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

Authors: Muhammad Naeem; Shahzaib Ashraf; Saleem Abdullah; F. M. AL‐Harbi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. 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 [8–10]. 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 et al. developed new models for decision making under Pythagorean hesitant fuzzy numbers. Khan et al. used the Dombi t-norms and t-conorms to Pythagorean fuzzy numbers and defined Pythagorean fuzzy Dombi aggregation operators [16].The 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. The 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.


## 2. 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 1.
(see [1]). A fuzzy set (FS) ℱ in arbitrary set U≠ϕ has the form(1)ℱ=υ,μυ|υ∈U,where μυ∈0,1 is represented by the positive membership grade.Definition 2.
(see [20]). An interval-valued fuzzy set (IVFS) ℱ in a universe set U is an object having the form(2)ℱ=υ,μ−υ,μ+υ|υ∈U,where μυ∈0,1 is represented by the positive membership grade.Definition 3.
(see [21]). A cubic set ℱ in a universe set U is an object having the form(3)ℱ=υ,μ−υ,μ+υ,μυ|υ∈U,where μ−υ,μ+υ⊆0,1∈IVFS and μυ∈0,1∈FS.Definition 4.
(see [3]). An intuitionistic FS ℱs in a universe set U is an object having the form(4)ℱs=υ,μυ,∂υ|υ∈U,where μυ∈0,1 are positive and ∂υ∈0,1 are negative membership grades, respectively. In addition, 0≤μυ+∂υ≤1, ∀υ∈U.Definition 5.
(see [5]). A Pythagorean FS ℱs in a universe set U is an object having the form(5)ℱs=υ,μυ,∂υ|υ∈U,where μυ∈0,1 are positive and ∂υ∈0,1 are negative membership grades, respectively. In addition, 0≤μ2υ+∂2υ≤1, ∀υ∈U.Definition 6.
(see [22]). An interval-valued Pythagorean FS (IVPFS) ℱs in a universe set U is an object having the form(6)ℱs=υ,μ−υ,μ+υ,∂−υ,∂+υ|υ∈U,where μ−υ,μ+υ⊆0,1 are positive and ∂−υ,∂+υ⊆0,1 are negative membership grades, respectively. In addition, 0≤μ+υ2+∂+υ2≤1,∀υ∈U.Definition 7.
(see [17]). A cubic Pythagorean fuzzy set ℱs in a universe set U is an object having the form(7)ℱ=υ,μ−υ,μ+υ,∂−υ,∂+υ,μυ,∂υ|υ∈U,where μ−υ,μ+υ,∂−υ,∂+υ⊆0,1∈IVPFS and μυ,∂υ∈0,1∈PFS.
Maclaurin symmetric mean (MSM) is established by Maclaurin [23] and defined as follows.Definition 8.
(see [23]). Take any collection of nonnegative elements hjj∈ℕ and k=1,2,…,n. If(8)MSMkh1,h2,…,hn=∑1≤i1≤i2≤⋯≤ik≤n∏j=1khijℂnk1/k,then MSMk is said to be MSM operator, where ℂnk is the binomial coefficient and i1,i2,…,ik traversal all the k-tuple combination of 1,2,…,n.
Dual Maclaurin symmetric mean (DMSM) is established by Wei et al. [24] and defined as follows.Definition 9.
(see [24]). Take any collection of nonnegative elements hjj∈ℕ and k=1,2,…,n. If(9)DMSMkh1,h2,…,hn=1k∏1≤i1≤i2≤⋯≤ik≤n∑j=1khij1/ℂnk,then MDSMk is said to be DMSM operator, where ℂnk is the binomial coefficient and i1,i2,…,ik traversal of all the k-tuple combination of 1,2,…,n.


## 3. Counter Section 2.3 of [19]

This 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(10)ℱ=υ,μ−4υ,μ+υ,μυ|υ∈U,where μ−υ,μ+υ⊆0,1 and μυ∈0,1 represent the positive and negative grades of memberships.
For whole study, the list of cubic Pythagorean fuzzy sets are represented byCPFSU. For simplicity, we write μ−υ,μ+υ,μυ for any cubic Pythagorean fuzzy number.
Definition 2.3.2 in [19] proposed the basic operational laws for CPFNs which are described as follows.Definition 11.
(see [19]). Let ℱcp1=μ1−υ,μ1+υ,μ1υ and ℱcp2=μ2−υ,μ2+υ,μ2υ∈CPFSU with ϖ>0. Then, the operational rules are described as(1)
ℱcp1⊕ℱcp2=μ1−2+μ2−2−μ1−2⋅μ2−2,μ1+2+μ2+2−μ1+2⋅μ2+2,μ1⋅μ2(2)
ℱcp1⊗ℱcp2=μ1−⋅μ2−,μ1+⋅μ2+,μ12+μ22−μ12⋅μ22(3)
ℱcp1ϖ=μ1−ϖ,μ1+ϖ,1−1−μ12ϖ(4)
ϖ⋅ℱcp1=1−1−μ1−2ϖ,1−1−μ1+2ϖ,μ1ϖ
Fahmi et al. [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.Definition 12.
(see [17]). A CPFS ℱcp in arbitrary set U≠ϕ has the form(11)ℱ=υ,μ−υ,μ+υ,υ−υ,υ+υ,μυ,υυ|υ∈U,where μ−υ,μ+υ,υ−υ,υ+υ⊆0,1 and μυ,υυ∈0,1 represent the positive membership grade in the form of interval valued Pythagorean fuzzy set and negative membership grade in the form of Pythagorean fuzzy set.
The operational laws for CPFNs are defined as follows.Definition 13.
Letℱcp1=μ1−υ,μ1+υ,υ1−υ,υ1+υ,μ1υ,υ1υ and ℱcp2=μ2−υ,μ2+υ,υ2−υ,υ2+υ,μ2υ,υ2υ∈CPFSU with ϖ>0. Then, the operational rules are described as(1)
ℱcp1⊕ℱcp2=μ1−2+μ2−2−μ1−2⋅μ2−2,μ1+2+μ2+2−μ1+2⋅μ2+2,υ1−⋅υ2−,υ1+⋅υ2+,μ12+μ22−μ12⋅μ22,υ1υ⋅υ2υ(2)
ℱcp1⊗ℱcp2=μ1−⋅μ2−,μ1+⋅μ2+,υ1−2+υ2−2−υ1−2⋅υ2−2,υ1+2+υ2+2−υ1+2⋅υ2+2,μ1υ⋅μ2υ,υ12+υ22−υ12⋅υ22(3)
ℱcp1ϖ=μ1−ϖ,μ1+ϖ,1−1−υ1−2ϖ,1−1−υ1+2ϖ,μ1ϖ,1−1−υ12ϖ(4)
ϖ⋅ℱcp1=1−1−μ1−2ϖ,1−1−μ1+2ϖ,υ1−ϖ,υ1+ϖ,1−1−μ12ϖ,υ1ϖ


