Redefined “Maclaurin Symmetric Mean Aggregation Operators Based on Cubic Pythagorean Linguistic Fuzzy Numbers”

In this study, we highlight the errors in Sections 2.3, 2.4, 3, and 4 in the article by Fahmi et al. (J Ambient Intell Human Comput (2020). https://doi.org/10.1007/s12652-020-02272-9) by counter definitions and theorems. We find that the definition of cubic Pythagorean fuzzy set (CPFS) (Definition 2.3.1) and operational laws (Definition 2.3.2) violates the rules to consider the membership and nonmembership functions, and then, we redefined the corrected definition and their operations for CPFS. Furthermore, we redefine the concept of cubic Pythagorean linguistic fuzzy set (CPLFS) and their basic operational laws. In addition, we find that Sections 3 and 4 (consist of a list of Maclaurin symmetric mean (MSM) and dual MSM aggregation operators) are invalid, and then, we redefined the list of updated MSM and dual MSM aggregation operators in correct way. Finally, we established the numerical application of the proposed improved algorithm using cubic Pythagorean linguistic fuzzy information to show the applicability and effectiveness of the proposed technique.


Introduction
Due to the increasing complexity of the system, it is difficult for the decision maker to select the best alternative/object from a set of attractive options in the real world. However, it is hard to summarize, but it is not incredible to achieve the best single objective. A large number of multicriteria decision-making problems exist in decision-making, where the criteria are found to be uncertain, ambiguous, imprecise, and vague. As a result, the crisp set appears to be ineffective in dealing with this uncertainty and imprecision in the data and can be easily dealt with by using fuzzy information. To deal with such uncertainty and ambiguity, Zadeh [1] presented the mathematical notion of fuzzy set (FS) which has been defined by using the membership function of the element. Various researchers have discovered the utility of the fuzzy set in a variety of fields, including decision-making, medical diagnosis, engineering, socioeconomic, and finance problems.
With the continuous process of human practice, decision-making problems have become more and more complicated, and many extended forms of fuzzy sets [1] have been proposed, such as the bipolar soft sets [2], the intuitionistic fuzzy (IF) sets [3], the interval-valued intuitionistic fuzzy sets [4], Pythagorean fuzzy (PyF) sets [5], picture fuzzy (PF) sets [6,7], and spherical fuzzy (SF) sets [8][9][10]. Many decision-making techniques under Pythagorean fuzzy information are established, for example, Wang et al. [11,12] presented the novel decision-making techniques under Pythagorean fuzzy interactive Hamacher power and interaction power Bonferroni mean aggregation operators are proposed and discussed their applications in multiple attribute decision-making problems. Khan et al. [13] presented the decision-making method based on probabilistic hesitant fuzzy rough information. In [14], Ashraf et al. worked on sine trigonometric aggregation operator for Pythagorean fuzzy numbers; in [15], Batool Khan et al. used the Dombi t-norms and t-conorms to Pythagorean fuzzy numbers and defined Pythagorean fuzzy Dombi aggregation operators [16]. e cubic Pythagorean fuzzy set (CPFS) is a well reputed structure of fuzzy sets, proposed by Abbas et al. [17] in 2019 to tackle the uncertainty in decision-making problems. Talukdar and Dutta [18] presented the distance measures under CPFS information. Fahmi et al. [19] proposed the decision-making technique using cubic Pythagorean linguistic fuzzy sets. e main objective of this note is to highlight the error in Sections 2.3, 2.4, 3, and 4 in the study by Fahmi et al. [19] by counterdefinitions and countertheorems.

Preliminaries
We initiate with rudimentary concept of fuzzy set, cubic set, intuitionistic fuzzy set, Pythagorean fuzzy set, and cubic Pythagorean fuzzy set that are required for the rest of this paper.
Definition 2 (see [20]). An interval-valued fuzzy set (IVFS) F in a universe set U is an object having the form where μ(υ) ∈ [0, 1] is represented by the positive membership grade.
Definition 7 (see [17]). A cubic Pythagorean fuzzy set F s in a universe set U is an object having the form Maclaurin symmetric mean (MSM) is established by Maclaurin [23] and defined as follows.
Dual Maclaurin symmetric mean (DMSM) is established by Wei et al. [24] and defined as follows.

