtain:

$$CLFOWA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)=\left({\Xi }_{u\left(1-{\displaystyle \prod }_{j=1}^u{\left(1-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_{o\left(j\right)}}^R}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)+i{\Xi }_{u\left(1-{\displaystyle \prod }_{j=1}^u{\left(1-\frac{{\mathfrak{k}}_{{\mathfrak{m}}_{o\left(j\right)}}^I}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)\right)$$

(23)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$.

Proof. 

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

Definition 10. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, the CLFHA operator is introduced by:

$$CLFHA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\mathsf{\Gamma }}_1{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(1\right)}}^{\ast }\oplus {\mathsf{\Gamma }}_2{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(2\right)}}^{\ast }\oplus {\mathsf{\Gamma }}_3{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(3\right)}}^{\ast }\oplus \dots \oplus {\mathsf{\Gamma }}_u{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(u\right)}}^{\ast }={\oplus }_{j=1}^u\left({\mathsf{\Gamma }}_j{\hat{\mathcal{Q}}}_{{𝓈}_{o\left(j\right)}}^{\ast }\right)$$

(24)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$, where $o\left(j\right)\le o\left(j-1\right)$and ${\hat{\mathcal{Q}}}_{{𝓈}_{o\left(j\right)}}^{\ast }=n{\mathsf{\Gamma }}_j^{\ast }{\hat{\mathcal{Q}}}_{{𝓈}_j}$where ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j{}^{\ast }=1$. The information in Equation (24) is superior to the information computed based on FS, CFS, and LTSs. This is because when we remove the linguistic terms, the information in Equation (24) is changed for CFSs, and similarly when we remove the linguistic term sets and imaginary parts, the information in Equation (24) is changed for FSs. Finally, when we remove only the imaginary part, the information in Equation (24) is changed for linguistic term sets, which are the particular cases of the introduced information.

Theorem 3. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then we evaluate Equation (24), and we obtain:

$$CLFHA\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)=\left({\Xi }_{u\left(1-{\displaystyle \prod }_{j=1}^u{\left(1-\frac{{\mathfrak{k}}^{\ast }{}_{{\mathfrak{m}}_{o\left(j\right)}}^R}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)+i{\Xi }_{u\left(1-{\displaystyle \prod }_{j=1}^u{\left(1-\frac{{\mathfrak{k}}^{\ast }{}_{{\mathfrak{m}}_{o\left(j\right)}}^I}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)\right)$$

(25)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$.

Proof. 

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

Definition 11. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, the CLFWG operator is introduced by:

$$CLFWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\hat{\mathcal{Q}}}_{{𝓈}_1}^{{\mathsf{\Gamma }}_1}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_2}^{{\mathsf{\Gamma }}_2}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_3}^{{\mathsf{\Gamma }}_3}\otimes \dots \otimes {\hat{\mathcal{Q}}}_{{𝓈}_u}^{{\mathsf{\Gamma }}_u}={\otimes }_{j=1}^u\left({\hat{\mathcal{Q}}}_{{𝓈}_j}^{{\mathsf{\Gamma }}_j}\right)$$

(26)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$. The information in Equation (26) is superior to the information computed based on FS, CFS, and LTSs. This is because when we remove the linguistic terms, the information in Equation (26) is changed for CFSs, and similarly when we remove the linguistic term sets and imaginary parts, the information in Equation (26) is changed for FSs. Finally, when we remove only the imaginary part, the information in Equation (26) is changed for linguistic term sets, which are the particular cases of the introduced information.

Theorem 4. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$ continuous LTS. Then, we evaluate Equation (26), and we obtain:

$$CLFWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)=\left({\Xi }_{u\left({\displaystyle \prod }_{j=1}^u{\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)+i{\Xi }_{u\left({\displaystyle \prod }_{j=1}^u{\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)\right)$$

(27)

where ${\mathsf{\Gamma }}_j$ represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$.

Proof. 