## 4. Counter Section 2.4 of [19]

This 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 14.
(see [19]). A CPLFS ℱ in arbitrary set U≠ϕ, with ℜ=ℓv0,ℓv1,ℓv2,…,ℓvt−1, is the linguistic term set, where t>0 is the odd cardinality, having the form(12)ℱ=ℓv,μ−υ,μ+υ,μυ|υ∈U,where ℓv∈ℜ and μ−υ,μ+υ⊆0,1 and μυ∈0,1 represent the positive and negative grades of memberships.
For the whole study, the list of cubic Pythagorean linguistic fuzzy sets are represented byCPLFSU. For simplicity, we write ℓv,μ−υ,μ+υ,μυ for any cubic Pythagorean linguistic fuzzy number.
Definition 2.4.2 in [19] proposed the basic operational laws for CPLFNs described as follows.Definition 15.
(see [19]). Let ℱcp1=ℓv1,μ1−υ,μ1+υ,μ1υ and ℱcp2=ℓv2,μ2−υ,μ2+υ,μ2υ∈CPLFSU with ϖ>0. Then, the operational rules are described as(1)
ℱcp1⊕ℱcp2=ℓv1+v2,μ1−2+μ2−2−μ1−2⋅μ2−2,μ1+2+μ2+2−μ1+2⋅μ2+2,μ1⋅μ2(2)
ℱcp1⊗ℱcp2=ℓv1×v2,μ1−⋅μ2−,μ1+⋅μ2+,μ12+μ22−μ12⋅μ22(3)
ℱcp1ϖ=ℓv1ϖ,μ1−ϖ,μ1+ϖ,1−1−μ12ϖ(4)
ϖ⋅ℱcp1=ℓv1×ϖ,1−1−μ1−2ϖ,1−1−μ1+2ϖ,μ1ϖ
Fahmi et al. [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.Definition 16.
A CLPFSℱcp in arbitrary set U≠ϕ, with ℜ=ℓv0,ℓv1,ℓv2,…,ℓvt−1, is the linguistic term set, where t>0 is the odd cardinality, having the form(13)ℱ=ℓv,μ−υ,μ+υ,υ−υ,υ+υ,μυ,υυ|υ∈U,where ℓv∈ℜ and μ−υ,μ+υ,υ−υ,υ+υ⊆0,1 and μυ,υυ∈0,1 represent the positive membership grade in the form of interval-valued Pythagorean fuzzy set and negative membership grade in the form of Pythagorean fuzzy set.
The operational laws for CPLFNs are defined as follows.Definition 17.
Letℱcp1=ℓv1,μ1−υ,μ1+υ,υ1−υ,υ1+υ,μ1υ,υ1υ and ℱcp2=ℓv2,μ2−υ,μ2+υ,υ2−υ,υ2+υ,μ2υ,υ2υ∈CPLFSU with ϖ>0. Then, the operational rules are described as(1)
ℱcp1⊕ℱcp2=ℓv1+v2,μ1−2+μ2−2−μ1−2⋅μ2−2,μ1+2+μ2+2−μ1+2⋅μ2+2,υ1−⋅υ2−,υ1+⋅υ2+,μ12+μ22−μ12⋅μ22,υ1υ⋅υ2υ(2)
ℱcp1⊗ℱcp2=ℓv1×v2,μ1−⋅μ2−,μ1+⋅μ2+,υ1−2+υ2−2−υ1−2⋅υ2−2,υ1+2+υ2+2−υ1+2⋅υ2+2,μ1υ⋅μ2υ,υ12+υ22−υ12⋅υ22(3)
ℱcp1ϖ=ℓv1ϖ,μ1−ϖ,μ1+ϖ,1−1−υ1−2ϖ,1−1−υ1+2ϖ,μ1ϖ,1−1−υ12ϖ(4)
ϖ⋅ℱcp1=ℓv1×ϖ,1−1−μ1−2ϖ,1−1−μ1+2ϖ,υ1−ϖ,υ1+ϖ,1−1−μ12ϖ,υ1ϖDefinition 18.
Let(14)ℱcpi=ℓvi,μi−,μi+,υi−,υi+,μi,υii∈ℕ∈CPLFSU⋅Sγℱcp1=μ1+2−υ1+2+μ12−υ12×v14,Sγℱcp2=μ2+2−υ2+2+μ22−υ22×v24,be the score values of CPLFNs. Also,(15)Ayℱcp1=μ1+2+υ1+2+μ12+υ12×v14,Ayℱcp2=μ2+2+υ2+2+μ22+υ22×v24,are the accuracy values of CPLFNs. If(a)
Sγℱcp1<Sγℱcp2⇒ℱcp1<ℱcp2(b)
Sγℱcp1=Sγℱcp2,Ayℱcp1<Ayℱcp2⇒ℱcp1<ℱcp2(c)
Sγℱcp1=Sγℱcp2,Ayℱcp1=Ayℱcp2⇒ℱcp1=ℱcp2