Counter Section 2.3 of [19]
is section recalls the discussion of cubic Pythagorean fuzzy numbers (CPFN) and their basic operations proposed by Fahmi et al. [19]. Definition 2.3.1 in [19] proposed the definition of CPFS which is described as follows.
Definition 10 (see [19]). A CPFS F in arbitrary set U ≠ ϕ has the form  [19] proposed the basic operational laws for CPFNs which are described as follows.
, μ 2 (υ)} ∈ CPFS(U) with ϖ > 0. en, the operational rules are described as [19] defined Definition 2.3.1 for CPFS and Definition 3 for cubic set which are the same. So, Fahmi et al. [19] presented the invalid definition of cubic Pythagorean fuzzy sets and also presented invalid operational laws for CPFSs. Now, we presented the valid definition and operational laws for CPFNs are as follows.
en, the operational rules are described as

Counter Section 2.4 of [19]
is section recalls the discussion of cubic Pythagorean linguistic fuzzy numbers (CPLFN) and their basic operations proposed by Fahmi et al. [19]. Definition 2.4.1 in [19] proposed the concept of CPLFS which is described as follows.
Definition 15 (see [19]). Let F cp 1 en, the operational rules are described as [19] presented Definition 2.4.1 for CPLFS and Definition 3 for cubic set which are the same. So, Fahmi et al. [19] presented the invalid definition of cubic Pythagorean linguistic fuzzy sets and also presented invalid operational laws for CPLFSs. Now, we presented the valid definition and operational laws for CPLFNs as follows.

Counter Section 3 of [19]
is section recalls the discussion of linguistic aggregation operators (AO) for CPLFN and their basic properties proposed by Fahmi et al. [19].

Weighted
Averaging. Definition 3.1.1 and eorem 3.1.2 in [19] proposed the weighted averaging operator using defined operational rules described as follows.
Definition 19 (see [19] . en, the CPLFWA operator is described as follows: where θ(i)(i ∈ N) represents any permutation and S n represents the total numbers of elements. Theorem 1 (see [19]).
en, using defined operational laws, CPLFWA operator is obtained as Definition 3.2.1 and eorem 3.2.2 in [19] proposed the generalized weighted averaging operator using defined operational rules described as follows.
Definition 20 (see [19] . en, the CGPLFWA operator is described as follows: where θ(i)(i ∈ N) represents any permutation and S n represents the total numbers of elements.
. en, using defined operational laws, the CGPLFWA operator is obtained as [19] proposed the weighted geometric operator using defined operational rules described as follows.

Weighted Geometric. Definition 3.3.1 and eorem 3.3.2 in
Definition 21 (see [19]). Let . en, the CPLFWG operator is described as follows: where θ(i)(i ∈ N) represents any permutation and S n represents the total numbers of elements.
en, using defined operational laws, the CPLFWG operator is obtained as

Updated Linguistic Cubic Pythagorean Fuzzy AO
In this section, utilizing valid Definition 16 of LCPFS and operational laws (Definition 17), we establish the updated operational laws to aggregate the uncertain data in the form of linguistic cubic Pythagorean fuzzy environment.

Updated Weighted Averaging AO
en, the CPLFWA operator is described as follows: where the weights (ρ 1 ,

en, using defined operational laws 17, CPLFWA operator is obtained as
where the weights (ρ 1 ,

Updated Weighted Geometric AO
. en, the CPLFWG operator is described as follows: 6 Mathematical Problems in Engineering en, using defined operational laws 17, the CPLFWG operator is obtained as where the weights (ρ 1 ,

Countersections 3.4 and 3.5 of [19]
is section recalls the discussion of linguistic MSM aggregation operators (AO) for CPLFN and their basic properties proposed by Fahmi et al. [19]. Definition 3.4.1 and eorem 3.4.2 in [19] proposed the MSM operator using defined operational rules (Definition 15) described as follows.
Definition 24 (see [19] vector having ρ ♮ ≥ 0 and n ♮�1 ρ ♮ � 1. en, the CPLFMSM operator is described as follows: where θ(i)(i ∈ N) represents any permutation and S n represents the total numbers of elements.
en, using defined operational laws, the CPLFMSM operator is obtained as Definition 3.5.1 and eorem 3.5.2 in [19] proposed the weighted MSM operator using defined operational rules (Definition 15) described as follows.
en, using defined operational laws, CPLFWMSM operator is obtained as

Updated Linguistic Cubic Pythagorean Fuzzy MSM AO
In this section, utilizing valid Definition 16 of LCPFS and operational laws (Definition 17), we establish the updated linguistic Maclaurin symmetric mean AO to aggregate the uncertain data in the form of linguistic cubic Pythagorean fuzzy environment.
en, the CPLFMSM operator is described as follows: where C k n is the binomial coefficient and (i 1 , i 2 , . . . , i k ) traversal of all the k-tuple combination of (1, 2, . . . , n).
en, using defined operational laws 17, the CPLFMSM operator is obtained as