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

Property 4 (Idempotency). 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. When ${\hat{\mathcal{Q}}}_{{𝓈}_j}={\hat{\mathcal{Q}}}_{𝓈}=\left({\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^I}\left(𝓏\right)\right)$, then:

$$CLFWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\hat{\mathcal{Q}}}_{𝓈}$$

(28)

Property 5 (Monotonicity). 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. When ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)\le {\hat{\mathcal{Q}}}_{{𝓈}_j}^{\ast }=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}^{\ast }\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}^{\ast }\left(𝓏\right)\right)$, then:

$$CLFWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)\le CLFWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1}^{\ast },{\hat{\mathcal{Q}}}_{{𝓈}_2}^{\ast },\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}^{\ast }\right)$$

(29)

Property 6 (Boundedness). 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. When ${\hat{\mathcal{Q}}}_{{𝓈}_j}^{-}=\left(\underset{j}{\min }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i\left(\underset{j}{\min }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)\right)\right)$and ${\hat{\mathcal{Q}}}_{{𝓈}_j}^{+}=\left(\underset{j}{\max }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i\left(\underset{j}{\max }{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)\right)\right)$, then:

$${\hat{\mathcal{Q}}}_{{𝓈}_j}^{-}\le CLFWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)\le {\hat{\mathcal{Q}}}_{{𝓈}_j}^{+}$$

(30)

Definition 12. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, the CLFOWG operator is introduced by:

$$CLFOWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\hat{\mathcal{Q}}}_{{𝓈}_{o\left(1\right)}}^{{\mathsf{\Gamma }}_1}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_{o\left(2\right)}}^{{\mathsf{\Gamma }}_2}\otimes {\hat{\mathcal{Q}}}_{{𝓈}_{o\left(3\right)}}^{{\mathsf{\Gamma }}_3}\otimes \dots \otimes {\hat{\mathcal{Q}}}_{{𝓈}_{o\left(u\right)}}^{{\mathsf{\Gamma }}_u}={\otimes }_{j=1}^u\left({\hat{\mathcal{Q}}}_{{𝓈}_{o\left(j\right)}}^{{\mathsf{\Gamma }}_j}\right)$$

(31)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$, where $o\left(j\right)\le o\left(j-1\right)$. The information in Equation (31) is superior to the information computed based on FS, CFS, and LTSs. This is because when we remove the linguistic terms, the information in Equation (31) is changed for CFSs, and similarly when we remove the linguistic term sets and imaginary parts, the information in Equation (31) is changed for FSs. Finally, when we remove only the imaginary part, the information in Equation (31) is changed for linguistic term sets, which are the particular cases of the introduced information.

Theorem 5. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, we evaluate Equation (31), and we obtain:

$$CLFOWG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)=\left({\Xi }_{u\left({\displaystyle \prod }_{j=1}^u{\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_{o\left(j\right)}}^R}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)+i{\Xi }_{u\left({\displaystyle \prod }_{j=1}^u{\left(\frac{{\mathfrak{k}}_{{\mathfrak{m}}_{o\left(j\right)}}^I}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)\right)$$

(32)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$.

Proof. 

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

Definition 13. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, the CLFHG operator is introduced by:

$$CLFHG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)={\widehat{{\mathcal{Q}}^{\ast }}}_{{𝓈}_{o\left(1\right)}}^{{\mathsf{\Gamma }}_1}\otimes {\widehat{{\mathcal{Q}}^{\ast }}}_{{𝓈}_{o\left(2\right)}}^{{\mathsf{\Gamma }}_2}\otimes {\widehat{{\mathcal{Q}}^{\ast }}}_{{𝓈}_{o\left(3\right)}}^{{\mathsf{\Gamma }}_3}\otimes \dots \otimes {\widehat{{\mathcal{Q}}^{\ast }}}_{{𝓈}_{o\left(u\right)}}^{{\mathsf{\Gamma }}_u}={\otimes }_{j=1}^u\left({\widehat{{\mathcal{Q}}^{\ast }}}_{{𝓈}_{o\left(j\right)}}^{{\mathsf{\Gamma }}_j}\right)$$

(33)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$, where $o\left(j\right)\le o\left(j-1\right)$and ${\hat{\mathcal{Q}}}_{{𝓈}_{o\left(j\right)}}^{\ast }=n{\mathsf{\Gamma }}_j^{\ast }{\hat{\mathcal{Q}}}_{{𝓈}_j}$ where ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j{}^{\ast }=1$. The information in Equation (33) is superior to the information computed based on FS, CFS, and LTSs. This is because when we remove the linguistic terms, the information in Equation (33) is changed for CFSs, and similarly when we remove the linguistic term sets and imaginary parts, the information in Equation (33) is changed for FSs. Finally, when we remove only the imaginary part, then the information in Equation (33) is changed for linguistic term sets, which are the particular cases of the introduced information.