## 5. Counter Section 3 of [19]

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


### 5.1. Weighted Averaging

Definition 3.1.1 and Theorem 3.1.2 in [19] proposed the weighted averaging operator using defined operational rules described as follows.Definition 19.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, the CPLFWA operator is described as follows:(16)CPLFWAℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 1 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, CPLFWA operator is obtained as(17)CPLFWAℱcp1,ℱcp2,…,ℱcpn=ℓ∑i=1nvi/n,1−1−∏i=1nμi−21/n!,1−1−∏i=1nμi+21/n!,∏i=1nμi21/n!.Definition 3.2.1 and Theorem 3.2.2 in [19] proposed the generalized weighted averaging operator using defined operational rules described as follows.Definition 20 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, the CGPLFWA operator is described as follows:(18)CGPLFWAℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi1/λ,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 2 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CGPLFSU. Then, using defined operational laws, the CGPLFWA operator is obtained as(19)CGPLFWAℱcp1,ℱcp2,…,ℱcpn=ℓ∑i=1nvi/n1/λ,1−1−∏i=1nμi−2λ1/n!1/λ,1−1−∏i=1nμi+2λ1/n!1/λ,1−1−∏i=1n1−1−μi2λ1/n!1/λ.


### 5.2. Weighted Geometric

Definition 3.3.1 and Theorem 3.3.2 in [19] proposed the weighted geometric operator using defined operational rules described as follows.Definition 21.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, the CPLFWG operator is described as follows:(20)CPLFWGℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 3 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFWG operator is obtained as(21)CPLFWGℱcp1,ℱcp2,…,ℱcpn=ℓ∑i=1nvi1/n/n,1−1−∏i=1nμi−2λ1/n!1/λ,1−1−∏i=1nμi+2λ1/n!1/λ1−∏i=1n1−μi21/n!.


## 5.1. Weighted Averaging

Definition 3.1.1 and Theorem 3.1.2 in [19] proposed the weighted averaging operator using defined operational rules described as follows.Definition 19.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, the CPLFWA operator is described as follows:(16)CPLFWAℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 1 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, CPLFWA operator is obtained as(17)CPLFWAℱcp1,ℱcp2,…,ℱcpn=ℓ∑i=1nvi/n,1−1−∏i=1nμi−21/n!,1−1−∏i=1nμi+21/n!,∏i=1nμi21/n!.Definition 3.2.1 and Theorem 3.2.2 in [19] proposed the generalized weighted averaging operator using defined operational rules described as follows.Definition 20 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, the CGPLFWA operator is described as follows:(18)CGPLFWAℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi1/λ,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 2 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CGPLFSU. Then, using defined operational laws, the CGPLFWA operator is obtained as(19)CGPLFWAℱcp1,ℱcp2,…,ℱcpn=ℓ∑i=1nvi/n1/λ,1−1−∏i=1nμi−2λ1/n!1/λ,1−1−∏i=1nμi+2λ1/n!1/λ,1−1−∏i=1n1−1−μi2λ1/n!1/λ.


## 5.2. Weighted Geometric

Definition 3.3.1 and Theorem 3.3.2 in [19] proposed the weighted geometric operator using defined operational rules described as follows.Definition 21.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, the CPLFWG operator is described as follows:(20)CPLFWGℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 3 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFWG operator is obtained as(21)CPLFWGℱcp1,ℱcp2,…,ℱcpn=ℓ∑i=1nvi1/n/n,1−1−∏i=1nμi−2λ1/n!1/λ,1−1−∏i=1nμi+2λ1/n!1/λ1−∏i=1n1−μi21/n!.


## 6. Updated Linguistic Cubic Pythagorean Fuzzy AO

In this section, utilizing valid Definition16 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.


### 6.1. Updated Weighted Averaging AO

Definition 22.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, the CPLFWA operator is described as follows:(22)CPLFWAℱcp1,ℱcp2,…,ℱcpn=ρ1ℱcp1⊕ρ2ℱcp2⊕⋯⊕ρnℱcpn=∑♮=1nρ♮ℱcp♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ have ρ♮≥0 and ∑♮=1nρ♮=1.Theorem 4.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws 17, CPLFWA operator is obtained as(23)CPLFWAℱcp1,ℱcp2,…,ℱcpn=∑♮=1nρ♮ℱcp♮=ℓ∑♮=1nρ♮×v♮,1−∏♮=1n1−μ♮−2ρ♮,1−∏♮=1n1−μ♮+2ρ♮,υ♮−ρ♮,υ♮+ρ♮,1−∏♮=1n1−μ♮2ρ♮,υ♮ρ♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ have ρ♮≥0 and ∑♮=1nρ♮=1.