Mathematical Problems in Engineering
Updated weighted MSM AO is defined as follows.
en, using defined operational laws, the CPLFDWMSMA operator is obtained as  [19] proposed the ordered weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.
en, using defined operational laws, the CPLFDOWMSMA operator is obtained as (37) Definition 4.3.1 and eorem 4.3.2 in [19] proposed the hybrid weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.
en, using defined operational laws, the CPLFDHWMSMA operator is obtained as  [19] proposed the weighted DMSM geometric operator using defined operational rules (Definition 15) described as follows.
Definition 31 (see [19]). Let en, the CPLFDWMSMG operator is described as follows: where θ(i)(i ∈ N) represents any permutation and S n represents total numbers of elements.
en, using defined operational laws, the CPLFDWMSMG operator is obtained as Definition 4.5.1 and eorem 4.5.2 in [19] proposed the ordered weighted DMSM geometric operator using defined operational rules (Definition 15) is described as Definition 32 (see [19] en, the CPLFDOWMSMG operator is described as follows: where θ(i)(i ∈ N) represents any permutation and S n represents the total numbers of elements. Theorem 14 (see [19]).
en, using defined operational laws, the CPLFDOWMSMG operator is obtained as Definition 4.6.1 and eorem 4.6.2 in [19] proposed the hybrid-weighted DMSM geometric operator using defined operational rules (Definition 15) described as follows.
en, using defined operational laws, the CPLFDHWMSMG operator is obtained as

Updated Dual MSM Operators
In this section, valid Definition 16 of LCPFS and operational laws (Definition 17) are utilized to establish the updated linguistic dual Maclaurin symmetric mean AO to aggregate the uncertain data in the form of linguistic cubic Pythagorean fuzzy environment.

Decision-Making Model Based on Updated MSM Operators
In this section, we propose a framework for solving multiattribute decision-making problems (DMPs) under cubic where Step 1: construct the CPLF decision matrix based on the experts' evaluations: where J represents the number of expert.
Step 2: exploit the established aggregation operators to achieve the CPLFNs F t (t � 1, 2, . . . , ♮) for the alternatives ℷ k , that is, the established operators to obtain the collective overall preference values of F t (t � 1, 2, . . . , ♮) for the alternatives ℷ k , where p � (p 1 , p 2 , . . . , p h ) is the weight vector of the attributes.
Step 3: after that, we compute the scores of all the overall values F t (t � 1, 2, . . . , ♮) for the alternatives ℷ k .

Numerical Application
e company of intranet is usually attacked by malicious intrusions. To enhance the security of the intranet, the company plans to purchase the firewall production and put it between the intranet and extranet for blocking illegal access. Basically, there are four types of firewall productions given to be considered, whose detailed is as follows: ∁ � {∁ 1 , ∁ 2 , ∁ 3 , ∁ 4 }. If the firewall production is chosen, the company pays attention to the factors, which are detailed as follows: G 1 ⟶ the promotion, G 2 ⟶ configuration simplicity, G 3 ⟶ security level, and G 4 ⟶ maintenance server level, whose weight vector is (0.2, 0.1, 0.3, 0.4) T . To examine the firewall production with respect to their factor, we consider the cubic Pythagorean linguistic fuzzy matrix, the decision matrix is given in the form of Table 1: Step 1: the evaluation result of the expert is listed in Table 1: Step 2: based on the proposed MSM operators, the collective CPLF information of each alternative is obtained as follows in Tables 2-4: Hence, we obtained similarly.  Here, we use the CPLFWDMSM operator to aggregate the expert evaluation in the form of CPLFSs. Without loss of generality, we take k � 2; then, using eorem 16, we obtain Hence, we obtained similarly in Table 4 utilizing eorem 16, Step 3: compute the score value of the each collective CPLF information of each alternative as in Table 5.

Mathematical Problems in Engineering
Step 4: select the optimal alternative according the maximum score value as in Table 6: From this above computational process, we can conclude that the alternative ∁ 1 is the best among the others, and hence, it is highly recommendable to select for the required task/ plan.

Conclusion
In this note, we discussed that Sections 2.3 and 2.4 in [16] incorrectively define the concept of cubic Pythagorean fuzzy set and their basic operational laws by violating to consider membership and nonmembership function, constructing counterdefinition and countertheorem, and then, we proposed the modified versions of operational laws to tackle the uncertain information in the form of CPFS in decision-making problems. Furthermore, we redefine the concept of cubic Pythagorean linguistic fuzzy set (CPLFS) and their basic operational laws and aim to establish the valid aggregation operators under CPLFS information. In addition, we find that Sections 3 and 4 that consist of list of Maclaurin symmetric mean (MSM) and dual MSM aggregation operators are invalid due to incorrect concept of cubic Pythagorean linguistic fuzzy set, and then, we redefined the list of updated MSM and dual MSM aggregation operators in a correct way. Finally, we proposed the improved algorithm-based numerical application to show the effectiveness and applicability of the valid aggregation operators under cubic Pythagorean linguistic fuzzy settings.

Data Availability
No data were used to support this study.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest
e authors declare that they have no conflicts of interest.