Theorem 6. 

Let ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, represent the family of CLF numbers and $\hat{\Xi }=\left\{{\Xi }_j:{\Xi }_0\le {\Xi }_j\le {\Xi }_u,j\in \left[0,u\right]\right\}$continuous LTS. Then, we evaluate Equation (33), and we obtain:

$$CLFHG\left({\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_u}\right)=\left({\Xi }_{u\left({\displaystyle \prod }_{j=1}^u{\left(\frac{{\mathfrak{k}}^{\ast }{}_{{\mathfrak{m}}_{o\left(j\right)}}^R}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)+i{\Xi }_{u\left({\displaystyle \prod }_{j=1}^u{\left(\frac{{\mathfrak{k}}^{\ast }{}_{{\mathfrak{m}}_{o\left(j\right)}}^I}{u}\right)}^{{\mathsf{\Gamma }}_j}\right)}\left(𝓏\right)\right)$$

(34)

where ${\mathsf{\Gamma }}_j$represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$.

Proof. 

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

5. Engineering Apps That Are Essential for All Engineers Using CLF Aggregation Operators

In this section, based on the presented operators, a technique for evaluating the MADM problems in the consideration of CLF sets is derived. The superiority of the derived technique is illustrated via a practical example, a set of experiments, and significant and qualitative comparisons. The illustration results indicate that the derived technique has the ability to be feasible and superior in evaluating CLF information. Further, it can be used for determining the interrelationships of attributes and the criteria of experts. Moreover, it is valuable and capable of evaluating the MADM problems using CLF numbers. For this, we considered the family of alternative ${\hat{\mathcal{Q}}}_{{𝓈}_1},{\hat{\mathcal{Q}}}_{{𝓈}_2},\dots ,{\hat{\mathcal{Q}}}_{{𝓈}_m}$ and their attributes ${\hat{A}}_{{𝓈}_1},{\hat{A}}_{{𝓈}_2},\dots ,{\hat{A}}_{{𝓈}_n}$, where ${\mathsf{\Gamma }}_j$ represents the weight vector with an important condition ${\displaystyle \sum }_{j=1}^u{\mathsf{\Gamma }}_j=1,{\mathsf{\Gamma }}_j\in \left[0,1\right]$. Further, using the CLF information we computed a matrix $D={\left[r_{ij}\right]}_{m\times n}$. The main idea of the CLF number is represented by: ${\hat{\mathcal{Q}}}_{{𝓈}_j}=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right)+i{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right)=\left({\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{{\mathfrak{m}}_j}^I}\left(𝓏\right)\right),j=1,2,\dots ,u$, where $\left({\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^R}\left(𝓏\right),{\Xi }_{{\mathfrak{k}}_{\mathfrak{m}}^I}\left(𝓏\right)\right)$ represents the complex linguistic fuzzy number with $0\le {\mathfrak{k}}_{\mathfrak{m}}^R\left(𝓏\right)\le 1,0\le {\mathfrak{k}}_{\mathfrak{m}}^I\left(𝓏\right)\le 1$. To resolve the above issue, we use the following procedure.

Stage 1: The main theme of this stage is to collect CLF information and write it in a closed bracket.

Stage 2: Using the CLFWA and CLFWG operators, we aggregate the collection of information into a singleton set.

Stage 3: Using the score value, we illustrate the score value of each aggregated value.

Stage 4: We rank all the decisions obtained from the score values and demonstrate the best.

To assess the practicality and feasibility of the above information, we use some practical information and try to evaluate it using the diagnosed operators to enhance the quality of the evaluated theory.