### 6.2. Updated Weighted Geometric AO

Definition 23.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, the CPLFWG operator is described as follows:(24)CPLFWGℱcp1,ℱcp2,…,ℱcpn=ℱcp1ρ1⊗ℱcp2ρ2⊗⋯⊗ℱcpnρn=∏♮=1nℱcp♮ρ♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ have ρ♮≥0 and ∑♮=1nρ♮=1.Theorem 5.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws 17, the CPLFWG operator is obtained as(25)CPLFWGℱcp1,ℱcp2,…,ℱ%cpn=∏♮=1nℱcp♮ρ♮=ℓ∏♮=1nv♮ρ♮,μ♮−ρ♮,μ♮+ρ♮,1−∏♮=1n1−υ♮−2ρ♮1−∏♮=1n1−υ♮+2ρ♮,μ♮ρ♮,1−∏♮=1n1−υ♮2ρ♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ having ρ♮≥0 and ∑♮=1nρ♮=1.


## 6.1. Updated Weighted Averaging AO

Definition 22.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, the CPLFWA operator is described as follows:(22)CPLFWAℱcp1,ℱcp2,…,ℱcpn=ρ1ℱcp1⊕ρ2ℱcp2⊕⋯⊕ρnℱcpn=∑♮=1nρ♮ℱcp♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ have ρ♮≥0 and ∑♮=1nρ♮=1.Theorem 4.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws 17, CPLFWA operator is obtained as(23)CPLFWAℱcp1,ℱcp2,…,ℱcpn=∑♮=1nρ♮ℱcp♮=ℓ∑♮=1nρ♮×v♮,1−∏♮=1n1−μ♮−2ρ♮,1−∏♮=1n1−μ♮+2ρ♮,υ♮−ρ♮,υ♮+ρ♮,1−∏♮=1n1−μ♮2ρ♮,υ♮ρ♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ have ρ♮≥0 and ∑♮=1nρ♮=1.


## 6.2. Updated Weighted Geometric AO

Definition 23.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, the CPLFWG operator is described as follows:(24)CPLFWGℱcp1,ℱcp2,…,ℱcpn=ℱcp1ρ1⊗ℱcp2ρ2⊗⋯⊗ℱcpnρn=∏♮=1nℱcp♮ρ♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ have ρ♮≥0 and ∑♮=1nρ♮=1.Theorem 5.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws 17, the CPLFWG operator is obtained as(25)CPLFWGℱcp1,ℱcp2,…,ℱ%cpn=∏♮=1nℱcp♮ρ♮=ℓ∏♮=1nv♮ρ♮,μ♮−ρ♮,μ♮+ρ♮,1−∏♮=1n1−υ♮−2ρ♮1−∏♮=1n1−υ♮+2ρ♮,μ♮ρ♮,1−∏♮=1n1−υ♮2ρ♮,where the weights ρ1,ρ2,…,ρ♮ of ℱ♮ having ρ♮≥0 and ∑♮=1nρ♮=1.


## 7. Countersections 3.4 and 3.5 of [19]

This 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 Theorem 3.4.2 in [19] proposed the MSM operator using defined operational rules (Definition 15) described as follows.Definition 24.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρ♮T is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFMSM operator is described as follows:(26)CPLFMSMℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi1/λ,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 6 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFMSM operator is obtained as(27)CPLFMSMℱcp1,ℱcp2,…,ℱcpn=ℓ1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏j=1khij1/λ,1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1nμi−21/ℂnk1/k,1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1nμi+21/ℂnk1/k,1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi21/ℂnk1/k.Definition 3.5.1 and Theorem 3.5.2 in [19] proposed the weighted MSM operator using defined operational rules (Definition 15) described as follows.Definition 25.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFWMSM operator is described as follows:(28)CPLFWMSMℱcp1,ℱcp2,…,ℱcpn=1n!∑θ∈Sn∏i=1nℱcpθi1/λ,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 7 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, CPLFWMSM operator is obtained as(29)CPLFWMSMℱcp1,ℱcp2,…,ℱcpn=ℓ1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏j=1khij1/k,1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi−2ρi1/ℂnk1/k,1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi+2ρi1/ℂnk1/k1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi2ρi1/ℂnk1/k.


## 8. Updated Linguistic Cubic Pythagorean Fuzzy MSM AO

In this section, utilizing valid Definition16 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.Definition 26.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, the CPLFMSM operator is described as follows:(30)CPLFMSMℱcp1,ℱcp2,…,ℱcpn=∑1≤i1≤i2≤⋯≤ik≤n∏j=1kℱcpijℂnk1/k,where ℂnk is the binomial coefficient and i1,i2,…,ik traversal of all the k-tuple combination of 1,2,…,n.Theorem 8.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws 17, the CPLFMSM operator is obtained as(31)CPLFMSMℱcp1,ℱcp2,…,ℱcpn=∑1≤i1≤i2≤⋯≤ik≤n∏j=1kℱcpijℂnk1/k=ℓ∑1≤i1≤i2≤⋯≤ik≤n∏j=1kℱcpij/ℂnk1/k,1−∏1≤i1≤i2≤⋯n≤ik≤n1−∏j=1nμij−21/ℂnk1/k,1−∏1≤i1≤i2≤⋯n≤ik≤n1−∏j=1nμij+21/ℂnk1/k,1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1n1−υij−21/ℂnk1/k,1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1n1−υij+21/ℂnk1/k,1−∏1≤i1≤i2≤⋯n≤ik≤n1−∏j=1nμij21/ℂnk1/k,1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1n1−υij21/ℂnk1/k.
Updated weighted MSM AO is defined as follows.Definition 27.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFWMSM operator is described as follows:(32)CPLFWMSMℱcp1,ℱcp2,…,ℱcpn=∑1≤i1≤i2≤⋯≤ik≤n∏j=1kℱcpijρijℂnk1/k,where ℂnk is the binomial coefficient and i1,i2,…,ik traversal of all the k-tuple combination of 1,2,…,n.Theorem 9.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws 17, the CPLFWMSM operator is obtained as(33)CPLFWMSMℱcp1,ℱcp2,…,ℱcpn=∑1≤i1≤i2≤⋯≤ik≤n∏j=1kℱcpijρijℂnk1/k=ℓ∑1≤i1≤i2≤…≤ik≤n∏j=1kℱcpijρij/ℂnk1/k,1−∏1≤i1≤i2≤…≤ik≤n1−∏j=1nμij−ρij21/ℂnk1/k,1−∏1≤i1≤i2≤…≤ik≤n1−∏j=1nμij+ρij21/ℂnk1/k,1−1−∏1≤i1≤i2≤…≤ik≤n1−∏j=1n1−υij−2ρij1/ℂnk1/k,1−1−∏1≤i1≤i2≤…≤ik≤n1−∏j=1n1−υij+2ρij1/ℂnk1/k,1−∏1≤i1≤i2≤…≤ik≤n1−∏j=1nμijρij21/ℂnk1/k,1−1−∏1≤i1≤i2≤…≤ik≤n1−∏j=1n1−υij2ρij1/ℂnk1/k.


## 9. Countersection 4 of [19]

This section recalls the discussion of DMSM AO for CPLFN and their basic properties proposed by Fahmi et al. [19].


### 9.1. Weighted DMSM Averaging

Definition 4.1.1 and Theorem 4.1.2 in [19] proposed the weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.Definition 28.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDWMSMA operator is described as follows:(34)CPLFDWMSMAℱcp1,ℱcp2,…,ℱcpn=1k∑θ∈Sn∑i=1nρjℱcpθi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 10 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDWMSMA operator is obtained as(35)CPLFDWMSMAℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi−2ρi1/ℂnk1/k1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi+2ρi1/ℂnk1/k1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−1−μi2ρi1/ℂnk1/k.Definition 4.2.1 and Theorem 4.2.2 in [19] proposed the ordered weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.Definition 29.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDOWMSMA operator is described as follows:(36)CPLFDOWMSMAℱcp1,ℱcp2,…,ℱcpn=1k∑θ∈Sn∑i=1nρjℱcpθi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 11 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDOWMSMA operator is obtained as(37)CPLFDOWMSMAℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi−2ρi1/ℂnk1/k,1−1−∏1≤i1≤i2≤...≤ik≤n1−∏i=1n1−μi+2ρi1ℂnk1/k1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−1−μi2ρi1/ℂnk1/k.Definition 4.3.1 and Theorem 4.3.2 in [19] proposed the hybrid weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.Definition 30.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is associated weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDHWMSMA operator is described as follows:(38)CPLFDHWMSMAℱcp1,ℱcp2,…,ℱcpn=1k∑θ∈Sn∑i=1nρjℱcpθi1/k,where θii∈ℕ represented the any permutation and Sn represented the total numbers of elements.Theorem 12 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDHWMSMA operator is obtained as(39)CPLFDHWMSMAℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi−2ρi1/ℂnk1/k1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi+2ρi1/ℂnk1/k1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−1−$μi2$ρi1/ℂnk1/k.


### 9.2. Weighted DMSM Geometric

Definition 4.4.1 and Theorem 4.4.2 in [19] proposed the weighted DMSM geometric operator using defined operational rules (Definition 15) described as follows.Definition 31.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDWMSMG operator is described as follows:(40)CPLFDWMSMGℱcp1,ℱcp2,…,ℱcpn=1k∏θ∈Sn∏i=1nℱcpθiρi1/k,where θii∈ℕ represents any permutation and Sn represents total numbers of elements.Theorem 13 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDWMSMG operator is obtained as(41)CPLFDWMSMGℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤.⋯≤ik≤n∏i=1khθiρi1/ℂnk,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi−2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi+2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi2ρi1/ℂnk1/k.Definition 4.5.1 and Theorem 4.5.2 in [19] proposed the ordered weighted DMSM geometric operator using defined operational rules (Definition 15) is described asDefinition 32 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDOWMSMG operator is described as follows:(42)CPLFDOWMSMGℱcp1,ℱcp2,…,ℱcpn=1k∏θ∈Sn∏i=1nℱcpθiρi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 14 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDOWMSMG operator is obtained as(43)CPLFDOWMSMGℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi−2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi+2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi2ρi1/ℂnk1/k.Definition 4.6.1 and Theorem 4.6.2 in [19] proposed the hybrid-weighted DMSM geometric operator using defined operational rules (Definition 15) described as follows.Definition 33.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDHWMSMG operator is described as follows:(44)CPLFDHWMSMGℱcp1,ℱcp2,…,ℱcpn=1k∏θ∈Sn∏i=1nℱcpθiρi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 15 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDHWMSMG operator is obtained as(45)CPLFDHWMSMGℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi−2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi+2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi2ρi1/ℂnk1/k.