5.1. Illustrated Example

Various people have developed different types of apps to evaluate complicated and awkward situations. For many years, engineers mostly used apps only for fun; however, as these have developed and gained academic attention, certain people have begun to use them for evaluating real-world problems. Some of the main beneficial engineering apps for Android and IOS include:

${\hat{\mathcal{Q}}}_{{𝓈}_1}$: AutoCAD 360;

${\hat{\mathcal{Q}}}_{{𝓈}_2}:$ Snip by MathPix;

${\hat{\mathcal{Q}}}_{{𝓈}_3}$: Finger CAD;

${\hat{\mathcal{Q}}}_{{𝓈}_4}$: Graphing calculator X84;

${\hat{\mathcal{Q}}}_{{𝓈}_5}$: Droid Tesla.

To determine the best of these under the consideration of classical information, certain applications have been developed by various scholars. For the above information, we use the weight vector 0.3, 0.4, 0.1, and 0.2. To resolve the above issue, we use the following procedure.

Stage 1: The main theme of this stage is to collect CLF information and write it in a closed bracket, see

Table 1. Complex linguistic fuzzy information matrix.

	${\hat{\mathit{A}}}_{{\mathit{𝓈}}_{\mathbf{1}}}$	${\hat{\mathit{A}}}_{{\mathit{𝓈}}_{\mathbf{1}}}$	${\hat{\mathit{A}}}_{{\mathit{𝓈}}_{\mathbf{1}}}$	${\hat{\mathit{A}}}_{{\mathit{𝓈}}_{\mathbf{1}}}$
${\widehat{\mathbf{𝓠}}}_{{\mathit{𝓈}}_{\mathbf{1}}}$	$\left({\Xi }_1,{\Xi }_2\right)$	$\left({\Xi }_3,{\Xi }_3\right)$	$\left({\Xi }_1,{\Xi }_3\right)$	$\left({\Xi }_4,{\Xi }_3\right)$
${\widehat{\mathbf{𝓠}}}_{{\mathit{𝓈}}_{\mathbf{2}}}$	$\left({\Xi }_3,{\Xi }_2\right)$	$\left({\Xi }_1,{\Xi }_4\right)$	$\left({\Xi }_2,{\Xi }_2\right)$	$\left({\Xi }_3,{\Xi }_2\right)$
${\widehat{\mathbf{𝓠}}}_{{\mathit{𝓈}}_{\mathbf{3}}}$	$\left({\Xi }_5,{\Xi }_2\right)$	$\left({\Xi }_2,{\Xi }_2\right)$	$\left({\Xi }_3,{\Xi }_4\right)$	$\left({\Xi }_2,{\Xi }_1\right)$
${\widehat{\mathbf{𝓠}}}_{{\mathit{𝓈}}_{\mathbf{4}}}$	$\left({\Xi }_3,{\Xi }_1\right)$	$\left({\Xi }_5,{\Xi }_1\right)$	$\left({\Xi }_4,{\Xi }_1\right)$	$\left({\Xi }_2,{\Xi }_3\right)$
${\widehat{\mathbf{𝓠}}}_{{\mathit{𝓈}}_{\mathbf{5}}}$	$\left({\Xi }_4,{\Xi }_2\right)$	$\left({\Xi }_3,{\Xi }_2\right)$	$\left({\Xi }_2,{\Xi }_3\right)$	$\left({\Xi }_1,{\Xi }_2\right)$

.

Stage 2: Using the CLFWA and CLFWG operators, we aggregate the collection of information into a singleton set; see

Table 2. Aggregated information matrix.

	$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{A}$	$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{G}$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_1}$	$\left({\Xi }_{2.6065},{\Xi }_{2.7295}\right)$	$\left({\Xi }_{2.0476},{\Xi }_{2.6564}\right)$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_2}$	$\left({\Xi }_{2.2124},{\Xi }_{2.9685}\right)$	$\left({\Xi }_{1.8563},{\Xi }_{2.6390}\right)$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_3}$	$\left({\Xi }_{3.4358},{\Xi }_{2.0975}\right)$	$\left({\Xi }_{2.7417},{\Xi }_{1.8660}\right)$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_4}$	$\left({\Xi }_{4.0336},{\Xi }_{1.4855}\right)$	$\left({\Xi }_{3.4925},{\Xi }_{1.2457}\right)$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_5}$	$\left({\Xi }_{2.9719},{\Xi }_{2.1134}\right)$	$\left({\Xi }_{2.5209},{\Xi }_{2.0827}\right)$

.