## 9.1. Weighted DMSM Averaging

Definition 4.1.1 and Theorem 4.1.2 in [19] proposed the weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.Definition 28.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDWMSMA operator is described as follows:(34)CPLFDWMSMAℱcp1,ℱcp2,…,ℱcpn=1k∑θ∈Sn∑i=1nρjℱcpθi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 10 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDWMSMA operator is obtained as(35)CPLFDWMSMAℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi−2ρi1/ℂnk1/k1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi+2ρi1/ℂnk1/k1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−1−μi2ρi1/ℂnk1/k.Definition 4.2.1 and Theorem 4.2.2 in [19] proposed the ordered weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.Definition 29.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDOWMSMA operator is described as follows:(36)CPLFDOWMSMAℱcp1,ℱcp2,…,ℱcpn=1k∑θ∈Sn∑i=1nρjℱcpθi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 11 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDOWMSMA operator is obtained as(37)CPLFDOWMSMAℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi−2ρi1/ℂnk1/k,1−1−∏1≤i1≤i2≤...≤ik≤n1−∏i=1n1−μi+2ρi1ℂnk1/k1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−1−μi2ρi1/ℂnk1/k.Definition 4.3.1 and Theorem 4.3.2 in [19] proposed the hybrid weighted DMSM averaging operator using defined operational rules (Definition 15) described as follows.Definition 30.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is associated weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDHWMSMA operator is described as follows:(38)CPLFDHWMSMAℱcp1,ℱcp2,…,ℱcpn=1k∑θ∈Sn∑i=1nρjℱcpθi1/k,where θii∈ℕ represented the any permutation and Sn represented the total numbers of elements.Theorem 12 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDHWMSMA operator is obtained as(39)CPLFDHWMSMAℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi−2ρi1/ℂnk1/k1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi+2ρi1/ℂnk1/k1−∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−1−$μi2$ρi1/ℂnk1/k.


## 9.2. Weighted DMSM Geometric

Definition 4.4.1 and Theorem 4.4.2 in [19] proposed the weighted DMSM geometric operator using defined operational rules (Definition 15) described as follows.Definition 31.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDWMSMG operator is described as follows:(40)CPLFDWMSMGℱcp1,ℱcp2,…,ℱcpn=1k∏θ∈Sn∏i=1nℱcpθiρi1/k,where θii∈ℕ represents any permutation and Sn represents total numbers of elements.Theorem 13 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDWMSMG operator is obtained as(41)CPLFDWMSMGℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤.⋯≤ik≤n∏i=1khθiρi1/ℂnk,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi−2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi+2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi2ρi1/ℂnk1/k.Definition 4.5.1 and Theorem 4.5.2 in [19] proposed the ordered weighted DMSM geometric operator using defined operational rules (Definition 15) is described asDefinition 32 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDOWMSMG operator is described as follows:(42)CPLFDOWMSMGℱcp1,ℱcp2,…,ℱcpn=1k∏θ∈Sn∏i=1nℱcpθiρi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 14 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDOWMSMG operator is obtained as(43)CPLFDOWMSMGℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi−2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi+2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi2ρi1/ℂnk1/k.Definition 4.6.1 and Theorem 4.6.2 in [19] proposed the hybrid-weighted DMSM geometric operator using defined operational rules (Definition 15) described as follows.Definition 33.
(see [19]). Let ℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFDHWMSMG operator is described as follows:(44)CPLFDHWMSMGℱcp1,ℱcp2,…,ℱcpn=1k∏θ∈Sn∏i=1nℱcpθiρi1/k,where θii∈ℕ represents any permutation and Sn represents the total numbers of elements.Theorem 15 (see [19]).
Letℱcpi=ℓvi,μi−υ,μi+υ,μiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws, the CPLFDHWMSMG operator is obtained as(45)CPLFDHWMSMGℱcp1,ℱcp2,…,ℱcpn=ℓ1/k1/ℂnk∑1≤i1≤i2≤⋯≤ik≤n∏i=1khθiρi1/ℂnk,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi−2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n∏i=1nμi+2ρi1/ℂnk1/k,∏1≤i1≤i2≤⋯≤ik≤n1−∏i=1n1−μi2ρi1/ℂnk1/k.


## 10. Updated Dual MSM Operators

In this section, valid Definition16 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.Definition 34.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU and ρ1,ρ2,…,ρnT is weight vector having ρ♮≥0 and ∑♮=1nρ♮=1. Then, the CPLFWDMSM operator is described as follows:(46)CPLFWDMSMℱcp1,ℱcp2,…,ℱcpn=1k∏1≤i1≤i2≤⋯≤ik≤n∑j=1kρij⊗ℱcpij1/ℂnk,where ℂnk is the binomial coefficient and i1,i2,…,ik traversal of all the k-tuple combination of 1,2,…,n.Theorem 16.
Letℱcpi=ℓvi,μi−υ,μi+υ,υi−υ,υi+υ,μiυ,υiυ;i∈ℕ∈CPLFSU. Then, using defined operational laws 17, the CPLFWDMSM operator is obtained as(47)CPLFWDMSMℱcp1,ℱcp2,…,ℱcpn=1k∏1≤i1≤i2≤⋯≤ik≤n∑j=1kρij⊗ℱcpij1/ℂnk=ℓ1/k∏1≤i1≤i2≤⋯≤ik≤n∑j=1kρij⊗ℱcpij1/ℂnk,1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1n1−μij−2ρij1/ℂnk1/k,1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1n1−μij+2ρij1/ℂnk1/k,1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1nυij−ρij21/ℂnk1/k,1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1nυij+ρij21/ℂnk1/k,1−1−∏1≤i1≤i2≤⋯≤ik≤n1−∏j=1n1−μij2ρij1/ℂnk1/k,1−∏1≤i1≤i2≤⋯%≤ik≤n1−∏j=1nυijρij21/ℂnk1/k.


## 11. Decision-Making Model Based on Updated MSM Operators

In this section, we propose a framework for solving multiattribute decision-making problems (DMPs) under cubic Pythagorean linguistic fuzzy (CPLF) information. Consider a MADM with a set ofm alternatives ℷ1,ℷ2,…,ℷ♮, and let y1,y2,…,yh be a set of attributes with weight vector p=p1,p2,…,ph, where pt∈0,1 and ∑t=1hpt=1. To assess the performance of kth alternative ℷk under the tth attribute yt, let D∘1,D∘2,…,D∘J be a set of decision makers and w^=w^1,w^2,…,w^J be the weighted vector of decision makers with w^s∈0,1 and ∑s=1Jw^s=1. The CPLF decision matrix can be written as(48)♭11♭12♭13⋯♭1h♭21♭22♭23⋯♭2h♭31♭32♭33⋯♭3h⋮⋮⋮⋱⋮♭♮1♭♮2♭♮3⋯♭♮h,where ♭ij=ℓvij,μij−υ,μij+υ,υij−υ,υij+υ,μijυ,υijυ and μ−υ,μ+υ,υ−υ,υ+υ⊆0,1 and μυ,υυ∈0,1 represented the positive membership grade in the form of interval-valued Pythagorean fuzzy set and negative membership grade in the form of Pythagorean fuzzy set. Key steps of the developed multiattribute decision-making (MADM) problem are described as follows.Step 1: construct the CPLF decision matrix based on the experts’ evaluations:(49)DJ^=♭11J♭12J♭13J⋯♭1hJ♭21J♭22J♭23J⋯♭2hJ♭31J♭32J♭33J⋯♭3hJ⋮⋮⋮⋱⋮♭♮1J♭♮2J♭♮3J⋯♭♮hJ,whereJ represents the number of expert.Step 2: exploit the established aggregation operators to achieve the CPLFNsℱtt=1,2,…,♮ for the alternatives ℷk, that is, the established operators to obtain the collective overall preference values of ℱtt=1,2,…,♮ for the alternatives ℷk, where p=p1,p2,…,ph is the weight vector of the attributes.Step 3: after that, we compute the scores of all the overall valuesℱtt=1,2,…,♮ for the alternatives ℷk.Step 4: according to Definition18, rank the alternatives ℷkk=1,2,…,♮ and select the best one having the greater value.