Stage 3: We illustrate the score value of each aggregated value; see

Table 3. Score value matrix.

	$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{A}$	$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{G}$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_1}$	$0.1659$	$0.3239$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_2}$	$0.2047$	$0.3761$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_3}$	$0.1166$	$0.3480$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_4}$	$0.1201$	$0.3154$
${\widehat{\mathbf{𝓠}}}_{{𝓈}_5}$	$0.2286$	$0.3490$

.

Stage 4: We rank all the decisions obtained from the score values and demonstrate the best; see

Table 4. Ranking information matrix.

Methods	Ranking Values
$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{A} \mathit{O}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}$	${\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_4}>{\hat{\mathcal{Q}}}_{{𝓈}_3}$
$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{G} \mathit{O}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}$	${\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_3}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_4}$

.

It can be noticed that the aggregation operators, called CLFWA operators, yields the optimal as ${\hat{\mathcal{Q}}}_{{𝓈}_5}$ and the CLFWG operator yields the best decision as ${\hat{\mathcal{Q}}}_{{𝓈}_2}$. If we consider the imaginary values will be zero, then, using the information in Table 1 (without imaginary parts), the ranking results are shown in

Table 5. Ranking information, using the data in Table 1, without imaginary parts.

Methods	Score Values	Ranking Values
$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{A} \mathit{O}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}$	0.8483, 0.9468, 0.6410, 0.4915, 0.7570	${\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_3}>{\hat{\mathcal{Q}}}_{{𝓈}_1}$
$\mathit{C}\mathit{L}\mathit{F}\mathit{W}\mathit{G} \mathit{O}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}$	0.9880, 1.0359, 0.8145, 0.6268, 0.8697	${\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_3}>{\hat{\mathcal{Q}}}_{{𝓈}_4}$

.

It can be noticed that the aggregation operators, called CLFWA operators, yield the optimal as ${\hat{\mathcal{Q}}}_{{𝓈}_2}$ and the CLFWG operator yields the best decision as ${\hat{\mathcal{Q}}}_{{𝓈}_2}$; this means that both operators are given the same ranking results. Further, we undertake the comparative analysis of the diagnosed operators with some existing operators with the help of the above examples.

5.2. Comparative Analysis

In the theory section, with the help of the information given in Table 1, we compare the diagnosed operators computed based on novel CLF information with some prevailing operators to show the reliability and feasibility of the evaluated approaches. To achieve this, it was necessary to select existing operators computed based on linguistic sets or complex fuzzy sets. For this, we considered the idea aggregation operators for linguistic variables evaluated by Xu [25] in 2008, arithmetic aggregation operators for CF information diagnosed by Bi et al. [26], and geometric aggregation operators for CF information introduced by Bi et al. [27], and compared these with the diagnosed operators. The main comparison is available in

Table 6. Comparative analysis for information in Table 1 (with imaginary parts).

Methods	Score Values	Ranking Values
Xu [25]	$Not evaluated feasibly$	$Not evaluated feasibly$
Bi et al. [26]	$Not evaluated feasibly$	$Not evaluated feasibly$
Bi et al. [27]	$Not evaluated feasibly$	$Not evaluated feasibly$
CLFWA operator	$0.1659,0.2047,0.1166,0.1201,0.2286$	${\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_4}>{\hat{\mathcal{Q}}}_{{𝓈}_3}$
CLFWG operator	$0.3239,0.3761,0.3480,0.3154,0.3490$	${\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_3}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_4}$

, based on the information in Table 1.