## 12. Numerical Application

The 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: G1⟶ the promotion, G2⟶ configuration simplicity, G3⟶ security level, and G4⟶ maintenance server level, whose weight vector is 0.2,0.1,0.3,0.4T. 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 Table1:Step 2: based on the proposed MSM operators, the collective CPLF information of each alternative is obtained as follows in Tables2–4:(50)♭11⊗♭12=ℓ8,0.15,0.48,0.48,0.750.21,0.44,♭11⊗♭13=ℓ2,0.10,0.32,0.48,0.680.14,0.22,♭11⊗♭14=ℓ10,0.20,0.40,0.64,0.750.42,0.52,♭12⊗♭13=ℓ4,0.06,0.24,0.54,0.820.06,0.41,♭12⊗♭14=ℓ20,0.12,0.30,0.68,0.860.18,0.60,♭13⊗♭14=ℓ5,0.08,0.20,0.68,0.820.12,0.50.Case 1: from Table1, we have ♭11=ℓ2,0.5,0.8,0.3,0.40.7,0.2, ♭12=ℓ4,0.3,0.6,0.4,0.70.3,0.4, ♭13=ℓ1,0.2,0.4,0.4,0.60.2,0.1, and ♭14=ℓ5,0.4,0.5,0.6,0.70.6,0.5∈CPLFSU. Here, we use CPLFMSM operator to aggregate the expert evaluation in the form of CPLFSs. Without loss of generality, we take k=2; then,Using Theorem8, we obtain(51)CPLFMSM2♭11,♭12,♭13,♭14=∑1≤i1≤i2≤4♭i1⊗♭i2ℂ421/2=ℓ1.52,0.242,0.438,0.352,0.5300.337,0.246.Hence, we obtained similarly.(52)♭110.2⊗♭120.1=ℓ1.319,0.77,0.90,0.01,0.040.82,0.01,♭110.2⊗♭130.3=ℓ1.148,0.53,0.73,0.03,0.060.57,0.004,♭110.2⊗♭140.4=ℓ2.186,0.60,0.72,0.05,0.080.75,0.02,♭120.1⊗♭130.3=ℓ1.148,0.54,0.72,0.02,0.090.54,0.007,♭120.1⊗♭140.4=ℓ2.186,0.61,0.72,0.05,0.120.72,0.04,♭130.3⊗♭140.4=ℓ1.903,0.42,0.57,0.09,0.170.50,0.01.(53)CPLFWMSM2♭11,♭12,♭13,♭14=∑1≤i1≤i2≤4♭i1ρi1⊗♭i2ρi2ℂ421/2=ℓ1.105,0.658,0.787,0.020,0.0460.725,0.007.(54)CPLFWDMSM2♭11,♭12,♭13,♭14=12∏1≤i1≤i2≤4ρi1⊗ℱcpi1⊕ρi2⊗ℱcpi21/ℂ42=ℓ0.657,0.179,0.302,0.821,0.8830.254,0.738.Case 2: from Table1, we have ♭11=ℓ2,0.5,0.8,0.3,0.40.7,0.2, ♭12=ℓ4,0.3,0.6,0.4,0.70.3,0.4, ♭13=ℓ1,0.2,0.4,0.4,0.60.2,0.1, and ♭14=ℓ5,0.4,0.5,0.6,0.70.6,0.5∈CPLFSU, and 0.2,0.1,0.3,0.4T is the weight vector. Here, we use CPLFWMSM operator to aggregate the expert evaluation in the form of CPLFSs. Without loss of generality, we take k=2; then,Using Theorem9, we obtainHence, we obtained similarly in Table3 utilizing Theorem 9,Case 3: From Table1, we have ♭11=ℓ2,0.5,0.8,0.3,0.40.7,0.2, ♭12=ℓ4,0.3,0.6,0.4,0.70.3,0.4, ♭13=ℓ1,0.2,0.4,0.4,0.60.2,0.1, and ♭14=ℓ5,0.4,0.5,0.6,0.70.6,0.5∈CPLFSU, and 0.2,0.1,0.3,0.4T is the weight vector. 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 Theorem 16, we obtainHence, we obtained similarly in Table4 utilizing Theorem 16,Step 3: compute the score value of the each collective CPLF information of each alternative as in Table5.Step 4: select the optimal alternative according the maximum score value as in Table6:Table 1
Expert 1 informationD1.
G1G2G3G4∁1ℓ2,0.5,0.8,0.3,0.40.7,0.2ℓ4,0.3,0.6,0.4,0.70.3,0.4ℓ1,0.2,0.4,0.4,0.60.2,0.1ℓ5,0.4,0.5,0.6,0.70.6,0.5∁2ℓ5,0.2,0.4,0.1,0.70.5,0.6ℓ1,0.4,0.8,0.2,0.50.6,0.2ℓ5,0.5,0.7,0.3,0.70.5,0.6ℓ2,0.3,0.6,0.4,0.70.4,0.2∁3ℓ1,0.3,0.6,0.5,0.80.6,0.4ℓ3,0.5,0.6,0.3,0.40.4,0.3ℓ3,0.4,0.4,0.5,0.60.3,0.8ℓ4,0.1,0.8,0.3,0.50.7,0.5∁4ℓ3,0.4,0.7,0.2,0.60.4,0.7ℓ5,0.2,0.8,0.3,0.50.7,0.1ℓ2,0.6,0.8,0.2,0.80.6,0.3ℓ1,0.2,0.4,0.5,0.60.3,0.8Table 2
CPLFMSM2 operator.
∁1ℓ1.520,0.242,0.438,0.352,0.5300.337,0.246∁2ℓ1.568,0.242,0.493,0.200,0.5780.362,0.339∁3ℓ1.482,0.231,0.464,0.329,0.5060.371,0.432∁4ℓ1.468,0.246,0.555,0.239,0.5530.372,0.431Table 3
CPLFWMSM2 operator.
∁1ℓ1.111,0.621,0.817,0.017,0.0410.762,0.027∁2ℓ1.118,0.655,0.812,0.005,0.0570.744,0.014∁3ℓ1.105,0.658,0.787,0.020,0.0460.725,0.007∁4ℓ1.061,0.665,0.841,0.008,0.0510.735,0.016Table 4
CPLFWDMSM2 operator.
∁1ℓ0.657,0.179,0.302,0.821,0.8830.254,0.738∁2ℓ0.796,0.179,0.339,0.721,0.9080.248,0.808∁3ℓ0.667,0.153,0.331,0.807,0.8780.275,0.858∁4ℓ0.522,0.187,0.364,0.752,0.9000.249,0.844Table 5
Score value.
Sγ∁1Sγ∁2Sγ∁3Sγ∁4CPLFMSM20.0908732-0.099975-0.033468-0.074122CPLFWMSM20.6550470.6373650.59151090.638794CPLFWDMSM2-0.501965-0.590133-0.550951-0.508489Table 6
Ranking results.
Score rankingBest alternativeCPLFMSM2Sγ∁1>Sγ∁3>Sγ∁4>Sγ∁2∁1CPLFWMSM2Sγ∁1>Sγ∁4>Sγ∁2>Sγ∁3∁1CPLFWDMSM2Sγ∁1>Sγ∁4>Sγ∁3>Sγ∁2∁1From 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.


## 13. 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.



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