Using both CLFWA and CLFWG operators, two different types of results were derived. The decision can then be left to the experts as to which kind of source to use. Based on the information in Table 6, it is clear that the prevailing operators based on the linguistic set [25], and complex fuzzy information [26,27] cannot effectively evaluate the suggested information. This is because, in Table 1, we choose the CLF type of information, and the prevailing operators are the special cases of the diagnosed operators. However, it can be noticed that the aggregation operators, called the CLFWA operator, yield the optimal as ${\hat{\mathcal{Q}}}_{{𝓈}_5}$ and the CLFWG operator yields the best decision as ${\hat{\mathcal{Q}}}_{{𝓈}_2}$. Similarly, the main comparison is available in

Table 7. Comparative analysis for information in Table 1 (without imaginary parts).

Methods	Score Values	Ranking Values
Xu [25]	$3.3934,3.7875,2.5641,1.9663,3.0280$	${\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_3}>{\hat{\mathcal{Q}}}_{{𝓈}_1}$
Bi et al. [26]	$Not evaluated feasibly$	$Not evaluated feasibly$
Bi et al. [27]	$Not evaluated feasibly$	$Not evaluated feasibly$
CLFWA operator	$0.8483,0.9468,0.6410,0.4915,0.7570$	${\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_3}>{\hat{\mathcal{Q}}}_{{𝓈}_1}$
CLFWG operator	$0.9880,1.0359,0.8145,0.6268,0.8697$	${\hat{\mathcal{Q}}}_{{𝓈}_2}>{\hat{\mathcal{Q}}}_{{𝓈}_1}>{\hat{\mathcal{Q}}}_{{𝓈}_5}>{\hat{\mathcal{Q}}}_{{𝓈}_3}>{\hat{\mathcal{Q}}}_{{𝓈}_4}$

, for the information in Table 1 (without imaginary parts).

From the information in Table 7, it is clear that the prevailing operators based on complex fuzzy information [26,27] cannot evaluate the suggested information, as in Table 1. For the section without imaginary parts, linguistic types of information were selected, and the prevailing operators as computed based on CF information are thus different from the diagnosed operators. However, it can be noticed that the aggregation operators, called CLFWA operators, yield the optimal as ${\hat{\mathcal{Q}}}_{{𝓈}_2}$, the CLFWG operator yields the best decision as ${\hat{\mathcal{Q}}}_{{𝓈}_2}$, and the information of Xu [25] also yields the best decision as ${\hat{\mathcal{Q}}}_{{𝓈}_2}$.

Therefore, the diagnosed operators computed based on CLF information are massively powerful and more feasible than the existing operators computed based on CF sets and linguistic sets.

6. Conclusions

The theory of complex fuzzy and linguistic information is highly valuable and superior for managing awkward and complicated information in real-life problems. Keeping the advantages of the existing information, the main impact of this analysis is the evaluation of the idea of CLF sets and their valuable laws. Furthermore, in the presence of CLF sets, we developed the CLFWA, CLFOWA, CLFHA, CLFWG, CLFOWG, and CLFHG operators, and derived their fundamental properties and some important results. Additionally, we enhanced the worth of the proposed analysis by evaluating the MADM technique based on the proposed operators. Finally, we confirmed the superiority of the proposed work with the help of a comparative analysis between the developed operators and some existing operators.

6.1. Limitation of the Proposed Approach

CLF information is a very valuable and dominant concept utilized in this manuscript; however, in some cases, the proposed idea has failed. For instance, when dealing with a voting system, we face two types of information: a vote cast in favor and a vote cast against a candidate. To manage this sort of problem, the theory of CLF sets has been neglected. For this, we will propose the idea of complex linguistic intuitionistic fuzzy sets, complex linguistic Pythagorean fuzzy sets, and complex linguistic q-rung orthopair fuzzy sets.

6.2. Future Directions

In upcoming research, to utilize some new ideas and operators, we will review the theory of complex intuitionistic fuzzy soft sets [28], complex spherical fuzzy sets [29], linear Diophantine fuzzy information [30,31], fuzzy N-soft sets [32], complex Pythagorean fuzzy N-soft sets [33], spherical fuzzy N-soft expert sets [34], and decision making [3,31,35,36,37], which are very important extensions in the field of fuzzy sets and their generalizations.